Experimental Investigations of Magnetohy

Experimental Investigations of Magnetohydrodynamic Plasma Jets - CaltechTHESIS
CaltechTHESIS
A Caltech Library Service
About
Browse
Deposit an Item
Instructions for Students
Experimental Investigations of Magnetohydrodynamic Plasma Jets
Citation
Kumar, Deepak
(2009)
Experimental Investigations of Magnetohydrodynamic Plasma Jets.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/ENZ7-QV92.
Abstract
This thesis primarily focuses on understanding the plasma behavior during the helicity injection stage of a pulsed spheromak experiment. Spheromak formation consists of a series of dynamic steps whereby highly localized plasma near the electrodes evolves toward a Taylor state equilibrium. The dynamical evolution stage has been modeled as a series of equilibrium states in the past. However, the experiments at the Caltech spheromak facility have revealed that unbalanced J x B forces drive non equilibrium Alfvénic flows during these preliminary stages.
The Caltech spheromak experiment uses coplanar electrodes to produce a collimated plasma jet flowing away from the electrodes. The jet formation stage precedes the spheromak formation and serves as a mechanism for feeding particles, magnetic helicity, energy, and toroidal flux into the system. Detailed density and flow velocity measurements of hydrogen and deuterium plasma jets have revealed that the jets are extremely dense with β [subscript thermal] ~1. Furthermore, the flow velocity was found to be Alfvénic with respect to the the toroidal magnetic field produced by the axial current within the plasma. An existing magnetohydrodynamics (MHD) model has been generalized to successfully predict the effect of plasma current on the jet's density and flow velocity. The behavior of these laboratory jets is in stark contrast to the often considered model for astrophysical jets describing them as equilibrium configurations with hollow density profiles.
Other contributions of this thesis include the following.
1. The thesis presents an analytical proof that resistive MHD equilibrium with closed flux tubes is not feasible. This implies that sustained spheromak experiments cannot maintain helicity while being in a strict equilibrium.
2. The thesis describes measurements to characterize the circuit parameters of the high voltage discharge circuit used in the Caltech spheromak experiment.
3. The thesis also describes the setup of novel He-Ne laser interferometers used to measure the density of plasma jets. The ease of alignment of these interferometers was greatly enhanced by having unequal path lengths of the scene and reference beams.
4. Finally, the thesis details the setup for a soft X-ray (SXR)/Vacuum ultra violet (VUV) imaging system. Some preliminary images of reconnecting flux tubes captured by the imaging setup are also presented.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
astrophysical jets; helicity injection; ignitron; interfereometer; pulsed power; resistive MHD equilibria; Spheromak; X-ray imaging
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Bellan, Paul Murray
Thesis Committee:
Gould, Roy Walter (chair)
Meier, David L.
Shepherd, Joseph E.
Bellan, Paul Murray
Defense Date:
9 March 2009
Record Number:
CaltechETD:etd-04092009-163047
Persistent URL:
DOI:
10.7907/ENZ7-QV92
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
1321
Collection:
CaltechTHESIS
Deposited By:
Imported from ETD-db
Deposited On:
13 May 2009
Last Modified:
26 Nov 2019 19:13
Thesis Files
Preview
PDF (Full thesis)
- Final Version
See Usage Policy.
57MB
Repository Staff Only:
item control page
CaltechTHESIS is powered by
EPrints 3.3
which is developed by the
School of Electronics and Computer Science
at the University of Southampton.
More information and software credits
Experimental Investigations of Magnetohydrodynamic
Plasma Jets
Thesis by
Deepak Kumar

In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2009

(Defended March 9, 2009)

c 2009
Deepak Kumar

iii

To the joy and beauty of life.

Acknowledgements
It may seem that completing one’s doctoral research is a great personal achievement. While
this is certainly the case and I would like to bask in the glory, I would also like to give credit
where it is due - to the numerous people who have influenced me in a positive way and
helped me during my stay at Caltech.
My stay at Caltech has been extremely enjoyable because of the people I have interacted
with, both on and off campus. I had the opportunity to learn karate at the Caltech Karate
Club - the first university karate club in the United States. Randy McClure and Pamela
Logan were my first instructors. They are champion instructors who made the trainings
difficult - and hence fun. The practices were invigorating and taught me the nuances of how
best to teach, communicate and motivate. A lot of these skills translated to other aspects of
my life. I also had the opportunity to train with many other excellent people related to SKA
(Shotokan Karate of America) at our practices, including Mr. Ohshima, Bruce Kanegai,
Tom Heyman, David Gabai, Paul Morgan, Dan Sakurai, Kevin Bench, Daniel McSween
and numerous others. I am thankful to my current instructor Ian Ferguson for training
me in class, especially for tournaments. Over the course of numerous karate practices, I
made a lot of friends. Unlike normal friends, they enjoyed hurting me, but I cherished them
nonetheless. They include Jon Hartzberg, Tom Livermore, Sasha Tsapin, Peter Ilott, Ted
Yu, Udi Vermesh, Charles Cohen, Maria, Michelle Blackway, Eric Wambach, Chris Finch,
Tom Mainiero, Rey Rodriguez, Dominic Hougham, Omar Facory, Alexey Solomatin, Raffi,
David Berry, Michael Busch, Chip Miller, and others. I would like to thank them all for
making practices enjoyable. I will always remember them because of the various bruises
and scars they gave me.
I would also like to thank the athletics department at Caltech (especially Howard, Jean
Fees, Wendell Jack, and Carlos Basulto) and the facilities management (Anthony Ford and
Delmy Emerson) for their constant support to the Caltech Karate Club.
During my stay at Caltech, I got in touch with a few alumni who were building a robot
car called “Golem.” As a team we participated in the DARPA Grand and Urban Challenges.
Even though, I had a very limited role in the team, it was a thrilling experience, and I learned
a lot of skills. The people involved with the project were smart and dedicated and made the

whole experience fun. I would like to thank Jim Radford, Richard Mason, Robb Walters,
Brian Fulkerson, Bill Caldwell, Dave Caldwell, Dima Kogan, Jim Swenson, Emilio Frazzoli,
Josh Arensberg, Eagle Jones, Michael Linderman, Roy Pollock, Ken Kappler, Brandyn
Webb, Brent Morgan, Jeff Elings, Izaak Giberson, and other members of the team.
Throughout my research, I often relied on help from representatives from other companies for the maintenance of equipment or getting a new equipment. I am grateful to the
following individuals (companies) for their help - Todd Rumbaugh and Frank Kosel (DRS
Imaging), Raj Korde (International Radiation Detectors), Dan Gorzen (X-ray and Speciality Instruments), Dr. Matthias Kirsch (Struck Innovative Systeme GmbH), Rolando (APD
Cryogenics), Del Munns (DV Manufacturing), and Mark Slattery (Berkeley Nucleonics).
One of the reasons why I enjoyed my time at Caltech was the excellent support from
members of the Caltech community. In particular, I would like to thank Rick Germond at
facilities stockroom, and Mike Gerfen at central engineering services. Mike was an excellent
help with designing and machining most of the parts I used for my experiments. I would
also like to thank the administrative assistants at the applied physics department at Caltech
- Eleonora Vorobieff, Connie Rodriguez, Irene Loera, Cierina Marks and Mary Metz. It was
extremely pleasing to be pampered, as they took care of all the administrative issues I was
bothered with. I would also like to thank Jim Endrizzi, Athena Trentin and Tina Lai from
the International Student Program office at Caltech. Their support is much appreciated.
I consider myself lucky to have had Dave Felt as an engineer in our lab. Experimental
research often involves troubleshooting and reconstructing equipment, and Dave has been
a tremendous help with it on numerous occasions. I would also like to thank Doug Strain
for writing the software interface for the timing sequencers used in the experiment.
The pace of experimental research is often slow at the beginning and relies on learning
good skills from mentors and collaborators. I am indebted to Setthivoine You and Shreekrihna Tripathi for being excellent mentors and helping me get started in the field of plasma
physics. They were the postdoctoral scholars in our research group when I joined and were
extremely diligent and methodical in upkeep of the experiment.
During my first couple of years at Caltech, I shared an office with senior graduate
students - Eli Jorné, Carlos Romero-Talamás and Steve Pracko. They ensured that the
office was a lively place - somewhere I looked forward to come to every morning. The office

vi
banter was lighthearted and fun, even though I was the target of most of the office jokes.
Over the last three to four years of my stay at Caltech, I also had the opportunity
to collaborate with new graduate student members of our research group - Rory Perkins,
Auna Moser, Eve Stenson, Mark Kendall, Bao Ha and Vernon Chaplin. It was wonderful
to collaborate with Rory on our quest to understand the origin of the X-rays in our experiment. I am also grateful to him for proof reading parts of this thesis and for other physics
discussions we have had over the years. Auna carried on the tradition of keeping the office
a lively place. I am thankful to her for the various upkeep and calibrations she performed
on the spheromak experiment. I would also like to acknowledge the feedback from Vernon
while he proofread most of this thesis. It was fun to get to know Eve, and I will certainly
miss the cookies she used to bring to the lab. Mark and Bao brought a young positive
energy to the group. Their help in maintaining the computing infrastructure in the group
is gratefully acknowledged.
I would like to acknowledge the helpful tips and guidance from other scientists during
my research - Freddy Hansen, Scott Hsu, Uri Shumlak, Brian Nelson, Raymond Golingo,
Tony Peebles, Terry Rhodes, Simon Woodruff, Richard Lovelace, Hui Li, Mike VanZeeland,
Carl Sovinec, William Bridges and Heun-Jin Lee.
I would also like to acknowledge Raffi Nazikian, Alan Hoffman, Cary Forest, Tom Jarboe,
Troy Carter, Walter Gekelman, and Dave Hammer for inviting me to present my research
during the last year. My presentation skills improved with each such opportunity as I
received positive feedback from the audience. I am also grateful to them for sharing ideas
and showing me their experiments.
I would also like to thank the members of my candidacy committee (Prof. Giapis, Prof.
Vahala, Prof. Rutledge, and Prof. Bellan) and the members of my thesis committee (Prof.
Gould, Prof. Shepherd, Dr. Meier, and Prof. Bellan) for their tips and feedback on the
thesis and presentation.
Completing the doctoral research is similar to running a marathon. It requires strong
determination and continuous effort for extended periods of time. During hard times, words
of support and encouragement help a lot. I would like to thank family and friends for their
continuous zealous support. I am blessed to have a lot of caring friends, but would like to
acknowledge a few of them in particular - my parents, Sonal, Sabiha, Ila and Zuma.

vii
Last but not least, I would like to acknowledge the mentor-ship of the two most helpful
people in my research - Paul Bellan and Gunsu Yun. Gunsu was a graduate student in our
lab. He is an excellent experimentalist, and I had the great fortune of collaborating with
him on many occasions. He single-handedly performed many upgrades to the experimental
setup. Paul has been a constant help throughout my stay at Caltech, and no words can do
justice to the positive impact he has had on me when it comes to doing research. I have
always been impressed by his knowledge and skills; however, his patience, diligence and
dedication have been even more motivating.

viii

Abstract
This thesis primarily focuses on understanding the plasma behavior during the helicity
injection stage of a pulsed spheromak experiment. Spheromak formation consists of a series
of dynamic steps whereby highly localized plasma near the electrodes evolves toward a
Taylor state equilibrium. The dynamical evolution stage has been modeled as a series of
equilibrium states in the past. However, the experiments at the Caltech spheromak facility
have revealed that unbalanced J × B forces drive non equilibrium Alfvénic flows during
these preliminary stages.
The Caltech spheromak experiment uses coplanar electrodes to produce a collimated
plasma jet flowing away from the electrodes. The jet formation stage precedes the spheromak formation and serves as a mechanism for feeding particles, magnetic helicity, energy,
and toroidal flux into the system. Detailed density and flow velocity measurements of hydrogen and deuterium plasma jets have revealed that the jets are extremely dense with
βthermal ∼ 1. Furthermore, the flow velocity was found to be Alfvénic with respect to the
the toroidal magnetic field produced by the axial current within the plasma. An existing
magnetohydrodynamics (MHD) model has been generalized to successfully predict the effect
of plasma current on the jet’s density and flow velocity. The behavior of these laboratory
jets is in stark contrast to the often considered model for astrophysical jets describing them
as equilibrium configurations with hollow density profiles.
Other contributions of this thesis include the following.
1. The thesis presents an analytical proof that resistive MHD equilibrium with closed
flux tubes is not feasible. This implies that sustained spheromak experiments cannot
maintain helicity while being in a strict equilibrium.
2. The thesis describes measurements to characterize the circuit parameters of the high
voltage discharge circuit used in the Caltech spheromak experiment.
3. The thesis also describes the setup of novel He-Ne laser interferometers used to measure the density of plasma jets. The ease of alignment of these interferometers was
greatly enhanced by having unequal path lengths of the scene and reference beams.
4. Finally, the thesis details the setup for a soft X-ray (SXR)/Vacuum ultra violet (VUV)

ix
imaging system. Some preliminary images of reconnecting flux tubes captured by the
imaging setup are also presented.

Contents

List of Figures

List of Tables

xvi

1 Introduction

1.1

Magnetohydrodynamics-MHD . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.1

Energy Conservation in MHD . . . . . . . . . . . . . . . . . . . . . .

1.2

Helicity Injection and Spheromak Formation . . . . . . . . . . . . . . . . . .

1.3

Overview of the Caltech Spheromak Experiment . . . . . . . . . . . . . . .

1.3.1

Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.2

Experimental Parameters and Dimensionless Numbers . . . . . . . .

2 On Magnetic Helicity Injection in a Steady State Scenario
2.1

Helicity Injection in Pulsed Spheromak Experiments . . . . . . . . . . . . .

15
19

3 Electrical Characterization of the Discharge Circuit of the Caltech Spheromak Experiment

3.1

Plasma Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.1

Plasma Parameters from Traces . . . . . . . . . . . . . . . . . . . . .

3.1.2

Plasma Parameters from Geometry . . . . . . . . . . . . . . . . . . .

3.2

Circuit Model for Spheromak Experiment . . . . . . . . . . . . . . . . . . .

3.3

Results and Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

Ignitron Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Interferometer for the Caltech Spheromak Experiment
4.1

Electromagnetic Wave Dispersion Relation in a Plasma

. . . . . . . . . . .

33
34

xi
4.2

4.3

4.4

4.5

Design Considerations for the Interferometer for the Caltech Spheromak Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Laser Phase Auto-correlation Function . . . . . . . . . . . . . . . . . . . . .

4.3.1

Frequency Spectrum of the Laser . . . . . . . . . . . . . . . . . . . .

4.3.2

Phase Auto-correlation Function Related to Power Spectrum . . . .

4.3.3

Measurement of Laser Phase Auto-correlation Function . . . . . . .

Homodyne Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.1

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.4

Procedural Details . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.5

Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.6

Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.7

Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Heterodyne Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.1

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.2

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.4

Procedural Details . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.5

Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.6

Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.7

Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

Future Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Non-equilibrium Alfvénic Plasma Jets Associated with Spheromak Formation

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1

Magnetic Field Structure in the Jets . . . . . . . . . . . . . . . . . .

5.2.2

Speed of the Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii
5.2.3

Density of the Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.4

Distribution of Neutrals in the Jet . . . . . . . . . . . . . . . . . . .

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1

Comparison of the Model with Experimental Results . . . . . . . . .

5.4

Energy Balance for Plasma Jets . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

Conclusion

5.3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 X-ray Imaging System for the Caltech Solar Coronal Loop Simulation
Experiment

6.1

Overview of the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.1

X-ray Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2.2

X-ray Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . .

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3.1

Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108

6.3

6.4

7 Summary

109

A Alignment of the Interferometers

111

A.1 Alignment Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

A.1.1 Ensuring a Constant Height of the Beam Above the Optical Table .

111

A.1.2 Steering the Heterodyne Interferometer’s Scene Beam through Sapphire Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

A.1.3 Combining the Scene and Reference Beams of the Heterodyne Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

A.2 Alignment Procedure for the Homodyne Interferometer . . . . . . . . . . . .

113

A.3 Alignment Procedure for the Heterodyne Interferometer . . . . . . . . . . .

118

Bibliography

131

xiii

List of Figures

1.1

Closed poloidal flux surfaces in an isolated spheromak. . . . . . . . . . . . .

1.2

Open and closed poloidal flux surfaces in an steady state spheromak. . . . .

1.3

Electrodes in the Caltech spheromak experiment. . . . . . . . . . . . . . . .

1.4

Cartoon showing the definition of the cylindrical coordinate system for the
Caltech spheromak experiment. . . . . . . . . . . . . . . . . . . . . . . . . .

1.5

Cartoon showing the location of bias field coil behind the cathode of the
Caltech spheromak experiment. . . . . . . . . . . . . . . . . . . . . . . . . .

1.6

Cartoon showing the sequence of events leading to plasma breakdown in the
Caltech spheromak experiment. . . . . . . . . . . . . . . . . . . . . . . . . .

1.7

Visual images of the three distinct stages of plasma evolution in the Caltech
spheromak experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

Cartoon of a driven spheromak showing open and closed flux tubes. . . . .

3.1

Schematic of the discharge circuit of the Caltech spheromak experiment. . .

3.2
3.3

The current and voltage traces measured across the electrodes for shot #8500. 22
Rt
Energy (= 0 V Idt) flowing into the plasma in shot #8500. . . . . . . . . .
22

3.4

A simplistic model of plasma jet. . . . . . . . . . . . . . . . . . . . . . . . .

3.5

A lumped circuit model for the spheromak discharge circuit. . . . . . . . . .

3.6

Measurements from discharging the high voltage capacitor charged to 2 kV
across the dummy load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

3.8

Current trace from various plasma shots confirming that the discharge circuit
acts as a current source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Current trace showing the ignitron stopped conducting briefly. . . . . . . .

xiv
4.1

Power spectrum of a laser showing discrete frequency resonance modes. . .

4.2

Michelson setup to measure phase auto correlation of laser. . . . . . . . . .

4.3

Envelope of the interference signal measured using the setup shown in figure
4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

Setup of the homodyne interferometer for the Caltech spheromak experiment. 45

4.5

Results from the homodyne interferometer for shot #7092.

4.6

Setup of the heterodyne interferometer for the Caltech spheromak experiment. 55

4.7

RF circuit for the heterodyne interferometer. . . . . . . . . . . . . . . . . .

4.8

Results from the heterodyne interferometer for shot #9114. . . . . . . . . .

4.9

Effect of misalignment on detector signal. . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

4.10 A plausible setup to alter the path length of the reference beam of a two-color
interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

False colored visible images depicting the formation of hydrogen plasma jet.

5.2

Poloidal current and flux surfaces of hydrogen plasma jets. . . . . . . . . . .

5.3

Typical interferometer density traces from the plasma jets.

. . . . . . . . .

5.4

Velocity of hydrogen plasma jets as a function of the maximum gun current.

5.5

Velocity of hydrogen and deuterium plasma jets as a function of the maximum
gun current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6

Thermal energy density as a function of toroidal magnetic field energy density
for hydrogen plasma jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.7

5.9

Thermal energy density as a function of toroidal magnetic field energy density
for deuterium plasma jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8

Density of the hydrogen plasma jet produced by gas valve pressurized to
100 psi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Cut out of a coaxial gun expanding against a spring. . . . . . . . . . . . . .

5.10 A cartoon of the jet showing three different regions of the plasma jet.

. . .

6.1

Electrodes for the dual prominence experiment. . . . . . . . . . . . . . . . .

6.2

Cartoon showing the setup of the single prominence experiment. . . . . . .

6.3

Cartoon showing the setup of the co-helicity merging experiment. . . . . . .

6.4

Cartoon showing the setup of the counter-helicity merging experiment. . . .

xv
6.5

Transmission characteristics of X-ray foil filters. . . . . . . . . . . . . . . . .

6.6

Schematic of the X-ray imaging setup. . . . . . . . . . . . . . . . . . . . . .

6.7

Setup of the X-ray imaging setup. . . . . . . . . . . . . . . . . . . . . . . .

6.8

Fast camera images from single prominence simulation experiment. Visible
band.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Current and voltage traces from single prominence simulation experiment. .

6.10 X-ray diode signals from single prominence simulation experiment. . . . . .

6.11 VUV/Soft X-ray images from single prominence simulation experiment. . .

6.12 Fast camera images from co-helicity merging experiment. . . . . . . . . . .

100

6.13 Current and voltage traces from co-helicity merging experiment. . . . . . .

101

6.14 X-ray diode signals from co-helicity merging experiment. . . . . . . . . . . .

101

6.15 VUV/Soft X-ray images from co-helicity merging experiment. . . . . . . . .

102

6.16 Fast camera images from counter-helicity merging experiment. . . . . . . .

103

6.17 Current and voltage traces from counter-helicity merging experiment. . . .

104

6.18 X-ray diode signals from counter-helicity merging experiment. . . . . . . . .

104

6.19 VUV/Soft X-ray images from counter-helicity merging experiment. . . . . .

105

A.1 Adjusting a laser to align the beam parallel to the optical table.

111

6.9

. . . . . .

A.2 Adjusting the scene beam of the heterodyne interferometer to pass through
the sapphire windows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

A.3 Alignment for the overlap of the scene and reference beams of the heterodyne
interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

A.4 Image of the 18” × 18” optical table showing the various optical components
of the homodyne interferometer. . . . . . . . . . . . . . . . . . . . . . . . .

114

A.5 The x − y “ellipse” from the signals of the two detectors of the homodyne
interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

A.6 Image of the 18” × 18” optical table showing the various optical components
of the heterodyne interferometer. . . . . . . . . . . . . . . . . . . . . . . . .

119

xvi

List of Tables

1.1

Measured parameters for the plasma jets . . . . . . . . . . . . . . . . . . . .

1.2

Some derived quantities for the plasma jets . . . . . . . . . . . . . . . . . .

3.1

Typical parameters of the discharge circuit of the Caltech spheromak experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

Description of some of the components used in the design of homodyne interferometer (refer to figure 4.4). . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Description of some of the components used in the design of the heterodyne
interferometer (refer to figure 4.6). . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1
Introduction

Plasma is an ionized gas. Some of the applications of plasma physics are fusion energy
research, plasma processing, arcs, space propulsion, and understanding many solar and
astrophysical phenomena.
The fusion reaction between deuterium and tritium has the highest reaction rate [1]
(∼ 10−22 m3 /s) at moderately high temperatures (∼ 10 keV). It yields Ereaction = 17.6 MeV
per reaction.
D + T → He4 (3.5 MeV) + n(14.1 MeV).
To achieve fusion, hot and dense D-T plasma has to be confined long enough for substantial
fusion reactions to occur. A fusion reactor will be profitable if the energy spent in confining
the hot plasma is less than the energy output from the fusion reaction. This argument is
used to derive the Lawson criterion
nτE &

kT
∼ 1020 − 1021 sec × m3 ,
Ereaction × reaction rate

(1.1)

where n is the plasma density, τE is the energy confinement time, and T is the temperature
of the plasma. A tokamak is a donut shaped device that confines hot plasma particles on
toroidal magnetic flux surfaces. The tokamak concept is the most widely pursued magnetic
fusion reactor design. The ITER device [2] being built in Cadarache, in the South of France
is an experimental reactor which is expected to demonstrate an energy efficiency of 10
by confining hot plasma (T ∼ 10 keV, n ∼ 1020 m−3 ) with confinement time of ∼ 4 s
(the experimental pulse will be substantially longer ∼ 500 s). However, there are huge

technological and monetary constraints for building tokamak reactors. The budget of ITER
is approximately 10 billion US dollars.
Spheromaks [3] are simply connected (and hence topologically simple) plasma configurations which may provide a cheaper alternative to other fusion reactor designs. The Sustained
Spheromak Physics Experiment (SSPX) at the Lawrence Livermore National Laboratory
has been the most successful spheromak experiment to date. It achieved plasma temperatures of few 100 eV and nτE approximately three orders of magnitude less than the Lawson
criterion [4] described in equation (1.1). Huge improvement in the spheromak performance
is required to make it a viable fusion reactor. The progress in the last three decades has
been exceptional and substantial progress is expected in near future.
Spheromaks configurations are constrained minimum energy states, and thus plasmas
have a natural tendency of evolving toward a spheromak equilibrium. Thus, concepts of
spheromak research have often been used to explain plasma behavior in many naturally
evolving plasma structures, for example solar prominences [5] and astrophysical jets [6].

1.1

Magnetohydrodynamics-MHD

Magnetohydrodynamics (MHD) is a description of the plasma which models it as an electrically conducting fluid [7, Chapter 2.6]. A MHD description of plasma behavior is valid
under the following assumptions:
1. Plasma is charge neutral. This is true when considering plasma behavior at length
scales much greater than the Debye length.
2. The plasma is collisional, or equivalently the collision times are much shorter than the
characteristic timescales of the experiment. This ensures that the particle distribution
is Maxwellian.
3. The plasma behavior under consideration has velocity much smaller than the speed
of light.
4. Either the timescales under consideration are much longer than the ion cyclotron
frequency or the electron cyclotron frequency is much smaller than the electron-ion
collision frequency.

5. The pressure and density gradients are parallel in the plasma.
The mass conservation equation in MHD is
∂ρ
+ ∇ · (ρu) = 0,
∂t

(1.2)

where ρ is the mass density of the plasma and u is its velocity. The MHD equation of
motion is

∂u
+ u · ∇u
∂t

= J × B − ∇P,

(1.3)

where J is the current density, B is the magnetic field and P the fluid pressure. Using
equation (1.2), the left hand side of equation (1.3) can be expressed as

∂u
+ u · ∇u
∂t

∂ (ρu)
∂ρ
+ ∇ · (ρuu) −
+ ∇ · (ρu) u
∂t
∂t
∂ (ρu)
+ ∇ · (ρuu) .
∂t

Thus, the MHD equation of motion can be expressed in the following alternative form
∂ (ρu)
+ ∇ · (ρuu) = J × B − ∇P.
∂t

(1.4)

E + u × B = ηJ,

(1.5)

MHD Ohm’s law is

where E is the electric field in the plasma and η is its resistivity. In a plasma with negligible
resistivity, the MHD Ohm’s law can be used to show that the magnetic flux linked by any
closed loop in the plasma is conserved.
The MHD heat transport equation [7, page 73],[8, section 10.6] is
∂t

NP
ρu2

(N + 2)P
ρu2
+∇· q+
u+
= J · E − Sl
= ηJ 2 − J · (u × B) − Sl , (1.6)

where N is the dimensionality of the system (usually N = 3), q is the heat flux, and Sl is
the rate of energy loss by radiation.

1.1.1

Energy Conservation in MHD

Dotting electric field E with the pre-Maxwell form of Ampere’s law gives
E · ∇ × B = µ◦ J · E.
Similarly, dotting the magnetic field with Faraday’s law gives
B·∇×E=−

∂B
· B.
∂t

Subtracting these two equations and using the vector identity ∇ · (E × B) = B · ∇ × E −
E · ∇ × B gives
∇·

E×B
µ◦

=−

∂ B2
− J · E.
∂t 2µ◦

Using equation (1.6) gives
∇·

(N + 2)P
E × B ρu2
u+
u+q
µ◦

=−
∂t

B2
ρu2 N P
2µ◦

− Sl .

This can be integrated over the entire volume of plasma and over time. By Gauss’ law, the
volume integral of the terms on the left-hand side turns into a surface integral. If the plasma
velocity is zero at the boundary of the volume, the surface integration terms involving the
plasma velocity are zero as well. Assuming that there is no heat flux at the boundary, we
get
Z t

Z
Z Z t
B×E
B2 3
ρu2 3
NP 3
d r+
d r+
d r+
Sl dt d3 r .
µ◦
2µ◦
{z
} | {z } | {z } | {z } |
{z

ds ·

Winput

Wmag

Wkin

Wth

(1.7)

Wradiation-loss

equation (1.7) is the MHD energy conservation equation. Note that ds is pointing outwards
from volume bounding the plasma. The left-hand term (Winput ) is the energy input into the
plasma and is the surface integral of the Poynting flux over time. If the plasma is bounded
Rt
by electrodes which link external current, then Winput can also be expressed as 0 V Idt.
V is the voltage at the electrodes, and I is the current linked by the electrodes. Wkin is
the kinetic energy in the plasma and Wth is the thermal energy. The thermal energy of
the plasma can be increased by adiabatic compression or from Ohmic heating from the ηJ 2

term in equation (1.6). Wradiation-loss accounts for the radiation losses in the plasma. The
energy lost in radiation also comes from the Ohmic term ηJ 2 . Wmag is the energy in the
magnetic field in the plasma. In case of azimuthically symmetric plasma configurations,
Wmag can be expressed as the sum of toroidal and poloidal field energies.
Bpol
B2 3
Btor
d r=
d r+
d3 r .
2µ◦
2µ◦
2µ◦
| {z } | {z } | {z }

Wmag

1.2

Wtor

(1.8)

Wpol

Helicity Injection and Spheromak Formation

Magnetic helicity in a plasma is defined as
K=

A · Bd3 r,

where A is the magnetic vector potential. Magnetic helicity as defined above is gauge invariant if there is no normal magnetic field component on the surface bounding the volume
under consideration. However, this condition is not satisfied in most spheromak experiments. For such scenarios, an alternate concept of relative helicity is used [3, Section 3.5].
Krel =

(A · B − Avac · Bvac ) d3 r,

where Avac and Bvac are the vacuum solutions to the magnetic vector potential and the
magnetic field inside the volume, and Bvac satisfies the same boundary conditions as B.
The dissipation of relative helicity by plasma resistivity and its implications on plasma
equilibrium is considered in chapter 2.
Woltjer [9] showed that the helicity is conserved in an ideal plasma, and conjectured that
in a slightly resistive plasma, magnetic helicity is nearly a constant in the time scale of decay
of magnetic energy. He considered a plasma with negligible thermal energy (P → 0), and
used a variational principle to show that the minimum energy state satisfying the constraint
of constant helicity is characterized by
∇ × B = λB,

(1.9)

z/h

0.2
−0.2
−0.5

r/a

0.5

Figure 1.1: Closed poloidal flux surfaces in an isolated spheromak in a cylindrical flux
conserver with aspect ratio r/z = 1.9.
where λ is constant throughout the volume. Plasmas defined by equation (1.9) are often
called force-free plasmas, as equation (1.3) shows that there are no forces in a uniform
plasma if current density is parallel to the magnetic field.
Taylor [10] argued that magnetic reconnection conserves the total magnetic helicity in a
plasma, but lowers the magnetic energy to a force-free plasma state. He used this hypothesis
to explain the spontaneous existence of a reversed toroidal field in a toroidal device called
a reversed field pinch (RFP).
The relative magnetic helicity conservation equation is
dKrel
dt

(2V B) · dn = −2
∂V

ηJ · Bd3 r.

It shows that magnetic helicity can be injected into a plasma by maintaining an electric
potential difference V across open magnetic field lines on its boundary ∂V. This is the most
common technique for helicity injection in spheromak plasmas and is called electrostatic
helicity injection. Initial spheromak experiments were non steady-state experiments which
involved electrostatic helicity injection and relaxation of the plasma into a force-free state
in a flux conserver. The eigenvalue λ in the force-free equation (1.9) is determined by the
shape of the flux conserver. Spheromak plasmas in the flux conserver can have closed flux
surfaces which are beneficial for particle confinement. Figure 1.1 shows an example of the
closed poloidal flux contours in a cylindrical flux conserver.
Helicity can also be injected in a spheromak experiment in steady state. Figure 1.2
shows the open and closed flux surfaces in a steady state spheromak. Particle confinement
is expected to be good on the closed flux surfaces, however chapter 2 shows that such a

1.4
1.2

z/h

0.8
0.6
0.4
0.2

−0.5

r/a

0.5

Figure 1.2: Open and closed poloidal flux surfaces in an steady state spheromak in a
cylindrical flux conserver with aspect ratio r/z = 0.67.
system can not be in a strict equilibrium. Magnetic helicity conservation in a steady state
spheromak was experimentally demonstrated by Barnes et al. [11].

1.3

Overview of the Caltech Spheromak Experiment

The Caltech spheromak experiment is a pulsed helicity injection experiment without a flux
conserver. The experiment employs a coaxial planar electrode design for helicity injection.
Figure 1.3 shows the design of the electrodes. The setup is installed inside a cylindrical
vacuum chamber with length ∼ 1.6 m and diameter ∼ 1.4 m. The inner disk cathode is
20 cm in diameter, and the outer annular anode has a diameter of 50 cm. Plasma is created
by discharging up to two high voltage 59 µF capacitors across the electrodes. Details of the
discharge circuit are presented in chapter 3.
Throughout this thesis, a cylindrical coordinate system {r, φ, z} is considered while
describing plasma dynamics in the spheromak experiment. The z axis is coming out of the
plane of the figure 1.3, r axis points radially outwards from the center of the electrodes,
and φ is the toroidal direction to form a right-handed coordinate system around the z axis.

Cathode (Radius = 10 cm)
Inner gas orifices(8)

Outer gas orifices(8)

Gas feed lines
Anode
Outer radius = 25 cm
Magnetic probe

Figure 1.3: Electrodes in the Caltech spheromak experiment.

Anode

z=0

Cathode

B pol,J pol

Anode

Apex of jet

B tor,J tor

Figure 1.4: Cartoon showing the definition of the cylindrical coordinate system for the
Caltech spheromak experiment. The red arrows represent the poloidal r − z direction, and
the green arrows represent the toroidal φ direction. The poloidal current in the experiment
is driven by an external capacitor bank linked to the electrodes. The poloidal current creates
a toroidal magnetic field. An external magnetic field coil (see figure 1.5) creates a poloidal
magnetic field in the experiment.

Anode
Bias field coil

Cathode
Fast gas puff valve

Figure 1.5: Cartoon showing the location of bias field coil behind the cathode of the Caltech
spheromak experiment. Also shown is the location of a fast gas puff valve. Image courtesy
of Paul Bellan.
The electrodes are located at z = 0. Vectors which lie in the r − z plane are referred
to as poloidal vectors. Vectors oriented along the φ̂ direction are referred to as toroidal
vectors. Poloidal and toroidal vectors are often referred to by subscripts “pol” and “tor”
respectively. Also, scalar quantities which are independent of φ are referred to as poloidal.
The coordinate system is described in figure 1.4.
A poloidal vacuum magnetic field created by a 2.8 mH coil behind the electrodes links
the two electrodes (see figure 1.5). The coil was powered by an electrically isolated 14.4 mF
capacitor bank. The voltage of the capacitor bank could be varied to create a magnetic flux
of up to 4.9 mWb at the cathode. The applied flux remains constant for the time scale of
the experiment and is referred to as the poloidal bias flux.1 Changing the direction of the
bias flux did not cause any change in the plasma behavior in the experiment.
Neutral gas was puffed near the electrodes using 16 orifices, eight each on the two
electrodes (see figure 1.5). Two fast gas puff valves [3, Chapter 14] each were used to
supply neutral gas at the anode and the cathode.
The following sequence of events was followed to create a plasma discharge (refer to

Calibration of the poloidal bias flux was done by Auna Moser, Gunsu Yun, and Deepak Kumar.

Anode

Poloidal field switched at t=−10ms

Gas puffed at t=−2ms

Ignitron switched at t=0

Cathode

Anode

Anode
Cathode
Anode

puffed near the electrodes, and (d) Spider leg formation (plasma is represented by red color) from neutral gas breakdown.

show: (a) Initial vacuum in front of the electrodes, (b) Poloidal flux surfaces created by the bias coil, (c) Neutral gas (green) being

Figure 1.6: Cartoon showing the sequence of events leading to plasma breakdown in the Caltech spheromak experiment. The figures

Initial Vacuum

Jet diameter ∼ 6 cm

0.50 µs
(a) Eight spider legs

5.00 µs
(b) Collimated jet

7.25 µs
(c) Kink unstable jet

Figure 1.7: Visual images of the three distinct stages of plasma evolution in the Caltech
spheromak experiment.
figure 1.6):
1. The poloidal flux power supply was triggered 10 ms before the plasma discharge. It
takes 10 ms for the magnetic field to reach its maximum and link the two electrodes.
2. Fast gas valves were triggered ∼ 2 ms before the plasma discharge. It takes ∼ 2 ms
for neutral gas to travel from the fast gas puff valves to the orifices.
3. The high voltage capacitor bank was discharged across the electrodes to create a
plasma. Initially the plasma links the gas nozzle along poloidal magnetic field lines.
The eight plasma filled flux tubes linking the two electrodes resemble the eight legs
of a spider as shown in figure 1.7(a). The capacitor bank drives a poloidal current
through the plasma (see figure 1.4), which creates a toroidal magnetic field.
4. The various diagnostics and the digitizers were triggered upon the neutral gas breakdown near the electrodes.
Figure 1.7, shows the typical stages in plasma evolution after the breakdown - eight
“spider legs” linking the electrodes, a collimated plasma jet, and a kink unstable plasma
column. The structure of the spider legs was investigated by You et al. [12]. The properties
of the collimated jet are described in chapter 5 of this thesis. The kink instability of the
plasma jets was studied by Hsu and Bellan [13].

1.3.1

Diagnostics

The following diagnostics were used in the experiment:
1. A Tektronix P6015 [14] high-voltage probe was used to measure the potential difference across the electrodes.
2. A Rogowski coil [3, Chapter 15] was used to measure the current flowing through the
high voltage capacitor bank.
3. Imacon 200-a high speed imaging camera manufactured by DRS Technologies [15] was used to take visible images of the plasma.
4. A He-Ne laser interferometer to measure the density of the plasma jets. Details of the
interferometer are described in Chapter 4.
5. A 60-element magnetic probe array [16] was used to measure the magnetic field in
the experiment (see figure 1.3). The probe measured magnetic field along {r, φ, z}
directions at 20 different radial locations separated by 2 cm. The axial position
(distance from the electrodes along the z direction) of the probe could be varied.
Assuming toroidal symmetry, the Ampere’s law can be used to calculate the poloidal
current from the toroidal magnetic field measurements:
I(r, z) =

2πrBφ (r, z)
µ◦

The poloidal current I(r, z) is calculated only at discrete radii, corresponding to the
location of the measurement coils in the magnetic probe.
The poloidal flux can be calculated using
Z r
ψ(r, z) =

Bz (r0 , z)2πr0 dr0 .

(1.10)

The integration over discrete radial locations in equation (1.10) may introduce substantial errors. Also, the calculation of poloidal current and flux assumes that the
magnetic probe is oriented along the radial direction. However, there is no mechanism to ensure this accurately.

1.3.2

Experimental Parameters and Dimensionless Numbers
Table 1.1: Measured parameters for the plasma jets
Symbol

Value

Comment

(Parameter)

5 − 10 µs

Experimentally observed.

∼ 0.3 m

Inferred from visual plasma images.

timescale of the jet
length of the jet

See figure 1.7(b).
∼ 0.03 m

radius of the jet

Inferred from visual plasma images.
See figure 1.7(b).

∼ 3 × 1022 m−3

Described in chapter 4.

plasma density

0.1 − 0.2 T

By magnetic probe measurements [16].

∼ 2 eV

By spectroscopic measurements [17,

magnetic field
Ti
ion temperature
Te

Page 79].
∼ 2 eV

electron temperature

The electron and ion temperatures are
expected to equilibrate because the
plasma is highly resistive. νei
τ −1
(Table 1.2).

∼ 40 km/s

Described in chapter 5.

jet axial velocity
Table 1.1 shows some of the experimentally measured parameters of the plasma jets.
These parameters were used to derive the quantities listed in Table 1.2.2

Rory Perkin’s help in formulating this table is greatly acknowledged.

Table 1.2: Some derived quantities for the plasma jets
Symbol

Parameter

vT i

ion thermal velocity

Formula
1

electron thermal velocity

λD

Debye length

particles in a Debye sphere

vT e

νei

collision rate

plasma resistivity

ωce

2kTi
mi

2kTe
me
ε◦ kT
nq 2

Value
∼ 20 km/s

∼ 800 km/s

∼ 0.1 µm

n 4π
3 λD

∼ 100

nq 4

∼ 300 GHz

2πε2◦ me2 (2kTe ) 2

electron cyclotron frequency

me νei
nq 2
qB
me

∼ 2 × 1010 rad/s

ωci

ion cyclotron frequency

qB
me

∼ 107 rad/s

ωpe

electron plasma frequency

nq 2
me ε◦

ωpi

ion plasma frequency

nq 2
mi ε◦

electron gyroradius

vT e
ωce

∼ 50 µm

ion gyroradius

vT i
ωci

∼ 0.2 cm

Bohm diffusion coefficient

1 kTe
16 qB

∼ 1 m2 /s

τB

Bohm time

r2
2DB

∼ 500 µs

δp

plasma skin depth

ωpe

∼ 30 µm

magnetic Reynold’s number

uz r

∼ 106

∼ 3 × 10−4 Ωm

∼ 1013 rad/s
∼ 2 × 1011 rad/s

Chapter 2
On Magnetic Helicity Injection in a Steady State
Scenario

Many driven plasma experiments consist of closed flux surfaces maintained by externally
linked currents along open field lines linking the electrodes. This chapter shows that it is
not possible to sustain magnetic helicity in such configurations with static magnetic fields.
Consider an externally driven resistive MHD plasma in a simply connected volume V
bounded by a perfect conductor with gaps (see figure 2.1). A driven configuration attempts
to sustain magnetic helicity in the plasma by having open magnetic field lines linking boundary surfaces at different potentials (electrostatic helicity injection) or by generating time
dependent surface potential and magnetic fields inductively (AC helicity injection). Such
systems have been used in many sustained spheromak experiments for both electrostatic
helicity injection [11, 18, 19] and AC helicity injection [20]. This chapter examines whether
a truly static equilibrium is possible for such driven systems.
Consider the relative magnetic helicity conservation equation in a resistive plasma [3,
Chapter 3]
dKrel
dt

(2V B) · dn = −2

∂V

ηJ · Bd3 r.

(2.1)

For further analysis, the volume of plasma is divided into open and closed flux tubes as
shown in figure 2.1. The open flux tube links electrodes having different potentials V+ and
V− respectively. An external source drives a current I◦ through the open flux tube. The
potential appearing at the electrodes is the cumulative effect of the resistive drop across the
open flux tube and also inductive voltages from the self inductance of the open flux tube and

ψ°

dl
ds
V+
V−

Figure 2.1: A cartoon showing an open flux tube (in blue) having a flux ψ◦ and an externally
linked current I◦ . The volume in red is a closed flux tube carrying a current I1 .
mutual inductance from the closed flux tube. For simplicity, we will consider just a single
open flux tube with infinitesimally small cross section, but the arguments presented in this
chapter can be extended to more than one open flux tube. Consider the right-hand-side
term of equation (2.1) within the volume V◦ of the open flux tube (blue region in figure
2.1):
−2

ηJ · Bd r = −2
V◦

ZV◦
= −2

(E + U × B) · Bd3 r
E · Bd3 r

V◦

= −2

(−∇V −
V◦

∂A
) · Bd3 r.
∂t

(2.2)

The first term in equation (2.2) can be expressed as

∇V · Bd r = 2

V◦

∇ · (V B) d r = 2
V◦

V B · dn,
∂V◦

which is exactly equal to the electrostatic helicity injection term in equation (2.1). The

17
second term in equation (2.2) can be expressed as

∂A
· Bd3 r = 2
∂t
V◦

∂A
· Bdl · ds = 2
V◦ ∂t

∂A
· dlB · ds = 2ψ◦
V◦ ∂t

∂A
· dl,
V◦ ∂t

(2.3)

where dl and ds are infinitesimal length and area elements along the open flux tube (refer
to Fig. 2.1) and ψ◦ = B · ds is the flux, which is constant throughout the length of the flux
tube. In deriving equation (2.3) we used the fact that dl, ds and B are parallel to each
other in the flux tube.
Thus equation (2.1) can be rewritten as
dKrel
− 2ψ◦
dt

∂A
· dl = −2
V◦ ∂t

ηJ · Bd3 r.

(2.4)

V−V◦

Note that the helicity source terms in equation (2.4) depend on a time-dependent magnetic
vector potential. This shows that helicity cannot be sustained in a driven plasma having
closed flux surfaces (V − V◦ 6= 0) with the plasma being in static equilibrium. Equation
(2.4) also shows that a resistive MHD equilibrium is not possible in a plasma containing
closed flux tubes (surfaces).
It is often considered that the rate of helicity injection into the plasma is proportional
to the voltage appearing across the electrodes with open field lines. However, equation (2.4)
clearly shows that meaningful helicity is injected only by the fluctuating voltage appearing
at the electrodes and not by the voltage caused by the resistive drop across the plasma.
A time-changing magnetic vector potential implies a time changing current distribution
in the plasma. Equivalently, it implies fluctuating topological changes in the plasma. High
node number (n 6= 0) modes and turbulent fluctuations have been observed in sustained
spheromak experiments during helicity sustainment [11, 18, 21].
At the SSPX experiment [21] it was found that once the gun current exceeded a soft
threshold limit, detached flux and current channels were formed (V − V◦ 6= 0). In this
regime, oscillations (10 − 100 kHz) in gun voltage and measured poloidal magnetic field
were observed. We argue that such oscillations are ubiquitous with helicity sustainment
against resistive decay. In fact the SSPX experiment was able to achieve a quiescent state
with higher electron temperatures Te and consequently lower resistive decay rates by altering
the bias magnetic field profile [19]. Since helicity injection was minimal during the quiescent

18
stage, the mode activity was suppressed. However, after the quiescent stage, when the gun
current started decaying, the mode activity increased.
The SPHEX experiment [18] found similar n = 1 oscillations at 20 kHz during the
sustainment stage and no oscillations in the resistive decay stage. The sense of oscillations
during the sustainment stage changed with the sign of total helicity being sustained.
Sustained spheromak configurations can be considered to be in a quasi-equilibrium with
small fluctuations. A naive consideration of equation (2.4) suggests that zero mean fluctuations in magnetic vector potential cannot balance the mean loss of helicity by the resistive
term on the right side. However, we now argue that this conclusion is false. Consider the
case that a current I◦ flows along the open flux tube and does not leak outside the open
flux tube. If so, then akin to Kirchoff’s voltage law, V◦ ∂A/∂t · dl can be expressed as a
sum of voltages induced by self inductance and mutual inductance
dKrel
− 2ψ◦
dt

∂(I◦ L◦ ) X ∂(Ii Mi )
∂t
∂t
i=1

= −2

ηJ · Bd3 r,

(2.5)

V−V◦

where L◦ is the self inductance of the current carrying open flux tube and Mi is the mutual
inductance between the open flux tube and ith closed current loop carrying a current Ii ,
where we have split the plasma volume into i = 1, . . . , N closed current loops. In order to
inject helicity into the plasma volume electrostatically, either the self inductance of the current carrying open flux tube should change or the mutual inductance should change. Most
magnetized gun driven pulsed spheromak experiments rely on an increasing self inductance
L◦ to inject helicity (see section 2.1). Even sustained spheromak experiments [21] rely on
increasing self inductance L◦ as a means of increasing helicity before plasma detachment.
Let us now examine a quasi equilibrium solution for equation (2.5) with fluctuating
quantities-L◦ , Mi s and Ii s. With only a fluctuating self inductance, DC helicity cannot be
injected into the plasma against resistive decay as h∂(I◦ L◦ )/∂ti = 0. Similarly, DC helicity
cannot be injected with only fluctuating mutual inductances (Mi s) and closed currents (Ii s).
However, a repetition of the following sequence of events can inject mean helicity through
fluctuating quantities: The expansion of the open flux loop increases the self inductance L◦
thereby injecting helicity into the plasma. Meanwhile the mutual inductance or the closed
currents in the plasma decay resistively. When the self inductance increases beyond a certain

19
limit, reconnection converts open flux to closed flux. The dynamical reconnection process
lowers the self inductance L◦ but increases the mutual inductance and closed currents.
The cycle thus continues and mean helicity is injected by fluctuating quantities. Such a
dynamo-like process is an essential requirement for sustained spheromaks. The essence of
this chapter is showing that this dynamo process cannot be time independent.

2.1

Helicity Injection in Pulsed Spheromak Experiments

Pulsed spheromak experiments [13, 22] do not have closed flux tubes during the initial
plasma ejection stage. For such systems, equation (2.5) can be expressed as
dKrel
∂(I◦ L◦ )
− 2ψ◦
= 0.
dt
∂t

(2.6)

Integrating equation (2.6) over time yields the injected helicity,
∆Krel = 2ψ◦ I◦ L◦ ,

(2.7)

which is twice the product of the imposed bias flux ψ◦ and the flux generated by the electrode
current L◦ I◦ . Thus, to comprehend the amount of helicity injected in pulsed spheromak
experiments, it is paramount to understand the dynamics leading to a change in the self
inductance L◦ of the open flux tubes. Chapter 5 focuses on understanding the mechanism
causing the change in self inductance L◦ of the open flux tube in the Caltech spheromak
experiment.

Chapter 3
Electrical Characterization of the Discharge
Circuit of the Caltech Spheromak Experiment

Figure 3.1 shows the schematic of the discharge circuit of the Caltech Spheromak Experiment. It consists of two high voltage (HV) capacitors (∼ 59 µF) each switched by a separate
ignitron [23–25]. The Caltech Spheromak Experiment uses size A GL-7703 ignitrons [26].
The ignitrons are each connected to the discharge electrodes by four low inductance coaxial
cables. A matched resistor (∼ 2.3 Ω) is connected across the electrodes. The purpose of
the matched resistor is two-fold:
• It serves as a safety dump resistor for the capacitors in the event of a misfire.
• Its resistance matches the characteristic impedance of the cables and thus prevents
reflections of the initial capacitor pulse when ignitron is switched.
This chapter describes methods for calculating the various internal resistances and inductances of the discharge circuit. These values are used to determine the temporal behavior
Cables

Ignitron trigger

HV capacitor

Cables

Ignitron
Matched
resistor

Ignitron trigger

Ignitron

Plasma
HV capacitor

Figure 3.1: Schematic of the discharge circuit of the Caltech spheromak experiment.

21
of the discharge circuit and also to account for all the energy losses in the circuit. Unless
otherwise mentioned, for the measurements and estimates reported in this chapter and in
this thesis, only one capacitor and ignitron was used in the discharge circuit (see figure 3.1).
This was done:
1. To remove jitter associated with triggering two ignitrons [27].
2. To prevent excessive currents from kinking the plasma jets studied in chapter 5.
However, as discussed in section 3.3, even with two capacitors being discharged, the characteristics of the discharge circuit were similar to that of a circuit with a single capacitor.
Section 3.1 describes various methods to calculate the plasma inductance and resistance.
Section 3.2 estimates the electrical resistances and inductances of the cables and the ignitron
in the discharge circuit. The results are interpreted in section 3.3 and the discharge circuit
is modeled as an under-damped current source.

3.1

Plasma Parameters

3.1.1

Plasma Parameters from Traces

Figure 3.2 shows the typical voltage and current traces measured at the electrodes across the
plasma. Note that the traces are almost out of phase, implying that the plasma is mostly
inductive. The voltage measured across the electrodes and the current flowing through the
electrodes are related by
V (t) = I(t)R(t) +
(I(t)L(t))
dt
dL(t)
dI(t)
= I(t) R(t) +
+ L(t)
dt
dt

(3.1)

where V is the voltage measured across the electrodes, I is the current flowing through the
electrodes and the plasma, L is the time-varying inductance of the plasma structure, and
R is the time-varying resistance of the plasma. At the plasma breakdown (see figure 3.2)
the inductance is due to the eight “spider legs” as shown in figure 1.7(a). As the plasma
current increases and a jet is formed, the plasma inductance is due to an outward moving
jet.

Gun current
Gun voltage

−50

−3

−10

20
30
Time (µs)

Gun voltage (kV)

Gun current (kA)

Breakdown

Figure 3.2: The current and voltage traces measured across the electrodes for shot #8500.

400

0.21

0.16

200

0.11

100

0.05

0.00

−10

Figure 3.3: Energy (=

20
30
Time (µs)

Fraction of Capacitor bank Energy

Energy (J)

∆E
300

0 V Idt) flowing into the plasma in shot #8500. A 59 µF capacitor

charged to 8 kV was discharged into a deuterium plasma. Total capacitor bank energy
= CV 2 /2 ≈ 1900 J.

23
The following estimates of plasma impedance can be made from figures 3.2 and 3.3:
• Consider the voltage and current traces from a typical plasma discharge as shown
in figure 3.2. A typical current waveform consists of approximately five half cycles.
Equation (3.1) indicates that the inductance of the plasma structure can be esti˙ Similar
mated at successive zero crossings (I ∼ 0) of the current traces by L ∼ V /I.
arguments have been used to estimate plasma inductances in other spheromak experiments [21, 28]. The plasma inductance at breakdown is estimated to be ∼ 50 nH.
The inductance for a fully developed plasma jet will be different from this estimate.
Note that the lifetime of the Caltech Spheromak is ∼ 10 µs, which corresponds to
ramping up of the current in the first half of the cycle. Thus after the initial ramping
of the current, each successive zero crossing corresponds to secondary breakdowns (or
secondary spheromak formations).
• Figure 3.3 plots the energy flowing into the plasma as a function of time. The final
steady state value of the energy is the total energy dissipated by the plasma by heating
and radiation. The steady state value of the energy dissipated is proportional to the
resistance of the plasma. However, the fluctuating part of the energy (∆E) is the
inductive energy sloshing back and forth between the plasma and the driving circuit.
The inductance of the plasma jet can be estimated as L ∼ 2∆E/I 2 . From figure 3.3,
∆E ∼ 70 J when a current of I ∼ 70 kA flows through the plasma (see figure 3.2).
Thus the typical inductance of a plasma jet is L ∼ 30 nH.
• Equation (3.1) also implies that (R + L̇) ∼ V /I when I(t) is at a local extrema.
Thus, the sum of the plasma resistance and the rate of change of inductance can be
estimated from figure 3.2 to be (R+ L̇) ∼ V /I ∼ 12 mΩ when I(t) is at its minimum at
t ∼ 4.3 µs. Note that we previously estimated the plasma jet to develop an inductance
of ∼ 30 nH in 4.3 µs, which implies L̇ ∼ 7 mΩ. Thus the jet resistance R and its rate
of change of inductance L̇ are comparable to each other.

3.1.2

Plasma Parameters from Geometry

Figure 3.4 shows a simple model of the plasma jet outflow. From such a model, the plasma
impedances can be estimated as shown below.

Electrodes

velocity (v)

Figure 3.4: A simplistic model of plasma jet. A current I flows through a central column
of radius r◦ . The current returns at radius r. The length of the jet is l and it is moving
outwards with velocity v.
The toroidal magnetic field for a typical plasma jet with l
r◦ shown in figure 3.4 is
given by
Bφ =

µ◦ I
2πr

and is non-zero only between the inner and the outer current channels of the jet. The flux
linked with the toroidal magnetic field is
Z r
Φ =

µ◦ I
ldr
r◦ 2πr
µ◦ Il
log( ).
2π
r◦

Thus the inductance is given by
L =

µ◦ l
log( ),
2π
r◦

(3.2)

Figure 3.5: A lumped circuit model for the spheromak discharge circuit. The time dependent
plasma resistance and inductance are denoted by Rp (t) and Lp (t) respectively. The resistor
Rl ∼ 2.3 Ω is a large load resistance matched to the cable impedance. The lumped resistance
of the cables is represented by Rc . The time dependent resistance and inductance of the
ignitron is denoted as Ri (t) and Li (t). The high voltage capacitor being discharged across
the electrodes is represented by Cb ∼ 59 µF.
and the rate of change of inductance is given by
L̇ =

µ◦ v
log( ).
2π
r◦

(3.3)

For typical D2 plasma jets, v ∼ 30 km/s, l ∼ 20 cm, r ∼ 25 cm, and r◦ ∼ 10 cm (see section
1.3.2). Using these values in equations (3.2) and (3.3) give, L ∼ 35 nH and L̇ ∼ 5.5 mΩ.
The plasma resistance can also be estimated by similar geometrical considerations. The
plasma ion temperature Ti is ∼ 2 eV [17, Pg 53]. Since the plasma is extremely collisional (see section 1.3.2), the plasma electron temperature Te should be equal to the
ion temperature Ti . Assuming a Coulomb logarithm of 10 implies a Spitzer resistivity
η ∼ 3.6 × 10−4 Ωm. For a plasma structure shown in figure 3.4, the resistance can be
estimated as R ∼ ηl/πr◦2 ∼ 2.5 mΩ. Due to the uncertainties involved with the Coulomb
logarithm and the plasma geometry, the plasma resistance is an extremely crude estimate.
The plasma impedances estimated in this section compare well with the values calculated
in section 3.1.1.

3.2

Circuit Model for Spheromak Experiment

Figure 3.5, shows the lumped circuit model for the spheromak discharge configuration. In
section 3.1 we estimated Rp (t) ∼ 2.5 − 5 mΩ and Lp (t) ∼ 30 − 50 nH. In this section we
describe how these parameters compare to the other impedances in the circuit.
To estimate the other impedances in the circuit, the following changes were made to the
circuit shown in figure 3.5:
• Rl is large compared to the plasma impedance and hence was temporarily removed.
• The “plasma” was replaced by a fixed dummy load1 of resistance 82 mΩ and inductance 1 µH.
A series of shots were done by charging the high voltage capacitor to 2 kV and discharging it across the dummy load. Voltages were measured at three different locations in the
circuit:
• Voltage was measured at location A (see figure 3.5). The energy flowing into the
dummy load is given by Eload = VA I dt.
• Voltage was measured at location B (see figure 3.5). The energy flowing into the
dummy load and the cables is given by Ecable+load = VB I dt.
• Voltage was measured at location C (see figure 3.5). The energy flowing into the
dummy load and the cables is given by Ecable+load+ignitron = VC I dt. Also, the
voltage across the ignitron is Vignitron = VB − VC .
Typical traces from the shots are shown in figure 3.6. The following parameters can be
estimated from these traces:
• Rc : Resistance of the cables. The total energy dissipated across the cable and the
R∞
load (see figure 3.6(c)) is 0 I 2 (Rc + Rp )dt. Since the energy dissipated across the
R∞
−Eload
load is 0 I 2 Rp dt, we get Rc = Rp cable+load
. Thus Rc ∼ 8 mΩ.
Eload
• Ri : Resistance of the ignitron. The ignitron resistance is expected to be time and
load dependent, but an average value can be estimated by a method similar to the

The dummy load was built by Auna Moser.

0.5

−1
−2

Volt (kV)

Current (kA)

−0.5

−3
−4
−5

−1

−1.5

−6
−7

−2

−8
−9
−10

20
30
Time (µs)

−2.5
−10

(a) Current flowing through the circuit. Average

20
30
Time (µs)

(b) Voltage across the ignitron.

of shots #8510 and #8513
120

100

Energy (J)

−20
−10

8513
8510
8506

20
30
Time (µs)

V I dt flowing into the dummy

load (point A in figure 3.5; blue), into the dummy
load and cables (point B in figure 3.5; red), and
into the dummy load, cables and ignitron (point
C in figure 3.5; black)

Figure 3.6: Measurements from discharging the high voltage capacitor charged to 2 kV
across the dummy load.

28
method described for estimating Rc above. Ri ∼ Rp

Ecable+load+ignitron −Eload
− Rc .
Eload

leads to a nominal estimate of Ri ∼ 22 mΩ.

ignitron
The ignitron resistance may also be estimated by Ri ∼ Iignitron
evaluated at time t1

from figures 3.6(a) and 3.6(b). This leads to an estimate Ri ∼ 21 mΩ.
• Li : Inductance of the ignitron. The ignitron inductance depends on the current
flowing through it. An average value of the ignitron inductance can be estimated by

Li = I˙ignitron , when Iignitron ∼ 0. From figure 3.6(a) Li ∼ 170 nH at time t0 . Similar
ignitron

estimates for Li were found by discharging slightly larger currents through the dummy
load. During normal operation of the spheromak experiment, there is almost a 10 fold
increase in the current through the ignitron. Thus, it is plausible that the inductance
of the ignitron may be slightly higher during normal operation because of the pinching
effect associated with higher currents. Also, from figure 3.6(a), Li ∼ 800 nH at the
turnoff time t2 .

3.3

Results and Interpretation

Table 3.1 summarizes the main estimates from this chapter. Note that Lp , Rp and L̇p were
estimated for deuterium jets. Lp and L̇p may be lower for heavier gases, but as described
later in this section, this will not change the characteristics of the discharge circuit. The
following conclusions can be drawn from these estimates:
1. Low energy coupling efficiency: During a plasma discharge only Rp /(Rp + Rc + Ri ) ∼
15% of the initial capacitor energy is dissipated into the plasma. This estimate is in
agreement with figure 3.3. It should be noted, however, that the energy fraction being
coupled into the plasma increases with an increase in plasma resistance or decrease in
ignitron resistance. Typically, it is observed that even with varying parameters not
more than 35%−40% of the energy is coupled to the plasma at the Caltech spheromak
experiment. The high resistance of the ignitron makes it a very inefficient technology
to couple power into the plasma. This was observed in other experiments as well [29].
2. Under-damped discharge circuit: For the typical parameters shown in Table 3.1, the
discharge circuit is under-damped [30, Chapter 9.6]. In an under-damped circuit the

Table 3.1: Typical parameters of the discharge circuit of the Caltech spheromak experiment.
Parameter

Approximate

Reference

estimate
Cb

59 µF

N/A

2.7 × 105 rad/sec

Figure 3.2

capacitance of bank
ωd a
damped frequency of discharge
Rp

5 mΩ (measured),

plasma resistance

2.5 mΩ (Spitzer)

30 nH (typical),

plasma inductance

50 nH (at breakdown)

ωd Lp

Section 3.1

8 − 13 mΩ

N/A

6 − 7 mΩ

Section 3.1

8 mΩ

Section 3.2

21 − 22 mΩ

Section 3.2

plasma inductive impedance
L̇p
rate of change of plasma inductance
Rc
cable resistance
Ri
ignitron resistance

170 nH (typical),

ignitron inductance

800 nH (at ignitron turnoff)

ωd Li

45 mΩ (typical),

ignitron inductive impedance

210 mΩ (at ignitron turnoff)

ωd = π/τ , where τ is the first zero crossing time in the current waveform.

Section 3.2

N/A

Current (kA)

−20

−40

−60

−60
#9057, Discharge Voltage=8 kV
#9114, Discharge Voltage=7 kV
#9157, Discharge voltage=6 kV

−80
−100
−10

20
Time (µs)

#9923, Poloidal flux=4.2 mWb
#9928, Poloidal flux=3.5 mWb
#9929, Poloidal flux=2.8 mWb

−80
−100
−10

20
Time (µs)

(a) Current traces from Hydrogen plasma shots (b) Current traces from Hydrogen plasma shots
with varying discharge voltages.

with varying stuffing flux.

Current (kA)

−20
−40

−40
−60

−60

−80

−80
−100
−120
−10

−20

#9057, Hydrogen plasma
#9300, Deuterium plasma

20
Time (µs)

−100
−120
−10

#8110, 2 Capacitors at 6 kV
#9057, 1 Capacitor at 8 kV

20
Time (µs)

with one and two capacitors in the discharge circuit.

Figure 3.7: Current trace from various plasma shots. Figures 3.7(a), 3.7(b) and 3.7(c) traces
show that even with varying plasma parameters, the temporal behavior of the discharge
current trace changes insignificantly, thus confirming that the discharge circuit acts as
a current source. Figure 3.7(d) shows that, even with two capacitors being discharged
across the plasma, the circuit was under-damped with similar temporal behavior. This is
because even though the capacitance in the circuit increased two-fold, the resistance and
the inductance also increased almost two-fold due to the extra ignitron.

31
current trace oscillates and reverses sign. The frequency of these damped oscillations,
R 2
ωd , is given by ωd = | 2L
− LC
|. Plugging in nominal values of C = 59 µF,
L = 200 nH, and R = 30 mΩ, we get ωd = 2.8 × 105 rad/sec, which is very close to
the measured frequency mentioned in Table 3.1.
3. Discharge circuit is a current source: Table 3.1 shows that the combined impedance of
the ignitron and the cables dominates the impedance of the plasma. Thus the plasma
impedance plays a negligible role in determining the profile of the current trace in the
circuit. It is observed that even when the parameters of plasma formation are varied,
the frequency of damped oscillations remains close to 2.7×105 rad/sec (see figure 3.7).
Thus, the Caltech spheromak discharge circuit can be modeled as a current source
driving an inductive plasma load.
4. Ignitron cannot be used as a crowbar device: For many inductive loads, a crowbar
device is placed across the inductive load to recycle the current through the load [31].
Switching devices like solid state diodes or ignitrons are used as crowbars to mantain
a high uni-directional current through the load and thus prevent high reverse voltage
on the main capacitors. However, an ignitron cannot be used as a crowbar for the
Caltech spheromak experiment as the ignitron impedance is much greater than the
plasma impedance. Previous attempts to use the ignitron as a crowbar device for the
Caltech spheromak experiment have been unsuccessful.
The energy efficiency and other electrical characteristics of the Caltech Spheromak Experiment are close to parameters of other pulsed plasma experiments [32]. For pulsed
experiments using spark gap switches instead of ignitrons, much higher energy coupling
efficiencies (∼ 90%) have been reported [33]. This is because spark gaps have a much lower
resistance (∼ 1 mΩ).

3.4

Ignitron Characterization

This section discusses a couple of characteristics of the ignitron GL-7703 [26] used in the
Caltech spheromak experiment.
1. The GL-7703 is a commercially available low inductance ignitron. The product data

32
15
10

Current (kA)

−5
−10
−15
−20
−25
−10

20
30
Time (µs)

Figure 3.8: The current trace measured across the electrodes for shot #8553 when the HV
capacitor was discharged across a dummy load. The ignitron stopped conducting briefly.
sheet [26] lists its approximate inductance to be 20 nH. However, our analysis estimates
the inductance to be 170 − 800 nH. The inductance of the ignitron depends on the
return path of the current outside the ignitron, but 20 nH is still an ultra optimistic
estimate.
2. The GL-7703 can remain conducting even if the current reverses across its terminals.
If the current does not reverse fast enough, the ignitron may turn off, as seen in
figure 3.8. The ignitron turn off may depend on factors like external temperature,
peak current flowing through the ignitron, and the reverse voltage across the ignitron
as the current approaches zero [24]. One of the most important factors determining
˙ the rate of current change as I ∼ 0. Figure 3.8 may be used
ignitron turn off is I,
to estimate a nominal rate of current change I˙ = 2 kA/µs required to prevent the
ignitron from turning off.2 However, a much lower cut off limit of I˙ ∼ 5 A/µs is often
cited for larger, high inductance ignitrons [24].

Note that the plasma experiments operated at much higher currents (> 60 kA), and so did not experience

ignitron turn off.

Chapter 4
Interferometer for the Caltech Spheromak
Experiment

Laser interferometry is an extensively used diagnostic for plasma experiments. Existing
plasma interferometers [34–39] are designed on the presumption that the scene and reference
beam path lengths have to be equal, a requirement that is costly in both the number of
optical components and the alignment complexity. It is shown in this chapter that having
equal path lengths is not necessary - instead what is required is that the path length
difference be an even multiple of the laser cavity length. This fact was used in the design of
a homodyne and a heterodyne laser interferometer for the Caltech spheromak experiment.
These interferometers measured typical line-average densities of ∼ 1021 /m2 with an error
of ∼ 1019 /m2 .
The homodyne interferometer was the first interferometer developed for the Caltech
spheromak experiment. It was later replaced by the heterodyne interferometer because
of specific advantages describes in Section 4.5.6. However, the uniqueness of both the
interferometers was that they operated at a large path length difference between the scene
and the reference beams - a feature which is often not utilized on existing interferometers.
This chapter is organized as follows. Section 4.1 describes the relation between the
density of the plasma and the induced phase change of an electromagnetic wave travelling
through it. Section 4.2 describes the design criteria for the interferometers. Section 4.3
shows that the laser phase auto-correlation function, a measure of the coherence, is a quasiperiodic function of the path length difference between the two beams of an interferometer.
Sections 4.4 and 4.5 describe the homodyne and heterodyne interferometers built for the

34
Caltech spheromak experiment. The alignment procedure for both the interferometers is
described in Appendix A.

4.1

Electromagnetic Wave Dispersion Relation in a Plasma

The dispersion relation of an electromagnetic wave travelling through the plasma is[7, Chapter 4]
ω 2 = ωpe
+ k 2 c2 ,

where the electron plasma frequency, ωpe , is given by
ωpe

ne e2
◦ me

If ω
ωpe , then the wavenumber k can be approximated as
!1

k =

ωpe
1− 2

ωpe
1− 2
2ω

Using the above relation, the phase shift in a beam traversing a length L through the plasma
is given by
Z L
φp =

k dx

Z L

ωpe
1− 2
2ω

dx.

The first term in the above integral is the phase shift experienced by a beam travelling
through vacuum. Thus the change in the phase shift caused by the plasma (second term) is
∆φp =

e2 λ
4πc2 ◦ me

Z L

n(x)dx
Z L
−15
= 2.8 × 10 λ
n(x)dx,

(4.1)

35
where the wavelength λ and the length L are expressed in m, and the density n(x) in m−3 .
Equation (4.1) can be used to calculate the line average plasma density from the measured
phase shift
n̄(x) =

4.2

∆φp
2.8 × 10−15 λL

Design Considerations for the Interferometer for the Caltech Spheromak Experiment

As discussed in Section 1.3.2, the plasma density in the jets produced by the Caltech
Spheromak Experiment is n ∼ 1022 m−3 . This corresponds to an electron plasma frequency
of ωpe ∼ 5 × 1012 rad/sec. The frequency of operation of a He-Ne laser is ω ∼ 3 × 1015 rad/s,
so the condition ω
ωpe is satisfied. The laser beam passes through a typical length of
L ∼ 0.1 m of plasma. This will result in an expected phase shift caused by the plasma on
the order of ∆φp ∼ 2 rad, which is of the order of a fringe shift and should be measurable
by the interferometer.
The interferometer for the Caltech Spheromak Experiment was designed as per the
following considerations:
• Due to space limitations and safety considerations, the interferometer could not be
placed close to the vacuum chamber.
• The large diameter of the vacuum chamber (∼ 1.5 m) ensured that the scene beam
had a long path length. The mirrors placed at the bottom and top of the chambers
have limited access, so aligning an interferometer in Mach Zehnder geometry for such
a setup would have been very costly and cumbersome. Thus the interferometer was
set up in a double pass geometry with a layout similar to that of a Michelson interferometer (for the homodyne interferometer refer to section 4.4.2) or a hybrid of the
Michelson and Mach-Zehnder interferometer (for the heterodyne interferometer refer
to section 4.5.2).
• Many interferometers used on existing plasma experiments are two-color interferometers [34–40] that decouple the phase shift caused by the plasma and by mechanical
vibrations. Because mechanical vibrations (kHz range) are unimportant for the fast

36
timescale (∼ 10 µs) of the Caltech plasma experiments, a single laser interferometer
is adequate. As discussed in sections 4.4.3 and 4.5.3, the effects of the vibrations can
be removed by low-pass filtering the detected phase. In fact, these vibrations were
used to “self”-calibrate the homodyne interferometer (refer section 4.4.3).
• Refractive bending of light may cause a spurious change in signal intensity that can be
incorrectly interpreted as a change in phase shift caused by the plasma. Interferometers for large plasma experiments have often used extra optics to counter refractive
bending caused by plasma [38, 41, 42]. However, at the Caltech spheromak experiment, refractive bending is not a concern as:
1. the spatial extent of the plasma is small.
2. the plasma is approximately azimuthically symmetric. Thus, the proposed path
of the beam will always be in the direction of ∇ne , the beam will not bend.

4.3

Laser Phase Auto-correlation Function

4.3.1

Frequency Spectrum of the Laser

A gas laser contains an active medium within a resonating optical cavity bounded by mirrors
on either end. The mirrors allow only those optical modes which traverse an integer number
of half-wavelengths within the cavity. The frequencies of these optical modes are
νq = q

q = 0, 1, 2 . . . ,

(4.2)

where c is the speed of light and d is the distance between the cavity mirrors. These discrete
frequencies are separated by νM = c/2d. For a typical He-Ne gas laser with a cavity length
of d ∼ 25 cm, the modes are separated by νM ∼ 600 MHz.
The active medium between the mirrors can be considered as a narrow-band optical
amplifier. The gain curve for this amplifier is centered around the frequency ν◦ , such that
hν◦ is the energy released by the atomic transition that emits the photon. Only a few of
the discrete frequencies given by equation (4.2) appear in the laser beam. These are the
amplified modes; the others are attenuated by the medium. For example, in a commercial

νM= c/2d

Doppler gain curve
Loss Line1
∆ν

Loss Line2
Resonance
modes

δν
ν0

Figure 4.1: Power spectrum of a laser showing discrete frequency resonance modes. Also
plotted are the Doppler gain curve and two possible levels of cavity loss.
red He-Ne laser, photons are emitted because of transition of Ne atoms from a 2p5 5s state
to 2p5 3p state, which corresponds to a center frequency of ν◦ ∼ 473 THz. The gain curve
is primarily Doppler broadened [43] by an amount
ν◦
∆ν ∼

2kT

where k is Boltzmann’s constant, T is the gas temperature and M is the molecular mass of
the radiating atom. For a collection of Ne atoms emitting light at the He-Ne wavelength
of λ◦ = 632.8 nm at room temperature, the Doppler width is ∼ 2 GHz. Thus, an amplifier
with gain width ∆ν ∼ 2 GHz allows about 4 modes separated by νM ∼ 600 MHz, as
sketched in figure 4.1 [44].
Power will build up from noise in modes for which the gain exceeds the losses. As the
power in modes builds up, modes will saturate and equilibrate, so that the gain balances the
losses. Modes for which the losses exceed the gain are severely attenuated. For example, if
loss-line 1 in figure 4.1 represents the losses in the system, a monochromatic wave will exist,
corresponding to the resonance mode closest to the peak of the amplifier gain function. On
the other hand, if the losses are represented by loss line 2, there will be 3 distinct modes in
the wave. Power in various modes is distributed according to the amplifier gain profile and
losses in the system [45].
The wave’s electric field in the polarization direction for an ideal laser can be represented

38
as:
E(t) =

1 X
Ẽq eiωq t
2π q
Z∞ X

2π

Ẽq δ(ω − ωq )eiωt dω,

(4.3a)

−∞

where ωq = 2πνq . Functions and variables in the frequency domain will be represented by
a “tilde.” Equation (4.3a) is just a Fourier transform relation. Thus the Fourier transform
of the electric field for an ideal laser is a series of delta functions, with the non-zero Fourier
coefficients Ẽq corresponding to the non-attenuated modes.
The discrete resonant frequency modes of a laser are each broadened by a small amount
δν, due to:
1. losses due to absorption and scattering within the medium [43]. These losses relate to
the finite photon decay time via the uncertainty relation between time and frequency.
2. imperfect reflection at the mirrors [43].
3. vibration of mirrors [46]. If the mirrors vibrate by an amount δd, the corresponding
broadening of the modes is given by, δν ∼ ν◦ δd/d.
In most commercial lasers, the frequency broadening thus produced is of the order of δν ∼
1 MHz, as sketched in figure 4.1. Typically, δν
νM , so the frequency broadened modes
do not overlap each other.
The Fourier transform of the electric field will now consist of a series of broadened
functions. Under the simplifying assumption that all the modes are broadened by the same
amount, the electric field Fourier transform can be represented as
Ẽ(ω) =

Ẽq F̃ (ω − ωq ),

where F̃ (ω) is a low-pass broadening function of width 2πδν. Because the modes are well
separated, the spectral power is
|Ẽ(ω)|2 =

|Ẽq |2 |F̃ (ω − ωq )|2 .

(4.4)

4.3.2

Phase Auto-correlation Function Related to Power Spectrum

The auto-correlation function of a laser is defined as
hE ∗ (t)E(t + τ )i
h|E(t)|2 i

G(τ ) =

(4.5)

where h·i denotes time average. The coherence time of a laser is defined as the time τ
at which the auto-correlation function G(τ ) falls significantly below 1, and the coherence
length of a laser is the coherence time scaled by c. It is traditionally assumed that if an
interferometer is set up with a path length difference greater than the coherence length, the
phases of the two waves will be uncorrelated, so no interference pattern will be observed.
However, this standard concept of coherence length is misleading because the phase autocorrelation function is an almost periodic function; for the purpose of interferometry, it is
sufficient to maintain a path length difference corresponding to a maximum of the autocorrelation function.
Using

Z∞

E(t) =
2π

Ẽ(ω)eiωt dω,

−∞

equation (4.5) can be expressed as
R∞ R∞
G(τ ) =

dω dω 0 Ẽ ∗ (ω)Ẽ(ω 0 )eiω τ hei(ω −ω)t i

−∞ −∞
R∞ R∞

(4.6)

dω dω 0 Ẽ ∗ (ω)Ẽ(ω 0 )hei(ω0 −ω)t i

−∞ −∞

Using the relation
i(ω 0 −ω)t

Z∞
i∼

ei(ω −ω)t dt ∼ δ(ω 0 − ω),

−∞

equation (4.6) reduces to
R∞
G(τ ) =

dω|Ẽ(ω)|2 eiωτ

−∞
R∞
−∞

dω|Ẽ(ω)|2

(4.7)

40
so

Z∞

G(τ ) =
2π

S̃(ω)eiωτ dω,

(4.8)

−∞

where S̃(ω) is the normalized spectral power defined by
2π|Ẽ(ω)|2
R∞
dω|Ẽ(ω)|2

S̃(ω) =

−∞

Equation (4.8) is the Wiener-Khinchin theorem [43] and shows that the auto-correlation
function and the normalized spectral power are Fourier transform pairs.
Using equation (4.4), the auto-correlation function has the dependence
Z∞
G(τ ) ∼

dω

|Ẽq |2 |F̃ (ω − ωq )|2 eiωτ .

−∞

Let
F̃(ω) = |F̃ (ω)|2 ,
and let F(τ ) be the Fourier inverse of F̃(ω) so
Z∞
F̃(ω) =

F(τ )e−iωτ dτ.

−∞

Since F̃(ω) has a spread of ∼ 2πδν, F(τ ) will have a spread of ∼ 1/δν.

(4.9)

41
From equation (4.9),
Z∞
G(τ ) ∼

dω
−∞
Z∞

−∞
Z∞

−∞

 ∞
F(τ 0 )e−i(ω−ωq )τ dτ 0  eiωτ
|Ẽq |2 

dτ 0

−∞

 ∞
|Ẽq |2 e+iωq τ F(τ 0 ) 
e−iω(τ −τ ) dω 

dτ 0

−∞

|Ẽq |2 eiωq τ F(τ 0 )δ(τ 0 − τ )

|Ẽq |2 eiωq τ F(τ )

= P(τ )F(τ ),
where
P(τ ) =

|Ẽq |2 eiωq τ

|Ẽq |2 ei2πqνM τ .

(4.10)

Each of the complex exponentials in equation (4.10) is periodic in τ , with a period of
νM .

Thus, P(τ ) is also periodic with the same period. For τ = 0, ν1M , ν2M , ν3M , · · · , all the

components add up constructively and so P(τ ) will be maximum at these values of τ . The
exact shape of P(τ ) will depend on the value of the coefficients |Ẽq |2 . For a laser with a
large number of modes (corresponding to many non-zero |Ẽq |2 ’s), P(τ ) may have a steep
decay away from its peaks.
For a typical laser with d ∼ 25 cm and δν ∼ 1 MHz, P(τ ) will be periodic with period
1.67 ns and F(τ ) will have a spread of 1 µs. It is convenient to scale time with c to express
P, F and G as functions of length. P(δL) is thus periodic with period 2d = 0.5 m and F(δL)
decreases with a 300 m scale length. G(δL) is thus the product of a slowly decaying envelope
function F(δL) and a periodic function P(δL). The interferometer at Caltech operates at
δL ∼ 8 m, corresponding to the 16th maximum of P(δL). We assume F(δL) to be Gaussian

∼ e−(δL) /2l , where l, the width of F, is approximately 300 m. If an interferometer is
operated at a path length difference corresponding to a maximum of P(δL), the strength

42
Isolator
He−Ne
Laser

Beam
Expander

M1
BS

Detector

L=0

Figure 4.2: Michelson setup to measure phase auto correlation of laser. BS stands for beam
splitter and M for mirror.
of interference signal will be proportional to F(δL). Thus for the Caltech interferometer,
operating at a path length difference of 8 m causes attenuation of the signal amplitude by
a factor F(δl = 8 m) = 0.9996. In other words, only 0.04% of power is lost due to unequal
path length effects, and phase coherence is maintained since P(δl) has the same value at
8 m as at 0 m.

4.3.3

Measurement of Laser Phase Auto-correlation Function

To test if the laser being used in the Caltech interferometer indeed has a periodic autocorrelation function, the laser was used in the Michelson interferometer setup shown in
figure 4.2. Mirror M4’s location, L, was varied with a linear translation stage at a constant
speed and the amplitude of the interference signal was plotted as a function of time, as
shown in figure 4.3. Interference is caused by ambient noise vibrating the mirrors. At
t = 0 s, L was 0, and thus the path lengths were approximately equal. The amplitude of
the interference signal is directly proportional to the phase auto-correlation function. Since
L was increased at a constant rate, the horizontal time axis in figure 4.3 is proportional
to path length difference 2L. As seen from figure 4.3, the phase auto-correlation function
is periodic. The difference between successive maxima corresponded to 2L ∼ 50 cm. The
amplitude decreased significantly beyond the third maximum because the interferometer
became misaligned with large motions of the mirror. Thus a significant decrease in contrast
ratio of the maximum and minimum of signal amplitude was observed beyond the third
maximum. However, this was an effect of misalignment and not path length difference.
The contrast ratio could be recovered by realigning the interferometer. The minima of the
amplitude of the interference signal were of the order of the noise level of the detector.

Interference signal
Envelope(V)

Corresponds to
2L ~50cm

0.2

0.1

−0.1

−0.2

Time(s)

Figure 4.3: Envelope of the interference signal measured using the setup shown in figure
4.2. The path length difference 2L was varied at a constant rate. The envelope magnitude
is directly proportional to the phase auto-correlation function.

4.4

Homodyne Interferometer

4.4.1

Theory

Quadrature phase information is generated in a homodyne interferometer by interfering a
linearly polarized scene beam and a circularly polarized reference beam [41]. Consider a local
coordinate system with the z-axis pointing towards the direction of the beam propagation
and the y-axis pointing upwards from the optical table. The electric field for the linearly
polarized scene beam is given by
Es = E0s (x̂ cos θ + ŷ sin θ) cos(kLs − ∆φp − ωt),
where θ is the polarization angle with respect to the x-axis, k is the vacuum wavenumber of
the laser beam, Ls is the length of the scene beam, ∆φp is the phase change caused by the
plasma and ω is the lasing frequency. The electric field for the circularly polarized reference
beam is given by
Er = E0r (x̂ cos(kLr − ωt) + ŷ sin(kLr − ωt)),

44
where Lr is the length of the reference beam. As shown in section 4.4.2, a Wollaston prism
is used to separately combine the x̂ and ŷ polarizations of the scene and reference beams.
Output power of the detector receiving the x̂ components of the beams is
Sx ∝ h(Erx + Esx )2 i
∝ h(E0r cos(kLr − ωt) + E0s cos θ cos(kLs − ∆φp − ωt))2 i
∝ E0r
hcos2 (kLr − ωt)i + E0s
cos2 θhcos2 (kLs − ∆φp − ωt)i

+2E0r E0s cos θhcos(kLr − ωt) cos(kLs − ∆φp − ωt)i
2 + E 2 cos2 θ
E0r
0s
+ E0r E0s cos θhcos(kδL − ∆φp )i
+E0r E0s hcos(k(Ls + Lr ) − ∆φp − 2ωt)i
S̃x ∝ E0r E0s cos θ cos(kδL − ∆φp ),
where δL = Ls − Lr is the path length difference between the scene and the reference beams
and S̃x is the AC component of the detector output power Sx . Similarly, the output of the
detector receiving the ŷ component of the beams is given by
S̃y ∝ E0r E0s sin θ cos(kδL − ∆φp ).
Thus, the AC output of the detectors is proportional to the sine and cosine of the phase
shift due to the plasma:
S̃x = α cos(∆φp − kδL),
S̃y = β sin(∆φp − kδL),

(4.11)

and the phase can be reconstructed as
∆φp = tan−1 (

α S̃y
) + kδL + nπ,
S̃x

(4.12)

where n is an integer. While performing the inverse tangent operation in equation (4.12),
actual signs of S̃x and S̃y can be used to lower the phase ambiguity from nπ to 2nπ.
The ratio αβ will be close to unity if the following criteria are met:

~1.9m
Plasma
Spherical
Mirror

~2.2m

Laser

HWP1
P2
Piezo Vibrating
Mirror

QWP Non Pol.
BS

HWP2
M1

Isolator

HWP3

Wollaston Prism
D2

D1
M3

Figure 4.4: Setup of the homodyne interferometer for the Caltech spheromak experiment.
The dotted beam signifies that the beam is coming out of the plane of the figure.
1. the angle of polarization, θ, of the linearly polarized scene beam is 45◦ ;
2. the sensitivities of both of the detectors are equal;
3. and the beams are properly aligned so that almost equal power is coupled to each of
the detectors.

4.4.2

Setup

46
Table 4.1: Description of some of the components used in the design of homodyne interferometer (refer to figure 4.4).

Component
Laser

Description
A 4 mW linearly polarized He-Ne laser with a cavity length of
25 cm. It produces a coherent beam of ∼ 2 mm diameter at
∼ 633 nm.

HWP1

Zero order half wave plate. It is used to rotate the polarization
vector of the incoming beam to be vertical or horizontal.

Isolator

Manufactured by Optics for Research [47] (part number IO-5660-LP). It prevents any reflected light from entering the laser.

HWP2

Zero order half wave plate. It transforms the beam coming
out of the isolator into a vertically polarized beam. The polarization of the vertically polarized reference beam is unaltered
upon reflection from mirrors or transmission through the beam
splitter.

A non-polarizing plate beam splitter was used to both split the
beams and then recombine them.

QWP

Zero order quarter wave plate used to make the reference beam
circularly polarized.

Piezo mirror

A mirror mounted on a piezo actuator. It could be vibrated
with frequencies ranging from ∼ 1 Hz to ∼ 1 kHz.

P1 and P2

Dichroic polarizers in the reference and scene beam paths respectively. P2 was used to make the scene beam vertically
polarized. P1 was used along with QWP to make the reference
beam circularly polarized.
Continued on next page

47
Table 4.1: continued from previous page
Component
SM

Description
The radius of curvature of the spherical mirror (SM) is 4 m,
the approximate distance the beam travels from the optical table to the spherical mirror, so the spherical mirror focuses the
beam back to almost its original size. The spherical mirror’s
position can be adjusted to ensure that the path length difference between the scene and reference beams is approximately
an even multiple of the laser cavity length.

HWP3

Zero order half wave plate, used to rotate the polarization angle
of scene beam to 45◦ .

Wollaston prism

Used to split the scene and reference beams into x̂ and ŷ polarization components.

Mirrors

Plane mirrors are labeled by the letter M followed by a number. These are 100 diameter mirrors manufactured by Newport
optics [48] (part number 10D10ER.1) and are used to steer the
beams. Mirrors M2 is mounted on a damped rod attached to
the vacuum chamber. It is used to direct the beam into the
vacuum chamber through sapphire windows.

Detectors

The low noise, high gain detector amplifiers (Model 712A-2,
from Analog Modules [49]) have a bandwidth of 200 Hz to
25 MHz.1 The detector amplifier modules were housed in a
RF shielded box and had a He-Ne filter in front.

A quadrature homodyne interferometer to measure plasma density was suggested by
Buchenauer and Jacobson [41]. However, the optical arrangement they suggested is difficult
to align over long distances. Hence the interferometer for the Caltech experiment was set
up in Michelson double pass geometry as shown in figure 4.4. By interfering beams with

Model 712A-2, being currently manufactured has a bandwidth of 250 Hz to 60 MHz.

48
a large path length difference, it was possible to locate most of the optical components on
a small and accessible optical bench (1800 × 1800 ). Table 4.1 describes the components used
in the homodyne interferometer. The process for aligning the homodyne interferometer is
describes in section A.2.

4.4.3

Results

Typical results from the homodyne interferometer are shown in figure 4.5.2 The two quadrature signals from the detectors shown in blue and red are plotted in figure 4.5(a). Note that
when one of the signals is at its maximum (or minimum), the other is passing through zero
- a consequence of being in quadrature. Plasma causes the sudden change in the signals
near 0 s.
The two signals in figure 4.5(a) are plotted as a Lissajous plot in figure 4.5(b). The data
set corresponding to plasma is plotted as a solid red line while the non-plasma times are
plotted in blue dots. The extent to which the signals are in quadrature can be estimated
from the extent to which the plot resembles a circle. Note that refractive bending diminished
the signal amplitude when the plasma intercepted the beam. The signal amplitude changed
by different amounts on different detectors and hence may have caused a slightly erroneous
measurement of density. Also note that the phase due to background vibrations changes by
around 40◦ during the time in which the phase due to plasma has changed by > 500◦ .
Figure 4.5(c) plots the interpreted line average density from the interferometer. The
slight drift in the signal is caused by mechanical vibrations of the mirrors and can be
accounted for by a polynomial fit (of 4th degree) to the phase corresponding to non-plasma
times. The polynomial fit is shown in green in figure 4.5(c). The plasma density after
subtracting the polynomial fit is shown in figure 4.5(d).
The mechanical vibrations of the mirror were used to calibrate the interferometer. Due
to the combined effect of the piezo vibrating mirror and the mechanical vibrations in other
mirrors, an interference signal with a bandwidth of a few kHz was observed in both the
detectors, as shown in figure 4.5(a). This kHz scale signal was used to estimate the signal
strengths α and β as described by equations (4.11). The ratio αβ , thus estimated was used
in equation (4.12). In other words, vibrations were used to calibrate the interferometer.

This particular shot had two capacitors (59 µF each) being discharged across the plasma.

Signal1

0.8

Signal2
0.6

0.4

0.2

Signal2(V)

Signal(V)

0.6

−0.2

−0.4

−0.6
−0.6

−0.8
−1

−0.8

Plasma intercepts the laser beam

200

400

600
800
Time(µ s)

1000

−1

1200

(a) Quadrature signals from the detectors.

−0.5

Signal1(V)

0.5

(b) Lissajous plot of the quadrature signals. The
data set corresponding to plasma intercepting the
laser beam is plotted as a solid red line while the
data corresponding to times with no plasma is plotted in blue dots.
21

x 10

508

406

305

203

102

−1

−102
80

−10

Line densityelectron/m2

508

Phase(°)

Line densityelectron/m2

x 10

406

305

203

102

−10

Phase(°)

Time(µs)

(d) Line average density with the effect of mechan-

nals. A polynomial fit to model the noise vibrations

ical vibrations compensated for.

is shown in green.

Figure 4.5: Results from the homodyne interferometer for shot #7092.

50
When inferring the phase from the two detector signals, the relative amplitude, not the
individual amplitude, of the signals is important. Quite often, refractive effects of plasma
caused the amplitudes of the signals to vary (when the overlap of the beams was altered, or
when beams fell on different parts of the detectors). These effects may increase or decrease
the signal amplitudes and may produce errors in the phase detection. As seen by the red
curve in figure 4.5(b), plasma effects caused the signal amplitude to change unequally in the
two detectors. Buchenauer and Jacobson [41] countered the effects of refractive bending by
focusing the scene beam at the center of plasma column.

4.4.4

Procedural Details

This section discusses certain procedural details for the homodyne interferometer.
1. The VME system introduced a fixed DC bias to the received signals. The bias was
estimated by recording a temporary data set with no inputs or by looking at the mean
of the maximum and the minimum of the quadrature signals.
2. The detectors were sensitive only to signals with frequencies greater than 200 Hz.
Consequently, any signal that varied asymmetrically about zero would become “wavy”
as the detectors filtered out its DC mean. In figure 4.5(a) the maxima (or minima)
of a particular signal had slightly varying values. This is a consequence of a filtered
DC mean. A changing DC mean corresponds to a changing position of the “center of
the circle” in the Lissajous plot of figure 4.5(b). This caused the signal data to form
more than one “circle.” The effect is usually more prominent than observed in figure
4.5. The varying DC mean makes it tricky to estimate the signal amplitude to get
the ratio αβ . The corresponding phase error, however, is minimal especially because
the phase change caused by the plasma is about 2π. It is difficult to get high-gain
photo-detector modules with bandwidth up to DC.
3. In a previous version of the interferometer, the mirror on top of the chamber was
attached to the ceiling via a mount. Unfortunately, the ceiling is prone to vibrations,
so the interferometer would misalign very frequently. Mounting the mirror on the
chamber diminished the effects of these vibrations. However, quite often, the signal
amplitude is seen to fluctuate with a period of about ∼ 1 s. This may be caused by

51
a fluctuating overlap of the scene and the reference beam or by a fluctuating overlap
of the beams on the detector.
4. After a few hours of interferometer operation, the signals were observed to fall out of
quadrature by a few degrees (< 5◦ ). The quarter wave plate QWP had to be adjusted
to bring the signals back into quadrature.
5. The laser had a switching power supply which operated at ∼ 18 kHz. If the piezo
mirror was set to vibrate at this frequency, strange resonances were introduced, and
the detector signals did not remain in quadrature. Thus the piezo vibrations were
maintained between 1 and 4 kHz.
6. If a non-polarizing plate beam splitter was used, fringes were observed from just
one beam due to reflections from the two surfaces. These were avoided by slightly
misaligning the beam splitter. Cube beam splitters do not suffer from this drawback
of self-interference, as they have just one surface for transmitting and reflecting.
7. Previously, & 50% power was lost by the scene beam due to multiple reflections while
passing through the sapphire windows. This power loss was minimized by an antireflection coating on the windows for the HeNe wavelength.
8. The data analysis relied heavily on the fact that the detector signals were in quadrature. Signal quadrature depended on the extent to which the beams were linearly
or circularly polarized. Since good polarization was of critical importance, dichroic
polarizers were used in the interferometer. These polarizers have a high extinction
ratio of 1 : 104 and a low transmittance of 36%. Consequently, substantial power was
lost in these polarizers.
9. The polarizer P2 ensures that the scene beam is vertically polarized before interfering
with the reference beam. Power loss in the scene beam is minimized if the optical axis
of the sapphire windows is oriented along the polarization direction.
10. Zero order wave plates were used in the interferometer, as their properties are relatively insensitive to ambient temperature variations.

52
11. The interferometer alignment was “relatively” simple as the beams traced back the
same path. This back reflection would have caused resonance modes in the laser if
not for the isolator next to the laser.

4.4.5

Error Analysis

As described by Buchenauer and Jacobson [41], the phase error of the interferometer is ∼ ασ ,
where σ is the standard deviation of the measured signal. This does not take into account
the phase error due to a varying DC mean. The standard deviation can be easily estimated
by smoothing the detector signal and taking its difference from the original measured signal.
Typically this phase error is on the order of 2◦ .

4.4.6

Advantages

As compared to the original design by Buchenauer and Jacobson [41], the design of the
interferometer for the Caltech Spheromak Experiment had the following advantages
1. Separate control of the phase quadrature and the relative amplitudes of two signals:
The error performance of the interferometer is better if the two signals have equal
amplitude. Their respective amplitude was adjusted by HWP3. Quadrature mismatch
between the signals was improved by adjusting P1.
2. Only one mirror to adjust at long distances: The interferometer was aligned in an
iterative scheme by first aligning all of its components alone on a small optical table
and then by just adjusting the spherical mirror so that the scene beam traced back
its path. Thus, as compared to the Mach-Zehnder geometry, only one mirror which
was not on the table required alignment.

4.4.7

Disadvantages

1. Poor signal to noise ratio (SNR): As compared to the heterodyne interferometer, the
homodyne interferometer has poor SNR performance.
2. Power inefficient: Due to stringent polarization requirements, significant amount of
power (& 75%) is lost in the polarizers.

4.5

Heterodyne Interferometer

Heterodyne interferometers are usually simpler in design and alignment than homodyne
interferometers. They also have better Signal to Noise Ratio (SNR). Thus the homodyne
interferometer at the Caltech experiment was replaced by a heterodyne interferometer. The
design for the interferometer was motivated by an interferometer built by Golingo [50].

4.5.1

Theory

Consider a vertically polarized reference beam with a frequency offset by an amount ∆ω
from the He-Ne frequency ω. The electric field of such a wave can be represented as:
Er = E0ry ŷ cos(kLr − (ω − ∆ω)t).
The scene beam can have an arbitrary polarization, as it passes through birefringent material
like the sapphire windows. The electric field of the scene beam may be represented as:
Es = E0sy ŷ cos(kLs − ∆φp − ωt) + E0sx x̂ cos(kLs − δφ − ∆φp − ωt).
The phase difference δφ between the 2 polarizations may be caused by birefringent materials.
If the scene and the reference beams interfere on a detector, the signal from the detector
will be proportional to
S ∝ h(Ery + Esy )2 + Esx

∝ h(E0ry cos(kLr − (ω − ∆ω)t) + E0sy cos(kLs − ∆φp − ωt))2
+(E0sx cos(kL
s − δφ − ∆φp − ωt)) i
∝ E0ry
hcos2 (kLr − (ω − ∆ω)t)i + E0sy
hcos2 (kLs − ∆φp − ωt)i

+2E0ry E0sy hcos (kLr − (ω − ∆ω)t) cos (kLs − ∆φp − ωt)i
+E0sx
hcos2 (kLs − δφ − ∆φp ωt)i
2 + E2 + E2
E0ry
0sy
0sx
+ E0ry E0sy hcos (∆ωt − kδL − ∆φp )i
+E0ry E0sy hcos (k(Ls + Lr ) − ∆φp − (2ω − ∆ω)t)i

S̃ ∝ E0ry E0sy cos (∆ωt − kδL − ∆φp ),

(4.13)

54
where S̃ represents the AC component of the signal. δL = Lr − Ls is the fluctuating change
in path lengths caused by mechanical vibrations. The detector is fast enough to track
changes occurring at frequency ∆ω ∼ 80 MHz but averages out any changes occurring at
He-Ne frequencies or higher. Refractive bending may cause the signal amplitude of the
detector to change as a function of time, so the detector signal can be expressed as
S̃ = F (t) cos (∆ωt − kδL − ∆φp ),

(4.14)

where the rate of change of F (t) is slow compared to ∆ω. Next, the signal from the detector
described by equation (4.14) is demodulated to recover the phase information. To do so,
the coupled signal from the radio frequency (RF) source at ∆ω = 80 MHz is split into
quadrature components: A cos(∆ωt + ϕ) and B sin(∆ωt + ϕ), where ϕ is the phase shift
caused by the transmission cables. The detector signal from equation (4.14) is split into
two equal parts and mixed (multiplied) by the quadrature signals. The resulting signals
coming out of the mixers are given by:
S1 =
S2 =

F (t)A
cos (∆ωt − kδL − ∆φp ) cos(∆ωt + ϕ)
F (t)A
(cos (kδL + ∆φp + ϕ) + cos (2∆ωt + ϕ − kδL − ∆φp ))
F (t)B
cos (∆ωt − kδL − ∆φp ) sin(∆ωt + ϕ)
F (t)B
(sin (kδL + ∆φp + ϕ) + sin (2∆ωt + ϕ − kδL − ∆φp ))

(4.15)

On low-pass filtering the above signals, we get quadrature signals:
S1 =
S2 =

F (t)A
cos(∆φp + ϕ0 ),
F (t)B
sin(∆φp + ϕ0 ),

where ϕ0 = ϕ + kδL is constant for the scale of the experiment. The phase shift caused by
the plasma can be calculated by taking the inverse tangent of the ratio of the two signals.
Note that taking the ratio of the signals eliminates the time dependence caused by refractive

Isolator

HWP2

HWP1

80 MHz

He−Ne
Laser

M0
M1

AOM

Non Pol.
BS1

Iris

Non Pol.
BS2
M2
M6

M5
SM

Plasma

~2.2m
~1.9m

Figure 4.6: Setup of the heterodyne interferometer for the Caltech spheromak experiment.
The dotted beam signifies that the beam is coming out of the plane of the figure.
bending F (t). The plasma phase shift can be estimated as
∆φp = tan−1 (

4.5.2

S2 A
) − ϕ0 + nπ.
S1 B

(4.16)

Setup

Table 4.2: Description of some of the components used in the design of the heterodyne
interferometer (refer to figure 4.6).

Component
Laser

Description
A 4 mW linearly polarized He-Ne laser with a cavity length of
25 cm. It produces a coherent beam of ∼ 2 mm diameter at
∼ 633 nm.

HWP1

Zero order half-wave plate. It is used to rotate the polarization
vector of the laser beam so that it aligns with the direction of
the polarizer at the input of the isolator.
Continued on next page

56
Table 4.2: continued from previous page
Component
Isolator

Description
Manufactured by Optics for Research [47] (part number IO-5660-LP). It prevents any reflected light from entering the laser.

HWP2

Zero order half-wave plate. It transforms the beam coming out
of the isolator into a vertically polarized beam. The polarization of the vertically polarized reference beam is unaltered
upon reflection from mirrors, beam splitter or from transmission through the acousto-optic modulator (AOM).

Non pol. BS1

Non-polarizing plate beam splitter is used to split the beam
into a scene and a reference beam.

AOM

The acousto-optic modulator (AOM). Up to 86% of the input
power to the AOM can be coupled into its first harmonic output. The iris obstructs all other beams except for the first
harmonic.

Mirrors

Plane mirrors are labeled by the letter M followed by a number.
These are 100 diameter mirrors manufactured by Newport optics
[48] (part number 10D10ER.1) and are used to steer the beams.
Mirrors M4 and M5 are mounted on a damped rod attached to
the vacuum chamber. They are used to direct the beam into
the vacuum chamber through sapphire windows.

Non pol. BS2

Non polarizing cube beam splitter used to recombine the scene
and reference beams.
Continued on next page

57
Table 4.2: continued from previous page
Component
SM

Description
The radius of curvature of the spherical mirror (SM) is 4 m, the
approximate distance the beam travels from the optical table to
the spherical mirror, so the spherical mirror focuses the beam
back to almost its original size. The spherical mirror position
can be adjusted to ensure that the path length difference between the scene and reference beams is approximately an even
multiple of the laser cavity length.

Detector

UDT Sensors part number HR040L [51] with bandwidth ∼
500 MHz used in a reverse biased mode.

RF electronics

All the components shown in figure 4.7 were off-the-shelf components from Mini Circuits. The components were selected
based on their power ratings.

Figure 4.6 shows the schematic for the interferometer. The RF electronics for demodulating the signal are shown in figure 4.7. The interferometer described in figure 4.6 is set up
in a double pass geometry. By interfering beams with a large path length difference, it was
possible to locate most of the optical components on a small and accessible optical bench
(1800 × 1800 ). Mirror M4 and the spherical mirror SM are mounted on the vacuum chamber.
They direct the laser beam through the plasma (via sapphire windows) and back to the optical bench. Sapphire is a birefringent material, and the windows are oriented to minimize
the change in the polarization of the scene beam. Note that equation (4.13) implies that
the interference signal strength is maximized if the polarization of the interfering beams is
the same.
A major advantage of the design shown in figure 4.6 is the ease of alignment. The two
beams are arranged to overlap each other simply by adjusting the cube beam splitter BS2,
and the mirror M2. Both these components are located on the optical bench and are easily
accessible. A cube beam splitter was used for combining the beams instead of a plate beam
splitter since a cube beam splitter does not introduce any lateral shift in the position of the

Detector

60 dB − Amplifier

80 MHz source

In
Cpl

To acousto−
optic modulator

Out

8dBm

19dBm

In o
2 way 90
Out1 Out2
16dBm
16dBm

In o
2 way 0
Out1 Out2
5dBm
5dBm

Mixers
Quadrature
Signals

Low Pass
11 MHz
Low Pass
11 MHz

1dBm
1dBm

Figure 4.7: RF circuit for the heterodyne interferometer. Typical signal power in dBm is
mentioned for each connection.

59
passing beam. The process for aligning the homodyne interferometer is describes in section
A.2.

4.5.3

Results

Typical results from the heterodyne interferometer are shown in figure 4.8. The two quadrature signals after demodulation are shown in blue and red in figure 4.8(a). Note that when
one of the signals is at its maximum (or minimum), the other is passing through zero - a
consequence of being in quadrature. Plasma causes the sudden change in the signals near
0 s.
The two signals plotted in figure 4.8(a) are plotted as a Lissajous plot in figure 4.8(b).
The data set corresponding to the beam passing through the plasma is plotted as a solid red
line while the non plasma times are plotted in blue dots. The extent to which the signals are
in quadrature can be estimated from the extent to which the plot resembles a circle. Note
that refractive bending intensified the signal amplitude when the plasma intercepted the
beam. Provided that the beams undergo only a “small” displacement because of refractive
bending, taking the ratio of the two signals removes the effects of refractive bending on
the phase inferred [42]. Also note that the phase due to background vibrations changes by
around 40◦ during the time in which the phase due to plasma changes by > 200◦ .
Figure 4.8(c) plots the interpreted line average density from the interferometer. The
slight drift in the signal is caused by mechanical vibrations of the mirrors, and can be
accounted for by a polynomial fit (of 4th degree) to the phase corresponding to non plasma
times. The polynomial fit is shown in green in figure 4.8(c). The plasma density after
subtracting the polynomial fit is shown in figure 4.8(d).

4.5.4

Procedural Details

In this section certain procedural details for the heterodyne interferometer are discussed.
1. The interferometer was aligned in a hybrid geometry motivated by the Michelson and
Mach-Zehnder designs. The beams did not trace back their paths, so did not come
back to the laser. Yet, the isolator was placed in front of the laser to remove any
possibility of light coming back to the laser and effecting its stability.

0.15

Signal1

0.1

Signal2

0.08

0.1

0.06
0.04
Signal2(V)

Signal(V)

0.05

−0.05

0.02
−0.02
−0.04
−0.06

−0.1

−0.08
Plasma intercepts
the laser beam

−600

−400

−200

200
Time(µ s)

400

−0.1

600

−0.1

800

(a) Quadrature signals after demodulation.

−0.05

Signal1(V)

0.05

0.1

(b) Lissajous plot of the quadrature signals. The
data set corresponding to plasma intercepting the
laser beam is plotted as a solid red line while the
data corresponding to the times with no plasma is
plotted in blue dots.
21

x 10

254

305

254

203

1.5

152

102

0.5

Line density

−0.5

203

1.5

152

102

0.5

electron

/m2

2.5

Phase( )

Line densityelectron/m2

2.5

x 10

Phase(°)

−51

Ignitron noise pickup
−1
−100

−80

−60

−40

−20

−102
100

−10

100

Time(µs)

(d) Line averaged density with the effect of mechan-

Polynomial fit to model the noise vibrations is shown

ical vibrations compensated for.

in green.

Figure 4.8: Results from the heterodyne interferometer for shot #9114.

Beam 2

Beam 1

dθ

Detector Surface
Figure 4.9: Effect of misalignment on detector signal.
2. As noted in section 4.4.4, the VME system introduced a DC bias to the measured
signals.
3. When aligning the homodyne interferometer, fringe patterns were observed when the
scene and reference beams overlapped reasonably well. This provided a visual feedback
on the degree of alignment. However, because of the 80 MHz modulation, fringe
patterns could not be observed for the heterodyne interferometer.
Let θ be the angle of misalignment of the beams on a detector with diameter d (see
figure 4.9). To ensure that the beams remain sufficiently coherent on the surface of the
detector, it is required that dθ
λ. For a detector of diameter ∼ 1000 µm, this implies
that the beams have to be aligned up to θ
0.03◦ . Thus, while a smaller detector has
better frequency response, it is undesirable when considering beam alignment. The
process for aligning the beams is described in section A.1.3.
4. In many heterodyne interferometers, the scene and the reference beams are the 0th
and the 1st harmonics coming out of the AOM [34, 36, 37, 39, 52, 53]. This approach
requires many mirrors to steer the beams long enough before the scene beam can be
directed to the vacuum chamber. Instead, as suggested by Kawano et al. [38, 40], a
beam splitter (BS1 in figure 4.6) was used to split the beams. However this approach
was slightly power inefficient as only ∼ 86% of the input beam power can be coupled
to the first harmonic by the AOM.
5. The 90◦ splitter in the demodulation circuit (figure 4.7) has slightly different gains
for each of its output channels. This resulted in a slight difference (∼ 3%) in the

62
signal amplitudes of the quadrature signals of the interferometer. Since the gain was
constant over time, it was compensated for in the software written to interpret the
phase from the quadrature signals.

4.5.5

Error Analysis

The phase ambiguity of the signals is given by σ/A [41], where σ is the rms error in the
signal and A is the strength of the signal. For typical data this was ∼ 1◦ , corresponding to
a density error of ∼ 1019 /m2 .

4.5.6

Advantages

As compared to the homodyne interferometer described in section 4.4, the heterodyne interferometer had the following advantages:
1. Bandwidth extending to DC: The bandwidth of the RF mixers shown in figure 4.7 extends to DC. Thus, unlike the homodyne interferometer, the heterodyne interferometer
can measure a steady phase difference. This also prevented a spurious introduction in
the mean value of the signal, as occurred for the homodyne interferometer (see section
4.4.4, point 1).
2. High SNR: The heterodyne interferometer had about a factor of two better noise
performance than the homodyne interferometer.
3. No drift in quadrature: As mentioned in point 4 of section 4.4.4, the homodyne signals
would drift out of quadrature with time. For the heterodyne interferometer, the phase
quadrature was generated by RF electronics, so no drift was observed.
4. Simpler alignment: Aligning the homodyne interferometer required adjusting the
spherical mirror SM (see figure 4.4), which was very inaccessible. However, aligning the heterodyne interferometer only involved adjusting the beam splitter BS2 and
the mirror M2, both of which were on the optical table and hence extremely accessible.
5. Unaffected by refractive bending of light: As explained in the discussion just prior to
equation (4.16), the phase detection of a heterodyne interferometer is unaffected by
refractive bending of light.

Two Beams

Dichroic
Mirror

Mirror

Mirror
One beam

Figure 4.10: A plausible setup to alter the path length of the reference beam of a two-color
interferometer. The dichroic mirrors reflect one beam and transmit the other.

4.5.7

Disadvantages

An extra mirror M4 was used in the heterodyne interferometer to steer the beam through the
vacuum chamber. The extra mirror lowered the alignment complexity of the interferometer.
However, it also increased the mechanical vibrations in the beam’s path.

4.6

Conclusion

He-Ne homodyne and heterodyne interferometers were developed for the Caltech spheromak
formation experiment. The designs were especially suited for fast plasma experiments with
time scales much smaller than the time scales of mechanical vibrations of the mirrors.
The interferometers operated well even though there is a path length difference of ∼ 8 m
between the scene and the reference beams. Operating at such a large path length difference
considerably reduced the number of optical components and also made alignment much
easier.
Line densities of the order of 5 × 1021 /m2 were observed in the experiment. Assuming a
double pass plasma length L ∼ 12 cm, as shown in the figure 1.7(b), corresponds to average
densities of ∼ 4 × 1022 /m3 . These results are in good agreement with the densities inferred
from Stark broadened spectral lines [54].
The idea of operating at a large path length difference could in principle be applied
to two-color interferometers as well. The beams in a two color interferometer might have
different periods for their phase auto correlation function. A plausible way of altering the

64
path length of reference beam is suggested in figure 4.10. For a two-color interferometer
with lasers with the same cavity length, or with a single laser [37], only a single normal
mirror would be needed to adjust the length of the reference beam.

4.7

Future Extension

The interferometer setup for the Caltech experiment could probe only one spatial location
corresponding to the first available view ports. To overcome this limitation, a future designs
of the interferometer should use fiber optics to couple light into and out of the vacuum
chamber [55, 56].

Chapter 5
Non-equilibrium Alfvénic Plasma Jets Associated
with Spheromak Formation

Strong MHD-driven flows have been observed over a wide range of scales from terrestrial
experiments (coaxial gun accelerators [57–59], plasma thrusters [60, 61], high-current arcs
[62], Z-pinch formation [63], spheromak formation [22] and sustainment [64]) to extraterrestrial phenomenon (solar coronal mass ejections [65] and astrophysical jets [66]).
One of the earliest experimental observation of flows during spheromak formation was
done by Uyama et al. [67] using Doppler shift measurements. Strong flows have also been
observed during the helicity injection stage in the Caltech spheromak experiment. These
flows are not predicted by Taylor’s relaxation theory [10], and so during this stage, the
plasma should not be considered to be evolving through a series of equilibrium stages, as
has been previously assumed [22].
It is shown in this chapter that MHD driven flows are generated because of the flaring
of the poloidal current channel profile, i.e., ∂I/∂z 6= 0. Reed [62] argued that the flow

velocities, u, should scale as I 2 in flared high current arcs, where I is the current in the arc.
This is in contrast to the u ∼ I scaling found for the jets in Caltech spheromak experiment.
Barnes et al. [64] also developed a model to predict plasma flow from the electrodes in a
steady state driven spheromak, but did not take into account the plasma pressure. They
argued that the plasma flows should be Alfvénic leading to a gun voltage which scales as
I 3 . Their experiments showed gun voltage scaling as I 2 , but no measurements of plasma
velocity were reported.
This chapter consists of five sections. Section 5.1 describes the sequence of plasma

66
dynamics leading to jet formation. Section 5.2 describes the experimental results showing
the magnetic field structure in the jet, the velocity scaling of the jet and also its pressure
scaling. A MHD model for these jets is presented in section 5.3. Section 5.4 presents an
energy balance argument for the jets and also justifies the observed small radial extent of
the plasma in these jets. Finally, section 5.5 concludes the chapter by summarizing its main
results.

5.1

Introduction

As shown in section 3.3, the plasma jet can be considered an inductive load. Figure 5.1
shows a series of plasma images which elucidate the sequence of plasma evolution leading
to a changing inductance. Initially (∼ 0.5 µs after breakdown) eight spider legs are formed
linking the gas nozzles on the two electrodes. The collimation and flow of plasma in the
spider legs was studied by You et al. [12]. As the current ramps up, the spider legs expand
due to hoop force and then merge to form a central column jet because of the pinch force
(∼ 3 µs after breakdown). This results in a slightly flaring plasma jet which drives plasma
from near the electrodes to the vacuum. The jet is extremely dense (β ∼ 1) and expands at
Alfvénic velocities [54]. As the jet evolves outward, it increases the plasma inductance and
thus acts as a helicity injection mechanism (see section 2.1). This chapter shows that MHD
driven flows act as the mechanism to drive the change in the inductance of the plasma jet.
As the jet expands towards vacuum, it eventually overcomes the Kruskal-Shafranov kink
instability [7, equation 10.190] condition (due to an increased axial length), and can detach
to form a spheromak like configuration. Kinking of the plasma jet, detachment from the
electrodes and spheromak formation has been studied earlier [13, 68].

5.2

Results

5.2.1

Magnetic Field Structure in the Jets

The magnetic probe described in section 1.3.1 was used to measure the magnetic field and
current distribution in the plasma jets. Figure 5.2 shows the typical poloidal current and
flux profiles in the jet. As discussed in section 1.3.1, the poloidal flux profile may not

0.50 µs

1.10 µs

1.70 µs

2.30 µs

2.90 µs

3.50 µs

4.10 µs

4.70 µs

5.30 µs

5.90 µs

6.50 µs

7.10 µs

Figure 5.1: False colored visible images depicting the formation of hydrogen plasma jet from
shot #9920 and #9923. The green vertical lines represent the path of the laser beam used
to measure plasma density.

−0.5

−10

−1

−20

−1.5

−30

−2

−40
I(r) (kA)

ψ (mWb)

−2.5

−50

−3

−60

−3.5

−70

−4

−5

−80

6.5 µs
7.5 µs
8.5 µs

−4.5
0.05

0.1

0.15

0.2
r (m)

0.25

0.3

0.35

6.5 µs
7.5 µs
8.5 µs

−90
−100

0.4

0.05

0.1

0.15

0.2
r (m)

0.25

0.3

0.35

0.4

(a) Poloidal flux surface observed in the plasma

(b) Axial current contours observed in the

jet from shot #9957 at z = 20 cm.

plasma jet from shot #9957 at z = 20 cm.

z=20 cm, #9957
z=12.5 cm, #9964

−0.5

z=20 cm, #9957
z=12.5 cm, #9964

−1
−20
I(r) (kA)

ψ (mWb)

−1.5
−2

−40

−2.5
−60

−3
−80

−3.5
−4

0.05

0.1

0.15

0.2
r (m)

0.25

0.3

0.35

0.4

−100

0.05

0.1

0.15

0.2
r (m)

0.25

0.3

0.35

0.4

(d) Axial current contour at two different loca-

tions z = 12.5 cm and z = 20 cm measured at

t = 7.5 µs after breakdown.

Figure 5.2: Poloidal current and flux surfaces of hydrogen plasma jets.

69
be accurate because of integration and alignment errors. The poloidal flux profile is still
expected to give an intuitive understanding.
Figure 5.2(a) shows the poloidal flux profile at different time instances for a single plasma
jet at a distance z = 20 cm from the electrode. The axial magnetic field Bz is strong where
the flux contours are steep, i.e., Bz has a radial extent of about 10 cm. Also, note that the
poloidal flux surfaces flare outward with time.
Figure 5.2(b) shows the poloidal current profile at different time instances for a single
plasma jet at a distance z = 20 cm from the electrode. From the figure it is noted that Jz
has a radial extent of about 10 cm and that that the axial current contours flare outward
with time (similar to the poloidal flux contours in figure 5.2(a)).
The poloidal flux in the jet at two different axial positions is shown in figure 5.2(c). The
figure is inconclusive regarding the flaring of the poloidal flux contours.
The poloidal current in the jet at two different axial positions is shown in figure 5.2(d).
It is clear from the data that the current channel flares and hence toroidal magnetic field
pressure decreases with increasing axial distance from the electrode. The radial extent of
the current channel increases from about 7 cm at z = 12.5 cm to about 9 cm at z = 20 cm.
Thus if the radius of the flaring current channel is modeled as a(z) = a◦ eκz , then κ ∼ 2 m−1 .

5.2.2

Speed of the Jets

The He-Ne interferometer [69] described in section 4.5 was used to measure the density
of the plasma jets. The interferometer beam (shown as a vertical green line in figure 5.1)
intercepted the plasma jet at a distance of 29 cm from the planar electrodes.
Figure 5.3 shows typical line-averaged density traces from the interferometer, for hydrogen and deuterium plasma jets. From visible images, the radius of the jet is estimated to
be about 3 cm (see figure 5.1) and so the nominal density of the jets is ∼ 3 × 1022 /m3 .
An obvious characteristic of the density traces shown in figure 5.3 is the extremely sharp
rise time in the observed density as the apex of the plasma jet traverses the path intercepted
by the laser beam. This sharp rise time gives a time of flight measurement and can be used
L=29cm
to estimate the average velocity of the plasma as v = time
of flight .

Figure 5.4 plots the time of flight velocity of hydrogen plasma jets as a function of the
maximum gun current flowing through the jet. Plasma experiments were done with fast gas

70
21

x 10

H2, 8 kv, 70 psi (9069)

H2, 7 kv, 70 psi (9114)

2.5

D2, 7 kv, 70 psi (9399)

D2, 8 kv, 70 psi (9358)

355
305
254

Phase (°)

Line densityelectron (/m2)

3.5

203

1.5

152

102

0.5

Time (µs)

Figure 5.3: Typical interferometer density traces from the plasma jets.
puff valves pressurized with H2 at either 70 or 100 psi. It is seen from figure 5.4 that plasma
jets with 70 psi gas valve pressure are faster. This demonstrates that the dominant flow
mechanism in the jet cannot be hydrodynamic, i.e., cannot be driven by pressure gradient
∇P , since if the jet were driven by ∇P , a 100 psi jet should move faster than a 70 psi jet.
Also, if the dominant flow mechanism were hydrodynamic, then the characteristic velocity
would be cs ∼ 2 km/s, which is much smaller than the velocity of the jets shown in figure 5.4.
Figure 5.5 plots the average velocity of hydrogen and deuterium plasma jets as a function
of the maximum gun current flowing through the plasma. This shows that the velocity of a
plasma jet is proportional to the current flowing through the plasma and that the hydrogen
plasma jets are faster than the deuterium ones. The high flow speed and its dependence on
mass and current indicate that the plasma jet’s behavior is in sharp contrast to a previously
considered model for astrophysical jets that describes their evolution as series of GradShafranov equilibria [7, Chapter 9.8.3] with boundary conditions determined by the twist
of poloidal field lines [70, 71]. Note also that in such models, a low density magnetically
dominated quasi-equilibrium jet expands against an external plasma with higher pressure,
whereas the laboratory jets in our experiment expand into a vacuum.

45
H2, 70 psi, circle
y=(0.45±0.03)x−(0.31±2.40)
residue=0.94 km/s

Velocity (km/s)

num. of points=87
40

35
H2, 100 psi, square
y=(0.38±0.03)x+(4.02±2.68)
residue=1.24 km/s
num. of points=114
30
70

100

105

Peak Current (kA)
Figure 5.4: Velocity of hydrogen plasma jets as a function of the maximum gun current.
Measurements from shots when the fast gas valves were pressurized to 70 and 100 psi are
plotted as circles and squares respectively. Here, and in subsequent figures, the cyan, green,
blue, black, and red data points refer to the gun discharge voltage of 6, 6.5, 7, 7.5 and 8 kV
respectively. Also, the linear fit to the respective data points is plotted in magenta, and the
equation for the linear fit and corresponding error is shown next to the lines. For the linear
fit, y represents the velocity in km/s, and x represents the peak current in kA.

72
45

Velocity (km/s)

H2, 70 psi, circle
y=(0.45±0.03)x−(0.31±2.40)
residue=0.94 km/s
num. of points=87

30
D2, 70 psi, square
y=(0.33±0.03)x+(1.62±2.86)
residue=1.43 km/s
num. of points=112

20
70

100

105

110

Peak Current (kA)
Figure 5.5: Velocity of plasma jets as a function of the maximum gun current for H2 and
D2 plasmas plotted in circles and squares respectively. The fast gas valves were pressurized
to 70 psi.

5.2.3

Density of the Jets

Figure 5.6 shows that plasma pressure is proportional to the toroidal magnetic field energy
density and that the plasma jets have βφ := BnkT
2 /2µ ∼ 0.5. For thermal energy density nkTi

in figure 5.6, ion temperature was assumed to be 2 eV [17] for all the shots, and density was
inferred from the peak of density traces (see figure 5.3) assuming a plasma radius of 10 cm.
Toroidal field energy density (Bφ2 /(2µ◦ )) was calculated using Bφ = µ◦ I/(2πa), where I is
the instantaneous current flowing through the plasma when the density was measured, and
a, the radius of current channel, was assumed to be 10 cm. Also the typical axial field in
the experiment was Bz ∼ 0.2 T, which corresponds to βz := BnkT
∼ 0.1 − 0.2. Plasma
2 /2µ

density was observed to scale directly with Bφ and inversely with Bz .
Figure 5.7 plots the plasma pressure as a function of toroidal magnetic field energy
density for D2 plasma jets. While the data has more scatter compared to figure 5.6, it is
seen that βφ ∼ 1 for shots involving higher poloidal current (the red and black squares).

βφ=0.65

Thermal energy density (kJ/m )

βφ=0.35

Toroidal field energy density (kJ/m3)

Figure 5.6: Thermal energy density as a function of toroidal magnetic field energy density
for hydrogen plasma jets.
4.5
βφ=1.0

Thermal energy density (kJ/m3)

3.5
2.5

β =0.45

1.5
0.5

Toroidal field energy density (kJ/m3)

Figure 5.7: Thermal energy density as a function of toroidal magnetic field energy density
for deuterium plasma jets.

74
30
25

Phase (°)

20
15

δφ∼3°

δ t∼0.1 µs

x 10

−5

Line densityelectron (/m2)

254

6.5

1.5

152

102

0.5

7.5

203

Time (µs)

Phase (°)

2.5

Time (µs)

Figure 5.8: Density of the hydrogen plasma jet produced by gas valve pressurized to 100 psi.
Shot #9205.

5.2.4

Distribution of Neutrals in the Jet

It is seen from figure 5.4 that plasma jets produced when the fast gas valves were pressurized
to 100 psi are slower. A typical density trace from such plasma shots is shown in figure 5.8.
The inset shows that just before the plasma intercepts the laser beam, there is a “negative
dip” in density. The negative dip corresponds to a phase change of δφ ∼ 3◦ , and has a
duration of δt ∼ 0.1 µs. The negative dip is from refractive index of neutrals in front of
the jet being pushed along. The refractive index of a dense neutral gas is given by the

75
Gladstone-Dale relation [72, Chapter 2]:
ρ = K(n − 1) = Kδn,
where ρ is the mass density, K is the proportionality constant, n is the refractive index of
the gas, and δn := n − 1 is the difference between the refractive index of the gas and the
refractive index of vacuum.
The phase difference measured by the interferometer because of neutral gas is
δφ =

δk dx =

2π
2πL
δn dx ∼
δn,

where λ is the free space wavelength of He-Ne laser and L ∼ 20 cm is the length of the path
of the laser beam within the dense neutral gas. Thus a phase change of δφ ∼ 3◦ corresponds
to δn ∼ 10−7 .
For hydrogen, δn ∼ 10−4 at NTP(normal temperature and pressure) [73, Page E-224].
Thus the number density of neutrals in front of the jet, nexperiment , is given by
δnexperiment
nexperiment
10−7
∼ −4 .
nNTP
δnNTP
10
Thus the number density of neutrals in front of the jets produced by gas valves pressurized to 100 psi is 1022 − 1023 /m3 . Since the jet is travelling at ∼ 40 km/s and the duration
of the negative dip is δt ∼ 0.1 µs, the thickness of the layer of neutral molecules is ∼ 0.4 cm.
The presence of a cloud of neutrals in front of the jets (for shots done with gas valves
pressurized to 100 psi), implies incomplete ionization of the neutrals at breakdown. The
Bohm time [74, section 5.10] for the plasma is ∼ 500 µs, which is much larger than the
lifetime of the plasma jets. Thus the ions in the plasma jet are unable to diffuse into the
neutral cloud in front of the jet.

5.3

Model

This section describes a model showing that the flow is driven by axial gradient in Bφ2
associated with the slight flaring of the jet. This model is a generalization of the model

76
presented by Bellan [75].
Consider a cylindrical coordinate system {r, φ, z} with the origin at the center of the
electrodes and z axis along the direction of the jet flow. The jet is assumed to be axisymmetric and slightly flared. Furthermore, the poloidal flux inside the jet is assumed to have
the simplest non-trivial physically relevant form
ψ(r, z) = ψ◦

r2
a(z)2

(5.1)

where
a(z) = a◦ eκz
describes the flaring of the jet of radius a(z). Here a◦ is the jet radius at the electrodes (refer
to figure 5.1) and the constant κ is determined from flaring in figure 5.1. We assume that
the pressure P (r, z) vanishes at r = a(z) which corresponds to assuming that Bz is nearly
uniform in the jet or equivalently that the radial scale length of Bz exceeds the radial scale
length for pressure. Plasma jets with a large aspect ratio (length
radius) are assumed to
be well described by equation (5.1).
Let
I(r, z) = I(ψ) =

λψ
λψ◦ r2
r2
µ◦
µ◦ a(z)2
a(z)2

(5.2)

where λ is a constant with dimension (length)−1 . The assumption I = I(ψ) implies that
current flows along flux surfaces so there is no torque (φ̂ · (J × B)) causing acceleration
in the φ direction [75]. Poloidal current described by equation (5.2) implies that the axial
current density Jz is independent of r.
The associated toroidal/poloidal magnetic fields and current densities are:
µ◦ I
∇φ,
2π
∇ψ × ∇φ,
2π
r2
= −
∇·
∇ψ ∇φ,
2πµ◦
r2
∇I × ∇φ,
2π

Btor =
Bpol
Jtor
Jpol

(5.3)

where ∇φ = φ̂/r. Note that a toroidal current density, Jtor , will only exist if the poloidal

77
flux function deviates from a vacuum field solution.
Due to the jet’s large aspect ratio, radial equilibrium is achieved much faster than the
axial equilibrium. This is evident from figure 5.1, where the radial profile of the jet hardly
changes as it evolves. MHD radial pressure balance (refer to equation (1.3)) then implies
∂P
∂r

= (Jpol × Btor )r + (Jtor × Bpol )r

µ◦ ∂ I 2
= −
(2πr)2 ∂r 2
∂ 1 ∂ψ
∂ 2 ψ ∂ψ
∂z 2 ∂r
(2πr)2 µ◦ ∂r r ∂r

Note that 1r ∂ψ
∂r = constant because of the ψ ∼ r dependence assumed in equation (5.1).

This implies
∂P
∂r

λ2
∂ψ
∂ 2 ψ ∂ψ
(2πr)2 µ◦ ∂r
(2πr)2 µ◦ ∂z 2 ∂r
λ2 + 4κ2 ∂ψ
= −
(2πr)2 µ◦ ∂r

= −

(5.4)

Equation (5.4) can be integrated radially to give
λ2 + 4κ2 ψ◦2
r2
P (r, z) =
(2πa)2 µ◦
a2

(5.5)

where the boundary condition is P (r, z) = 0 at r = a(z).
Current and hence λ are time dependent in the experiment (see figure 3.2). However,
using an average value of current I ∼ 75 kA and applied poloidal flux ψ◦ ∼ 4 mWb yields
a nominal value of λ ∼ 20 m−1 . From the visual images (see figure 5.1) the flaring of the
jet corresponds to κ ∼ 2 m−1 . Thus λ2
4κ2 and hence 4κ2 can be neglected in equation
(5.5). Thus, P (r, z) is predominantly determined by the axial current (or equivalently the
toroidal magnetic field). For such a jet, the plasma toroidal beta βφ should be of order
unity, as shown in figure 5.6.
The jet is in radial force balance, but there is no such balance along the z-axis. Consider

78
the z component of the MHD equation of motion (refer to equation (1.4)) in steady-state
∂P
∂z

(5.6)

#
λ2 + 4κ2 ψ◦2
r2
1− 2
(2πa)2 µ◦
2a

(5.7)

[∇ · (ρuu)]z = (Jpol × Btor )z + (Jtor × Bpol )z −
Using equations (5.1)-(5.3) and (5.5), equation (5.6) becomes
[∇ · (ρuu)]z = −
∂z

Using λ2
4κ2 , equation (5.7) can be rewritten as

λ2 ψ◦2
r2
= −
1− 2
∂z (2πa)2 µ◦
2a

∂ µ◦ I◦
= −
1− 2
∂z 4π 2 a2
2a
#
∂ Bφ,a
r2
= −
1− 2
∂z µ◦
2a

[∇ · (ρuu)]z

(5.8)

where Bφ,a = µ◦ I◦ /2πa is the toroidal magnetic field evaluated at r = a(z). The plasma
velocity is predominantly oriented along the z direction. So equation (5.8) can be simplified
as
∂ 2
ρuz ≃ −
∂z
∂z

Bφ,a

µ◦

#
r2
1− 2
2a

(5.9)

Evaluating it at r = 0 gives
∂z

ρu2z +

Bφ,a

µ◦

= 0,

(5.10)

r=0

which is similar to the Bernoulli equation. Integrating equation (5.10) along the length of
the jet gives
ρu2z +

Bφ,a

µ◦

r=0,z=0

ρu2z +

Bφ,a

µ◦

(5.11)

r=0,z=L

At the electrodes (z = 0), uz ∼ 0, and Bφ,a = µ◦ I◦ /2πa◦ . Far from the electrodes (z = L)
of the jet, Bφ,a is small and can be neglected. Thus the flow velocity at the jet tip can be

79
estimated as
ur=0,z=L ∼
ur=0,z=L ∼

Bφ,a

ρµ◦

2πa◦

µ◦
I◦ .

(5.12)

Equation (5.6) assumed the plasma jet to be in steady-state. If the time dependent
inertial term ∂ (ρu) /∂t was included in the analysis, the final result of equation (5.12)
could still be derived by considering dimensionless scaling of the equations. In such a case,
the equations can be cast in a dimensionless form with a characteristic velocity given by
equation (5.12). Also, the steady-state assumption is justified for the bulk of the plasma
jet, except for the dynamic apex.

5.3.1

Comparison of the Model with Experimental Results

We now compare the experimental results to the quantitative predictions of the theory.
Using the typical parameters of a hydrogen plasma jet (ne ∼ 3 × 1022 /m3 , a◦ ∼ 3 − 10 cm),
equation (5.12) predicts that the slope of the u vs I linear fit for hydrogen plasmas in
figure 5.5 should be 0.25 − 0.84 m/sA−1 . The experimentally observed linear dependence
has a slope of 0.45 m/sA−1 . Equation (5.12) also suggests that the slope for the linear
fits in figure 5.5 should scale inversely with the square root of the mass of ions. Thus it
predicts that the ratio of slopes of the linear fits in figure 5.5 for hydrogen and deuterium
plasmas should be 12 = 0.707. From the experiments the ratio of slopes is measured as
0.33±0.03
0.45±0.03 = 0.73±0.08. Thus the experimentally measured value agrees reasonably well with

the predicted ion mass dependence. The results clearly show that the phenomenon causing
bulk plasma motion is non-equilibrium dynamics as the response of the plasma is inversely
related to the square root of the ion mass.

5.4

Energy Balance for Plasma Jets

An energy balance argument for the plasma jets is presented in this section. A coaxial
gun expanding against a restraining spring (see figure 5.9) is a simple analogue to the
Caltech spheromak experiment. The current flowing between the outer (brown) and the

Figure 5.9: Cut out of a coaxial gun expanding against a spring.
inner (blue) conductor in figure 5.9 produces a toroidal magnetic field. The toroidal field
pressure accelerates the conducting red disk outwards, thus increasing the inductance of
the rail gun. Thus the toroidal field pressure acts in increasing the inductance in both
the rail gun and in the Caltech spheromak experiment. As the rail gun moves outward, it
deforms the spring and thus stores potential energy in the compressed spring. In the Caltech
spheromak experiment, as the jet expands, it deforms the applied poloidal magnetic field
R B2
there by storing energy in the stretched poloidal field Wpol = 2µpol◦ d3 r.
We now consider the energy balance in the coaxial rail gun. Neglecting the resistance
of the rail gun, the input energy is given by the integral of Poynting flux at the electrodes
Z t
Winput =

Z t
V Idt =

(LI) Idt,
0 dt

where V is the voltage appearing across the electrodes, I is the current linked by the
electrodes and L is the inductance of the jet. Since the current is assumed to be constant,
the input energy can be approximated as
Z t
Winput =

dL
dt = LI 2 .
dt

81
The input energy is split into the the energy in the toroidal field, the potential energy in
the spring and the kinetic energy of the disk.
Winput = Wtor + Wpot + Wkin ,
|{z} | {z }
| {z }
= 12 LI 2

= 12 kx2

(5.13)

= 12 mv 2

where k is the spring constant, x is the deformation of the spring, m is the mass of the disk,
and v is the velocity of the disk.
Note that the energy in the toroidal field is half of the input energy. Thus, in the absence
of a restraining spring, the energy input from the current source is distributed equally into
the disk’s kinetic energy and the toroidal magnetic field energy in the rail gun.
However, a more physically relevant situation is when the kinetic energy term in equation
(5.13) can be neglected. When the kinetic energy of the disk is negligible, the input energy
is split evenly between the toroidal field energy and the spring potential energy. The kinetic
energy of the rail gun can be neglected if:
1. The restraining force from the spring is almost equal to the outward force from the
toroidal magnetic field pressure. The velocity gained by the disk will be negligible
and the rail gun can be considered to be almost in equilibrium. The analog of this
situation for the Caltech spheromak experiment is if the plasma jet were to evolve in
force-free equilibrium states, thereby having equal amounts of toroidal and poloidal
field energies (see [3, section 4.4.1]).
2. The mass, m, of the disk is small, and hence the kinetic energy is negligible.
We will now extend these ideas developed for the energy balance in the rail gun to
the energy balance in the plasma jet. Consider an ideal plasma jet that is driven by an
electrode with constant current. The energy flowing into the plasma is the time integral of
the Poynting flux at the electrodes
Z t

Z t

Winput =
V Idt =
(LI) Idt +
0 dt

Z t

I 2 Rdt,

where V is the voltage appearing across the electrodes, I is the current linked by the
electrodes, L is the inductance of the jet, and R is the resistance of the jet. Since the

82
current is assumed to be constant, the input energy can be approximated as
Z t
Winput =

dL
dt +
dt

Z t

I 2 Rdt = LI 2 +

Z t

I 2 Rdt.

(5.14)

According to the definitions introduced in section 1.1.1, let Wtor , Wpol , Wth and Wkin
be the energy in the toroidal magnetic field, the energy in the poloidal magnetic field, the
thermal energy of the jet and the kinetic energy of the jet. The corresponding energy
densities are represented by w and the appropriate subscript. Thus
Wtor =

wtor d r,

Wpol =

wpol d r,

Wth =

wth d r,

Wkin =

wkin d3 r.

By conservation of energy (see equation (1.7))
Winput = Wtor + Wpol + Wth + Wkin + Wradiation-loss .
In low temperature plasmas the energy input from the resistive term in equation (5.14) is
expected to be almost completely lost in line emission (Wradiation-loss ) [76]. Thus, the plasma
jets are expected to gain minimal thermal energy from Ohmic heating. This gives
LI 2 ∼ Wtor + Wpol + Wth + Wkin .
Since, Wtor = LI 2 /2, we get
Wpol + Wth + Wkin ∼ Wtor .

(5.15)

Had the jet been force free, then Wth = Wkin = 0 and the energies in the toroidal and
poloidal fields would have been equal [3, section 4.4.1]. However, with the non-equilibrium
jets at the Caltech spheromak experiment all the quantities in equation (5.15) are positive definite. At the Caltech spheromak experiment, a strong poloidal field is applied for
substantial helicity injection and also for ensuring kink stability. Thus the energies in the
toroidal and poloidal magnetic fields are comparable to each other. Using equation (5.15),
this implies Wpol . Wtor , and also that the thermal and kinetic energy content of the jet is

Vacuum like

Anode

50 cm

20 cm

Dense plasma

Cathode
6 cm

Force−free

Vacuum like

Anode

Force−free

Figure 5.10: A cartoon of the jet showing three different regions of the plasma jet.
significantly less than the toroidal field energy. Equivalently
Wkin
Wtor ,
Wth
Wtor .

(5.16)

The results in sections 5.3 and 5.2 indicate that
wth ∼ wkin ∼ wtor .

(5.17)

Equations (5.16) and (5.17) can be simultaneously satisfied only if the spatial extent of the
plasma is small compared to the spatial extent of the toroidal magnetic field. This fact is
validated by observation of the plasma jet radius of ∼ 3 cm in figure 5.1 and a radius of
∼ 10 cm of the toroidal field densities in figure 5.2. Even equation (5.5) predicts that the
radial extent of plasma is smaller than the radius of the current channel.
These inferences can be summarized as follows:
1. The center of the jet (represented by the yellow in figure 5.10) consists of dense plasma.
It has a radial extent of ∼ 3 cm.

84
2. Outside the central region (represented by the cyan region in figure 5.10), the plasma
density is negligible. Hence this region is expected to be in a force free state described
by J × B ∼ 0. The radial extent of the force free region is ∼ 10 cm. The toroidal
magnetic pressure in this region is comparable to the plasma thermal energy density
and kinetic energy density in the plasma central region.
3. Outside the force free region, the magnetic fields are vacuum fields until the return
currents to the anode are encountered. this region is represented by green in figure
5.10. The return currents are located after a radius of ∼ 25 cm.

5.5

Conclusion

Alfvénic flows based on toroidal magnetic field pressure have been observed in earlier coaxial
gun experiments [58]. However, this chapter shows that even in the presence of a substantial
external poloidal magnetic field required for helicity injection, the plasma flow is predominantly Alfvénic with respect to the toroidal magnetic field, provided there is only a slight
flare in the poloidal current channel of the plasma jet.
The following are the main results from this chapter:
1. MHD based non-equilibrium slightly flaring plasma jets are observed during helicity
injection.
2. These jets emanate outwards from the electrodes towards the vacuum with Alfvénic
velocities. The speeds of these jets are not based on neutral pressure gradients, but
on the axial gradient in the toroidal magnetic field energy density.
3. The pressure in the jets is balanced by the toroidal magnetic field. Hence these jets
have βφ ∼ 1.
4. The radial scale for pressure in these jets is smaller than the radial extent of poloidal
current or poloidal flux.
5. When the jets are formed by neutral gas fed from fast gas valves pressurized at 100 psi,
then there is a layer of neutrals of density 1022 − 1023 /m3 and thickness ∼ 0.5 cm in
front of the jet.

Chapter 6
X-ray Imaging System for the Caltech Solar
Coronal Loop Simulation Experiment

Energetic particles and radiation have been observed in reconnecting plasmas in both laboratory [77] and extra-terrestrial plasmas [78]. The Caltech solar coronal loop simulation
experiment [79] is designed to study reconnecting magnetic flux tubes and the relevant
physics of solar prominences [5, 80]. This chapter focuses on characterizing the soft X-ray
(SXR) and Vacuum Ultra Violet (VUV) radiation observed from the experiment.
The chapter is organized as follows. Section 6.1 describes the Caltech solar coronal
loop simulation experiment. It also explains the three different modes of operation of the
experiment. Section 6.2 describes the diagnostics used in the experiment. The X-ray diodes
and the X-ray imaging system are described in detail in sections 6.2.1 and 6.2.2 respectively.
Section 6.3 compares the various modes of operating the experiment (single prominence,
co-helicity and counter-helicity merging), focusing on the production of X-ray photons.
Finally, section 6.4 suggests some ideas for future research on the experiment.

6.1

Overview of the Experiment

The Caltech solar prominence simulation experiment uses the “Mark IV” electrode design,
described in detail by Hansen [79], Chapter 5. The setup consists of four electrodes each
with a gas injection orifice (see figure 6.1). The upper electrodes are the cathode and the
lower electrodes are the anode. The electrode polarity is fixed and is usually not changed.
During a plasma discharge, the electrodes are floating with respect to the chamber ground.

86
Cathodes
Radius = 12.7 cm

8.1 cm

Gas injection
foot point

Anodes

Figure 6.1: Electrodes for the dual prominence experiment.
A 59 µF capacitor bank is switched by a size A GL-7703 ignitron to achieve the plasma
discharge. For all the results presented in this chapter, the capacitor was charged to 6 kV.
Four magnetic field coils of inductance ∼ 280 µH each are located behind each of the gas
injection points. Each coil creates a bias magnetic field either out of the plane of the picture
in figure 6.1 (represented by “North - N” in subsequent figures) or into the plane of the
picture (represented by “South - S” in subsequent figures). The polarity of each magnetic
field bias coil can be controlled independently. The applied magnetic field (referred to
as the toroidal bias field in this chapter) is constant for the duration of experiment. The
magnetic coils behind the cathode foot points are powered by an electrically isolated 9.9 mF
capacitor bank and the magnetic coils behind the anode foot points are powered by another
electrically isolated 9.6 mF capacitor bank. For all the measurements reported in this
chapter, the magnetic field coils were discharged at 200 V, thereby creating a toroidal flux
of ∼ 0.2 mWb at each foot point.1
Neutral gas was puffed into the chamber at the gas injection foot points shown in the
picture. The two orifices in the cathode were fed by a fast gas puff valve [3, chapter 14].
The two orifices behind the anode were fed by a similar gas puff valve.
The following sequence of events was followed to create a plasma discharge:

Calibration of the toroidal flux was performed by Rory Perkins and Eve Stenson.

87
1. The toroidal field power supplies were triggered 4.5 ms before the plasma discharge.
It takes 4.5 ms for the magnetic field to reach its maximum and link adjacent foot
points on the cathode and the anode.
2. Fast gas valves were triggered 2 ms before the plasma discharge. It takes about 2 ms
for neutral gas to travel from the fast gas puff valves to the orifices at the foot points.
The fast gas valves were pressurized with deuterium at 100 psi for all the experiments
reported in this chapter.
3. The 59 µF capacitor was discharged across the electrodes to create a plasma.
4. The various diagnostics and the digitizers were triggered upon the neutral gas breakdown near the electrodes.
The experiment can be run in the following modes:
1. Single prominence: The setup for the single prominence experiment is shown in figure
6.2. Only the left electrodes are used to create the plasma discharge. The bias
toroidal field is not created at the right electrodes. Neutral gas is also not injected at
the orifices in the right electrodes. As the plasma flux tube expands due to the hoop
force, it kinks. The shape of the flux tube resembles the shape of a typical helical
field line shown in the figure i.e., a dip is observed from the side view and a reversed
“S” shape from the top view.
2. Co-helicity merging: The setup for the co-helicity merging experiment is shown in
figure 6.3. Two similar magnetic flux tubes are created by each of the electrodes.
Since the same amount of helicity is injected into both the flux tubes, they behave in
an identical manner. For example, they have identical shapes when looked at from
the top view. The plasma flux tubes expand due to the hoop force and also merge
with each other due to the attraction between parallel currents.
3. Counter-helicity merging: The setup for the counter-helicity merging experiment is
shown in figure 6.4. Two magnetic flux tubes are created by each of the electrodes. The
toroidal bias field is in opposite direction in the flux tubes. Thus, equal but opposite
amounts of helicity are injected into each of the flux tubes. Since the handedness of

S+

N−

in black showing the characteristic “dip” in the side view and the reversed “S” shape in the top view.

arrow and the field generated by the gun current is represented by the green arrow. A representative magnetic field line is plotted

Figure 6.2: Cartoon showing the setup of the single prominence experiment. The applied toroidal field is represented by the black

(b) Side view.

S+

(a) Setup for single prominence.

N−

S+

(b) Side view.

S+

N−

S+

N−

S+

N−

in black showing the characteristic “dip” in the side view and the reversed “S” shapes in the top view.

arrow and the field generated by the gun current is represented by the green arrow. Representative magnetic field lines are plotted

Figure 6.3: Cartoon showing the setup of the co-helicity merging experiment. The applied toroidal field is represented by the black

(a) Setup for co-helicity merging.

N−

N+

S+

(b) Side view.

S+

N−

N+

S−

plotted in black showing the characteristic “dip” in the side view and the “S” and reversed ”S” shapes in the top view.

black arrow and the field generated by the gun current is represented by the green arrow. Representative magnetic field lines are

Figure 6.4: Cartoon showing the setup of the counter-helicity merging experiment. The applied toroidal field is represented by the

(a) Setup for counter-helicity merging.

S−

N−

91
the magnetic field is opposite in each of the flux tubes, an “S” and a reversed “S”
shaped flux tube is observed from the top view.
The origin of the “S” and the reversed “S” shape of the plasma in the experiments can be
explained in terms of a kink in the flux tube (as shown in figures 6.2, 6.3, 6.4 and explained
by Rust and Kumar [81]) or by considering plasma in a Taylor state equilibrium [5].

6.2

Diagnostics

The following commercially available diagnostics were used in the experiment:
1. A Tektronix P6015 [14] high voltage probe was used to measure the potential difference
across the electrodes.
2. A Rogowski coil [3, chapter 15] was used to measure the current flowing through the
high voltage capacitor bank.
3. Imacon 200 - a high speed imaging camera manufactured by DRS Technologies [15] was used to take visible images of the plasma.
In addition to the diagnostics mentioned above, two more diagnostics were upgraded for
the experiment. These are described below.

6.2.1

X-ray Diodes

Four high speed X-ray sensitive diodes (Part number AXUV-HS5) manufactured by International Radiation Detectors [82] were used to diagnose the evolution of plasma-filled flux
tubes in the experiment.2 The diodes were reverse biased to 45 V, to achieve maximal
efficiency and the best rise time performance (< 1 ns). The diodes are sensitive to photons
with energies & 10 eV, and their gain increases by ∼ 17% per eV increase in photon energy.
Three out of the four diodes were covered with thin metal foil filters manufactured by
Lebow Company[83]. The filters placed in front of the diodes were Ti 50 nm, Al 200 nm
and Ti 500 nm. The transmission properties of the filters is plotted in figure 6.5 [84, 85].

The mechanical assembly of the diodes and the electronics were designed by Paul Bellan.

92
Ti 50 nm
Al 200nm
Ti 500 nm

0.9
0.8

Filter transmission

0.7
0.6
0.5
0.4
0.3
0.2
0.1

100

150
200
250
Photon energy (eV)

300

350

400

Figure 6.5: Transmission characteristics of X-ray foil filters.
It is shown in section 6.3 that no output was detected by the diode behind the Ti 500 nm
filter. This shows that X-ray photons produced by the experiments have energies . 200 eV.
To understand the energy spectrum of the photons produced in the experiment, the Ti 500 nm
filter should be replaced by a filter whose bandwidth is limited to 75 − 200 eV.

6.2.2

X-ray Imaging System

Photons in the VUV (Vacuum Ultra Violet) to SXR (Soft X-ray) band are severely attenuated by glass. Thus X-ray scintillators are usually placed so that they share the same
vacuum as the main experiment, or special windows are used to ensure transmission of the
photons. The X-ray diode array built by Snider et al. [86] used Beryllium vacuum windows
to prevent X-ray attenuation. They also used pressurized neutral gas as an X-ray energy
filter.
The VUV/SXR imaging system built for the Caltech solar coronal loop experiment was
installed in vacuum. figure 6.6 shows the setup of the imaging system. A pin hole of
diameter 200 µm was used in the setup. The pin hole was placed ∼ 5 cm in front of the
Micro Channel Plate (MCP). The position of the pinhole can be adjusted to change the

Figure 6.6: Schematic of the X-ray imaging setup.

Meade DSI Pro II

lens

Mechanical mount
(MCP assembly inside)

2.75" conflat vacuum bellows
10" conflat

tilt
adjustment
gas lines used to
break vacuum or
flush dry nitrogen

linear
translation

6" gate valve

(a) Setup of the X-ray imaging camera showing parts outside the vacuum.

(b) Pin hole used for X-ray imaging

Figure 6.7: Setup of the X-ray imaging setup.

95
zoom level of the imaging system. There was no foil filter used for the images presented
in this chapter. The MCP and phosphor screen assembly (Part number XUV-2018) was
manufactured by X-ray and Speciality Instruments [87]. The MCP had a diameter ∼ 1 inch
and was coated with CsI. It was sensitive to radiation with energies & 6 eV. The MCP
assembly was mounted on a 2.75” conflat vacuum flange. The MCP assembly was attached
to the main vacuum chamber with flexible vacuum bellows which could be tilted to change
the region under view. However, the tilt was not adjusted for the results presented in this
chapter.
A −900 V pulse generated by a Berkeley Nucleonics high voltage pulse generator Model
310H [88] was connected to the front end of the MCP. The pulse allowed the creation of
secondary electrons by the MCP when X-ray photons were incident on it. Thus the pulse
acted as an effective shutter mechanism for the imaging system. The camera was sensitive
with 10 ns pulse duration, but sharper images were obtained by pulses of duration 20 ns.
The pulse duration was maintained at 20 ns for all the X-ray images presented in this
chapter. The electrons from the MCP were accelerated by a 3 kV DC bias applied between
the phosphor screen and the MCP (see figure 6.6). The electrons caused the phosphor
screen to fluorescence. The image on the phosphor screen lasted for several milliseconds
(∼ 50 ms) and was imaged by a Meade DSI Pro II CCD camera [89]. Data from the Meade
camera was communicated to a desktop PC with a USB 2.0 cable. The Meade camera was
mounted on an optical mount with tilt and linear translation capability (see figure 6.7(a)).3
Sample images from the X-ray imaging system are shown in section 6.3. However, the
following changes may improve the X-ray imaging system:
1. The DC voltage bias to the phosphor screen was limited to 3 kV by the existing power
supply. Increasing it to 5 kV will improve the contrast of the images.
2. The high voltage pulse from Berkley Nucleonics 310H was relayed to the MCP via a
∼ 10 m long coaxial cable. The cable was terminated by its characteristic impedance
of 50 Ω, yet the long length of the cable caused a slight resistive loss of the pulse
amplitude and also altered the shape of the pulse. Moving the 310H module closer to
the MCP and using a shorter cable will remove these problems and may improve the

The mount for the Meade camera was modified from an initial design by Hyungmin Park.

96
contrast of the images.
3. For the images shown in this chapter, no foil filter was used in the setup (see figure
6.6). Using a foil filter may help localize the high energy radiation observed only in
the counter-helicity merging experiment.
4. Different pin holes may be used in the setup of the camera. The effect of changing
the pin holes on the intensity and sharpness of the images has not been investigated.

6.3

Results

The X-ray production and other differences between the three modes of the experiment
have been previously studied semi-quantitatively by Hansen et al. [90]. In their work, they
found:
1. Counter-helicity experiments produced significantly more X-rays than the co-helicity
and single prominence experiments.
2. A bright region (the dip) at the apex of the flux tubes was observed in counter-helicity
merging experiment.
The results presented in this section confirm the above observations and also offer more
specific insights. The results and the dynamics of the plasma filled flux tubes for each of
the three modes are described below:
1. Single prominence: It is seen from figure 6.8 that the plasma-filled flux tube expands
due to the hoop force and starts writhing at ∼ 1 µs. As the flux tube expands further,
the plasma becomes extremely diffuse and detaches from the electrodes at ∼ 2.8 µs.
The voltage and current traces (figure 6.9) across the electrodes show discrete voltage
jumps and current fluctuations at ∼ 3 − 4 µs after breakdown. This is a characteristic
of the plasma flux tube detaching from the electrodes, thereby causing a change in
plasma inductance. The X-ray diode signals from the single prominence simulation
experiment (figure 6.10) indicate that almost no signal is measured by the diodes
with foil filters in front. Thus the radiation from the single prominence experiment
is limited to . 10 eV. The measured radiation has a distinct temporal behavior. It

0.28 µs

0.57 µs

0.88 µs

1.18 µs

1.48 µs

1.77 µs

2.08 µs

2.38 µs

2.67 µs

2.98 µs

3.27 µs

3.58 µs

Figure 6.8: Fast camera images from single prominence simulation experiment. Visible
band. Shot #6343.

Smoothed gun current
Gun current
Gun voltage

−50

−1

Gun voltage (kV)

Gun current (kA)

Discrete voltage jumps

Current fluctuations

−100
−5

Time (µs)

Figure 6.9: Current and voltage traces from single prominence simulation experiment. Shot
# 6343.
300
No filter
Ti 50 nm
Al 200nm
Ti 500 nm

250

Signal (mV)

200

150

100

−50
−5

Time (µs)

Figure 6.10: X-ray diode signals from single prominence simulation experiment. Shot
#6343.

4.40 µs
Shot #6339

3.40 µs
Shot #6335

Figure 6.11: VUV/Soft X-ray images from single prominence simulation experiment.

1.40 µs
Shot #6326

0.40 µs
Shot #6318

5.40 µs
Shot #6344

2.40 µs
Shot #6329
99

100

0.28 µs

0.57 µs

0.88 µs

1.18 µs

1.48 µs

1.77 µs

2.08 µs

2.38 µs

2.67 µs

2.98 µs

3.27 µs

3.58 µs

Figure 6.12: Fast camera images from co-helicity merging experiment. Visible band. Shot
#6296.

101

Smoothed gun current
Gun current
Gun voltage

−50

−1

Gun voltage (kV)

Gun current (kA)

Discrete voltage jumps

Current fluctuations

−100
−5

Time (µs)

Figure 6.13: Current and voltage traces from co-helicity merging experiment. Shot #6296.

300
No filter
Ti 50 nm
Al 200nm
Ti 500 nm

250

Signal (mV)

200

150

100

−50
−5

Time (µs)

Figure 6.14: X-ray diode signals from co-helicity merging experiment. Shot #6296.

4.40 µs
Shot #6293

3.40 µs
Shot #6297

Figure 6.15: VUV/Soft X-ray images from co-helicity merging experiment.

1.40 µs
Shot #6307

0.40 µs
Shot #6312

5.40 µs
Shot #6287

2.40 µs
Shot #6302
102

103

0.28 µs

0.57 µs

0.88 µs

1.18 µs

1.48 µs

1.77 µs

2.08 µs

2.38 µs

2.67 µs

2.98 µs

3.27 µs

3.58 µs

Figure 6.16: Fast camera images from counter-helicity merging experiment. Visible band.
Shot #6273.

104

Smoothed gun current
Gun current
Gun voltage

−50

−1

Gun voltage (kV)

Gun current (kA)

Discrete voltage jumps

Current fluctuations

−100
−5

Time (µs)

Figure 6.17: Current and voltage traces from counter-helicity merging experiment. Shot
#6273.

300
No filter
Ti 50 nm
Al 200nm
Ti 500 nm

250

Signal (mV)

200

150

100

−50
−5

Time (µs)

Figure 6.18: X-ray diode signals from counter-helicity merging experiment. Shot #6273.

4.40 µs
Shot #6259

3.40 µs
Shot #6262

Figure 6.19: VUV/Soft X-ray images from counter-helicity merging experiment.

1.40 µs
Shot #6272

0.40 µs
Shot #6276

5.40 µs
Shot #6283

2.40 µs
Shot #6267
105

106
consists of a slowly varying background signal of ∼ 30−50 mV along with sharp bursts
of duration . 1 µs. These bursts usually range from 100 − 200 mV and occur when
the plasma is detaching form the electrodes (between 3.5 − 4.5 µs after breakdown).
The bursts occur at the same time as the discrete jumps are seen in the voltage across
the electrodes.
The contrast of each individual VUV/SXR image from the single prominence experiment (figure 6.11) has been manually enhanced. It shows an expanding flux tube
which detaches from the electrodes between 2.4 − 3.4 µs.
2. Co-helicity: It is seen from figure 6.12 that the two plasma filled flux tubes expand due
to the hoop force. They start writhing at ∼ 1 µs and also tend to merge at ∼ 2 µs.
The slightly merged flux tubes detach from the electrodes at ∼ 3 µs. The voltage
and current traces (figure 6.13) across the electrodes show discrete voltage jumps and
current fluctuations at ∼ 3 − 4 µs after breakdown, indicating plasma detachment.
The X-ray diode signals from the co-helicity merging experiment (figure 6.14) again
indicate that the radiation from the co-helicity merging experiment is limited to .
10 eV (similar to the single prominence experiment). Usually one (or sometimes
two) bursts are observed in the signal from the unfiltered diode. The first burst
usually occurs at ∼ 3.4 µs after breakdown. As compared to the single prominence
experiment, the first X-ray burst (corresponding to the plasma detachment from the
electrodes) occurs slightly earlier.
The contrast-enhanced VUV/SXR images from the co-helicity merging experiment
(figure 6.15) show two distinct flux tubes until 2.4 µs after the discharge. Images
from after 4 µs show bright arcing between electrodes, unlike the single prominence
experiment.
3. Counter-helicity: It is seen from figure 6.16 that the two plasma filled flux tubes
expand due to the hoop force. They start writhing at ∼ 1 µs and also tend to merge
at ∼ 2 µs because of the pinch force from the parallel currents in adjacent flux tubes.
The central dip at the apex of the flux tubes is brighter than the apex from the cohelicity merging experiment (figure 6.12). The voltage and current traces (figure 6.17)
across the electrodes show discrete voltage jumps and current fluctuations at ∼ 3−5 µs

107
after breakdown, indicating plasma detachment. The X-ray diode signals from the
counter-helicity merging experiment (figure 6.18) are significantly different from the
signals from co-helicity merging experiment. Usually 3 − 4 bursts are observed in the
X-ray diode signals, compared to just one in the co-helicity experiment. Compared to
the co-helicity experiment, these bursts are higher in magnitude (100 − 700 mV). The
sharp peaks in the Ti 50 nm and Al 200 nm traces in figure 6.18 indicate that the the
photon energies are up to ∼ 75 eV, which is considerably higher than photon energies
in the co-helicity merging experiment. Also, the first burst in the X-ray diode signal
is usually 3 µs after the breakdown, which is approximately 0.3 − 0.4 µs before the
first burst observed in the co-helicity merging experiment. This indicates that the
merged flux tubes in the counter-helicity experiment tend to go kink unstable and
detach from the electrodes earlier than the corresponding merged flux tubes in the
co-helicity experiment.
The contrast-enhanced VUV/SXR images from the counter-helicity merging experiment (figure 6.19) are also significantly different than the images from the co-helicity
experiment (figure 6.15). Comparing the images at 2.4 µs indicates that the flux tubes
have already merged in the counter-helicity experiment and are still distinct in the
co-helicity experiment. The intensity of the radiation from the bulk of the flux tubes
in the counter-helicity experiment is also greater than the radiation observed from
the co-helicity experiment. When the merged flux tubes detach from the electrodes
(images at 2.4 and 3.4 µs in figure 6.19), an intense spot is observed near the cathodes.
Such a bright region is not observed in the co-helicity experiment.

6.3.1

Interpretation

There are two distinct reconnection events in the co- and counter-helicity merging experiments. The first is when the adjacent flux tubes merge, and the second is when the flux
tubes detach from the electrodes.
During the merging of the flux tubes in the counter-helicity experiment, the oppositely
directed toroidal magnetic fields in adjacent flux tubes are annihilated. The annihilation of
the toroidal field releases more energy than the merging of the flux tubes in the co-helicity
experiment. Thus the bulk plasma is brighter in the counter-helicity experiment than in

108
the co-helicity experiment. Since the bulk plasma mostly radiates in the VUV range, the
radiation is most likely from line emission. This can be investigated by VUV spectroscopy.
Annihilation of the toroidal magnetic field also makes the flux tube in counter-helicity
experiment more susceptible to kink instability. Thus the merged flux tube in the counter
helicity experiment goes unstable and detaches from the electrode before the corresponding
flux tube in the co-helicity experiment. The reason why the detachment leads to significantly higher energy photons in the counter-helicity experiment than in the co-helicity
experiment is still unclear. The bursts in X-rays from the counter-helicity experiment are
in a significantly higher energy band (up to 80 eV) and might be Bremsstrahlung.

6.4

Future Work

The results presented in this chapter provided some insight into the dynamics of flux tube
merging in the experiment and the production of high energy photons. The following steps
are suggested to further enhance the understanding of the experiments:
1. The gas orifice foot points at the cathode are much brighter than the anode foot points.
This can be seen in all the visual and X-ray images from section 6.3. It is certainly
a non-MHD effect, and the spectra of the emitted photons should be diagnosed using
different energy filters.
2. The X-ray images from the counter-helicity experiment were taken at 1 µs intervals
(see figure 6.19). Images should be taken at much more frequent intervals and with
different foil filters to help understand the origin and spectra of the X-ray bursts.
3. The application of an external strapping field [80] will inhibit the detachment of the
flux tubes from the electrodes. Such a scenario will be beneficial to focus only on the
reconnection of the adjacent flux tubes before they detach from the electrodes.
4. The bursts of X-rays from the experiment corresponded to the flux tubes detaching
from the electrodes. However, such bursts were absent when the kinked plasma jets
detached from the spheromak electrodes (see chapter 5). The reason for this is not
understood.

109

Chapter 7
Summary

Chapter 1 described the concept of spheromaks as viable magnetic fusion reactors. Spheromaks rely on the conservation of magnetic helicity as plasmas relax to an equilibrium state.
Chapter 2 considered the problem of sustaining magnetic helicity in a steady state driven
spheromak. It was shown that resistive MHD equilibrium is not possible when a plasma
has closed flux surfaces and thus a true equilibrium is not possible in a driven spheromak.
Furthermore, it was shown that a time dependent change of open flux to closed flux is
essential to maintain helicity in a spheromak in quasi equilibrium.
Chapter 3 described the discharge circuit of the Caltech spheromak experiment. Various
resistances and inductances in the discharge circuit were found, and it was shown that the
inductance and resistance of the ignitron are the most dominant. It was also shown that the
discharge circuit was an under damped current source with low energy coupling efficiency.
A homodyne and a heterodyne interferometer were built to measure the plasma density
in the Caltech spheromak experiment. These interferometers were described in chapter 4.
The heterodyne interferometer had about a factor of two better signal to noise ratio as
compared to the homodyne interferometer, and was also much easier to align. It measured
typical line-average densities of ∼ 1021 /m2 with an error of ∼ 1019 /m2 . Chapter 4 also
showed that the phase auto correlation function of a laser is periodic in length. Thus the
traditionally assumed requirement of keeping the path lengths equal in an interferometer is
not necessary. This fact was utilized to simplify the design of both the homodyne and the
heterodyne interferometers.
The planar electrode structure of the Caltech spheromak provides an excellent oppor-

110
tunity to study the dynamics of magnetic helicity injection. Strong collimated jets have
been observed previously in the experiment. The density and velocity measurements from
these jets were described in detail in chapter 5. These flows are generated by by MHD
forces because of the slight flaring of the plasma jet. The flow velocity was found to be
Alfvénic with respect to the toroidal magnetic field. Also, the thermal pressure in the jets
was balance by the toroidal magnetic field energy density.
The design of a VUV/SXR imaging system for the Caltech solar coronal loop simulation
experiment was described in chapter 6. It was found that the bright energetic photons were
radiated when the plasma flux tube detached from the electrodes. In the counter-helicity
experiment, the detachment caused the plasma flux tube near the electrodes to be be
extremely bright in the VUV/SXR images. In general, the plasma flux tubes from the
counter-helicity experiment were found to be brighter in VUV/SXR range, and also more
susceptible to being kink unstable.

111

Appendix A
Alignment of the Interferometers

The appendix describes the alignment procedure of the two interferometers. Section A.1
describes in detail certain critical procedures in the alignment process. Sections A.2 and A.3
describes the steps in aligning the homodyne and heterodyne interferometers respectively.

A.1

Alignment Techniques

This section describes in detail certain techniques to align the interferometers described
earlier. The techniques described in this section were motivated by the work of Galvez [91].

A.1.1

Ensuring a Constant Height of the Beam Above the Optical Table

The first and the most critical step in aligning the interferometers is to ensure that the
beam coming out of the laser is parallel to the optical table. The height of the beam was
chosen to be the height of the detector. The technique for adjusting the laser for a parallel
beam is shown in figure A.1.
The desired height of the beam was marked on the screen. The screen was then placed
close to the laser at position #1 (see figure A.1). The laser mount was adjusted to change

Laser

Beam

Screen position #1
Adjust vertical height of laser

Screen position #2
Adjust pitch of laser

Figure A.1: Adjusting a laser to align the beam parallel to the optical table.

112

S1
M4
M5

Figure A.2: Adjusting the scene beam of the heterodyne interferometer to pass through the
sapphire windows.
the height of the laser so that the beam struck the screen at the desired height. The screen
was then placed at a long distance from the laser (position #2). The pitch of the laser was
adjusted to ensure that the beam struck the screen again at the desired height. The whole
process was repeated a few times to align the beam parallel to the optical table.

A.1.2

Steering the Heterodyne Interferometer’s Scene Beam through
Sapphire Windows

Figure A.2 shows the arrangement used to steer the scene beam through the sapphire
windows for the heterodyne interferometer (see figure 4.6). Mirror M5 is located directly
below and in close proximity to the lower sapphire window S1. The first step involved
adjusting mirror M4 to ensure that the scene beam was reflected by M5 to the approximate
center of S1. For the second step, M5 was adjusted to steer the beam through the lower
sapphire window to the approximate center of the upper sapphire window S2. The two

113

Reference beam
To wall
Screen
M2
M6

BS2

Scene beam
Figure A.3: Alignment for the overlap of the scene and reference beams of the heterodyne
interferometer.
steps were repeated iteratively. If, however, the beam was obstructed by the chamber walls
during the second step, then the alignment process was restarted at the first step.

A.1.3

Combining the Scene and Reference Beams of the Heterodyne Interferometer

As discussed in point 3 of section 4.5.4, the requirements for the overlap of the scene and
the reference beams for the heterodyne interferometer are very challenging. The alignment
procedure for overlapping the two beams is shown in figure A.3. For the first step, the
screen was placed next to the cube beam splitter BS2, and the mirror M2 was adjusted so
that the beam spots from the scene and reference beams overlapped on the screen. For the
second step, the screen was removed and the mirror M6 was used to steer the beams to
a distant wall a few meters away. Next, BS2 was adjusted to ensure that the beam spots
from the scene and reference beams overlapped on the distant wall. The steps were repeated
until a perfect overlap was achieved.

A.2

Alignment Procedure for the Homodyne Interferometer

Figure A.4 shows the arrangement of the various optical components of the heterodyne interferometer on the optical bench. The homodyne interferometer for the Caltech spheromak
experiment was aligned by the following sequence of techniques (refer to figure 4.4):

114

QWP

Piezo mirror

Laser

HWP1
HWP2

Temporary
mirror

Isolator
M1

Extra
mirror

HWP3

Wollaston
prism

D1
D2

Figure A.4: Image of the 18” × 18” optical table showing the various optical components of
the homodyne interferometer.

115
1. The laser beam was set up to be parallel to the optical table.
2. Half wave plate HWP1 was oriented to set the polarization of the laser beam to
be either vertical or horizontal. Vertically or horizontally polarized light is desired
because such a polarization is unaltered by reflections. A Wollaston prism can be
used to check if the beam is horizontally or vertically polarized.
3. The mirror M1 was adjusted to align the beam approximately parallel to the grid on
the optical table.
4. The two polarizers on the ends of the isolator were adjusted as per the manufacturer’s
specification to ensure near perfect isolation of the laser beam.
5. The isolator was placed to intercept the beam. The isolator was rotated about its
post and the mirror M1 was adjusted to ensure that the beam at the output of the
isolator was strongest and unobstructed.
6. The polarization vector of the light coming out of the isolator is rotated. The half
wave plate HWP2 was adjusted to again polarize the laser beam either horizontally
or vertically.
7. The non polarizing beam splitter was adjusted so that the split beams propagated
almost orthogonally. The tilt of the beam splitter was adjusted so that the scene beam
was parallel to the ground and hits the mirror M2 at the bottom of the chamber.
8. The piezo vibrating mirror was placed and adjusted so that the reflected reference
beam follows the path of the incoming beam. It was helpful to ensure that the beam
spots were coinciding on HWP2.
9. The optical axis of P1 was aligned vertically. The optical axis of the QWP was
adjusted to make a 45◦ angle with P1’s axis. The axis of P1 was found by noting that
when the axis was horizontal, P1 would completely obstruct an incident vertically
polarized beam. Also, the optical axis of any wave plate can be found by noting that
if the wave plate intercepts a linearly polarized beam with the polarization angle along
the optical axis of the wave plate, then the polarization of the beam is unaffected.

116
10. The second polarizer P2 was placed in the scene beam’s path with its optical axis
vertical.
11. It was advantageous to align the whole interferometer on the optical bench before
worrying about the passage of the beam through the plasma. A mirror was temporarily
placed on the optical bench to intercept the scene beam’s path after P2. The path
lengths of the scene and reference beams were maintained to be approximately equal.
The temporary mirror was adjusted so that the reflected beam traces the incoming
beam’s path. Again, it was helpful to align the spots on HWP2.
12. The beams on the other end of the beam splitter must be perfectly combined. The
beams were steered to a distant screen. When perfectly aligned, the beam spots on
the screen were still overlapping. If not, the piezo mirror and/or the extra mirror
placed in the previous step were adjusted.
13. Once perfectly aligned, the circularly polarized reference beam and will be interfered
with the vertically polarized scene beam.
14. As discussed after equation (4.12), if the polarization angle of the linearly polarized
beam makes an angle of 45◦ with the geometric axis of the Wollaston prism, the output
of the two detectors will be almost equal in magnitude. To achieve this, another half
wave plate HWP3 was placed in the path of the interfering beams, with the optical

axis of the wave plate making an angle of 22 21 with the vertical direction. Note that
HWP3 does not effect the circularly polarized beam.
15. The Wollaston prism was placed in the path of the interfering beams after the half
wave plate.
16. An optical post was mounted at the likely position of each detector (D1 and D2). The
mirrors M3 and M4 were adjusted so that the beams strike the center of the posts
at a height equal to the height of the detectors. The piezo mirror was vibrated at
∼ 1 Hz, and visible fringes were observed on the posts.
17. The posts were replaced by the detectors. The piezo mirror was vibrated at a frequency
of ∼ 1 kHz and the signals from the detectors were monitored on the oscilloscope.

117

Figure A.5: The x − y “ellipse” from the signals of the two detectors of the homodyne
interferometer.
18. Mirrors M3 and M4 were adjusted to maximize the signal strength.
19. If needed, the overlap of the scene and reference beams was improved by finely adjusting the piezo mirror and the temporary mirror. This increased the signal strengths of
the detectors.
20. If the outputs of the detectors had unequal amplitude, HWP3 was finely adjusted so
that the signal amplitudes were approximately equal.
21. On the oscilloscope, the two signals from the detectors were observed in an x − y
plot or Lissajous figure. If the signals were in quadrature then the plot resembled
a circle. The voltage setting of one of the channels was set to an extreme so that
the circle turned to an ellipse resembling figure A.5. If the major axis of the ellipse
was not entirely horizontal or vertical, the signals were not in quadrature. To achieve
quadrature, the quarter wave plate QWP was finely adjusted until the major axis of
the ellipse became horizontal or vertical.
The previous step completes the calibration of the interferometer. The next steps

118
describe the process of aligning the scene beam through the vacuum chamber.
22. The temporary mirror in the path of the scene beam was removed, and the mirror
M2 at the bottom of the chamber was adjusted to steer the scene beam through the
sapphire windows.
23. The spherical mirror on top of the chamber was adjusted so that the reflected beam
traced the incoming beam’s path. It was helpful to place a temporary translucent
screen in the scene beam’s path when adjusting the spherical mirror to align the
reflected beam. When the spherical mirror was adjusted, quadrature signals were
observed on the oscilloscope.
24. If the signal observed on the oscilloscope was low in amplitude, then the interferometer
might have been operating at a minimum of the phase auto-correlation function of
the laser as discussed in section 4.3. If this is the case, increasing the path length of
the scene beam changed the signal amplitude. This could be achieved by moving the
optical table back and forth or changing the location of the spherical mirror on its
mount.

A.3

Alignment Procedure for the Heterodyne Interferometer

Figure A.6 shows the arrangement of the various optical components of the heterodyne interferometer on the optical bench. The heterodyne interferometer for the Caltech spheromak
experiment was aligned by the following sequence of techniques (refer to figure 4.6):
1. The laser beam was set up to be parallel to the optical table.
2. The isolator was placed so that it intercepted the laser beam. Its position was adjusted
so that the beam came through the center of the polarizer at the end and its shape
was not distorted.
3. Half wave plate HWP1 was placed between the laser and the isolator. To align its
optical axis, it was rotated so that the output from the isolator went to a minimum.
At this point the polarization of the beam entering the isolator was orthogonal to the

119

Detector

BS2

Iris

M1
He−Ne
Laser

HWP1

HWP2

AOM
M0

BS1

Isolator

Figure A.6: Image of the 18” × 18” optical table showing the various optical components of
the heterodyne interferometer.

120
axis of the input polarizer of the isolator. To align the polarization, the axis of HWP1
was rotated by an extra 45◦ .
4. The axis of half wave plate HWP2 was adjusted to vertically polarize the laser beam.
The polarization angle was checked using a Wollaston prism.
5. The mirror M0 and beam splitter BS1 were placed at their respective positions.
6. The acousto-optic modulator (AOM) was positioned so that it intercepted the reference beam. It was rotated so that maximum power was coupled into the first harmonic
or equivalently so that the first harmonic was the brightest.
7. The iris was placed so that it blocked all beams except for the first harmonic.
8. The mirrors M1, M2 and M3 and beam splitter BS2 were adjusted to steer the beam.
Mirror M3 was adjusted to steer the beam onto the center of the aperture of mirror
M4.
9. The mirrors M4 and M5 were adjusted to steer the beam through the sapphire windows
onto the spherical mirror SM. The process of steering the beam through the windows
was easier than the homodyne design due to the extra mirror beneath the chamber
(refer to figure 4.4).
10. The spherical mirror SM on top of the chamber was adjusted so that the beam fell
back on BS2 as shown in the figure 4.6.
11. Mirror M2 and beam splitter BS2 were adjusted iteratively so that the scene and
the reference beams overlapped almost perfectly. This was achieved by the technique
described in section A.1.3.
12. Mirror M6 was aligned to direct the interfering beams onto the detector. The RF
electronics were switched on, and the quadrature signals were monitored on the oscilloscope. M5 was adjusted to maximize the signal strength.
13. If the signal observed on the oscilloscope was low in amplitude, then the interferometer
might have been operating at a minimum of the phase auto-correlation function of
the laser as discussed in section 4.3. If so, increasing the path length of the scene

121
beam changed the signal amplitude. This could be achieved by moving the optical
table back and forth or by changing the location of the spherical mirror on its mount.

122

Bibliography

[1] J. D. Huba. NRL Plasma Formulary. 2002.
[2] ITER, March 2009. URL http://www.iter.org/.
[3] Paul M. Bellan.

Spheromaks: A Practical Application of Magnetohydrodynamic

Dynamos and Plasma Self-Organization. Imperial College Press, 2000.
[4] FESAC Panel Staff. Report of the FESAC toroidal alternates panel. Technical report,
November 2008. URL http://fusion.gat.com/tap.
[5] P. M. Bellan and J. F. Hansen. Laboratory simulations of solar prominence eruptions.
volume 5, pages 1991–2000. AIP, 1998. doi: 10.1063/1.872870. URL http://link.
aip.org/link/?PHP/5/1991/1.
[6] S. C. Hsu and P. M. Bellan. A laboratory plasma experiment for studying magnetic
dynamics of accretion discs and jets. Monthly Notices of the Royal Astronomical
Society, 334, 2002. doi: 10.1046/j.1365-8711.2002.05422.x. URL http://dx.doi.org/
10.1046/j.1365-8711.2002.05422.x.
[7] P. M. Bellan. Fundamentals of Plasma Physics. Cambridge University Press, 2006.
[8] Jeffrey P. Freidberg. Plasma Physics and Fusion Energy. Cambridge University Press,
2007.
[9] L. Woltjer. A theorem on force-free magnetic fields. Proceedings of the National
Academy of Sciences of the United States of America, 44(6):489–491, 1958. URL

123
[10] J. B. Taylor. Relaxation of toroidal plasma and generation of reverse magnetic fields.
Physical Review Letters, 33(19):1139–1141, Nov 1974. doi: 10.1103/PhysRevLett.33.
1139. URL http://link.aps.org/doi/10.1103/PhysRevLett.33.1139.
[11] Cris W. Barnes, J. C. Fernández, I. Henins, H. W. Hoida, T. R. Jarboe, S. O. Knox,
G. J. Marklin, and K. F. McKenna. Experimental determination of the conservation of
magnetic helicity from the balance between source and spheromak. Physics of Fluids,
29(10):3415–3432, 1986. doi: 10.1063/1.865858. URL http://link.aip.org/link/
?PFL/29/3415/1.
[12] S. You, G. S. Yun, and P. M. Bellan. Dynamic and stagnating plasma flow leading to magnetic-flux-tube collimation. Physical Review Letters, 95(4):045002, Jul
2005. doi: 10.1103/PhysRevLett.95.045002. URL http://link.aps.org/doi/10.
1103/PhysRevLett.95.045002.
[13] S. C. Hsu and P. M. Bellan. Experimental identification of the kink instability as a
poloidal flux amplification mechanism for coaxial gun spheromak formation. Physical
Review Letters, 90(21):215002, May 2003. doi: 10.1103/PhysRevLett.90.215002. URL
[14] Tektronix, Inc., March 2009. URL http://www.tek.com/.
[15] DRS Technologies, Inc., March 2009. URL http://www.drsdi.com/.
[16] C. A. Romero-Talamás, P. M. Bellan, and S. C. Hsu. Multielement magnetic probe
using commercial chip inductors. Review of Scientific Instruments, 75(8):2664–2667,
2004. doi: 10.1063/1.1771483. URL http://link.aip.org/link/?RSI/75/2664/1.
[17] Gunsu Soonshin Yun. Dynamics of Plasma Structures Interacting with External and
Self-Generated Magnetic Fields. PhD thesis, California Institute of Technology, July
2007. URL http://resolver.caltech.edu/CaltechETD:etd-07242007-162442.
[18] P. K. Browning, G. Cunningham, S. J. Gee, K. J. Gibson, A. al Karkhy, D. A. Kitson,
R. Martin, and M. G. Rusbridge. Power flow in a gun-injected spheromak plasma.
Physical Review Letters, 68(11):1718–1721, Mar 1992. doi: 10.1103/PhysRevLett.68.
1718. URL http://link.aps.org/doi/10.1103/PhysRevLett.68.1718.

124
[19] H. S. McLean, S. Woodruff, E. B. Hooper, R. H. Bulmer, D. N. Hill, C. Holcomb,
J. Moller, B. W. Stallard, R. D. Wood, and Z. Wang. Suppression of MHD fluctuations
leading to improved confinement in a gun-driven spheromak. Physical Review Letters,
88(12):125004, Mar 2002. doi: 10.1103/PhysRevLett.88.125004. URL http://link.
aps.org/doi/10.1103/PhysRevLett.88.125004.
[20] T. R. Jarboe, W. T. Hamp, G. J. Marklin, B. A. Nelson, R. G. O’Neill, A. J. Redd, P. E.
Sieck, R. J. Smith, and J. S. Wrobel. Spheromak formation by steady inductive helicity
injection. Physical Review Letters, 97(11):115003, 2006. doi: 10.1103/PhysRevLett.
97.115003. URL http://link.aps.org/abstract/PRL/v97/e115003.
[21] B. W. Stallard, E. B. Hooper, S. Woodruff, R. H. Bulmer, D. N. Hill, H. S.
McLean, R. D. Wood, and SSPX Team. Magnetic helicity balance in the sustained
spheromak plasma experiment.

Physics of Plasmas, 10(7):2912–2924, 2003.

doi:

10.1063/1.1580121. URL http://link.aip.org/link/?PHP/10/2912/1.
[22] W. C. Turner, G. C. Goldenbaum, E. H. A. Granneman, J. H. Hammer, C. W. Hartman, D. S. Prono, and J. Taska. Investigations of the magnetic structure and the decay
of a plasma-gun-generated compact torus. Physics of Fluids, 26(7):1965–1986, 1983.
doi: 10.1063/1.864345. URL http://link.aip.org/link/?PFL/26/1965/1.
[23] Ignitron, February 2008. URL http://en.wikipedia.org/wiki/Ignitron.
[24] Ignitrons for capacitor discharge and crowbar applications. Personal Copy.
[25] W. James Sarjeant and R. E. Dollinger. High-Power Electronics, chapter 7, pages
231–237. TAB professional and Reference Books, 1989.
[26] GL-7703 product information, March 2009.

URL http://catalog.rell.com/

rellecom/Images/Objects/8200/8103.PDF.
[27] M. Stanway and R. Seddon. Ignitron firing time and jitter. Personal Copy.
[28] S. Woodruff, B. I. Cohen, E. B. Hooper, H. S. Mclean, B. W. Stallard, D. N. Hill, C. T.
Holcomb, C. Romero-Talamas, R. D. Wood, G. Cone, and C. R. Sovinec. Controlled
and spontaneous magnetic field generation in a gun-driven spheromak. Physics of

125
Plasmas, 12(5):052502, 2005. doi: 10.1063/1.1878772. URL http://link.aip.org/
link/?PHP/12/052502/1.
[29] Charles J. Michels and Fred F. Terdan. Characteristics of a 5-kilojoule, ignitronswitched, fast-capacitor bank. Technical report, Lewis Research Center, Ohio, 1965.
[30] Richard C. Dorf and James A. Svoboda. Introduction to electric circuits. John Wiley
and Sons, Inc., 2006.
[31] Yun-Sik Jin, Hong-Sik Lee, Jong-Soo Kim, Young-Bae Kim, and G.-H. Rim. Novel
crowbar circuit for compact 50-kJ capacitor bank. Plasma Science, IEEE Transactions
on, 32(2):525–530, April 2004. ISSN 0093-3813. doi: 10.1109/TPS.2004.826028. URL
[32] Zhehui Wang, Paul D. Beinke, Cris W. Barnes, Michael W. Martin, Edward Mignardot,
Glen A. Wurden, Scott C. Hsu, Thomas P. Intrator, and Carter P. Munson. A penningassisted subkilovolt coaxial plasma source. Review of Scientific Instruments, 76(3):
033501, 2005. doi: 10.1063/1.1855071. URL http://link.aip.org/link/?RSI/76/
033501/1.
[33] M.E. Savage. Final results from the high-current, high-action closing switch test program at Sandia National Laboratories. Plasma Science, IEEE Transactions on, 28
(5):1451–1455, Oct 2000. ISSN 0093-3813. doi: 10.1109/27.901213. URL http:
//dx.doi.org/10.1109/27.901213.
[34] P. Acedo, H. Lamela, T. Estrada, and J. Sánchez. Operation of a CO2 -HeNe laser
heterodyne interferometer in the TJ-II stellarator. volume 24B, pages 1252–1255. 27th
EPS Conference on Controlled Fusion and Plasma Physics, June 2000. URL http:
//web.gat.com/conferences/meetings/eps00/start.htm.
[35] Dan R. Baker and Shu-Tso Lee. Dual laser interferometer for plasma density measurements on large tokamaks. Review of Scientific Instruments, 49(7):919–922, 1978. doi:
10.1063/1.1135492. URL http://link.aip.org/link/?RSI/49/919/1.
[36] T. N. Carlstrom, D. R. Ahlgren, and J. Crosbie. Real-time, vibration-compensated
CO2 interferometer operation on the DIII-D tokamak. Review of Scientific Instruments,

126
59(7):1063–1066, 1988. doi: 10.1063/1.1139726. URL http://link.aip.org/link/
?RSI/59/1063/1.
[37] J. Irby, R. Murray, P. Acedo, and H. Lamela. A two-color interferometer using a
frequency doubled diode pumped laser for electron density measurements. Review
of Scientific Instruments, 70(1):699–702, 1999. doi: 10.1063/1.1149489. URL http:
//link.aip.org/link/?RSI/70/699/1.
[38] Yasunori Kawano, Akira Nagashima, Katsushiko Tsuchiya, So-ichi Gunji, Shin-ichi
Chiba, and Takaki Hatae. Tangential CO2 laser interferometer for large tokamaks.
Journal of plasma and fusion research, 73(8):870–891, June 1997. ISSN 09187928.
URL http://ci.nii.ac.jp/naid/110003826925/.
[39] Horacio Lamela, Pablo Acedo, the Optoelectronics and Laser Technology Group, and
James Irby. Laser interferometric experiments for the TJ-II stellarator electron-density
measurements. Review of Scientific Instruments, 72(1):96–102, 2001. doi: 10.1063/1.
1333040. URL http://link.aip.org/link/?RSI/72/96/1.
[40] Yasunori Kawano, Akira Nagashima, Shinichi Ishida, Takeshi Fukuda, and Tohru Matoba. CO2 laser interferometer for electron density measurement in JT-60U tokamak.
Review of Scientific Instruments, 63(10):4971–4973, 1992. doi: 10.1063/1.1143515.
URL http://link.aip.org/link/?RSI/63/4971/1.
[41] C. J. Buchenauer and A. R. Jacobson. Quadrature interferometer for plasma density
measurements. Review of Scientific Instruments, 48(7):769–774, 1977. doi: 10.1063/1.
1135146. URL http://link.aip.org/link/?RSI/48/769/1.
[42] D. D. Lowenthal and A. L. Hoffman. Quasi-quadrature interferometer for plasma
density radial profile measurements. Review of Scientific Instruments, 50(7):835–843,
1979. doi: 10.1063/1.1135935. URL http://link.aip.org/link/?RSI/50/835/1.
[43] Bahaa E. A. Saleh and Malvin Carl Teich. Fundamentals of Photonics. John Wiley
and Sons, 1991.
[44] W. B. Bridges. Light sources for optical holography. Private communication.

127
[45] Anthony E. Seigman. Lasers. University Science Books, 1986.
[46] Robert J. Collier, Christoph B. Burckhardt, and Lawrence H. Lin. Optical Holography.
Academic Press, Inc., 1971.
[47] Optics for Research Inc., March 2009. URL http://www.ofr.com.
[48] Newport Corporation, March 2009. URL http://newport.com.
[49] Analog Modules, Inc., March 2009. URL http://www.analogmodules.com/.
[50] Raymond Peter Golingo. Formation of a sheared flow Z-pinch. PhD thesis, University
of Washington, 2003.
[51] OSI Optoelectronics, March 2009. URL http://www.osioptoelectronics.com.
[52] B. V. Weber and D. D. Hinshelwood. He-Ne interferometer for density measurements
in plasma opening switch experiments. Review of Scientific Instruments, 63(10):5199–
5201, 1992. doi: 10.1063/1.1143428. URL http://link.aip.org/link/?RSI/63/
5199/1.
[53] R. Kristal and R. W. Peterson. Bragg cell heterodyne interferometry of fast plasma
events. Review of Scientific Instruments, 47(11):1357–1359, 1976. doi: 10.1063/1.
1134545. URL http://link.aip.org/link/?RSI/47/1357/1.
[54] Gunsu S. Yun, Setthivoine You, and Paul M. Bellan. Large density amplification
measured on jets ejected from a magnetized plasma gun. Nuclear Fusion, 47(3):181–
188, 2007. URL http://stacks.iop.org/0029-5515/47/181.
[55] L. M. Smith, D. R. Keefer, and N. W. Wright. A fiber-optic interferometer for in
situ measurements of plasma number density in pulsed-power applications. Review of
Scientific Instruments, 74(7):3324–3328, 2003. doi: 10.1063/1.1582389. URL http:
//link.aip.org/link/?RSI/74/3324/1.
[56] M. A. Van Zeeland, R. L. Boivin, T. N. Carlstrom, T. Deterly, and D. K. Finkenthal. Fiber optic two-color vibration compensated interferometer for plasma density measurements.

Review of Scientific Instruments, 77(10):10F325, 2006.

10.1063/1.2336437. URL http://link.aip.org/link/?RSI/77/10F325/1.

doi:

128
[57] John Marshall. Performance of a hydromagnetic plasma gun. Physics of Fluids, 3(1):
134–135, 1960. doi: 10.1063/1.1705989. URL http://link.aip.org/link/?PFL/3/
134/2.
[58] T. D. Butler, I. Henins, F. C. Jahoda, J. Marshall, and R. L. Morse. Coaxial snowplow
discharge. Physics of Fluids, 12(9):1904–1916, 1969. doi: 10.1063/1.1692758. URL
[59] Kurt F. Schoenberg, Richard A. Gerwin, Jr. Ronald W. Moses, Jay T. Scheuer, and
Henri P. Wagner. Magnetohydrodynamic flow physics of magnetically nozzled plasma
accelerators with applications to advanced manufacturing. Physics of Plasmas, 5(5):
2090–2104, 1998. doi: 10.1063/1.872880. URL http://link.aip.org/link/?PHP/5/
2090/1.
[60] K.F. Schoenberg, R.A. Gerwin, I. Henins, R.M. Mayo, J.T. Scheuer, and G.A. Wurden. Preliminary investigation of power flow and performance phenomena in a multimegawatt coaxial plasma thruster.
(6):625–644, Dec 1993.

Plasma Science, IEEE Transactions on, 21

ISSN 0093-3813.

doi: 10.1109/27.256783.

URL http:

//dx.doi.org/10.1109/27.256783.
[61] M. Zuin, R. Cavazzana, E. Martines, G. Serianni, V. Antoni, M. Bagatin, M. Andrenucci, F. Paganucci, and P. Rossetti. Kink instability in applied-field magnetoplasma-dynamic thrusters.

Physical Review Letters, 92(22):225003, Jun 2004.

doi: 10.1103/PhysRevLett.92.225003.

URL http://link.aps.org/doi/10.1103/

PhysRevLett.92.225003.
[62] T. B. Reed. Determination of streaming velocity and the flow of heat and mass in
high-current arcs. Journal of Applied Physics, 31(11):2048–2052, 1960. doi: 10.1063/
1.1735494. URL http://link.aip.org/link/?JAP/31/2048/1.
[63] U. Shumlak, B. A. Nelson, and B. Balick.
flow Z-pinch.

Plasma jet studies via the

Astrophysics and Space Science,

doi: 10.1007/s10509-006-9218-5.
h724757274241160.

307(1):41–45,

Jan 2007.

URL http://www.springerlink.com/content/

129
[64] Cris W. Barnes, T. R. Jarboe, G. J. Marklin, S. O. Knox, and I. Henins. The impedance
and energy efficiency of a coaxial magnetized plasma source used for spheromak formation and sustainment. Physics of Fluids B: Plasma Physics, 2(8):1871–1888, 1990.
doi: 10.1063/1.859459. URL http://link.aip.org/link/?PFB/2/1871/1.
[65] James Chen and Jonathan Krall. Acceleration of coronal mass ejections. Journal
of Geophysical Research, 108(A11), 2003. doi: 10.1029/2003JA009849. URL http:
//dx.doi.org/10.1029/2003JA009849.
[66] Masanori Nakamura, Hui Li, and Shengtai Li. Stability properties of magnetic tower
jets. The Astrophysical Journal, 656(2):721–732, 2007. URL http://stacks.iop.
org/0004-637X/656/721.
[67] T. Uyama, Y. Honda, M. Nagata, M. Nishikawa, A. ozaki, N. Satomi, and K. Watanabe.
Temporal evolution of the decaying spheromak in the CTCC-I experiment. Nuclear
Fusion, 27(5):799–813, 1987.
[68] S. C. Hsu and P. M. Bellan. On the jets, kinks, and spheromaks formed by a planar
magnetized coaxial gun. Physics of Plasmas, 12(3):032103, 2005. doi: 10.1063/1.
1850921. URL http://link.aip.org/link/?PHP/12/032103/1.
[69] Deepak Kumar and Paul M. Bellan. Heterodyne interferometer with unequal path
lengths. Review of Scientific Instruments, 77(8):083503, 2006. doi: 10.1063/1.2336769.
URL http://link.aip.org/link/?RSI/77/083503/1.
[70] H. Li, R. V. E. Lovelace, J. M. Finn, and S. A. Colgate. Magnetic helix formation
driven by keplerian disk rotation in an external plasma pressure: The initial expansion
stage. The Astrophysical Journal, 561(2):915–923, 2001. URL http://stacks.iop.
org/0004-637X/561/915.
[71] D. Lynden-Bell. On why discs generate magnetic towers and collimate jets. Monthly
Notice of the Royal Astronomical Society, 341(4):1360–1372, 2003. doi: 10.1046/
j.1365-8711.2003.06506.x.

URL http://dx.doi.org/10.1046/j.1365-8711.2003.

06506.x.
[72] F. J. Weinberg. Optics of Flames. Butterworth, 1963.

130
[73] Robert C. Weast, editor. Handbook of Chemistry and Physics. CRC Press, 56th edition,
1975.
[74] Francis F. Chen. Introduction to Plasma Physics and Controlled Fusion. Plenum press,
1983.
[75] P. M. Bellan. Why current-carrying magnetic flux tubes gobble up plasma and become
thin as a result. Physics of Plasmas, 10(5):1999–2008, 2003. doi: 10.1063/1.1558275.
URL http://link.aip.org/link/?PHP/10/1999/1.
[76] C. Cote.

Power Balance and Characterization of Impurities in the Maryland

Spheromak. PhD thesis, The University of Maryland, College Park, 1993.
[77] V. H. Chaplin, M. R. Brown, D. H. Cohen, T. Gray, and C. D. Cothran. Spectroscopic measurements of temperature and plasma impurity concentration during magnetic reconnection at the Swarthmore spheromak experiment. Physics of Plasmas, 16
(4):042505, 2009. doi: 10.1063/1.3099603. URL http://link.aip.org/link/?PHP/
16/042505/1.
[78] S. Masuda, T. Kosugi, H. Hara, S. Tsuneta, and Y. Ogawara. A loop-top hard x-ray
source in a compact solar flare as evidence for magnetic reconnection. Nature, 371:
495–497, October 1994. doi: 10.1038/371495a0. URL http://dx.doi.org/10.1038/
371495a0.
[79] J. Freddy Hansen. Laboratory Simulations of Solar Prominences. PhD thesis, California Institute of Technology, May 2001. URL http://resolver.caltech.edu/
CaltechETD:etd-08192008-160303.
[80] J. Freddy Hansen and Paul M. Bellan. Experimental demonstration of how strapping
fields can inhibit solar prominence eruptions. The Astrophysical Journal Letters, 563
(2):L183–L186, 2001. URL http://stacks.iop.org/1538-4357/563/L183.
[81] D. M. Rust and A. Kumar. Evidence for helically kinked magnetic flux ropes in solar
eruptions. The Astrophysical Journal Letters, 464(2):L199–L202, 1996. URL http:
//stacks.iop.org/1538-4357/464/L199.

131
[82] International Radiation Detectors, Inc., March 2009. URL http://www.ird-inc.com/.
[83] Lebow Company, March 2009. URL http://lebowcompany.com.
[84] Eric Gullikson. X-ray interactions with matter, March 2009. URL http://henke.
lbl.gov/optical_constants/filter2.html.
[85] B. L. Henke, E. M. Gullikson, and J. C. Davis. X-ray interactions: Photoabsorption,
scattering, transmission, and reflection at E=50 − 30000 eV, Z=1 − 92. Atomic Data
and Nuclear Data Tables, 54(2):181–342, July 1993.
[86] R. T. Snider, R. Evanko, and J. Haskovec. Toroidal and poloidal soft X-ray imaging
system on the DIII-D tokamak. Review of Scientific Instruments, 59(8):1807–1809,
1988. doi: 10.1063/1.1140119. URL http://link.aip.org/link/?RSI/59/1807/1.
[87] X-Ray and Specialty Instruments, March 2009. URL http://xsiinc.com/.
[88] Berkeley

Nucleonics

Corporation,

March

2009.

URL

berkeleynucleonics.com/.
[89] Meade Corporation, March 2009. URL http://meade.com/.
[90] J. F. Hansen, S. K. P. Tripathi, and P. M. Bellan. Co- and counter-helicity interaction
between two adjacent laboratory prominences. Physics of Plasmas, 11(6):3177–3185,
2004. doi: 10.1063/1.1724831. URL http://link.aip.org/link/?PHP/11/3177/1.
[91] E. J. Galvez, March 2009.

URL http://departments.colgate.edu/physics/

research/Photon/root/Apparatus/alignment.htm.