uv led based charge control for the lisa …
TRANSCRIPT
UV LED BASED CHARGE CONTROL FOR THE LISA GRAVITATIONAL REFERENCE SENSOR
By
TAIWO JANET OLATUNDE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2018
© 2018 Taiwo Janet Olatunde
To Yomi. Ore mi. Ololufe.
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ACKNOWLEDGMENTS
When I started at the University of Florida in the spring of 2012, it was my first
time outside of my country of birth and I was in great need of a support network to help
me thrive. I met Dr. John Conklin at the end of the Fall 2012 semester and started
working with him and other members of the Precision Space Systems Lab (PSSL)
group in 2013. What a blessing that turned out to be! He not only provided the guidance
I needed as far as translating theoretical classroom knowledge into real hands-on
situations in the lab, he was also very supportive. He wanted me to succeed.
Many thanks to the other members of My PhD dissertation Committee; Dr. Peter
Wass, Dr. George Sawyer and Dr. Guido Mueller for their valuable feedback and for
letting me use resources available to them when I needed to.
I want to express my appreciation for the financial support that made the work
presented in this Dissertation possible, specifically, the NASA N.G. Roman Tech
Fellowship, grant number NNX15AF26G.
I would like to thank members of the PSSL group and the LISA group for the
warm, welcoming work environment and their friendship. I must mention Dr. Guido
Mueller, Paul Serra, Nathan Barnwell, Seth Nydam, Stephen Apple, Tyler Ritz, Daniel
Hillsberry, Samantha Parry, Deep JariwaIa, Thida Pechsiri, Benjamin Letson, Henri
Inchauspe and Myles Clark. Special thanks to Paul Serra for spending several weeks
with me on UV LED fiber coupling and the design of the UV LED driver board,
Samantha Parry for helping with further board testing, Stephen Apple for helping me get
a better understanding of the pendulum and for helping to analyze charge measurement
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data. I am grateful to Nicholas Turetta and Henri Inchauspe for working with me on the
charge control model.
To my husband: Yomi, I need you. Thank you for your patience and sacrifice my
love. To my children: Mayowa Katherine and Ayomide Joanne, I am always excited to
see you after a day of hard work. Thank you for brightening every aspect of my
existence. Life was certainly not this interesting before you both came along. I also want
to acknowledge my parents, Esther Binsola Talabi and Olubiyi Joseph Talabi for their
words of encouragement.
I am fortunate to have had the opportunity to experience the wonderful and
inclusive environment that the University of Florida provides.
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TABLE OF CONTENTS
ACKNOWLEDGMENTS .................................................................................................. 4
LIST OF TABLES ............................................................................................................ 8
LIST OF FIGURES .......................................................................................................... 9
LIST OF ABBREVIATIONS ........................................................................................... 13
ABSTRACT ................................................................................................................... 14
CHAPTER
1 INTRODUCTION .................................................................................................... 16
1.1 LISA- A Developing Gravitational Wave Observatory ....................................... 19 1.2 LISA Requirements ........................................................................................... 21
1.3 Gravitational Reference Sensor ........................................................................ 23 1.4 Charge Control on LISA Test Masses ............................................................... 24 1.5 Summary of Chapters ....................................................................................... 25
1.6 Contributions ..................................................................................................... 28
2 UV LEDS FOR LISA TEST MASS CHARGE CONTROL ....................................... 30
2.1 Hg Lamps vs. UV LEDs .................................................................................... 30 2.2 UV LED Testing ................................................................................................ 31
2.2.1 UV LED Fiber Coupling and Electronics Driver Board for Charge Control........................................................................................................... 33
2.2.2 Proposed Heat Dissipation Management ................................................ 35
2.3 Performance of UV LED Driver Board .............................................................. 35
3 QUANTUM YIELD AND REFLECTIVITY OF LISA TEST MASSES ...................... 37
3.1 Motivation for Measuring QY and Reflectivity ................................................... 37 3.2 Reflectivity Analysis for Two Infinite Parallel Plates .......................................... 37 3.3 QY Experimental Set up and Apparatus Description ........................................ 41
3.3.1 QY Dependence on Energy of Emitted UV Light .............................. 42
3.3.2 QY Measurements and Wavelength ................................................. 44 3.3.3 QY Measurements vs. Time .............................................................. 45 3.3.4 QY After a Bake-out .......................................................................... 46
3.3.5 Sample Preparation, Cleaning and Storage ...................................... 49 3.3.6 Evaluating the QYs of Commercially Polished Au Surfaces .............. 52 3.3.7 Investigating QY Dependence on Pressure ...................................... 53
3.5 Summary of QY Measurements ........................................................................ 59 3.6 Alternative Coatings Explored ........................................................................... 64
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3.7 X-ray Photoelectron Spectroscopy Measurements ........................................... 65
3.8 Conclusions ...................................................................................................... 69
4 DEMONSTRATION OF DC AND AC UV LED CHARGE CONTROL ON THE UF TORSION PENDULUM ..................................................................................... 72
4.1 Pendulum Description ....................................................................................... 72 4.1.1 Capacitive and Interferometric Readout .................................................. 76 4.1.2 Measured Time Series of The Pendulum Rotation Angle ........................ 77 4.1.3 Pendulum Acceleration Noise Performance ............................................ 79
4.2 Electrostatic Force acting on the Pendulum’s Test Masses and Charge Measurement ....................................................................................................... 81
4.3 Analytical Charge Control Model for the LISA GRS ...................................... 84 4.3.1 The Pendulum GRS as a Simplified Parallel Plate Circuit ....................... 85
4.3.2 Photoelectron Energy Distribution ........................................................... 88 4.3.3 Implementing The Charge Control Model in MATLAB ............................. 94
4.4 DC Charge Control Demonstration on the UF Torsion Pendulum .................... 95 4.5 AC Charge Control With The UF Torsion Pendulum......................................... 96
4.5.1 AC Charge Control Experiments With Varying UV Phase ................... 99 4.5.2 Measuring Saturation Voltage as a Function of UV Light Injection ........ 100 4.5.3 Charge Rate .......................................................................................... 103
4.6 Fitting Analytical Charge Control Model to Data .......................................... 105
4.6.1 Case 1: 𝒏𝒕𝒐𝒕 (𝑽𝒊𝒏𝒋 , 𝑽𝑻𝑴 = 𝟎) .............................................................. 105
4.6.2 Case 2: 𝑽𝑻 (𝑽𝒊𝒏𝒋 , 𝒕 = ∞) ...................................................................... 110
4.6.3 Case 3: 𝒏𝒕𝒐𝒕 (𝑽𝒊𝒏𝒋 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕, 𝑽𝑻𝑴) ............................................... 112
4.7 Conclusions .................................................................................................... 115
5 SUMMARY OF CONCLUSIONS INCLUDING RECOMMENDATIONS FOR FUTURE WORK ................................................................................................... 117
APPENDIX
A DISTRIBUTION OF ELECTRON ENERGIES AS A FUNCTION OF APPLIED BIAS VOLTAGE .................................................................................................... 121
B SOLUTIONS TO PHOTOELECTRON FLUX EQUATIONS .................................. 123
C CHARGE MEASUREMENT EQUATIONS ........................................................... 127
LIST OF REFERENCES ............................................................................................. 131
BIOGRAPHICAL SKETCH .......................................................................................... 134
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LIST OF TABLES
Table page 3-1 Quantum Yield Measurements from Au samples coated from March 2013
through August 2017, refer to Figures 3-4 through 3-14 ..................................... 59
3-2 XPS measurements for bare Au sample cleaned with isopropanol ......................... 66
3-3 XPS measurements for Au sample treated with Thiol. ............................................ 67
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LIST OF FIGURES
Figure page 1-1 A gravitational wave propagating orthogonally to the detector plane and
linearly polarized parallel to the 4-km optical cavities will have the effect of lengthening one 4-km arm and shortening the other during one half-cycle of the wave; these length changes are reversed during the other half-cycle. The output photodetector records these differential cavity length variations.. ........... 18
1-2 Key results of the analysis of GW150914, comparing the reconstructed gravitational-wave strain with the predictions of the best matching waveform computed from general relativity, over the three stages of the event; inspiral, merger and ring down. Also shown are the separation and velocity of the black holes, and how they change as the merger event unfolds ........................ 19
1-3 LISA’s nominal configuration .............................................................................. 20
1-4 The LISA technology package (ESA/ATG media lab, LISA L3 mission proposal) ............................................................................................................ 22
1-5 Sensitivity curve showing gravitational wave sources in the LISA frequency range compared with its sensitivity for a three arm configuration ....................... 23
1-6 Flight-like GRS ................................................................................................... 24
2-1 Left: fiber coupled TO18 UV LEDs from SETi, Right: Intensity spectrum of 240 and 250 nm UV LEDs compared with the work function of Au and the Hg lamp discharge wavelength ................................................................................ 30
2-2 Left: IV curve for TO18 UV LED, Right: PI curve for TO18 UV LED.................... 32
2-3 Left: IV curve for 240 nm TO39 BL and HS, Right: PI curve for 240 nm TO39 BL and HS .......................................................................................................... 32
2-4 Left: 240 nm TO39 BL UV LEDs, Right: 250 nm TO39 BL UV LEDs .................. 32
2-5 Output optical power of TO39 UV LED after being mechanically fiber coupled . 33
2-6 Heat dissipation management design ................................................................ 34
2-7 UV LED driver board assembled with the UV LED inside a box ........................ 34
2-8 Average power output of UV LED driver board for a 5 – 20 mA current setting input and a constant 5 % duty cycle ................................................................... 36
2-9 Average power output of UV LED driver board for various duty cycle settings .. 36
3-1 Reflections between two surfaces with different QYs and similar reflectivities ... 38
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3-2 Reflections between two surfaces with different QYs and different reflectivities ......................................................................................................... 40
3-3 QY measurement apparatus set up in large vacuum at ~3 × 10 − 6 Torr, QY apparatus was subsequently set up in the small vacuum chamber with a leak valve for pressure adjustments ........................................................................... 42
3-4 QY of Au samples at 240 nm > QY at 250 nm ................................................... 45
3-5 Successive QY measurements with sample 1DP held at 10 − 5 Torr using a UV wavelength of 240 nm .................................................................................. 46
3-6 Measured QY on Au sample 2BP with bake-out at 130˚C and 240 nm .............. 46
3-7 Fraction of light reflected as a function of angle of incidence (in degrees) for
254 nm wavelength showing the two polarization components Rp and Rs comparing results described by Daniel Hollington (Dissertation 2011) and Johnson and Christy (1972). .............................................................................. 48
3-8 QY with detailed cleaning procedure and bake-out at 130°C using 240 nm UV LEDs for sample 1CP ......................................................................................... 51
3-9 QY with detailed cleaning procedure and bake-out at 130°C with UV wavelengths of 240 nm vs. 250 nm for sample 1AP ........................................... 51
3-10 QY of Commercially polished sample 4AP measured at 130°C with a 240 nm
at ~2x10 − 6 Torr ................................................................................................ 53
3-11 QY of sample 4AP measured between 10-8 and 10-6 Torr ................................ 57
3-12 QY of sample 4BP measured between 10-8 and 10-6 Torr ................................ 57
3-13 QY of sample 4CP measured between 10-8 and 10-6 Torr ................................ 58
3-14 QY of sample 4DP measured between 10-8 and 10-6 Torr ................................ 58
3-15 Long term QY measurements of sample 4CP at 10-8 Torr ................................ 59
3-16 Results of QY as a function of pressure at 240 nm ............................................. 60
3-17 Results of QY as a function of pressure at 250 nm ............................................. 60
3-18 Left to right; Aluminum substrates, hand-polished and coated with Au, TiC and SiC ............................................................................................................... 64
3-19 Results of QY measurements for Au, TiC and SiC coatings .............................. 67
3-20 The XPS measurements of bare Au cleaned with isopropanol .......................... 69
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3-21 XPS measurement spectrum of Au after being treated in thiol solution for 24 hours .................................................................................................................. 68
4-1 Pendulum inertial member surrounded by electrostatic shields, Simplified electrode Housing, CAD model of the torsion pendulum, Torsion pendulum assembly enclosed inside the vacuum chamber ................................................ 73
4-2 UV light injection geometry for the simplified GRS, Left: UV light illuminates either the TM or the electrodes, Right: UV light simultaneously illuminates both the TM and injection electrodes .................................................................. 74
4-3 Test mass rotation and test mass swing ........................................................... 75
4-4 Capacitive and Interferometric read-out scheme, Actuation and capacitive sensing scheme for a pair of electrodes enclosing a single TM, Electrostatic actuation electronics, Interferometric readout scheme for TM position, Layout of interferometer components in the torsion pendulum ....................................... 77
4-5 Pendulum angle as a function of time as measured both capacitively and interferometrically ............................................................................................... 78
4-6 UF torsion pendulum acceleration noise spectrum ............................................ 79
4-7 LISA requirements compared with the acceleration noise performance of the torsion pendulum facility ..................................................................................... 80
4-8 Left: TM position with respect to the electrodes, Right: Suspended pendulum
showing the force Fx in the direction of the sensitive axis ................................... 81
4-9 Pendulum angle as a function of time during charge measurement on the pendulum ............................................................................................................ 84
4-10 Simplified circuit model for charge control ........................................................... 88
4-11 Electron exchange in a parallel plate configuration ............................................ 89
4-12 Distribution of energies of emitted electrons in a parallel plate configuration ..... 90
4-13 Straight line approximation of the distribution of energies .................................. 91
4-14 Pendulum charge during DC charge control operations on the UF torsion pendulum ............................................................................................................ 96
4-15 Left: UV LED emission out of phase with respect to the injection signal Right: UV LED emission in phase with respect to the injection signal .......................... 96
4-16 Test mass potential during AC charge control experiment ................................. 98
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4-17 TM Potential while illuminating the AC port with pulsed UV light while varying the phase with respect to the injection voltage. ................................................ 100
4-18 Saturation voltage Vs. UV light phase using the electrodes port ...................... 101
4-19 Saturation voltage Vs. UV light phase using the TM port ................................. 102
4-20 Saturation voltage Vs. Phase using the TM-Injection electrodes port .............. 102
4-21 TM voltage as a function of time when using the TM port and EH port to move the voltage due to charge on the TM positive and negative .................... 104
4-22 UV light injection geometry and circuit model for calculating 𝑛𝑡𝑜𝑡 with varying 𝑉𝑖𝑛𝑗 and 𝑉𝑇𝑀 = 0 ............................................................................... 105
4-23 Measured test mass charge rate versus the potential barrier and the best fit model for the electrodes port ............................................................................ 106
4-24 Measured test mass charge flow rate as a function potential barrier for each surface when UV light is directed towards the electrodes ................................ 107
4-25 Measured charge rate vs. phase using the TM port and best fit model to the data .................................................................................................................. 109
4-26 Measured charge flow rate for each surface from TM port and best fit model . 109
4-27 TM Voltage at equilibrium for different values of the potential barrier on the electrodes port .................................................................................................. 111
4-28 TM Voltage at equilibrium for different values of potential barrier when using the TM port ....................................................................................................... 111
4-29 Circuit model showing the TM voltage as a contribution of the voltage due to charge and the TM polarization due to the injection voltage ............................. 112
4-30 Voltage time series as TM voltage approaches saturation from below ............. 112
4-31 Voltage time series as TM voltage approaches saturation from above ............. 112
4-32 Measured charge flow rate and best fit model as test mass voltage changes at a constant potential bias when using the electrodes housing port ................ 114
4-33 Charge flow rate as the test mass voltage changes at a constant potential bias when using the TM port .................................................................................... 115
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LIST OF ABBREVIATIONS
Au
BL
EH
ESA
FOC
GRS
Hg
HS
LISA
LPF
NASA
ODE
TEC
TM
UV LED
Gold
Ball Lens
Electrode Housing
European Space Agency
Fiber Optic Cable
Gravitational Reference Sensor
Mercury
Hemispherical Lens
Laser Interferometer Space Antenna
LISA Pathfinder
National Aeronautics and Space Administration
Ordinary Differential Equation
Thermo Electric Cooler
Test mass
Ultra Violet Light Emitting Diode
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
UV LED BASED CHARGE CONTROL FOR THE LISA GRAVITATIONAL REFERENCE
SENSOR
By
Taiwo Janet Olatunde
December 2018
Chair: John W. Conklin Major: Aerospace Engineering
Test masses in the Laser Interferometer Space Antenna (LISA) must maintain
residual test mass accelerations under 3 fm/s2/√Hz at all frequencies between 0.1 mHz
and 3 mHz. Charge build-up on the test masses couples to stray electrical potentials and
external electromagnetic fields to contribute to unwanted force noise which mask
gravitational wave signals. The currently proposed discharge system for LISA follows
the LISA Pathfinder (LPF) discharge system which used the photoelectric effect to
transfer electrons from the test masses to the space craft and vice versa. In LPF the
required UV-photons were generated by a Hg-discharge lamp (254 nm) while the plan
for LISA is to use Ultra Violet Light Emitting Diodes (UV LEDs). These UV LEDs have a
lower mass, higher power efficiency and their photon energy often exceeds the work
function of pure gold.
The Quantum Yield, calculated as the number of electrons ejected from the
surface divided by the number of incident photons, is one of the essential properties
controlling the charge control system. Evaluated spectral properties for fiber coupled UV
LEDs manufactured by Sensor Electronic Technology (SETi), performed at the
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University of Florida shows that the wavelength of the 240 nm UV LED peaks at a value
that is closer to the work function of Au and is less sensitive to surface properties.
Experimental results of long term yield measurements on Au samples as a
function of wavelength showed 240 nm UV LEDs consistently outperforming 250 nm UV
LEDs in Quantum Yield by more than a factor of two with or without a bake-out. Using
240 nm UV LEDs for the charge control scheme in LISA would reduce the need to rely
on probable contamination of the Au samples for a lowering of work functions which
allow higher wavelengths and lower energy UV LEDs to be used.
The discharging of the test masses in LPF was done at DC. This involved
Illuminating the test mass or the electrode housing irrespective of voltages applied on
the electrodes. Illuminating the test mass resulted in a more positive test mass charge
while illuminating the electrode housing made the test mass charge more negative. AC
charge control is a new scheme being studied as an alternative to the DC charge
control system. In this case, the output of the UV light is modulated with respect to the
voltage on the electrodes. This work addresses the results of AC and DC charge control
modes demonstrated on the University of Florida torsion pendulum.
For AC charge control in the torsion pendulum, the output of a 240 nm UV LED
was pulsed with respect to the 100 KHz injection signal already present in the simplified
Gravitational Reference Sensor (GRS). Key results show that test mass charge can be
controlled at AC by setting the phase of the UV LED output at a particular value with
respect to the injection. The set phase depends on whether the UV light is illuminating
the TM, the EH or both simultaneously. The need to have three different ports for the
UV Injection scheme in LISA is removed but could still be useful for redundancy.
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CHAPTER 1 INTRODUCTION
Albert Einstein concluded in 1915 that spacetime deforms in the presence of
massive objects by bending like an elastic fabric, a lighter object moving along a path
nearby does not just continue, it is rerouted and pulled into the curve created by the
heavier object. The solar system gives a perfect example of the curvature of space
mainly due to the Sun; the other planets are not being pulled towards it, they just follow
the curved space time distortion it creates. When the curvature created in space time by
a heavy object such as a star becomes extreme, the gravitational pull, which is
proportional to the inverse square of the distance between them (GMm
r2 ), is very strong for
objects that are close enough. They are pulled in and never escape. Objects at longer
distances experience a weaker gravitational pull since the distance between them is
increased by a factor raised to the second power. For instance, a factor of four increase
in separation distance will lead to a reduction in the gravitational pull by a factor of
sixteen.
Two objects orbiting each other create a dynamic curvature in space-time by
their movements, this results in little ripples through the universe known as gravitational
waves. The waves cause variations in the apparent distance between free-falling
objects with a very small relative amplitude that makes them difficult to detect. The
violent collision or merger of two massive objects such as black holes or neutron stars
produces the strongest gravitational waves. At typical cosmological distances, even the
most powerful gravitational waves create strain amplitudes which are only of the order
of ~10-21. If the distance between two reference test masses in free fall is measured by
means of an interferometer, a gravitational wave moving in a direction that is not aligned
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with the space between them would alter the amplitude of their separation with a
displacement comparable to the amplitude of the gravitational wave.
The discovery in 1974, of a pulsar in orbit with a neutron star by Russell Hulse
and Joseph Taylor (PSR1913+16) [1] was taken as an indirect evidence of the
existence of gravitational waves. On September 14, 2015, the first direct detection of
gravitational waves was made by the Laser Interferometer Gravitational-wave
Observatory (LIGO) at Hanford, Washington and Livingston, Louisiana. Both
observatories detected gravitational wave signal GW150914, produced by the merger of
two black holes each several times the mass of the sun. LIGO detectors are modified
versions of the Michelson Interferometer [25]. In the original version of the Michelson
Interferometer, light from a source is passed through a beam splitter to create two
beams of light moving towards mirrors that reflect them back such that they recombine
at the beam splitter. The recombined beams form an interference pattern which
depends on the arm length difference. Any change of the arm length difference will
modulate the interference pattern and create a measurable change in the intensity
leaving the interferometer. Included in each LIGO arm are two 40 kg fused silica mirrors
with longer separation distances of 4 km apart and ‘Fabry Perot Cavities’ added for
increased sensitivity. The optical system in LIGO is installed in a vacuum system
covering about 16 km at a pressure of 10−8 Torr to eliminate likely disturbances from
dust, sound waves, temperature variations and air currents which would make
gravitational wave detection impossible [26]. Figure 1-1 shows the key elements of the
LIGO detector.
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Figure 1-1. A gravitational wave propagating orthogonally to the detector plane and linearly polarized parallel to the 4-km optical cavities will have the effect of lengthening one 4-km arm and shortening the other during one half-cycle of the wave; these length changes are reversed during the other half-cycle. The output photodetector records these differential cavity length variations. Inset (a): Location and orientation of the LIGO detectors at Hanford, WA (H1) and Livingston, LA (L1). Inset (b): The linear spectral density of the equivalent gravitational wave strain noise near the time of the first detection.
The need for LIGO to be able to detect the motion of the test masses due to a
passing gravitational wave requires that environmental interferences be eliminated. This
is done through active systems that correct disturbing movements in real time and
passive damping systems that hold the mirror test masses via a suspension system
made up of four pendulums. Laser beams are incident on one side of the system while
the other side helps to steady the mirrors for disturbances not associated with space.
This ensures that the mirrors remain in the free fall condition required for gravitational
19
wave detection while sensors and actuators control the position and orientation of the
test masses [27]. Figure 1-2 gives the analysis of GW150914. Both pictures were
obtained from the LIGO scientific collaboration [2].
Figure 1-2. Key results of the analysis of GW150914, comparing the reconstructed gravitational-wave strain with the predictions of the best matching waveform computed from general relativity, over the three stages of the event; inspiral, merger and ring down. Also shown are the separation and velocity of the black holes, and how they change as the merger event unfolds
1.1 LISA- A Developing Gravitational Wave Observatory
LIGO is limited to observing objects in the frequency range of 10 Hz to 1000 Hz.
A detector capable of observing lower frequency gravitational waves with longer
wavelengths makes the case for a space based detector such as the Laser
20
Interferometer Space Antenna (LISA) which will observe in the low frequency range of
the gravitational wave spectrum between 0.1 mHz and 1 Hz.
Figure 1-3. LISA’s nominal configuration
LISA will be a space based gravitational wave observatory that will use laser
interferometry to measure changes in separations between isolated test masses caused
by gravitational wave induced strains in space time. LISA requires test mass (TM)
accelerations to be suppressed below 3 fm /𝑠2 /√Hz at all frequencies between 0.1 mHz
and 3 mHz. It is made up of three spacecraft flying in equilateral triangle formation as
illustrated in Figure 1-3 [3]. There are two test masses per spacecraft, free in one axis
each and acting as end mirrors for the interferometry measurements. Arm lengths; L1,
L2 and L3 are interferometer arms with lengths of 2.5 million kilometers each.
21
Each TM in the spacecraft is enclosed in an electrode housing that supports the
sensing electrodes and protects the TM from many disturbances including solar wind
and high energy particles from galactic and solar rays. However, some of them can still
penetrate and charge the TMs. The caging and uncaging process may also leave a
charge on the TMs. Charge accumulation on the TM will cause its potential to vary and
result in an electrostatic attraction towards the surrounding [14]. Test mass charge also
couples to imbalanced DC biases on opposing conductors of the housing to produce
random walk force noise [4, 5].
1.2 LISA Requirements
Strains produced by the strongest gravitational waves are on the order of ~10−21
therefore LISA relies on the ability of laser interferometers to accurately measure the
displacement between two free falling test masses in opposing spacecraft on the order
of a picometer. An obvious challenge to this is the elimination of unwanted forces acting
on the TM that could mask TM accelerations that are only due to gravitational waves.
LISA Pathfinder (LPF), shown in Figure 1-4 was launched on December 3 2015.
It is a joint European Space Agency (ESA) and National Aeronautics Space
Administration (NASA) mission intended to demonstrate many of the technologies
required for LISA such as drag free control, inertial sensors and precision laser
interferometry [6]. One key difference between LPF and LISA is that the sensitive axes
of the two TMs in LPF lie on the same straight line and since they are both located in a
single spacecraft which cannot follow the trajectories of both TM along the same degree
of freedom, one of them has to be electrostatically controlled to follow the other [32].
The TMs in LISA will be located in two separate spacecrafts 2.5 million kilometers apart.
22
Figure 1-4. The LISA technology package (ESA/ATG media lab, LISA L3 mission proposal)
LISA pathfinder has demonstrated that a TM can be maintained in the free fall
condition needed for gravitational wave detection. The performance is related to the low
frequency side of the sensitivity curve and is related to acceleration noise. Acceleration
noise in the LISA GRS is due to various uncontrolled forces that could mimic or mask a
gravitational wave signal. For this reason, acceleration noise measurements are used to
assess the performance of the system and allow for the identification and mitigation of
possible sources of noise.
23
Figure 1-5. Sensitivity curve showing gravitational wave sources in the LISA frequency range compared with its sensitivity for a three arm configuration
The characteristic strain referenced on the y- axis in Figure 1-5 refers to the ratio
of the change in length of the interferometer arms and its original length. The
performance of LISA will be limited by acceleration noise demonstrated by LPF below 3
mHz and limited by interferometric noise above this frequency [16].
1.3 Gravitational Reference Sensor
The Gravitational Reference Sensor (GRS) is comprised of the test mass,
electrode housing, and other associated components as shown in Figure 1-6. The
electrode housing completely surrounds the free floating TM and protects it from
external forces produced by the spacecraft or the space environment. The host
spacecraft uses a drag free system to maintain the position of the TM by using sensors
24
that constantly determine the position of the freely floating TM with respect to the
housing. Information about the position of the TM is sent to a control system which then
commands the spacecraft to reposition itself in such a way that the TM is always
centered within the housing. The spacecraft therefore follows the orbit of the TM. Since
the actuation is not done directly on the TM, the TM avoids any noise associated with
the actuation itself, any residual acceleration noise of the TM will therefore be due to
local forces related to the spacecraft and external forces that are not already absorbed
by the spacecraft [31].
Figure 1-6. Flight-like GRS
1.4 Charge Control on LISA Test Masses
One possible substantial source of acceleration noise for the LISA GRS is
charge build up on the TMs. Charge increases electrostatic stiffness and interacts with
stray electric fields to induce additional force noise. Test masses in LISA must float
freely and the system of discharging must not involve any form of contact therefore
charge build up has to be controlled with UV light using the photoelectric effect. One
way to shine UV light on the test masses is through the use of fiber optic cables. It can
25
be done at DC, which involves illuminating either the gold coated TMs or electrode
housing surfaces to generate a net flow of electrons in the desired direction. This was
first demonstrated on the Gravity probe B mission [7]. It can also be done at AC, where
the UV light is pulsed and synchronized with the 100 KHz AC voltage already present in
the inertial sensor. The UV light would then be switched on only when the voltages
present in the sensor support the flow of electrons in the desired direction and suppress
the flow in the unwanted direction. When the energy of the absorbed light is equal to the
material work function, photoemission occurs for those electrons occupying the highest
energy level.
UV photons require energy above the work function of Au to liberate electrons.
The minimum required illuminating wavelength for a pure Au surface with work function
5.1 eV is ~243 nm. However, contamination generally leads to a lowering of the work
function of Au and allows higher wavelengths to be used [6, 7]. If no bias voltages are
applied to the electrodes surrounding the TM, the illumination of the TM should cause
electron flow from the TM to the housing. However a large fraction of the light will be
reflected to the housing which gives rise to unwanted electron flow from the electrode
housing to the TM. If on the other hand, the housing is illuminated, electrons would flow
from the housing to the TM, light is reflected to the TM and gives rise to unwanted
electron flow from TM to electrode housing. A positive test mass discharge rate is
obtained if the electron flow from the TM to the housing is greater than the electron flow
from the housing to the TM.
1.5 Summary of Chapters
Surface properties as well as preparation and handling that could affect the
Quantum Yields (QY) of Au surfaces intended for use in LISA are addressed in this
26
work. A demonstration of UV LED charge control at AC on a LISA test bed such as the
torsion pendulum described in chapter four is a big step forward in advancing a key part
of one of the technologies required for LISA. AC Charge control was not demonstrated
on LPF. An analytical model that explains the behavior of the pendulum when charge
control is being demonstrated is also important and has been developed.
The characteristics of the UV LEDs used in the QY and pendulum charge control
measurements were examined as detailed in Chapter two. A spectrum analysis was
carried out on both fiber coupled TO18 and non-fiber coupled TO39 UV LEDs from
Sensor Electronic Technology (SETi). Measurements of optical power were also made.
A method for mechanically fiber coupling TO39 UV LEDs was discussed as well. A
driving electronics board was designed and developed to enable pulse width modulation
of the TO39 UV LEDs. This was subsequently used to demonstrate AC charge control
on the pendulum.
Chapter three is dedicated to Investigating Quantum Yield as a function of
several parameters that could affect charge control. A simple analysis of the resulting
QY after multiple reflections showed that the QY of the illuminated surface needs to be
within a factor determined by the reflectivities, R, of both surfaces if they have a
common reflectivity or just the reflectivity, R2, of the surface adjacent to the illuminated
one if their reflectivities differ. A typical Au sample examined for surface composition
revealed a hydrocarbon with oxygen functionalities to be present. No attempt was made
to rid the samples of hydrocarbons, but a bake-out was done to eliminate the effect of
probable water adsorption. After bake-out, repeated measurements on individual
27
samples showed that variations in QY went from a factor of 10 to consistently less than
a factor of 2. A QY dependence on pressure was also shown to be possibility.
In Chapter four, the UF Torsion pendulum is described including its capabilities
for capacitively and interferometrically reading out the position and orientation of the
TMs within the simplified GRS. The acceleration noise performance of the pendulum is
also presented and compared with LISA requirements. An analytical charge control
model developed by representing the relevant surfaces of the GRS as parallel plate
capacitors allowed the distribution of normal energies of the photoelectrons involved in
electron exchange between the plates to be modelled as a function of bias voltages on
the electrodes. Thus an expression for the total electron flow rate for all interfaces was
found. The model adequately explained the voltage due to charge on the test mass
when controlled using DC and AC schemes. One major finding is that the reflectivity of
the gold surfaces allow AC charge control to be performed on any of the three existing
UV injection ports. Another finding is that the TM voltage due to charge can be kept
close to zero as long as the phase of the UV light is set appropriately rather than the
completely “in phase” and “out of phase” configurations usually proposed.
Chapter five discusses how the results obtained impact the LISA design and
operation. Appendix A gives the details the distribution of electron energies as a
function of applied bias voltage. Appendix B completely describes how solutions to
photoelectron flux equations were obtained and Appendix C explains how the charge
measurement equations used on the pendulum are obtained.
28
1.6 Contributions
Finding a method that will make the Quantum Yields of Au samples used in LISA
more uniform and reproducible as well as examining environmental factors that affect
QY was one of the major goals I set out to achieve. With Dr. John Conklin’s supervision,
I designed and implemented a system that enabled Au samples to be set up in a
vacuum chamber in a way that allowed long term QY measurements as a function of
temperature, pressure, and different illuminating UV wavelengths to be made. I also
processed the data from these measurements and analyzed the results as presented in
this dissertation.
Another goal was to demonstrate UV LED charge control on a LISA test bed
such as the UF Torsion pendulum. This was a team effort led by an advisory team
made up of Dr. John Conklin (University of Florida), Dr. Giacomo Ciani (University of
Padua), Dr. Guido Mueller (University of Florida) and Dr. Peter Wass (University of
Florida). Other members of the research team involved in the charge control effort apart
from myself include; Stephen Apple (University of Florida), Andrew Chilton (University of
Florida), Paul Serra (Massachusetts Institute of Technology), Samantha Parry
(University of Florida), Nicholas Turreta (University of Padua) and Henri Inchauspe
(Post Doc. University of Florida).
Paul Serra and I worked together on mechanically fiber coupling 240 nm and 250
nm TO39 UV LEDs for use in Quantum Yield measurements and charge control
experiments on the pendulum. Paul came up with the schematic and design of the
driver board used to synchronize the UV LED with the 100 KHz signal in the pendulum
for AC charge control, I implemented the circuit design for this driver board in Altium
software and sent it to Sierra Circuits to be printed. I also populated the required
29
components on to the Printed Circuit Board (PCB) and incorporated the mechanically
fiber coupled 240 nm UV LED with the driving board in a box to make a complete
charge control unit. Testing the functionalities of the board and fixing major problems
was mostly done by Paul and I. Samantha Parry later worked with Paul to build an
external voltage follower needed to scale the injected voltage signal as needed and
helped with further testing of the board itself.
In addition to implementing an interferometer in the pendulum, Andrew Chilton
designed and implemented the electronics used in data acquisition for capacitive and
interferometric readout on the pendulum. Stephen Apple and I set up and made charge
control measurements at DC and AC on the pendulum. Stephen Apple analyzed the
data afterwards and produced several useful post processed data sets. I worked with
Nicholas Turetta on developing an initial version of the analytical charge control model
which he implemented in Mathematica. I later worked with Henri Inchauspe to
implement a modified and more robust version of the analytical model in MATLAB. I
focused on developing, understanding and implementing the analytical version of the
model as presented in this dissertation but Henri implemented independent versions of
same.
30
CHAPTER 2 UV LEDS FOR LISA TEST MASS CHARGE CONTROL
2.1 Hg Lamps vs. UV LEDs
LISA Pathfinder discharge system exploits the photoelectric effect using UV light
produced by Hg lamps. A proposed discharge system for LISA could follow the same
principle, but replace Hg lamps with UV LEDs which produce light at wavelengths as
low as 240 nm. Their lower mass, better power efficiency and small size make them an
ideal replacement for Hg lamps. Gold samples that are without contamination will not be
discharged of electrons by mercury lamps producing light at 254 nm because the use of
the lamps rely on several unpredictable altered properties of the contaminated samples,
a lowering of the work function being one. Attempts to reduce surface contamination
and purify gold surfaces are therefore not favorable when Hg lamps are intended for
use in discharge. When the spectra of TO18 UV LEDs from SETi were examined, the
240 nm and 250 nm wavelengths peaked at approximately 239 nm (closer to the work
function of pure gold) and 254 nm respectively as shown in Figure 2-1.
Figure 2-1.Left: fiber coupled TO18 UV LEDs from SETi, Right: Intensity spectrum of 240 and 250 nm UV LEDs compared with the work function of Au and the Hg lamp discharge wavelength
31
The TO18 PIV characteristics in Figure 2-2 were found by driving the UV LEDS
with an ILX Lightwave LDX 3200 precision current source and measuring the optical
power with a Thorlabs PM100D power meter with a S120VC detector head.
Figure 2-2. Left: IV curve for TO18 UV LED, Right: PI curve for TO18 UV LED
2.2 UV LED Testing
Further specifications for the TO18 package, indicate that operating the UV LEDs
at 10 ˚C above their maximum operating temperature of 30 ˚C significantly reduces the
number of operating hours [16]. The commercially fiber coupled TO18 package UV
LEDs also failed launch shock vibration tests at Stanford / NASA Ames. It was therefore
decided that a custom, fiber coupler with active thermal control was needed. Two new
non fiber coupled 240 nm UV LEDs from SETi were tested as shown in Figure 2-3; the
TO39 Ball Lens (BL) and the TO39 Hemispherical Lens (HL). Their output optical
powers were found to be ~295 μW and ~200 μW respectively at a maximum current of
24 mA.
32
Figure 2-3. Left: IV curve for 240 nm TO39 BL and HS, Right: PI curve for 240 nm TO39 BL and HS
Figure 2-4. Left: 240 nm TO39 BL UV LEDs, Right: 250 nm TO39 BL UV LEDs
Fiber coupling a TO39 UV LED in the lab as further described in section 2.2.1,
involves collimating the beam with a fused silica bi-convex lens, placing it against a 600
µm core fiber optic cable which has already been glued to a fused silica ferrule, and
mechanically securing the set-up as shown in Figure 2-5.
A choice of the TO39 BL was made for the proposed heat dissipation
management design in Figure 2-6 because the resulting coupling efficiency for the BL
was ~7 % while the HS resulted in a coupling efficiency of ~3%. Also, lifetime tests
33
already carried out on TO39 UV LEDs resulted in > 30000 hours in vacuum and >
12000 hours in Nitrogen [5].
Figure 2-5. Output optical power of TO39 UV LED after being mechanically fiber coupled
2.2.1 UV LED Fiber Coupling and Electronics Driver Board for Charge Control
A UV LED fiber coupler with built-in temperature control was developed for
future space applications at the University of Florida. The design of the fiber coupler as
depicted in Figure 2-6 shows how light from the TO39 UV LED is coupled to the 600 µm
core of a fiber optic cable (FOC) glued inside a fused silica ferrule by a UV curable low
outgassing epoxy. The coupling of the UV light is done by placing a 10 mm focal length
fused silica bi-convex lens between the UV LED and the FOC/ferrule combination. A
mount made out of copper, Aluminum and polymer material is used to hold the unit in
place. The copper holder securing the UV LED before the lens will eventually
incorporate a Thermo-Electric Cooler (TEC) driven by a set of electronics [15].
34
Figure 2-6. Heat dissipation management design
Figure 2-7. UV LED driver board assembled with the UV LED inside a box
In the normal DC operation of the UV LED, a feedback loop on the driving
electronics board depicted in Figure 2-7 regulates any input signal between 0 - 3 V to
output a 0 - 3 V signal that is proportional to a current between 0 and 30 mA. When the
operation of the board goes from DC operation to pulse width modulation (PWM) of the
UV LED, parameters such as phase, amplitude voltage and duty cycle can be adjusted
35
by using variable resistors. Efforts will be made to space qualify the final version of the
driver board which will incorporate digital control electronics. Thermal vibration and
shock tests will be performed to verify its compatibility with the space environment.
2.2.2 Proposed Heat Dissipation Management
Heat from the TEC due to contact with the base of the UV LED will conduct
through the copper piece to the body of the assembly to a heat sink such as the
Aluminum box containing the unit in Figure 2-7. The TEC will be driven by an ADN8831
controller which incorporates a proportional integral differential (PID) feedback
mechanism that will stabilize the temperature of the unit by feeding back the sensed
temperature to complete a “closed thermal control loop of the TEC” [24].
2.3 Performance of UV LED Driver Board
Testing of the driver board included validating the power output of the UV LED
when driven at inputs of 0 - 3 V (corresponding to driving currents of 0- 30 mA) and duty
cycles of 5 % to 15 %. Charge control experiments on the torsion pendulum were
mostly carried out by setting the UV LED at a 5 % duty cycle with respect to the
injection because this value is small enough for a constant average injection voltage to
be assumed during the UV light pulse. Figure 2-8 shows the average UV power output
while driving the UV LED at 5-20 mA, increasing almost linearly and is stable with
standard deviations mostly below 0.1% of the average power output at each voltage.
Varying the duty cycle while driving the UV LED at 10 mA gives the results shown in
Figure 2-9. The power output here is also stable with a standard deviation of less than
0.1 % of the output for each duty cycle.
36
Figure 2-8. Average power output of UV LED driver board for a 5 – 20 mA current setting input and a constant 5 % duty cycle
Figure 2-9. Average power output of UV LED driver board for various duty cycle settings
37
CHAPTER 3 QUANTUM YIELD AND REFLECTIVITY OF LISA TEST MASSES
3.1 Motivation for Measuring QY and Reflectivity
One of the critical properties governing the performance of a charge
management system is the apparent Quantum Yield (QY). The QY is calculated as the
number of photoelectrons emitted from the gold surface per incident UV photon. It is
important to know the amount of UV light absorbed by opposing surfaces under
illumination since in the LISA Pathfinder GRS design; both the TM and the electrode
housing are gold coated. When UV light illuminates the TM directly, some of the light
reflects back and is absorbed by the housing. This can create photoelectrons that flow
in opposition to the intended direction. Large QY differences in the surfaces involved in
charge control may also result in electron flow in only one direction irrespective of which
one is illuminated. Therefore the relationship between the QYs and reflectivities of
surfaces involved in charge control should be understood. Studies show that the TM
absorbs about 70% of the total light when it is illuminated and about 20% is absorbed by
the housing. When illuminating the housing, around 25% is absorbed by it and about
15% by the TM [Hollington, Daniel. 2011], the remaining percentage is lost in crevices
and gaps in the electrode housing that are not involved in the charge control process.
3.2 Reflectivity Analysis for Two Infinite Parallel Plates
Adjacent surfaces intended for use in charge control should ideally have identical
Quantum yields so that when one of the surfaces is illuminated, all the electrons will
make the transition in the desired direction. However reflections make this nearly
impossible. Maximum deviations from the ideal that still allow for effective charge
control should follow the approximate rules derived in the following two cases;
38
The number of electrons 𝑁𝑒 ejected by a surface with incident UV light photons
𝑁𝑣 can be represented as;
𝑁𝑒 = 𝑁𝑣𝑄𝑌 (3-1)
The surfaces in Figure 3-1 with quantum yields QY1 and QY2 and equal
reflectivities, R with UV light directed at surface 1, can have the total number of
photoelectrons due to multiple reflections represented as follows;
Figure 3-1. Reflections between two surfaces with different QYs and similar reflectivities
Assuming N reflections between surfaces 1 and 2, the net number of electrons
flowing from surface 1 to 2 is,
𝑁𝑒 = 𝑁𝑣𝑄𝑌1 + 𝑁𝑣𝑄𝑌1 𝑅2 + 𝑁𝑣𝑄𝑌1 𝑅
4 + ⋯ − 𝑁𝑣𝑄𝑌2 𝑅 − 𝑁𝑣𝑄𝑌2 𝑅3 − 𝑁𝑣𝑄𝑌2 𝑅
5 − ⋯
(3-2)
Combining the terms gives,
𝑁𝑒 = 𝑁𝑣 ∑ (𝑄𝑌1
𝑁𝑛=0 𝑅2𝑛 − 𝑄𝑌2 𝑅
2𝑛+1 ) (3-3)
𝑁𝑒 = 𝑁𝑣 ∑ 𝑅2𝑛𝑁
𝑛=0 (𝑄𝑌1 − 𝑄𝑌2 𝑅) (3-4)
For an infinite number of reflections, N ∞,
39
∑ 𝑅2𝑛𝑁𝑛=0 =
1
1−𝑅2 (3-5)
And therefore, the net number of electrons becomes,
𝑁𝑒 = 𝑁𝑣1
1−𝑅2(𝑄𝑌1 − 𝑄𝑌2 𝑅) (3-6)
If 𝑁𝑒 > 0, then,
𝑅 <𝑄𝑌1
𝑄𝑌2
< 𝑅 (3-7)
Therefore, the net flow of electrons away from the illuminated surface will be
positive if the QY of that surface is greater than R.QY of the opposing surface. If a 36%
reflectivity is assumed based on previous work done on Au surfaces [8] then Equation
3-7 shows that for two surfaces involved in charge control with different QYs and similar
reflectivities, the net flow of electrons will be away from the illuminated surface if the QY
of that surface is greater than the QY of the opposing surface by a factor of 0.36 or
more.
If the reflectivities of the two surfaces are different, say 𝑅1 and 𝑅2 as in Figure 3-
2, the total number of electrons flowing away from surface 1 is given by,
𝑁𝑒 = 𝑄𝑌1𝑁𝑣 + 𝑄𝑌1𝑁𝑣𝑅2𝑅1 + 𝑄𝑌1𝑁𝑣𝑅2𝑅1 + ⋯
−𝑄𝑌2𝑁𝑣𝑅2 − 𝑄𝑌2𝑁𝑣𝑅22𝑅1 − 𝑄𝑌2𝑁𝑣𝑅2
3𝑅12 − ⋯ (3-8)
Combining terms gives,
40
Figure 3-2. Reflections between two surfaces with different QYs and different reflectivities
= 𝑁𝑣(𝑄𝑌1(1 + 𝑅2𝑅1 + 𝑅22𝑅1
2 + ⋯ ) − 𝑄𝑌2(𝑅2 + 𝑅22𝑅1 + 𝑅2
3𝑅12 + ⋯ )) (3-9)
And
= 𝑁𝑣(𝑄𝑌1(∑ (𝑅2𝑅1)𝑛) − 𝑄𝑌2𝑅2(∑ (𝑅2𝑅1)𝑛𝑁𝑛=0
𝑁𝑛=0 )) (3-10)
If N ∞, then the following holds
∑ (𝑅2𝑅1)𝑛 = 1
1−𝑅2𝑅1 𝑁
𝑛=0 (3-11)
= 𝑁𝑣(𝑄𝑌1(1
1−𝑅2𝑅1) − 𝑄𝑌1𝑅2(
1
1−𝑅2𝑅1)) (3-12)
= 𝑁𝑣
1−𝑅2𝑅1(𝑄𝑌1 − 𝑄𝑌2𝑅2) (3-13)
For 𝑁𝑒 > 0
𝑅2 <𝑄𝑌1
𝑄𝑌2< 𝑅1 (3-14)
So the QY of the illuminated surface must be greater than the QY of the opposing one
by a factor determined by the reflectivity of that opposing surface.
In reality, the number of reflections N, between GRS surfaces is not infinite but
the above derivations follow a reasonable “rule of thumb”. Actual required QY ratio
depends on detailed light distribution over a more complex geometry.
41
3.3 QY Experimental Set up and Apparatus Description
The quantum yields of several Au samples have been quantified by isolating a
2 in × 2 in gold sample inside a hollow 3.5 in diameter sphere using Ultem holders
which have a resistivity of 1017 Ωm [5]. In the set-up in Figure 3-3, the sphere is biased
positively (+9 V) and the sample negatively (−9 V), this was done by using 9 V batteries
to reduce ground loops. An ultra-high vacuum UV fiber optic cable with a 600 μm core is
used to direct the light from the UV LED to the sample at a zero degree incident angle.
The UV LED current is then varied over a range of 10 mA-24 mA, using an ILX
Lightwave LDX-3200 precision current source. This range of current corresponds to 6
μW-20 μW of optical power with only 0.6 μW-2 μW directly incident on the Au sample in
the chamber after accounting for losses through fiber couplings. At a maximum current
draw of 24 mA, both the 240 nm and 250 nm UV LEDs consume roughly 180 mW of
electrical power. The resulting photocurrent is measured using a Keithley 6485
Picoammeter. This is then converted to number of photoelectrons and divided by the
number of incident illuminating UV photons [16]. Even though the Aluminum sphere
absorbs some of the UV light reflected by the gold sample, any electrons ejected from
its surface is trapped by the large 18 V retarding potential barrier and does not factor
into the Quantum Yield calculations. The quantum yield measurement results presented
in this work were made at pressures of 10−6 Torr to 10−8 Torr by using both vacuum
chambers shown in Figure 3-3.
A local bake-out of the sample can be carried out by attaching a 28 V, 5 W
polyimide film strip heater to the back of the Au sample. The heater is connected to an
LM2575T regulator which provides a differential 5 V input to an LT1006 operational
42
amplifier and an IRF3708 MOSFET for switching the output voltage connected to a
PT1000 sensor also attached to the sample on and off. A standard data sheet gives the
sensor temperatures and corresponding resistance values. These resistances are used
to determine the voltages to be set by an incorporated potentiometer.
Figure 3-3. QY measurement apparatus set up in large vacuum at ~3 × 10−6 Torr, QY apparatus was subsequently set up in the small vacuum chamber with a leak valve for pressure adjustments
3.3.1 QY Dependence on Energy of Emitted UV Light
The quantum yields of most metals have a linear dependence on the wavelength
of the incident UV light with lower wavelengths resulting in higher quantum yields and
vice versa [19]. The energy of the emitted photoelectrons has a value between 0 eV and
43
a maximum energy given by (hν − ɸ) eV [17]. The maximum energy is related to the
quantum yield through the following relationship.
𝑄𝑌 ∝ (ℎ𝜈 − ɸ)2 (3-15)
The QY experiments carried out in this work are done at 240 nm and 250 nm. A
rough estimate of the validity of the results obtained can be found by considering two
different wavelength UV LEDs and representing Equation 3-15 as,
𝑄𝑌𝑙𝑤 = 𝛽 (ℎ𝑐
𝜆𝑙𝑤− ɸ𝐴𝑢)2 (3-16)
𝑄𝑌ℎ𝑤 = 𝛽 (ℎ𝑐
𝜆ℎ𝑤− ɸ𝐴𝑢)2 (3-17)
Here, 𝑄𝑌𝑙𝑤 and 𝑄𝑌ℎ𝑤 represent the respective QY values obtained at the lower
and higher wavelengths while λlw and λhw are the corresponding wavelengths in nm, h is
Planck’s constant (4.1357 × 10−15 eV s) β is a constant, c is the speed of light (3 × 108
m/s) and ɸAu is the work function of the Au sample under experiment. Dividing Equation
3-16 by Equation 3-17 results in,
ɸ𝐴𝑢 = − ℎ𝑐
(1−√𝑟)(
√𝑟
𝜆ℎ𝑤−
1
𝜆𝑙𝑤) (3-18)
When more than one UV wavelength is used in the QY measurements, a rough estimate
of the work functions of the measured Au samples can be found from Equation 3-18.
Previous work function measurements carried out on several Au samples at the
University of Modena have values between 3.6 eV to 5.1 eV with an average of 4.3 ±
0.4 eV [8]. If the results from Equation 3-18 fall within this range, then the QY results
obtained would seem reasonable.
44
3.3.2 QY Measurements and Wavelength
One of the factors influencing the measured QYs of Au surfaces include the
energy of the illuminating photons from the UV light. The work here focuses on two
different UV LED wavelengths, the 240 nm UV LED with a corresponding energy of
~ 5.17 eV and the 250 nm UV LED corresponding to an energy of ~4.96 eV. Regardless
of surface preparation and handling, the closer the work function of the surface is to the
energy of the illuminating photons, the more likely it is that photoelectrons will be ejected
from that surface.
The QYs of several random Au samples have been quantified as shown in Figure
3-4. Samples 1 (A, B, C, D) were coated in March 2013 with a gold layer 200 nm thick
and samples 2 (A, B) with a gold layer 500 nm thick were coated in September 2013.
The letter P indicates a polished surface and the letter U, an unpolished surface so that,
1AP is the measurement made on a polished surface coated in the first batch of samples
and 2BU is that made on an unpolished surface coated with the second batch. The
measurements were made in 2014. The dates are as indicated. The vacuum chamber
was vented back to atmospheric pressure in between measurements. As expected and
shown, the 240 nm UV LEDs result in a higher QY than the 250 nm ones. Both UV LEDs
are similar in terms of power consumption and cost.
The QY measurements vary by as much as a factor of 20 across the different
samples with individual samples varying by factors of 1.2 to 10. An example is sample
1CP which produces QYs varying by a factor of ~2 for four successive measurements
made within a period of three weeks without breaking vacuum. There is an overall lack of
repeatability in the measurements which is believed to be due in part to atmospheric
45
contaminants which adsorb on the sample surface while in storage and from the walls of
the vacuum chamber during the experiment.
Figure 3-4. QY of Au samples at 240 nm > QY at 250 nm
The effect of water adsorption on the sample surface is an increase in the work
function [21]. Previous work has shown that unlike water, carbon compounds cannot be
gotten rid of by baking out the sample, but argon sputtering successfully removed carbon
compounds and led to an increase in work function [22] meaning that the presence of
carbon compounds may have led to a reduction of same.
3.3.3 QY Measurements vs. Time
The measured QYs of some of the Au samples in Figure 3-4 appear to be
inconsistent with no indication of what the actual values could be. It might be possible to
obtain more information about this behavior with longer term measurements. For this
purpose, sample 1DP was measured, using a UV wavelength of 240 nm under vacuum
46
for two weeks. The vacuum chamber was briefly vented up to atmosphere once during
this time period in order to see what the effect of possible atmospheric compounds
interacting with the sample surface could be. Apart from a slight decrease in the QY after
the vacuum chamber was pumped down again, the QY continued to increase until it was
up to a factor of 2 over 300 hours as shown in Figure 3-5. This suggests that even
though all samples were cleaned to remove surface impurities just before testing, they
still come in contact with contaminants such as water and hydrocarbons adsorbed
through contact with the atmosphere before being set up in vacuum.
Figure 3-5. Successive QY measurements with sample 1DP held at 10−5 Torr using a UV wavelength of 240 nm
3.3.4 QY After a Bake-out
One way to reduce variations in QY may be to get rid of contaminants on the
sample surface by baking it out in vacuum [8]. A bake-out of sample 2BP was carried out
in vacuum at 130 ˚C over a period of 550 days. The QY measurements, using a
47
wavelength of 240 nm are shown in Figure 3-6. During this process, just before the bake-
out was started on the sample, a QY measurement was made. Another was made
immediately after the sample reached 130 ˚C and others while the sample was being
baked further. Within the first few hours, an increase in the measured QY, from ~9 ×
10−7 to ~2 × 10−5 was observed indicating that impurities evaporating from the sample
surface probably lowered the work function. The QY eventually stabilized at ~ 7 × 10−6
(previously measured as ~ 4 × 10−7 in Figure 3-4) for three repeated cycles of
measurements made after the heater was turned off and the sample allowed to cool
down to room temperature without venting the chamber, indicated by the points in green
(these are the only points considered in the ongoing discussion). These QY
measurements appear to be fairly uniform and are within a factor of 1.3 of one another
for the 0° UV incidence angle after three cycles of repeated measurements on the same
sample.
The UF torsion pendulum is a test bed for LISA GRS technology which will be
discussed in chapter 4. In the current version of the pendulum’s simplified GRS, UV light
incident on the TM and EH are each at a 45 ° incidence angle. This prompted the need
to make QY measurements with the UV LED inclined at 45° to sample 2BP. The
resulting QYs lie within a factor of 1.08 for repeated measurements. These are slightly
lower than those made at 0 °by about a factor of 1.2. At 0°, more of the UV light is likely
absorbed by the sample surface than at 45° which could explain the slightly higher
quantum yield. From previous work done, the amount of reflected light for 0° and 45°
are approximately 36 % and 37 % respectively for unpolarized UV light [8,23]. This is
illustrated in Figure 3-7.
48
Figure 3-6. Measured QY on Au sample 2BP with bake-out at 130˚C and 240 nm
Figure 3-7. Fraction of light reflected as a function of angle of incidence (in degrees) for 254 nm wavelength showing the two polarization components Rp and Rs
comparing results described by Daniel Hollington (Dissertation 2011) and Johnson and Christy (1972).
Measurement before bake-out with UV light at 45 degrees Measurement during bake-out with UV at 45 degrees
49
The green points after the heater is turned off and the sample has been allowed
to cool are important because the LISA GRS will be baked out before launch to provide
more uniform quantum yields required for effective charge control. Points in yellow are
those taken before the bake-out measurements at 45˚.
3.3.5 Sample Preparation, Cleaning and Storage
The samples discussed so far were generally hand polished and stored in
designated cabinets in air at room temperature with no special cleaning procedure.
However, care was taken to make sure that protective gloves were worn when handling
them. In an effort to investigate if the handling of the samples had any significant effects
on the QY, a dedicated cleaning procedure for Au samples was obtained from Imperial
College London (ICL). The research group at ICL was responsible for the charge
management system for LISA Pathfinder. The procedure is outlined as follows,
Step 1: Ten minutes ultrasound immersed in acetone
Step 2: Rinse with de-ionized water
Step 3: Ten minutes ultrasound immersed in ethanol
Step 4: Rinse with de-ionized water
Step 5: Ten minutes ultrasound immersed in propan-2-ol
Step 6: Warm propane-2-ol containing sample in warm water bath
50
Step 7: Remove sample slowly allowing alcohol to drain and evaporate as the sample is
lifted
Step 8: Ensure that the surface has no tide marks
Step 9: Remove any remaining droplets with clean dry Nitrogen
Step 10: Attach connectors, mount in vacuum system and begin pumping down
immediately
The above cleaning steps were applied to two separate Au samples, 1CP and
1AP, before doing a bake-out at 130 °C. Figure 3-8 shows the QY of sample 1CP at 240
nm stabilizing at ~ 8 × 10−6 (average of green points) over 700 hours, this a factor of 6
increase over the average of ~1.3 × 10−6 that was obtained for the same sample without
a bake-out shown in Figure 3-4 using 240 nm UV LEDs. The overall result does remains
fairly consistent (in the 10−6 range) with those obtained from other samples from this
batch after a bake-out without any special consideration for cleaning and handling. The
same procedure applied to sample 1AP in Figure 3-9 follows the same pattern as 1CP
with additional measurements made simultaneously at 250 nm.
The QY for the 240 nm wavelength averages at 1.2 × 10−5 after baking out
(green points); this is a factor of 1.5 above the 0.8 × 10−5 average obtained from Figure
3-4 with no bake-out. The measurements at 250 nm resulted in a lower value of 0.23 ×
10−5, a factor of ~2 increase over the values obtained in Figure 3-4 at 250 nm.
51
Figure 3-8. QY with detailed cleaning procedure and bake-out at 130°C using 240 nm UV LEDs for sample 1CP
Figure 3-9. QY with detailed cleaning procedure and bake-out at 130°C with UV wavelengths of 240 nm vs. 250 nm for sample 1AP
QY at 240 nm with 130˚C bake-out
QY at 250 nm with 130˚C bake-out
Measurement before bake-out Measurement after turning heater off Measurement after vacuum is vented and pumped down again
52
3.3.6 Evaluating the QYs of Commercially Polished Au Surfaces
Apart from atmospheric contaminants which could be reduced by baking out the
sample, another likely reason for the variation in measured QY is that the samples were
not prepared in a consistent manner. All the Au samples discussed so far were hand
polished Aluminum substrates, gold-coated by Teer coatings in the UK and tested with
the fiber coupled TO18 UV LEDs described in section 2.1.
The new samples called 4AP, 4BP, 4CP, 4DP were Aluminum substrates
commercially polished in August 2017 by Cabot Microelectronics. The polishing was
done by diamond fly cutting to a specification of less than 10 µm for flatness and 12-16
nm for surface roughness before being sent off to Teer Coatings to be gold coated.
These samples were not cleaned in any way before QY measurements. They were
simply stored in clear all-glass petri dishes inside a shelf in the lab after they were coated
and received from Teer Coatings UK. The samples were evaluated with TO39 UV LEDs
fiber coupled as described in section 2.2. These 240 nm and 250 nm TO39 UV LEDs
peaked at 247 nm and 255 nm respectively.
As shown in Figure 3-10, three cycles of measurements were made on sample
4AP at a pressure of ~2 × 10−6 Torr over 500 hours with a 130 ˚C bake-out. As with
previous bake-out measurements, the blue points are the measurements made after the
vacuum chamber is pumped down before the bake-out is started. The red points
represent the measured QYs while the sample is being held at 130˚C. The points in
green are measurements made after the heater is switched off and the sample is allowed
to cool down to room temperature. The cyan points represent measurements made
immediately after the vacuum is pumped down again after venting. The pressure
adjustments are shown in grey. Measurements revealed a fairly consistent QY with
53
values of ~7 × 10−5 after the heater was switched off and the sample was allowed to
cool.
Figure 3-10. QY of Commercially polished sample 4AP measured at 130°C with a 240
nm at ~2x10−6 Torr
Only one sample can be set up in the chamber at a time and prolonged
measurements such as the one shown in Figure 3-10 could take several weeks.
Previous bake out results show that the average of the green points do not change
significantly with repeated measurements. This is why an average of three cycles was
chosen for the measurements on the commercially prepared samples.
3.3.7 Investigating QY Dependence on Pressure
The UF torsion pendulum which will be discussed in Chapter 4 operates at
~10−6 Torr but the LISA GRS will operate around 10−8 Torr. This motivates the need to
54
investigate possible QY dependence on pressure. The current vacuum chamber in
Figure 3-3 uses O-rings which do not provide as good of a vacuum seal as copper
gaskets. The lowest vacuum pressure obtained with the O-ring seals averaged at
~ 3 × 10−6 Torr. A set up capable of reaching pressures as low as ~2 × 10−8 Torr range
was achieved by designing a smaller vacuum chamber with all feedthroughs sealed with
copper gaskets. The smaller vacuum chamber was also baked out to help get rid of
substances which might out gas and contribute to a lowering of overall vacuum quality.
Variations in vacuum pressure in the new vacuum chamber were obtained with a
variable leak valve from Agilent Technologies, which offers leak rates as small as
1 × 10−10 Torr-liters per second and almost precisely controls the size of the leak.
Samples 4AP, 4BP, 4CP and 4DP were investigated at three different pressures and a
130 °C bake-out temperature as shown in Figures 3-11, 3-12, 3-13 and 3-14. The first
noticeable detail is that 4AP has lower QY values than the other three samples;
averaging between 4 − 7 × 10−5 at 240 nm and 2 − 4 × 10−5 at 250 nm shown in
Figures 3-16 and 3-17 respectively. This may be because the whole chamber along with
this particular sample set up inside, was baked out to temperatures as high as 200°C to
help drive the vacuum chamber pressures into the proposed LISA range. The actual
local bake-out on the sample was done afterwards at 130°C before cooling down to
make QY measurements. It is likely that the surface properties of the sample were
already altered by the 200 ˚C bake-out so that making measurements after this bake-out
at a temperature below this value did not have the same effect on the quantum yields
compared to the other three samples. This is also evident in the standard deviations from
measurement to measurement on this sample. Overall, sample 4AP shows no significant
55
variation with pressure since the standard deviations mostly overlap. The 200° C initial
bake-out does not apply to the other three samples in this discussion.
Figure 3-11. QY of sample 4AP measured between 10-8 and 10-6 Torr
Figure 3-12. QY of sample 4BP measured between 10-8 and 10-6 Torr
56
Figure 3-13. QY of sample 4CP measured between 10-8 and 10-6 Torr
Figure 3-14. QY of sample 4DP measured between 10-8 and 10-6 Torr
Additional measurements of more than 1000 hours were made on sample 4 CP at
10−8 Torr after concluding the third measurement cycle without breaking vacuum. As
shown in Figure 3-15, the measured QYs reduced to a factor of two below what the
average was at 10−6 Torr. This is important because LISA will undergo a long integration
57
process where pressures could rise to values as high as 10−5 Torr after intermittently
breaking vacuum. This pressure will be adjusted to 10−8 Torr after it is released into the
space environment. It is important to take changes as a result of expected pressure
variations into account.
Figure 3-15. Long term QY measurements of sample 4CP at 10-8 Torr
All the measured QYs represented by the green points in Figures 3-11 to Figure
3-14 were plotted as shown in Figure 3-16 and Figure 3-17. Samples 4BP and 4CP are
directly comparable in how they were handled because at the time of measurement, the
pressure in the chamber easily reached the 10−8 Torr range and could be increased by
factors of ten when needed. However, this was not the case for 4DP where pressures in
the lower regions of 10−8 Torr could not be easily accessed probably due to a leak
somewhere in the vacuum system.
The QYs of samples 4BP and 4CP do not appear to have a significant pressure
dependence at 10−8 and 10−7 Torr since the QYs for these samples fall within similar
58
ranges at these two pressures. At 10−6 Torr however, the mean QYs for both samples
reduce by as much as a factor of 3. This could be because; for one, at 10−6 Torr, water is
present in a much higher quantity on the sample surface than at the starting pressure of
10−8 Torr. This most likely led to an increase in the work functions and resulted in lower
QYs. Sample 4DP followed a similar pattern to 4BP and 4CP at pressures of 10−8 and
10−7 Torr. There was no meaningful QY change at 10−6 Torr.
Figure 3-16. Results of QY as a function of pressure at 240 nm
Figure 3-17. Results of QY as a function of pressure at 250 nm
59
3.5 Summary of QY Measurements
A summary of all QY measurements made so far is provided in Table 3-1. Some
of the information in the table includes the approximate date the samples were coated,
the way they were processed and the UV wavelengths used in the measurements.
Table 3-1 Quantum Yield Measurements from Au samples coated from March 2013
through August 2017, refer to Figures 3-4 through 3-14
Parameters Average QY at 240 nm
Average QY at 250 nm
Sample#: 1AP Date of coating: March 2013 Processing: Hand polished 8 × 10−6 1.2 × 10−6 Cleaning method: Wiping with Isopropanol Figure 3-4 Figure 3-4 UV light angle with respect to the sample: 0° Temperature: Ambient Pressure: 3E-6 Torr
Sample#: 1BP Date of coating: March 2013 Processing: Hand polished 2.5 × 10−6 1.5 × 10−7 Cleaning method: Wiping with Isopropanol Figure 3-4 Figure 3-4 UV light angle with respect to the sample: 0° Temperature: Ambient Pressure: 3E-6 Torr Sample#: 1CP Date of coating: March 2013 Processing: Hand polished Cleaning method: Wiping with Isopropanol UV light angle with respect to the sample: 0°
Temperature: Ambient Pressure: 3E-6 Torr Sample#: 1DP Date of coating: March 2013 Processing: Hand polished Cleaning method: Wiping with Isopropanol UV light angle with respect to the sample: 0°
Temperature: Ambient Pressure: 3E-6 Torr
1 × 10−6 Figure 3-4
1.2 × 10−5
Figure 3-4
4.6 × 10−7 Figure 3-4
6 × 10−7
Figure 3-4
60
Table 3-1. Continued
Parameters Average QY at 240 nm
Average QY at 250 nm
Sample#: 2AP Date of coating: September 2013
Processing: Hand polished 9 × 10−6 5 × 10−6 Cleaning method: Wiping with Isopropanol Figure 3-4 Figure 3-4 UV light angle with respect to the sample: 0° Temperature: Ambient Pressure: 3E-6 Torr Sample#: 2BP Date of coating: September 2013
Processing: Hand polished 4 × 10−7 -
Cleaning method: Wiping with Isopropanol Figure 3-4 UV light angle with respect to the sample: 0° Temperature: Ambient Pressure: 3E-6 Torr Sample#: 2BU Date of coating: September 2013 Processing: Unpolished Cleaning method: Wiping with Isopropanol UV light angle with respect to the sample: 0°
Temperature: Ambient Pressure: 3E-6 Torr
1.6 × 10−7
Figure 3-4 8 × 10−8
Figure 3-4
Sample#: 2BP Date of coating: September 2013 Processing: Hand polished Cleaning method: Wiping with Isopropanol UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 3E-6 Torr
Stable green points after cooling
7 × 10−6
Figure 3-6
-
Sample#: 2BP Date of coating: September 2013 Processing: Hand polished Cleaning method: Wiping with Isopropanol UV light angle with respect to the sample: 45°
Temperature: Baked out at 130 °C Pressure: 3E-6 Torr
5.5 × 10−6
Stable green points after cooling
Figure 3-6
-
61
Table 3-1. Continued
Parameters Average QY at 240 nm
Average QY at 250 nm
Sample#: 1CP Date of coating: March 2013 Processing: Hand polished Cleaning method: Procedure from ICL UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 3E-6 Torr
Stable green points after
cooling
8 × 10−6
Figure 3-8
-
Sample#: 1AP Date of coating: March 2013 Processing: Hand polished Cleaning method: Procedure from ICL UV light angle with respect to the sample: 0°
Temperature: baked out at 130 °C Pressure: 3E-6 Torr
Stable green points after
cooling
1.2 × 10−5 Figure 3-9
Stable green points after
cooling
2.3 × 10−6 Figure 3-9
Sample#: 4AP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 1.5E-6 Torr
Stable green points after
cooling
7 × 10−5 Figure 3-10
-
Sample#: 4AP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 2E-8 Torr
Stable green points after
cooling
5.9 × 10−5 Figure 3-11
Stable green points after
cooling
3.9 × 10−5 Figure 3-11
Sample#: 4AP Date of coating: August 2017 Processing : Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 2.5E-7 Torr
Stable green points after
cooling
6.5 × 10−5 Figure 3-11
Stable green points after
cooling
3.8 × 10−5 Figure 3-11
62
Table 3-1. Continued
Parameters Average QY at 240 nm
Average QY at 250 nm
Sample#: 4AP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 2E-6 Torr Sample#: 4BP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 3E-8 Torr
Stable green
points after cooling
6.3 × 10−5 Figure 3-11
Stable green
points after cooling
2.5 × 10−4 Figure 3-12
Stable green points after
cooling
3.5 × 10−5 Figure 3-11
Stable green points after
cooling
1.3 × 10−4 Figure 3-12
Sample#: 4BP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 2E-7 Torr
Stable green
points after cooling
2 × 10−4 Figure 3-12
Stable green points after
cooling
1 × 10−4 Figure 3-12
Sample#: 4BP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 4E-6 Torr
Stable green points after cooling
8 × 10−5 Figure 3-12
Stable green points after
cooling
3.8 × 10−5 Figure 3-12
Sample#: 4CP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 6E-8 Torr
Stable green points after cooling
3.1 × 10−4 Figure 3-13
Stable green points after
cooling
1.5 × 10−4 Figure 3-13
63
Table 3-1. Continued
Parameters Average QY at 240 nm
Average QY at 240 nm
Sample#: 4CP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 3E-7 Torr Sample#: 4CP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 4E-6 Torr
Stable green points
after cooling
3 × 10−4 Figure 3-13
Stable green points
after cooling
2 × 10−4 Figure 3-13
Stable green points after
cooling
1.6 × 10−4 Figure 3-13
Stable green points after
cooling
7 × 10−5 Figure 3-13
Sample#: 4DP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 6E-8Torr
Stable green points
after cooling
2.1 × 10−4 Figure 3-14
Stable green points after
cooling
9.2 × 10−5 Figure 3-14
Sample#: 4DP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 1.3E-7 Torr
Stable green points
after cooling
2.1 × 10−4 Figure 3-14
Stable green points after
cooling
9.6 × 10−5 Figure 3-14
Sample#: 4DP Date of coating: August 2017 Processing: Diamond turned polishing Cleaning method: None UV light angle with respect to the sample: 0°
Temperature: Baked out at 130 °C Pressure: 1.8E-6 Torr
Stable green points
after cooling
2.1 × 10−4 Figure 3-14
Stable green points after
cooling
8 × 10−5 Figure 3-14
64
3.6 Alternative Coatings Explored
Gold is the default coating material for the LISA test mass and housing because
of its non-magnetic properties and uniform surface potential which reduces spurious
force noise on the test mass. Gold is reflective at 1064 nm, the wavelength of the laser
interferometer, and is suitable for UV photoemission-based charge control. It is also soft
and prone to sticking and scratching when not properly handled and could complicate
the caging and release process. Carbide coatings on the other hand have been
considered as potential alternatives, therefore the QYs of some alternate carbide
coatings have been explored. Four extra samples each of Au, TiC and SiC were coated
at TEER coatings as shown in Figure 3-18. They were measured at 240 nm UV
wavelength without doing a bake-out. The coatings applied were, Ti (50 nm, 99.5%
purity) followed by Au (500 nm, 99.99% purity), Ti (50nm, 99.5% purity) followed by TiC
(500 nm, 99.5% purity) and Si (50 nm, 99.99% purity) followed by SiC (500 nm, 99.99%
purity).
Figure 3-18. Left to right; Aluminum substrates, hand-polished and coated with Au, TiC and SiC
Figure 3-19 shows the QYs for Au as the highest of the three varying between
~2 × 10−6 and ~1.3 × 10−5. TiC with work function 3.80 eV [9] is the most stable
between ~3.5 × 10−7 and ~6 × 10−7. SiC with work function 4.80 eV [10] results in QY
values from ~2.1 × 10−7 to ~1 × 10−6.
65
Figure 3-19. Results of QY measurements for Au, TiC, and SiC coatings
The reflectivities calculated for TiC and SiC at 255 nm were found to be 15% and
12% respectively [5], meaning that they would absorb sufficient amount of photons and
would be suitable alternative materials for charge control. Since the time of these
measurements LISA Pathfinder successfully demonstrated UV charge control and
caging and release, solidifying the use of Au as the coating choice for LISA. However,
other missions could consider these materials in the future.
3.7 X-ray Photoelectron Spectroscopy Measurements
The prevalence of non-reproducible QY measurements on non-annealed gold
samples motivated the need to determine the surface composition of the gold samples
used in QY measurements and find a process that would make the surface properties
more uniform for consistent QYs across samples.
Au, TiC, SiC
66
X-ray Photoelectron Spectroscopy (XPS) measurements, carried out at the
University of Florida in the Ultrahigh-Vacuum Lab at the department of materials science
and engineering were used to analyze and determine the chemical composition on the
surfaces of two gold samples. Both samples were 0.7 in × 0.6 in steel plates, first coated
with Ti (50 nm, 99.5% purity) for non-magnetism followed by Au (500 nm, 99.99% purity)
by Teer coatings. XPS uses the principle of photoelectric effect which involves the
excitation of electrons by photons. The setup includes a monochromatic X-ray source, a
sample chamber and an electron analyzer, all in ultra-high vacuum [33]. The first Au
sample was cleaned with isopropanol in the same way some regular Au samples were
cleaned before QY measurements. This sample was transferred to XPS vacuum within
one hour. The elements present on the surface of the gold sample are identified in Table
3.2 while Figure 3-20 shows the XPS measured spectrum of the sample.
Table 3-2: XPS measurements for bare Au sample cleaned with isopropanol
Peak Intensity (CPS) Atomic Ratio (%) Sensitivity factor
Au 4f 194340 0.671625695 5.24 O1s 1214 0.030920427 0.711 C1s 4862 0.297453878 0.296 S2p 0 0 0.57
The second Au sample was treated in 1-Dodecanethiol solution for 24 hours. The
molecules of the thiol solution adsorb and form protective, well-arranged, chemically
stable and self-assembled hydrocarbon monolayers on an inert substrate like Au [18].
67
Figure 3-20. The XPS measurements of bare Au cleaned with isopropanol
The layer assembled in this case was reproducible with a thickness of 1.5 - 2.5 nm. The
sample was rinsed with ethanol and dried with N2 before being transferred to the XPS
vacuum within the hour. Figure 3-21 shows the spectrum of the electrons in counts per
second (CPS) plotted against the binding energy of the electrons on the material surface.
Each XPS peak identifies the elements that are present on the surface of the gold
sample which are listed in Table 3-3.
Table 3-3. XPS measurements for Au sample treated with Thiol.
Peak Intensity (CPS) Atomic Ratio (%) Sensitivity factor
Au 4f 66040 0.357304464 5.24 O1s 0 0 0.711 C1s 6490 0.621606648 0.296 S2p 424 0.021088887 0.57
68
Figure 3-21. XPS measurement spectrum of Au after being treated in thiol solution for 24 hours
The results obtained from the XPS measurements are consistent in terms of the
composition of what would be expected for samples prepared in these ways. It was
further determined that the sample cleaned with isopropanol had at least, a 1 nm thick
film of an unknown hydrocarbon with oxygen functionalities that would be unavoidable in
a QY measurement where the sample preparation process involves cleaning with
isopropanol only.
Repeated QY measurements made on a third 2 in × 2 in Au sample prepared with
the self-assembled monolayer gave results between 2.75 × 10−4 and 2.9 × 10−4, a
factor of 1.05 in maximum variation. This is a significant improvement from the samples
69
in Figure 3-4, where repeated measurements on individual samples varied by factors of
1.2 to 10. These measurements are comparable since they were made at the same
pressure but are not conclusive because only one sample with the monolayer was tested
for QY.
3.8 Conclusions
The results obtained in this chapter are summarized here with conclusions and
recommendations useful for the LISA mission.
Baking out the Au test samples at 130 ˚C had the effect of reducing likely
contamination from probable water adsorption on the sample surface and resulted in
more uniform QYs. Before a bake-out, individual samples varied in QY by factors of 1.2
to 10 for repeated measurements at 10−6 Torr and UV wavelengths of 240 nm and 250
nm. After baking out, the QYs of these individual samples consistently varied only by
factors of 2 or less.
QY results at 0˚ incidence angles were higher than those at 45˚ by about a factor
of 1.2. This could be because the reflectivity of Au at a 0˚ incidence angle is ~1% less
than the reflectivity at 45˚ meaning that a little more light is absorbed at 0˚ that
contributes to the QY.
Commercially polished samples were generally cleaner with a more uniform
surface and did not result in initial increases in QY at 10−6 Torr when a bake-out is first
started on them as in the case of the hand polished samples (see Figures 3-6, 3-8, and
3-9). The hand polished ones likely experienced a quick evaporation of certain
impurities at 130 °C that probably caused an increase of the work function and led to
initially high quantum yields which settled over time.
70
QY measurements made at 240 nm are consistently higher than those made at
250 nm by an average factor of 2.3 ± 0.3 for the samples that were commercially
prepared as shown in Figures 3-11 to 3-14. The resulting Au work function, calculated
using Equation 3-18 was (4.56 ± 0.07) eV, which agrees with expected range of Au
work functions [8].
The 240 nm QY measurements in Figure 3-9 for the hand polished sample 1AP
were higher than those at 250 nm by a factor of ~ 5. A value of ~4.8 eV was estimated
as the work function when Equation 3-18 was used (this particular sample, initially
measured as shown in Figure 3-4, gave inconsistent values varying by factors of 1.2 to
10 when no bake-out was done beforehand in Figure 3-4).
QY measurements were made while pressures in the vacuum chamber were
purposefully varied between the 1 × 10−6 Torr and 3 × 10−8 Torr in order to investigate
if there was a dependence on pressure. No significant change in QY was observed
between 10−8 and 10−7 Torr for all four samples considered, but at 10−6 Torr, two of the
samples showed a reduction in QY of at least a factor of 3. Some sort of balancing effect
seems to be at play as these surfaces are likely to be contaminated with water and
carbon compounds, no effort was made to get rid of the latter except for the bake-out for
these measurements, but one thing that can be said is that water is present in higher
quantities on the surface at 10−6 Torr leading to increased work functions and a lower
QY than at 10−8 Torr. Hence only two out of four samples tested showed a noticeable
dependence on pressure.
Tested alternative coatings; SiC and TiC seem to have properties that would
make them suitable alternatives to Au. Reflectivity values obtained for these coatings [5]
71
indicate that they would be able to absorb sufficient photons and be adequate
alternatives for charge control but their average QYs are lower than that of the Au
samples prepared in the same way by more than a factor of ten with TiC exhibiting the
best stability. Au is the coating choice for LISA because LISA Pathfinder has
successfully demonstrated UV charge control and caging and release by using Au. Other
missions could consider these alternative coating materials in future.
XPS measurements performed on an Au sample cleaned with isopropanol
revealed the likely composition of the surface contaminants to be oxygen and
hydrocarbons. Hydrocarbons are a challenge to get rid of since they are not removed by
a bake-out and have to be subjected to Argon sputtering if there is an absolute need to
get rid of them [21]. Another Au sample surface was treated with a chemically stable self-
assembled monolayer from Dodecanethiol. The purpose of this monolayer was to create
a uniform surface on Au that could result in more repeatable QY measurements. Further
studies would be needed to determine whether this would be useful for LISA or not. The
introduced monolayer described in this chapter did result in QY values in the 10−4 range
and significantly reduced QY variation of a factor of 1.05 with repeated measurements.
Simple reflectivity analyses described in section 3.2 shows that the QYs of
adjacent surfaces used in charge control should be within a factor of three of each other.
This means, for example, that sample 4AP measured as described in Figures 3-11, 3-16
and 3-17 would not be a suitable adjacent match for 4BP, 4CP or 4DP since the QYs of
these samples exhibits more than a factor of three difference at LISA-like pressures of
10−8 Torr.
72
CHAPTER 4 DEMONSTRATION OF DC AND AC UV LED CHARGE CONTROL ON THE UF
TORSION PENDULUM
4.1 Pendulum Description
Investigating the operational TM charge control modes proposed for LISA
requires ground testing. The UF Torsion Pendulum is a laboratory test bed for testing
the operation and performance of GRS technology on ground. The UF torsion pendulum
design is based on the University of Trento pendulum. Key differences for the UF
torsion pendulum include longer inertial members and an incorporated laser
interferometric position readout in addition to the capacitive readout. The capacitive and
interferometric readouts measure the TM displacement with sensitivities of 30 nm/√Hz
and 0.5 nm/√Hz respectively at 1 Hz [29].
The pendulum consists of an inertial member made up of 4 TMs affixed to the
ends of a cross shaped design and suspended by a 1 m long, 50 μm diameter fiber as
shown in Figure 4-1(c). The distance between the center of each TM and the center of
the crossbar is 22.2 cm. The total mass of the suspension is ~ 477 g. The suspended
test masses are hollow, gold-coated 46 mm aluminum cubes, two of which are isolated
inside separate gold-coated aluminum housings. The use of hollow TMs ensures a
reduction in the mass of the pendulum and the required thickness of the suspension
fiber which in turn leads to reduced thermal noise. Accelerations and displacements due
to all the surface forces acting on the pendulum are increased because of the reduced
total mass of the structure which enables them to be better studied and understood [11].
73
Figure 4-1. (a) Pendulum inertial member surrounded by electrostatic shields, (b) Simplified electrode Housing, (c) CAD model of the torsion pendulum, (d) Torsion pendulum assembly enclosed inside the vacuum chamber
The pendulum electrode housing shown in 4-1(b) is a simplified version of the
LISA pathfinder GRS. There are 6 total electrodes with each one centered on the
interior faces of the housing. Ceramic bushings are used to electrically isolate the
electrodes from the structure of the housing. The electrodes are for sensing and
actuating the TMs in three translational degrees of freedom. In the sensitive direction in
which the pendulum rotates about its axis as shown in Figure 4-1(a), the gaps between
the electrodes and the test mass surface are 8 mm. This gap size was chosen in
contrast to the 4 mm gap size in the LPF GRS to minimize interactions between the
GRS and the TM. The 4 mm gap in LPF offers a balance between minimizing the
effects of possible noise due to potentials on the TM surface and the capacitive sensing
74
requirements of 2 nm/√Hz [30]. The 8 mm gap utilized in the torsion pendulum is
designed to minimize these effects even further.
The whole pendulum structure is housed within a 60 cm diameter vacuum
chamber that includes a hollow tube for suspending the TM assembly from the fiber.
The fiber is attached at the top of the tube to both a rotational manipulator to allow for
adjusting the orientation of the pendulum, and a translational XYZ manipulator, to allow
for calibration and centering.
Figure 4-2 shows three UV optical fiber ports incorporated on each electrode
housing to allow for AC and DC charge control on the TMs via UV-light. One port
illuminates the test mass, one illuminates the electrodes, and one illuminates both the
test mass and the electrodes simultaneously.
Figure 4-2. UV light injection geometry for the simplified GRS, Left: UV light illuminates either the TM or the electrodes, Right: UV light simultaneously illuminates both the TM and injection electrodes
The set of electrodes installed on the electrode housing (EH) together with the
TM make up the simplified pendulum GRS. The electrode housing completely
surrounds the TMs. An AC signal can be injected into two of the electrodes to polarize
75
the TM and the polarization is measured by another set of electrodes used for sensing.
The GRS also allows systematic actuation to be carried out on the TMs. If the system of
TMs is decoupled from the gravity of the Earth, it is possible to identify and quantify
many sources of noise in the sensor.
As shown in figure 4-3 (a), for small rotations, the differential displacements of
the two enclosed test masses, x1 and x2 with respect to their individual housings is
defined as 𝜑 through the following equation,
Figure 4-3. (a) Test mass rotation, (b) Test mass swing
(𝑥1− 𝑥2)
2 𝐿=
𝜑 𝐿−(−𝜑 𝐿)
2 𝐿=
2 𝜑 𝐿
2 𝐿= 𝜑 (4-1)
Where 2 L is the total length of the pendulum arm and the z coordinate is defined out of
the page.
The translation of the inertial member in the x direction as shown in Figure 4-3
(b) is defined as the “swing” motion of the pendulum which is calculated as the average
of the corresponding displacements, x1 and x2,
76
x1+ x2
2 (4-2)
4.1.1 Capacitive and Interferometric Readout
The simplified GRS measures the position of the TM by measuring the
capacitance of a set of surrounding electrodes towards the TM. The TMs are polarized
by a 100 kHz sine wave generated by a data acquisition board applied to injection
electrodes on opposite sides of the housing as shown in Figure 4-4. The differential
current measured through the two sensing electrodes is proportional to the
displacement of the TM from the center. The measured current is converted into a
voltage with a transimpedance amplifier which grounds the sensing electrodes with
respect to the TM to ensure that the voltage drop between the TM and electrodes is
only due to the injection voltage. The capacitive sensing and actuation scheme includes
digital to analog electronics (DAC) for generating injection and actuation signals and
analog to digital electronics (ADC) for acquiring readout signals. The digitized signal
from the readout is used to multiply the 100 KHz injection used to polarize the TM. This
digitized signal is also used to multiply the same 100 kHz signal shifted in phase by 90˚.
Cascaded Integrator Comb (CIC) filters are subsequently used to downsample and
integrate the multiplied signals. The In-phase (I) and Quadrature (Q) components of the
resulting vector R (θ), defined by the pendulum’s motion can be modified in such a way
that all the required information is contained in only one of the components of the vector
[12]. Another method of reading out the pendulum’s position is by the use of the
interferometer which measures the angular displacement of the two opposite
unenclosed TMs. The interferometer passes 45° polarized light through a beam splitter
77
(BS) and reflects the beams off them before recombining at a power beam splitter
(PBS) outside the chamber that sends the beam into a polarizing beam splitter and
separates them on two different photodiodes [14].
Figure 4-4. Capacitive and Interferometric read-out scheme, (a) Actuation and capacitive sensing scheme for a pair of electrodes enclosing a single TM, (b) Electrostatic actuation electronics, (c) Interferometric readout scheme for TM position, (d) Layout of interferometer components in the torsion pendulum
4.1.2 Measured Time Series of the Pendulum Rotation Angle
The pendulum angle obtained from the natural free oscillations of the pendulum
as a function of time is shown in Figure 4-5. Sensors record the displacements of the
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test masses with respect to the equilibrium point of the pendulum and the differential
motion associated with the torsional rotation of the fiber is scaled by the arm length and
expressed as an angle as in Equation 4-1. The natural rotational period of the pendulum
is about 48 minutes.
Figure 4-5. Pendulum angle as a function of time as measured both capacitively and interferometrically
A typical noise run set up overnight yields more than 50000 seconds of data.
Longer stretches are reserved for the weekends and typically yield up to 200000
seconds. The time series from both the capacitive and interferometric readout are
represented in Figure 4-5. A close up view of a section of the time series shows a
sudden change in the pendulum motion due to some fairly regular disturbance torques.
Investigating this disturbance did not yield any interdependence with changes in
temperature, magnetic fields or charge build up on the pendulum test masses but a
correlation was found with sudden increases in pressure as indicated on the pressure
79
gauge. As a result of the sudden change in pendulum motion showing up in the time
series, data analyzed was limited to the segments in between the disturbances.
4.1.3 Pendulum Acceleration Noise Performance
A PID routine first damps the natural oscillation of the pendulum to within tens of
microrads before it is left to oscillate freely without applying any actuation. An LTPDA
algorithm is used to solve the equation of motion of the pendulum and calculate the
corresponding external torque. The torque is divided by the length of the pendulum arm
and the mass of the LISA TMs. The spectrum is then taken to get the equivalent LISA
TM acceleration noise. Another method of finding the acceleration noise would be to
divide the spectrum of phi by the pendulum transfer function to get the torque and then
divide this by the length of the pendulum arm and the arm of the LISA TMs.
Figure 4-6. UF torsion pendulum acceleration noise spectrum
80
The acceleration noise performance for both the interferometric and capacitive
readout with and without the regular disturbances in Figure 4-5 are shown in Figure 4-6,
as well as the fiber thermal noise. Segments between the disturbances provide a
shorter stretch of data than that from a complete weekend run hence the average is
limited to a minimum frequency value of 0.3 mHz.
An acceleration noise plot with lower frequency data obtained from an entire
stretch of weekend data with disturbances included and higher frequency data obtained
from the data in between the disturbances is shown Figure 4-7.
Figure 4-7. LISA requirements compared with the acceleration noise performance of the torsion pendulum facility
The disturbances occur between 1 × 10−4 Hz and 3.5 × 10−4 Hz as indicated by
the gap between the red curve and the blue curve in Figure 4-7. The acceleration noise
estimated for the capacitive readout is about 2 × 10−12 ms−2/√Hz at 0.5 mHz while that
for the interferometric readout is around 4 × 10−13 ms−2/√Hz at 2 mHz. The
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interferometric result is still a factor of 100 above what is required for LISA and a factor
of 20 above the fiber thermal noise, but similar to other similar facilities, including the
one at the University of Trento.
4.2 Electrostatic Force acting on the Pendulum’s Test Masses and Charge
Measurement
Apart from deliberately polarizing the TMs with bias voltages on the housing
electrodes to induce a charge on them, TM contact with adjacent internal electrode
housing surfaces can impart a potential to the TMs and the ion gauge used to determine
the pressure of the system intermittently can leave the TMs charged as well.
The electrostatic force Fx acting on the TM along the x axis as defined in Figure
4-8, is represented by the formula for the force of attraction between the parallel plates
of a capacitor.
Figure 4-8. Left: TM position with respect to the electrodes, Right: Suspended pendulum
showing the force Fx in the direction of the sensitive axis
𝐹𝑥 = 1
2∑
∂C𝑒
∂𝑥𝑒 (𝑉𝑇𝑀 − 𝑉𝑒)2
(4-3)
Here, 𝐶𝑒 is the capacitance between the TM and the e-th housing electrode (e = 1, 2, 3
…), 𝑉𝑒 is e-th electrode voltage, and 𝑉𝑇𝑀 is the TM potential defined by the combination
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of the TM voltage due to charge given by 𝑞
𝐶𝑇 and the polarization induced on it due to the
surrounding voltages, ∑𝐶𝑒𝑉𝑒
𝐶𝑇 [11] ;
𝑉𝑇𝑀 = 𝑞
𝐶𝑇 +
1
𝐶𝑇∑ 𝐶𝑒𝑉𝑒𝑒 (4-4)
In Equation 4-4, q is the net charge on the TM and CT is the total capacitance of the TM
towards surrounding surfaces. Considering equal and opposite electrode voltages as
shown in Figure 4-8, 𝑉1 = 𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡) and 𝑉2 = − 𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡). Equation 4-4 becomes,
𝑉𝑇𝑀 = 𝑞
𝐶𝑇+
1
𝐶𝑇 (𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡)(𝐶1 − 𝐶2)) (4-5)
If 𝑋+ and 𝑋− indicate the movement of the TM in the positive and negative X direction
respectively with corresponding capacitances of 𝐶𝑋+ and 𝐶𝑋−
towards an opposite
surface with area A, then for a TM displaced from the center position by 𝑥 as described
by Giacomo Ciani (Diss. Univ. of Trento, 2008)
𝐶𝑋±=
𝜖0𝐴
𝑑∓𝑥 (4-6)
Then,
𝜕𝐶𝑋+
𝜕𝑥|
𝑥=0=
𝐶𝑋+
𝑑 ,
𝜕𝐶𝑋−
𝜕𝑥|
𝑥=0=
−𝐶𝑋−
𝑑 (4-7)
If 𝐶1 = 𝐶𝑋− and 𝐶2 = 𝐶𝑋+
and 𝐶1 = 𝐶2, then ∂C2
∂x=
C
d and
∂C1
∂x=
−C
d .
After substituting for VTM and 𝐶1 and 𝐶2 in Equation 4-3, the force on the pendulum in
the direction of the sensitive axis as derived in appendix C is given by;
𝐹𝑥 = 2𝐶𝑞𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡)
𝑑𝐶𝑇 (4-8)
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Here, A is the amplitude of sinusoidal voltages applied to the electrodes and f is the
frequency of the applied signal. The electrostatic force and the TM motion are related to
the equation of motion of the pendulum for small rotations as given by the second order
ODE,
�� = −𝜔02𝜑 +
𝐿
𝐼𝐹𝑥 (4-9)
Where, ω0 is the natural frequency of the pendulum at (1
48 min), L is the distance
between the point of suspension and the test mass (~0.22 mm), and I is the moment of
inertia(1.63 × 10−2kgm2). Since the pendulum is subjected to an external periodic force,
the particular solution to Equation 4-9 in Appendix C accounts for the forced motion of
the pendulum. It can be represented as,
𝜑𝑝(𝑡) =
2𝐶𝑞𝐴 𝐿
𝑑𝐶𝑇𝐼
𝜔02− (2𝜋𝑓)2 𝑐𝑜𝑠 (2𝜋𝑓𝑡) (4-10)
Where d is the distance between the TM and EH surfaces.
If the measured amplitude of the oscillations in Equation 4-10 is defined as B,
then the resulting net charge q, on the TM can be found as,
𝑞 = 𝐵 𝐶𝑇𝐼 (𝜔0
2− (2𝜋𝑓)2)
2𝐶𝑑𝐴𝐿 (4-11)
In the pendulum, the GRS measures the position of the TM by measuring the
capacitance of the set of electrodes towards the TM which depends on the spacing
between them. Applying a sinusoidal signal directly to the pair of electrodes in the GRS
as shown in Figure 4-8 makes the TM potential oscillate at the same frequency as that
of the driving signal with an amplitude proportional to the amount of charge on the
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pendulum. For a perfectly centered TM, the capacitances and the currents flowing in
both branches would be equal so that the TM is not attracted towards either of the two
opposing electrodes. A charged TM develops a net force proportional to the TM charge
q, towards the electrodes. A plot of pendulum angle as a function of time is provided in
Figure 4-9.
Figure 4-9. Pendulum angle as a function of time during charge measurement on the pendulum
4.3 Analytical Charge Control Model for the LISA GRS
A variation in the amount of charge on the TM corresponds to a change in its
potential (𝑉𝑇𝑀). As VTM changes, the electron flow rate also changes. It is desirable to
have 𝑉𝑇𝑀 ≈ 0 V to minimize the force noise on the TM, so there is a need to find the
electron flow rate at this value. After considering the surfaces involved in electron
exchange and specifying parameters related to the TM-GRS area illuminated by the UV
85
LED, the analytical model presented here explains the behavior of the TM voltage due
to charge for three cases,
1. It calculates the total electron flow rate ��𝑡𝑜𝑡 when the TM voltage VTM is close to
zero while varying the injection voltage 𝑉𝑖𝑛𝑗 on the electrodes; ��𝑡𝑜𝑡(𝑉𝑖𝑛𝑗 , 𝑉𝑇𝑀 = 0).
2. It determines the electron flow rate at saturation. Here, the voltage on the test
mass is allowed to go to equilibrium while the injection voltage is changing;
𝑉𝑇𝑀(𝑉𝑖𝑛𝑗 , 𝑡 = ∞)
3. It calculates the total electron flow rate when the injection voltage is at a constant
value and the voltage on the TM is changing; ��𝑡𝑜𝑡 (𝑉𝑖𝑛𝑗 = Constant, 𝑉𝑇𝑀).
Sections 4.3.1 and 4.3.2 describe how the model was developed
4.3.1 The Pendulum GRS as a Simplified Parallel Plate Circuit
The electrode housing, the electrodes and the test mass configuration in the
torsion pendulum’s simplified GRS system can be modelled as a simple circuit made up
of parallel plate capacitors. The capacitor plates represent the different surfaces where
the UV light used in charge control is incident. The TM interfaces with both the EH and
injection electrodes surfaces as shown in Figure 4-10. The different surfaces are
labelled i to l:
i. Active injection electrodes surface that faces the TM
j. TM surface that faces the active injection electrodes
k. EH surface that faces the TM
l. TM surface that faces the EH
86
Figure 4-10. Simplified circuit model for charge control
The EH surface which is chosen as the zero reference voltage includes the
grounded non-electrode surfaces. The orientation of the UV light injection ports is
designed to reduce light leakages towards non-zero voltage surfaces that could add
undesirable electron flows. This means that most of the radiation is absorbed after a
certain number of reflections and that transmission effects and light leakages through
non-gold surfaces are neglected. The relatively small gaps between the TM and the EH
of around 8 mm prevent electron absorption by gas ionization in the high vacuum
environment and border effects are minimized because of the design of the electrodes.
Other assumptions are that no electron flow is lost, no border effects at the TM
edges exist and there is no electron cross talk between EH and injection electrodes
since the electrode layout and sizing are optimized to maintain the overall extension
inside a safe-frame smaller than the TM face. In this way, each electrode capacitance
toward the TM does not change appreciably when the TM moves parallel to the
electrode itself. The electrodes are separated from the structure of the housing by a 1
mm groove around each electrode and isolated by ceramic spacers.
87
Surfaces i and j have a controlled electric field from the injection bias on i so that
in addition to the voltage due to charge 𝑞
𝐶𝑇 induced on the TM, an additional offset of
1
𝐶𝑇∑ 𝐶𝑖𝑉𝑖𝑖 due to polarization from the injection surface is included. No polarization
component from the k surface is added since it is grounded and voltages there are zero.
The capacitance of the injection electrodes towards the TM is 𝐶𝑖, 𝐶𝑇 is the total
capacitance and 𝑉𝑖 ≡ 𝑉𝑖𝑛𝑗. This is depicted in Figure 4-11.
Figure 4-11. Electron exchange in a parallel plate configuration
Depending on the value of the injection voltage, the electrons from a surface may
or may not have enough energy to overcome the potential barrier and be successfully
transferred to the other side. If they do have sufficient energy, then the electron flow in
that direction becomes dominant while the other direction is completely suppressed.
The resulting total charge rate on each of the different surfaces therefore includes two
extremes with a transition point in the region where the applied potential barrier is close
to zero.
88
4.3.2 Photoelectron Energy Distribution
The successful extraction and transfer of an electron from one surface to another
depends on the kinetic energy perpendicular to the plates overcoming the potential
barrier ∆V as shown in Figure 4-12.
Figure 4-12. Distribution of energies of emitted electrons in a parallel plate configuration
The energy distributions for electrons emanating from different experimental
geometries have been investigated by Hechenblaikner et al [17] and the distribution of
the normal energies as a function of the applied bias voltage for a parallel plate
configuration was found to be of the form,
𝑓 (∆𝑉) 𝑑𝛥𝑉 = [𝐴𝑛 𝑙𝑜𝑔 (𝑒𝑥𝑝(−∆𝑉+ 𝑉𝑚 )𝑒
𝐾𝑏𝑇 + 1) ] 𝑑𝛥𝑉 (4-12)
Here, An is a constant, e is the charge of the electron, 𝐾𝑏 is Boltzmann’s
constant, ∆𝑉 is the potential difference of the two adjacent surfaces and 𝑉𝑚 = ℎ𝜈−𝜙
𝑒 is
the maximum energy an electron can have after being extracted by a photon of energy
hν from a surface that has a work function of ϕ. Equation 4-12 can be rewritten as,
𝑓 (∆𝑉) ∝ 𝑙𝑜𝑔 (𝑒𝑥𝑝(−∆𝑉+ 𝑉𝑚 )𝑒
𝐾𝑏𝑇 + 1). (4-13)
89
As shown in Figure 4-13, the sign of the potential barrier before V = 0 favors the
movement of electrons in one direction because the electric field is directed in the sense
of that flow, the change in the sign of the potential causes the field to work against the
direction of travel of the electrons and they begin to lose energy until most of them are
unable to overcome the potential barrier. This is referred to as the stopping voltage Vm.
With the exception of the region around the stopping voltage, the function in Equation 4-
13, shown in blue, is well approximated by two straight lines defined in ∆𝑉 ≪ 𝑉𝑚 with a
negative slope and ∆𝑉 ≫ 𝑉𝑚 which is a constant.
The approximation for electrons moving in direction i to j or k to l becomes,
𝑓 (∆𝑉) ∝ −𝑒
𝐾𝑏𝑇(∆𝑉 − 𝑉𝑚) (4-14)
While those electrons moving in direction j to i or l to k are represented as;
𝑓 (∆𝑉) ∝ −𝑒
𝐾𝑏𝑇(−∆𝑉 − 𝑉𝑚). (4-15)
A full derivation of Equation 4-14 and Equation 4-15 can be found in Appendix A.
Figure 4-13. Straight line approximation of the distribution of energies
90
Not all the light sent through the UV light ports is absorbed by the intended
areas. Multiple reflections distribute the total light intensity among the different surfaces.
The TM surfaces are treated as generic surfaces and for a given set of parameters, it
will be straightforward to adapt photoelectron flow rate equations. If positive
contributions are assigned to electrons moving from i to j and negative contributions are
assigned to electrons moving from j to i, then for all the reference surfaces described in
the circuit model in Figure 4-10, the generic expression for the total electron flow rate is,
��𝑡𝑜𝑡 = ��𝑖𝑗 − ��𝑗𝑖 + ��𝑘𝑙 − ��𝑙𝑘 (4-16)
The term ��𝑡𝑜𝑡 is the sum of electron flow rate contributions extended to all
interfaces. ��𝑖𝑗 is the electron flow rate from the i-th surface to j-th surface expressed
in e−
s and so on. The single electron flow rates must be expressed in terms of accessible
quantities. For instance, the general form of nij is related the electrons extracted from
the i-th surface moving towards the j-th surface and can be written as,
��𝑖𝑗 = 𝛼𝑖 𝑎𝑖 𝑄𝑌𝑖(𝜈) 𝐼𝑖 ∙
∫ 𝑔𝑖(𝜈) 𝑑𝜈+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖ℎ
∫ 𝑓𝑖(𝜙𝑖,∆𝑉𝑖𝑗,𝜈,𝑇)
ℎ𝜈−𝜙𝑖𝑒
∆𝑉𝑖𝑗𝑑∆𝑉𝑖𝑗
∫ 𝑔𝑖(𝜈) 𝑑𝜈+∞
𝜙𝑖ℎ
∫ 𝑓𝑖(𝜙𝑖,∆𝑉𝑖𝑗,𝜈,𝑇) 𝑑∆𝑉𝑖𝑗
ℎ𝜈−𝜙𝑖𝑒
0
(4-17)
Where αi the fraction of the total light intensity is absorbed by the i-th surface and
𝑎𝑖 is the i-th surface area in m2. 𝑄𝑌𝑖(𝜈) is the quantum yield of the i-th surface in e−
γ. This
quantity is assumed to be independent on the voltage applied to the surface but
depends on the surface composition and optical properties. The quantity, 𝐼𝑖 is the total
number of photons per square meter per second (γ
s∙m2) sent to a specific surface with
energy higher than the work function of that surface. The energy distribution of the
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photons from the UV LED is given by 𝑔(𝜈), the measurement unit is seconds (s). The
radiation frequency ν is in s−1, h is Plancks constant in J ∙ s. The effective work function
of the i-th surface 𝜙𝑖 is in Joules. The energy distribution of the electrons extracted from
the i-th surface at the ij interface as a function of the voltage difference is represented
as 𝑓𝑖. Finally, e is the charge of an electron in Coulombs.
With respect to Equation 4-17, the potential barrier ∆Vij is a function of the
injection electrodes voltage and the TM voltage which can be represented as;
∆𝑉𝑖𝑗 = 𝑉𝑖 − 𝑉𝑗 (4-18)
Where the voltage difference between the two surfaces that the electrons have to
overcome to be successfully transferred to the other side is ∆V and Vi is the potential of
the i-th surface (in Volts),
The resolution of the integrals in Equation 4-17 gives a time differential equation
in 𝑉𝑖 which makes it possible to solve the equation for the voltage due to charge on that
surface. The denominator in Equation 4-17 can be defined as the integral, 𝐷𝑓,𝑖 which
gives the total number of photoelectrons produced with or without enough energy to
overcome the potential barrier. It can be written as,
𝐷𝑓,𝑖 = ∫ 𝑔𝑖(𝜈) 𝑑𝜈+∞
𝜙𝑖ℎ
∫ 𝑓𝑖(𝜙𝑖, ∆𝑉𝑖𝑗, 𝜈, 𝑇) 𝑑∆𝑉𝑖𝑗
ℎ𝜈−𝜙𝑖𝑒
0 (4-19)
If the function 𝑔(𝜈) is known from the photon source characteristics, then for
surface i, the rate of absorption of photons per unit area is calculated according to the
following,
𝐼𝑖 = 𝐼0,𝑖 ∫ 𝑔𝑖(𝜈)𝑑𝜈∞
𝜙𝑖ℎ
(4-20)
92
Where 𝐼0 (in photons per square meter per second) is the total integral area of the
original spectrum
A photon must be absorbed for an electron to be successfully extracted. The
energy distribution of the photons out of the UV LED, g(ν) represents the UV LED
emission spectrum and is well approximated by the Gaussian fit shown in Figure 2-1.
The minimum normal energy the ejected electrons have to overcome to cross the gap
depends on the potential of the originating surface and its work function given by,
|𝑒∆𝑉+𝜙
ℎ|. The combination of the normal energy distribution of extracted electrons
f(𝜙, 𝛥𝑉, 𝜈, 𝑇) and the energy distribution of the photons 𝑔(𝜈) in the numerator, is the
function that counts the number of UV photons that produce electrons with enough
normal energy to overcome the potential barrier. Each illuminated surface considered
receives a different amount of light α with the same frequency distribution. If the system
is closed and no light is transmitted out of it then the injected UV light is completely
absorbed in different portions by the surfaces after consecutive reflections. The ratio of
the double integrals gives the proportion of electrons out of all the extracted
photoelectrons that obtain enough normal energies to overcome the potential barrier
along their path towards or away from the TM surface. The number of electrons
extracted is proportional to the absorbed fraction of light α, the illuminated surface area
A, the UV light intensity 𝑔(𝜈), and the Quantum Yield QY, which takes all the other
effects into account. The quantum yield is a constant with respect to the photon
frequency but it must be integrated along with the other expressions in the double
integral if an analytical expression is found for it.
93
If Equation 4-14 is substituted into Equation 4-17, the equation for the electron
flow rate nij can be written as;
��𝑖𝑗 = 𝛼𝑖 ∙ 𝑎𝑖 ∙ 𝑄𝑌𝑖(𝜈) ∙ 𝐼𝑖
∫ 𝑔(𝜈) 𝑑𝜈+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖ℎ
∫− 𝑒
𝐾𝑏𝑇(∆𝑉𝑖𝑗−𝑉𝑚)𝑑∆𝑉𝑖𝑗
ℎ𝜈−𝜙𝑖𝑒
∆𝑉𝑖𝑗
∫ 𝑔(𝜈) 𝑑𝜈+∞
𝜙𝑖ℎ
∫− 𝑒
𝐾𝑏𝑇(∆𝑉𝑖𝑗−𝑉𝑚)𝑑∆𝑉𝑖𝑗
ℎ𝜈−𝜙𝑖𝑒
0
(4-21)
The photon distribution g(ν) follows a normal distribution with,
𝑔(𝜈) = 1
𝜎√2𝜋𝑒
− (𝜈− 𝜈)2
2𝜎2 (4-22)
Here, ν and 𝜎 are the mean and standard deviation of the distribution respectively. After
evaluating the double integral in Equation 4-21 as shown in Appendix B, the following
substitutions can be made for the final expression in the numerator (for surface i or j
when defined accordingly),
𝐴𝑖 = ∫ 𝑔(𝜈) +∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
𝑑𝜈 (4-23)
𝐵𝑖 = ∫ 𝜈 𝑔(𝜈)+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
𝑑𝜈 (4-24)
𝐶𝑖 = ∫ 𝜈2𝑔(𝜈)+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
𝑑𝜈 (4-25)
For the denominator:
𝐴𝑖𝑑 = ∫ 𝑔(𝜈) +∞
𝜙𝑖ℎ
𝑑𝜈 (4-26)
𝐵𝑖𝑑 = ∫ 𝜈 𝑔(𝜈)+∞
𝜙𝑖ℎ
𝑑𝜈 (4-27)
𝐶𝑖𝑑 = ∫ 𝜈2𝑔(𝜈)+∞
𝜙𝑖ℎ
𝑑𝜈 (4-28)
The terms, A, B, and C will use −𝑒∆𝑉+𝜙
ℎ in the integral limits for electrons going in the
reverse direction. Equation 4-21 can then be written as:
94
��𝑖𝑗 = 𝛼𝑖 ∙ 𝑎𝑖 ∙ 𝑄𝑌𝑖(𝜈) ∙ 𝐼𝑖.𝑒2∆𝑉𝑖𝑗
2𝐴𝑖+(2𝑒𝜙𝑖𝐴𝑖−2𝑒ℎ𝐵𝑖)∆𝑉𝑖𝑗+ℎ2𝐶𝑖−2ℎ𝜙𝑖𝐵𝑖+𝜙𝑖2𝐴𝑖
ℎ2𝐶𝑖𝑑− 2ℎ𝜙𝑖𝐵𝑖𝑑+ 𝜙𝑖2𝐴𝑖𝑑
(4-29)
If Equation 4-15 is substituted into the expression for ��𝑗𝑖 derived in the appendix, the
expression for ��𝑗𝑖 becomes:
��𝑗𝑖 = 𝛼𝑗 ∙ 𝑎𝑗 ∙ 𝑄𝑌𝑗(𝜈) ∙ 𝐼𝑗 ∙
∫ 𝑔𝑗(𝜈) 𝑑𝜈+∞
−𝑒∆𝑉𝑗𝑖+𝜙𝑗ℎ
∫ − 𝑒
𝐾𝑏𝑇(−∆𝑉𝑗𝑖− 𝑉𝑚)
∆𝑉𝑗𝑖
−(ℎ𝜈−𝜙𝑗)
𝑒
𝑑∆𝑉𝑗𝑖
∫ 𝑔𝑗(𝜈) 𝑑𝜈+∞
𝜑𝑗ℎ
∫ − 𝑒
𝐾𝑏𝑇(−∆𝑉𝑗𝑖− 𝑉𝑚)
ℎ𝜈−𝜙𝑗𝑒
0𝑑∆𝑉𝑗𝑖
(4-30)
After substituting the modified versions of Equation 4-23 through Equation 4-26 into
Equation 4-30, nji can finally be written as;
��𝑗𝑖 = 𝛼𝑗 ∙ 𝑎𝑗 ∙ 𝑄𝑌𝑗(𝜈) ∙ 𝐼𝑗 𝑒2∆𝑉𝑗𝑖
2𝐴𝑗 −(2𝑒𝜙𝑗 𝐴𝑗 −2𝑒ℎ𝐵𝑗 )∆𝑉𝑗𝑖+ℎ2𝐶𝑗−2ℎ𝜙𝑗𝐵𝑗+𝜙𝑗2 𝐴𝑗
ℎ2𝐶𝑗𝑑− 2ℎ𝜙𝑗𝐵𝑗𝑑 + 𝜙𝑗2𝐴𝑗𝑑
(4-31)
Similar equations can be developed for nkl and nlk by following the same process.
4.3.3 Implementing The Charge Control Model in MATLAB
Experimental data from charge control measurements on the pendulum was
imported into a MATLAB script for comparing with the calculated total electron flow rate
developed from the model in Equation 4-16. This script calls individual functions that
calculate electron flow rates on the individual surfaces described in section 4.3.1. A
likelihood function for determining if the model could explain the actual data was used to
fit parameters such as the area of the GRS / TM system illuminated by the UV LED and
the work function after providing an initial best guess close to the actual expected value
for the required fitting parameters. The likelihood function is finally maximized to
improve the chances of finding fitting parameters that make the observed experimental
results more probable.
95
4.4 DC Charge Control Demonstration on the UF Torsion Pendulum
For the TM charge measurements and control, a LabVIEW Virtual Instrument
(VI) was created with an interface that allowed an amplitude and frequency to be
specified for the equal and opposite sinusoidal voltages applied to opposing x-injection
electrodes. These voltages polarize the TMs with a force described by Equation 4-8. If
the TM is centered and discharged within the GRS, attractive forces from the TM
towards the two opposite electrodes are equal and they cancel out but if this is not the
case, the result is a force that causes a displacement of the TM. The amplitude of this
displacement is proportional to TM charge. Demodulating the TM displacement allows a
measurement of the TM charge to be obtained. Another VI allows the position of the TM
with respect to the center or equilibrium position to be read out.
In the case of charge control at DC, If UV light is used to illuminate the TM,
electrons are transferred from the TM to the electrode housing and the TM voltage due
to charge becomes more positive. The reverse is the case if the electrode housing is
illuminated. Preliminary DC charge control experiments carried out on the pendulum’s
TM prove that UV LED charge control can be done on a LISA test bed as illustrated by
actual measurements presented in Figure 4-14. The initial voltage due to charge was
0.8 V. The electrodes were then illuminated with the UV LED driven at 0.8 mA
equivalent to less than 1 µW of UV power to enhance the flow of photoelectrons from
electrodes to TM. The voltage on the TM was − 2.4 V over 30 minutes and the process
is reversed by illuminating the test mass with 0.8 mA until equilibrium is reached at 0.8
V. The whole procedure can be repeated until the desired voltage due to charge within
the bounds of saturation is reached. Spikes in the data are usually due to someone
96
physically approaching the pendulum or making adjustments to the current driver while
regulating the intensity of the UV LED.
Figure 4-14. Pendulum charge during DC charge control operations on the UF torsion pendulum
4.5 AC Charge Control With The UF Torsion Pendulum
In the case of AC charge control, only one UV light injector would be needed to
illuminate either the TM, the injection electrode or both. The UV LEDs are switched on
only when the electric field in the illuminated section supports the preferred direction of
discharge and prevents the movement of photoelectrons in the other direction [5].
Figure 4-15. Left: UV LED emission out of phase with respect to the injection signal Right: UV LED emission in phase with respect to the injection signal
97
For negative charge transfer, when the 100 kHz signal on the injection electrodes
is in the negative half of the period as shown on the left in Figure 4-15, the UV light is
switched on (out of phase) and the electrons generated within that interval experience a
negative offset that repel the electrons towards the TM so that it becomes negatively
charged. In the case of positive charge transfer, the output of the UV light is shifted in
phase resulting in the electrons being repelled away from the TM. The duty cycle is
mostly kept at a minimum of 5% so that the voltages at a particular set phase with
respect to the injection cycle can be considered as roughly constant as the phase is
varied from 0˚ to -180˚ (right to left).
In the AC charge control demonstration on the torsion pendulum provided in
Figure 4-16, the amplitude of the UV LED voltage with respect to the injection was set in
phase at 0° while being driven at 10 mA (~180 nW), and a 5% duty cycle at the
beginning of the experiment using the UV port illuminating both the TM and the
electrodes. This caused the charge on the TM to become more positive. Using the
same parameters but out of phase at −180° made the voltage due to charge go in the
opposite direction.
98
Figure 4-16. Test mass potential during AC charge control experiment
There is presently a limit on the ability to push the voltage due to charge both
positive and negative while using the charge control port that illuminates both the TM
and electrodes for the following likely reasons; the injection voltage on the pendulum
which was previously set at an amplitude of ± 10 V is now currently set at an amplitude
of ± 3.5 V. This is because the maximum value of the injection voltage sets the limit on
the voltage due to charge acquired by the test mass. This has resulted in equilibrium
voltages as high as ± 4.5 V that cause the pendulum to go unstable due to positive or
negative charge on the TM relative to the housing. Another probable reason is the area
illuminated by UV light being too shallow such that directed UV photons are not
satisfactorily absorbed by intended surfaces. Variations in quantum yield on the
opposing illuminated surfaces also likely contribute to this effect since the present TM
surfaces and opposing electrodes and housing surfaces were not polished to the same
99
specification and probably have different surface properties. These issues are being
addressed in a new, more flight-like GRS to be installed around one of the TMs. It
includes provision for the UV light to be injected at a steeper angle. The surfaces of
both the TMs and the inside of the new GRS are also polished to the same
specification. One of the two current gravitational reference sensors will be left in place
so that its performance can be compared with the new one.
4.5.1 AC Charge Control Experiments With Varying UV Phase
If the duty cycle of the UV LED output with respect to the injection voltage is small
enough, the injection voltage can be considered as equivalent to the set phase of the
UV LED through the following relationship,
𝑉𝑖𝑛𝑗 = 𝐴 ∗ 𝑐𝑜𝑠(𝑃ℎ𝑎𝑠𝑒) (4-32)
Where A is the amplitude of the injection voltage set to a value of 3.5 V
If the output of the UV LED is varied in phase with respect to the injection voltage
while keeping the duty cycle and amplitude constant, the voltage due to charge on the
TM becomes more positive until it reaches equilibrium. On the other hand, the TM
voltage becomes more negative if the output of the UV LED is out of phase with respect
to the injection. Interestingly, the equilibrium voltage due to charge settles at different
values for each phase adjustment. Figure 4-17 shows that the TM saturation voltages
when the UV LED is driven at 15 mA (~ 300 nW) and a 5% duty cycle, go to different
values for 0° phase (equivalent to an injection voltage of 3.5 V), − 22° phase (equivalent
to an injection voltage of 3.24 V) and − 58° phase (equivalent to an injection voltage of
1.85 V) while using the TM-EH UV port for AC charge control measurements. The final
saturation voltage therefore depends on the initial potential barrier (∆V).
100
Figure 4-17. TM Potential while illuminating the AC port with pulsed UV light while varying the phase with respect to the injection voltage.
4.5.2 Measuring Saturation Voltage as a Function of UV Light Injection
In the effort to continue investigating why the TM-Injection electrodes UV port did
not allow for charge to be moved both positive and negative, AC charge control
experiments were carried out by also using the UV ports meant to shine light individually
at either the electrodes or the TM. For measurements made by using the TM port, TM
potential remained in the positive region but stayed in the negative region when the
injection electrodes port was used. In either case, AC charge control is possible
regardless of which of the surfaces is illuminated. This means that reflections between
opposing surfaces illuminated by the UV LED turn out to be an advantage, because the
intended Au surface reflects a percentage of the incident UV light towards the opposing
surface and depending on the work function of that surface, electrons are emitted. This
works out in the way that the TM–Injection electrodes port was originally intended to
101
work because the whole idea was to take advantage of the available 100 kHz injection
field to move the charge from one surface to the other while two surfaces are
simultaneously illuminated.
Test mass saturation voltages obtained were subsequently plotted as a function
of phase for the three existing UV ports as shown in Figures 4-18, 4-19 and 4-20. These
plots were obtained by doing fits to voltage due to charge time series data such as is
shown in Figure 4-17 to determine their eventual saturation values while the phase was
varied from 0˚ to −180˚ ( + 3.5 V to − 3.5 V injection voltage) in 18˚ adjustments. In
plotting Figures 4-18, 4-19 and 4-20, most of the data with saturation voltages that
converged close to 0 V did not require a fit since the UV light could be used to illuminate
the surface long enough for the saturation voltage to be easily reached.
Figure 4-18. Saturation voltage Vs. UV light phase using the electrodes port
102
Figure 4-19. Saturation voltage Vs. UV light phase using the TM port
Figure 4-20. Saturation voltage Vs. Phase using the TM-Injection electrodes port
103
It is interesting to note that the voltage due to charge on the TM can be kept close
to zero by just setting the phase of the UV light at a certain value depending on which
UV port is being utilized. Setting the phase of the UV light from 0° to −90° (which is
equivalent to a potential barrier of 3.5 V to 0 V through Equation 4-32) for both the TM-
Injection electrodes port and the electrodes port ensured that the TM voltages were
close to zero all the time. Setting the phase from − 90° to −180° (0 V to − 3.5 V
injection voltage) for the TM port gave similar results.
4.5.3 Charge Rate
The charge rate can be calculated from the voltage time series plots obtained
during charge measurements. This is especially important for the region where the
voltage on the TM crosses zero because LISA TMs will be kept close to 0 V all the time.
Since the ability to move the TM potential to both positive and negative values on the
pendulum by using a single UV injection port is currently limited (as discussed in section
4.5) by the possibility of differences in QY between the TM and the electrode housing
surfaces and the inability of the directed UV light to be absorbed by the intended
surfaces, the only way to handle the process was by going through the following steps
as demonstrated in Figure 4-21,
1. Start the measurements when the TM is at a negative voltage and then use the
TM port to illuminate the TM to enable the voltage eventually settle at the
saturation value on the positive side for the set phase. The rate of change of the
TM voltage when the voltage crosses zero is then the charge rate for the TM port
at that phase.
104
2. When using the electrodes port, the measurements were started with the voltage
on the TM positive. The electrodes were then illuminated at a particular phase
and the TM voltage allowed to settle at the saturation voltage on the negative
side. The rate of change of the TM voltage when it crosses zero is the charge
rate for the electrode housing port at that set phase.
Figure 4-21. TM voltage as a function of time when using the TM port and EH port to move the voltage due to charge on the TM positive and negative
A combination of the above methods can be used to get the TM voltage due to
charge to move positive or negative by using the TM and electrodes housing port.
Quadratic and exponential fits were done to each of the resulting curves to determine
the charge rate at VTM = 0. The resulting charge rates for the three UV ports with the
105
injection voltage varying from 0˚ through −180˚ are fitted to the analytical model as
discussed in the following sections.
4.6 Fitting Analytical Charge Control Model to Data
The analytical charge control model described in section 4.3 was fit to charge
measurement data obtained from AC charge control experiments using the pendulum’s
UV injection ports as described in section 4.5. Cases one to three below show the
model compared with measurements of voltage due to charge on the test masses.
Phase values from 0˚ to −180˚ were converted to Vinj through Equation 4-32.
4.6.1 Case 1: ��𝒕𝒐𝒕 (𝑽𝒊𝒏𝒋 , 𝑽𝑻𝑴 = 𝟎)
Figure 4-22. UV light injection geometry and circuit model for calculating ��𝑡𝑜𝑡 with
varying 𝑉𝑖𝑛𝑗 and 𝑉𝑇𝑀 = 0
The charge flow rate (V/s) when the voltage due to charge on the TM is zero is
calculated at different values of Vinj when the UV light is used on the electrodes housing
port as shown Figure 4-22. The term Vinj is equivalent to the potential barrier between
the TM and the injection surface when the voltage on the TM is only due to the induced
106
polarization offset. A plot of the charge rate of the pendulum test mass as a function of
the applied potential barrier is presented in Figure 4-23. This plot also includes the
analytical model fitted to the data. The fit was done as described in section 4.3.3.
Figure 4-23. Measured test mass charge rate versus the potential barrier and the best fit model for the electrodes port
Figure 4-24 shows the electron flow rates on the individual surfaces in the system
as described by the analytical model when fitted to the experimental data from the
pendulum. In the negative half of the cycle where the potential barrier is between
− 3.5 V and 0 V, all the electrons from the electrodes surface (i) make it to the TM (j)
because the electrical field is directed in the sense of the flow and encourages the
movement of the electrons. The change in the sign of the potential begins to work
against the direction of electron motion when ∆𝑉 > 0 V. The electrons begin to have
insufficient energy and slowdown from 0 V until they are completely suppressed and
107
none of them make it to the TM by the time the potential is 2 V. The reverse is the case
for the movement of electrons from the TM (j) to the electrodes (i) in the negative half of
the cycle. The electrons here are mostly suppressed between − 3.5 V and −2 V. Some
of them begin to have enough energy to make it to the electrodes surface after ∆𝑉 =
2 V. As soon as the potential barrier becomes 0 V or greater, the sign of the potential
barrier helps all the photoelectrons make it to the electrodes surface in the positive half
of the cycle between 0 V and 3.5 V.
Figure 4-24. Measured test mass charge flow rate as a function potential barrier for each surface when UV light is directed towards the electrodes
There is no applied electric field in the region between the EH surface (k) and the
adjacent TM surface (l), which represents the small area of the grounded EH that faces
the TM which is illuminated along with the injection electrodes. Hence, in both the
negative and positive half of the potential barrier as shown in Figure 4-24, electrons
emitted from the EH remain mostly suppressed to a negligible value. Those produced
by the TM are successfully transferred in the negative half. The electric field working
-4 -3 -2 -1 0 1 2 3 4-6
-5
-4
-3
-2
-1
0
1
2
3
4x 10
-3
deltaV (V)
dV
/dt
ij, INJ-TM
ji, TM-INJ
kl, EH-TM
lk, TM-EH
Total electron flow rate
108
against the direction of flow in the positive half causes the electrons to gradually have
insufficient energy. The term, 1
𝐶𝑇∑ 𝐶𝑖𝑉𝑖𝑖 dominates in this case and is responsible for
the more gradual slope and offset.
The sum total of the electron flow rate which was compared to experimental data
in Figure 4-23 makes sense from an AC charge control standpoint because as expected
in the negative half of the cycle corresponding to when the UV LED is out of phase with
respect to the injection, the movement of electrons from the injection electrodes surface
to the TM surface is dominant while those moving from the TM to the electrodes are
mostly suppressed. The opposite is the case in the positive half of the cycle.
Figure 4-25 shows the charge rate (V/s) as a function of the applied potential
when the analytical model is fitted to AC charge control data from the TM injection port.
The contributions of the electron flow rates from the individual surfaces in the system as
described by the analytical model is shown in Figure 4-26. In the negative half of the
cycle, the electric field works against the direction of the flow of electrons moving from
TM (j) to electrodes (i) but when the sign of the potential changes, the field favors the
direction of travel of the electrons such that all electrons emitted from the TM (j) make it
to the electrodes (i). The reverse is true for the electrons emitted from the electrodes (i)
which have an advantage due to the electric field in the first half of the cycle when
∆ 𝑉 < 0 but gradually lose energy in the second half after the sign of the potential
changes. Electrons moving from the EH (k) to the TM (l) are suppressed in the negative
half of the cycle but all the electrons from the TM (l) make it to the EH (k). When the
sign of the potential changes, all electrons from k make it to l while those moving from l
to k are gradually suppressed.
109
Figure 4-25. Measured charge rate vs. phase using the TM port and best fit model to the data
Figure 4-26. Measured charge flow rate for each surface from TM port and best fit model
-4 -3 -2 -1 0 1 2 3
0
2
4
6
8
10
12
14
x 10-3
deltaV (V)
dV
/dt
ij, INJ-TM
ji, TM-INJ
kl, EH-TM
lk, TM-EH
Total electron flow rate
110
The total electron flow rate which was compared to experimental data in Figure
4-25 makes sense in this case as well because, as expected, in the negative half of the
cycle corresponding to when the UV LED is out of phase with respect to the injection,
the movement of electrons from the injection to the TM surface is dominant while those
moving from the TM to the injection electrodes are mostly suppressed. The opposite is
the case in the positive half of the cycle.
4.6.2 Case 2: 𝑽𝑻𝑴 (𝑽𝒊𝒏𝒋 , 𝒕 = ∞)
Here, the model is used to describe the case where the voltage on the TM is
allowed to go to saturation for changing values of the potential barrier while the
electrodes housing is illuminated. In the first half of the cycle where the potential barrier
is between −3.5 V and 0 V in Figure 4-27, the electric field favors the movement of
electrons towards the TM until the equilibrium voltage for the corresponding value of Vinj
is reached (or until total charge flow rate equals zero at the set Vinj). The voltage on the
TM is maintained close to 0 V after the sign of the potential barrier changes. Therefore,
the total charge flow rate is close to zero for all Vinj in the positive half of the cycle.
When the TM is illuminated, the voltage due to charge on the TM remains close
to 0 V in the negative half of the cycle because the electric field favors the movement of
electrons away from the TM. Therefore, the charge flow rate at equilibrium is similar for
all values of ∆ V in the negative half of the cycle, as shown in Figure 4-28. When the
sign of the potential changes, the TM begins to accumulate excess electrons and the
charge flow rate at equilibrium is not similar for all values of Vinj in the positive half of the
cycle.
111
Figure 4-27. TM Voltage at equilibrium for different values of the potential barrier on the electrodes port
Figure 4-28. TM Voltage at equilibrium for different values of potential barrier when using the TM port
112
4.6.3 Case 3: ��𝒕𝒐𝒕 (𝑽𝒊𝒏𝒋 = 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕, 𝑽𝑻𝑴)
This section examines the charge flow rate (V/s) when the voltage due to charge,
𝑞
𝐶𝑇 on the TM is allowed to go to a saturation value determined by a known constant
potential bias as described in section 4.6.2. The total test mass voltage becomes a
contribution of the voltage due to charge 𝑞
𝐶𝑇 and the polarization from the injection,
1
𝐶𝑇∑ 𝐶𝑖𝑉𝑖𝑖 as shown in Figure 4-29.
Figure 4-29.Circuit model showing the TM voltage as a contribution of the voltage due to charge and the TM polarization due to the injection voltage
The measured voltage time series shown in Figures 4-30 and 4-31 respectively,
provide the charge flow rate calculated as the TM voltage approaches saturation from
both the negative and positive regions when the electrodes port is used. In Figure 4-30,
the slope of the curve from −1.6 V all the way to − 0.6 V is calculated. At −0.6 V the TM
has reached its saturation voltage and the charge rate is zero. The charge rates from
+1.8 V to − 0.6 V in Figure 4-31 was also found by calculating the charge rates along
the curve as the voltage moved to its saturation value.
113
Figure 4-30. Voltage time series as TM voltage approaches saturation from below
Figure 4-31. Voltage time series as TM voltage approaches saturation from above
All the slopes derived in this way are plotted with their corresponding TM
voltages shown in Figure 4-32 with the corresponding model fit to the data. The side of
114
the plot to the left of − 0.6 𝑉 has a more gradual slope because the TM approaches
saturation gradually over 25000 seconds as shown in Figure 4-30. Therefore the
change in the charge rate (V/s) converted to apparent yield (electrons per photon) from
the starting TM voltage to saturation at − 0.6 𝑉 is only about a factor of 0.3. The
apparent yields to the right of − 0.6 V were calculated from the data in Figure 4-31. The
charge rates here change more drastically because the TM approaches saturation more
quickly in about 10000 seconds with the change in apparent yield being more than a
factor of 3.
Figure 4-32. Measured charge flow rate and best fit model as test mass voltage changes at a constant potential bias when using the electrodes housing port
A similar behavior is observed when the TM UV port is used. Figure 4-33 shows
the best fit model to the measured data for this case. The saturation voltage on the test
115
mass is around 1.16 V. The apparent yield above this value shows a pronounced
change in apparent yield from the starting point to the equilibrium value of about a factor
of 1.5. This is similar to the change in apparent yield obtained below the saturation
value.
Figure 4-33. Charge flow rate as the test mass voltage changes at a constant potential bias when using the TM port
4.7 Conclusions
Charge build-up on the TMs in the Torsion pendulum was successfully controlled
in two ways: by illuminating either the electrodes housing or the TM with constant (DC)
UV light, and by synchronizing the output of the UV LED with the 100 KHz injection
signal (AC charge control).
An analytical model was developed for calculating the total electron flow rate
within the GRS as a function of the applied potential on the injection electrodes. This
116
model can be modified for the case when the voltage on the TM is zero while
accounting only for the polarization induced on the TMs from the electrodes. It can also
be modified for the case when the voltage on the TM is allowed to evolve until it reaches
equilibrium. The model adequately explained the AC charge control data obtained from
the pendulum within a reasonable margin of error. Further developments to this model
will include ray tracing to determine the area illuminated by the UV light inside the
sensor and track the number of photoelectrons. The results of the ray tracing will then
be incorporated into a numerical charge control model
The results obtained indicated that the UV port illuminating both the TM and the
electrodes housing were likely so shallow that the photoelectrons seemed to pass
though the region without hitting the intended surfaces. A new flight-like GRS will be
installed to compare with one of the simplified ones to help with obtaining results more
compatible with LISA expectations.
117
CHAPTER 5 SUMMARY OF CONCLUSIONS INCLUDING RECOMMENDATIONS FOR FUTURE
WORK
The proposed charge control scheme for LISA will use the principle of
photoelectric effect to remove excess test mass charge by illuminating the relevant
surfaces with UV light in order to generate a flow of electrons in the desired direction.
The effectiveness of this process depends on the quantum yield (QY) of the surfaces.
The QY depends on the energy of the illuminating photons, the work function of the
illuminated surface, the reflectivity of the surfaces and the number of electrons that
successfully cross the gap between the illuminated surface and the opposite surface.
One of the arguments for using UV LEDs in charge control is their availability at lower
wavelengths of 240 nm and 250 nm compared with Hg lamps which have a wavelength
of 254 nm. A part of this work focuses on quantifying the QYs of several Au samples at
240 nm and 250 nm as a function of time, temperature, and vacuum pressure. The LISA
test plan can then account for possible QY changes related to these parameters.
After several Au samples were repeatedly baked-out at 130 °C, their QYs went
from varying by a factor of ten before baking out to varying by less than a factor of two
afterwards. This is useful, because after the LISA GRS is baked out in vacuum before
integration, surface contamination will be reduced and the quantum yield will be
stabilized.
The simplified GRSs in the torsion pendulum assembly have been operating at
the base pressure of around 10−6 Torr. This is a factor of one hundred above what LISA
requires. The new flight like GRS to be installed will also likely be operated at the same
pressure. It is therefore important to investigate if the QYs of tested Au samples have a
significant dependence on pressure. Four Au samples which were prepared in the same
118
way were each measured at three different vacuum pressures. Results showed that a
pressure increase from 10−8 Torr to 10−6 Torr resulted in a decrease in QY of at least a
factor of three for two out of the four samples. After concluding measurements at 10−6
Torr on one of the Au samples tested, additional long term measurements of more than
1000 hours were conducted on the same sample at 10−8 Torr without breaking vacuum.
Results show that the QYs for long term measurements at 10−8 Torr dropped by a
factor of two below what the average was at 10−6 Torr. While this measurement is only
for one sample, this is important because LISA pressures could rise to values as high
as 10−5 Torr before launch due to vacuum leaks. The pressure is expected to drop back
to 10−8 Torr after it is vented to the space environment. It is important to take changes
as a result of expected pressure variations into account.
Adjacent surfaces participating in electron exchange in LISA may differ too much
in QY if care is not taken. If this is the case, electrons may move in the unintended
direction and complicate the charge control process. The results of simple reflectivity
analysis on adjacent Au surfaces described in this work show that the QY of the
illuminated surface should be greater than that of the adjacent surface by a factor
determined by the common reflectivity of both surfaces if their reflectivities are equal. If
the reflectivities are different, then the factor is determined by the reflectivity of the
adjacent surface.
It is reasonable for the GRS system in LISA to be represented as a circuit
composed of parallel plate capacitors for describing individual surfaces that undergo
illumination during charge control. The GRS system in the torsion pendulum test bed
119
was modelled as such to provide a means of describing the photon energy distribution
and electron exchange process during charge control.
DC charge control experiments were carried out by using 240 nm UV LEDs to
either illuminate the TM or the electrodes in the pendulum’s simplified GRS system. The
direction of illumination would be determined by the initial voltage on the TM. The
results discussed in section 4.4 prove that bipolar discharge is possible for LISA, using
UV LEDs.
For AC charge control, the power output of the UV LED was synchronized with
the 100 kHz injection voltage already present in the GRS for capacitively reading out the
positions of the TMs with respect to the housing. AC charge control experiments carried
out by using all available UV ports demonstrate that the voltage due to charge on the
TM can be controlled depending on the phase, amplitude and duty cycle of the UV
power output with respect to the injection signal.
An analytical charge model was developed for calculating the total charge flow
rate as a function of applied potential. The model could be modified for three different
cases. One of which is the calculation of the charge flow rate when the voltage on the
TM is close to zero. This is important because the aim is to keep LISA test masses
close to zero volts at all times. Despite the GRS being a simplified version of what is
required for LISA, the model adequately explained data obtained from the different UV
injection ports with the margin of error increasing for situations where the voltage on the
TM is far from zero.
A new flight-like GRS system has been designed and developed at the University
of Florida. This will be installed in the pendulum to compare with one of the existing
120
simplified ones. Further developments to the analytical model will include multi physics
finite element analysis and ray tracing to more accurately determine the area illuminated
by the UV light inside the GRS and track the number of photoelectrons. The results of
this analysis will then be incorporated into a numerical model to obtain results that are
more representative for LISA.
121
APPENDIX A DISTRIBUTION OF ELECTRON ENERGIES AS A FUNCTION OF APPLIED
BIAS VOLTAGE
The distribution of the normal energies as a function of the applied bias
voltage for a parallel plate configuration is of the form,
𝑓 (∆𝑉) 𝑑𝛥𝑉 = [𝐴𝑛 𝑙𝑜𝑔 (𝑒𝑥𝑝(−∆𝑉+ 𝑉𝑚 )𝑒
𝐾𝑏𝑇 + 1) ] 𝑑𝛥𝑉 (A-1)
Here, 𝐴𝑛 is a constant, e is the charge of the electron, 𝐾𝑏 is Boltzmann’s
constant, ∆𝑉 is the potential difference of the two adjacent surfaces
𝑉𝑚 = ℎ𝜈−𝜙
𝑒 (A-2)
The expression for 𝑉𝑚 represents the maximum energy an electron can have
after being extracted by a photon of energy ℎ𝜈 from a surface that has a work
function of 𝜙. Equation A-1 can be rewritten as,
𝑓 (∆𝑉) ∝ 𝑙𝑜𝑔 (𝑒𝑥𝑝(−∆𝑉+ 𝑉𝑚 )𝑒
𝐾𝑏𝑇 + 1) (A-3)
𝑑𝑓(∆𝑉)
𝑑∆𝑉 = −
𝑒
𝐾𝑏𝑇 (
𝑒𝑥𝑝
−(∆𝑉−𝑉𝑚)𝑒𝐾𝑏𝑇
𝑒𝑥𝑝
−(∆𝑉−𝑉𝑚)𝑒𝐾𝑏𝑇 +1
) (A-4)
If ∆V>>𝑉𝑚 or ∆V+∞
𝑑𝑓(∆𝑉)
𝑑∆𝑉= 0 (A-5)
If ∆V<<𝑉𝑚 or ∆V-∞
𝑑𝑓(∆𝑉)
𝑑∆𝑉= −
𝑒
𝐾𝑏𝑇∗ 1 (A-6)
The function in Equation A-1, shown in blue, is well approximated by two
straight lines defined as ∆𝑉 ≪ 𝑉𝑚 with a negative slope and ∆𝑉 ≫ 𝑉𝑚 which is a
constant.
122
Figure A-1. Illustration of straight line approximation of Equation A-1
If the equation of the given straight line with intercept β is written as,
𝑓 (∆𝑉) = −𝑒
𝐾𝑏𝑇∆𝑉 + 𝛽 (A-7)
And
𝑓 (𝑉𝑚) = −𝑒
𝐾𝑏𝑇𝑉𝑚 + 𝛽 = 0 (A-8)
Find intercept β = 𝑒
𝐾𝑏𝑇𝑉𝑚
For electrons moving from i to j;
𝑓 (∆𝑉) = −𝑒
𝐾𝑏𝑇(∆𝑉 − 𝑉𝑚) (A-9)
And for electrons moving from j to i;
𝑓 (∆𝑉) = −𝑒
𝐾𝑏𝑇(−∆𝑉 − 𝑉𝑚) (A-10)
123
APPENDIX B SOLUTIONS TO PHOTOELECTRON FLUX EQUATIONS
The general form of ��𝑖𝑗, which is related the electrons extracted from the i-
th surface towards the j-th surface can be written as,
��𝑖𝑗 = 𝛼𝑖 ∙ 𝑎𝑖 ∙ 𝑄𝑌𝑖(𝜈) ∙ 𝐼𝑖
∫ 𝑔(𝜈) 𝑑𝜈+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖ℎ
∫− 𝑒
𝐾𝑏𝑇(∆𝑉𝑖𝑗−𝑉𝑚)𝑑∆𝑉𝑖𝑗
ℎ𝜈−𝜙𝑖𝑒
∆𝑉𝑖𝑗
∫ 𝑔(𝜈) 𝑑𝜈+∞
𝜙𝑖ℎ
∫− 𝑒
𝐾𝑏𝑇(∆𝑉𝑖𝑗−𝑉𝑚)𝑑∆𝑉𝑖𝑗
ℎ𝜈−𝜙𝑖𝑒
0
(B-1)
Numerator of the integral ratio in Equation B-1;
1
2𝐾𝑏𝑇𝑒 (ℎ2 ∫ 𝜈2𝑔(𝜈) 𝑑𝜈
+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
− 2ℎ𝜙𝑖 ∫ 𝜈 𝑔(𝜈) 𝑑𝜈+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
+ 𝜙𝑖2 ∫ 𝑔(𝜈) 𝑑𝜈
+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
+
𝑒2∆𝑉𝑖𝑗2 ∫ 𝑔(𝜈) 𝑑𝜈
+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
− 2𝑒∆𝑉𝑖𝑗ℎ ∫ 𝜈 𝑔(𝜈) 𝑑𝜈 + 2𝑒∆𝑉𝑖𝑗𝜙𝑖
+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
∫ 𝑔(𝜈) 𝑑𝜈+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
)
(B-2)
Denominator of the integral ratio in Equation B-1;
1
2𝐾𝑏𝑇𝑒 (ℎ2 ∫ 𝜈2𝑔(𝜈) 𝑑𝜈
+∞𝜙𝑖ℎ
− 2ℎ𝜙𝑖 ∫ 𝜈 𝑔(𝜈) 𝑑𝜈+∞
𝜙𝑖ℎ
+ 𝜙𝑖2
∫ 𝑔(𝜈) 𝑑𝜈+∞
𝜙𝑖ℎ
(B-3)
Define Ai, Bi and Ci for Equation B-2 and Aid, Bid and Cid for Equation B-3 as,
𝐴𝑖 = ∫ 𝑔(𝜈) +∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
𝑑𝜈 (B-4)
𝐵𝑖 = ∫ 𝜈 𝑔(𝜈)+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
𝑑𝜈 (B-5)
𝐶𝑖 = ∫ 𝜈2𝑔(𝜈)+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
𝑑𝜈 (B-6)
𝐴𝑖𝑑 = ∫ 𝑔(𝜈) +∞
𝜙𝑖ℎ
𝑑𝜈 (B-7)
𝐵𝑖𝑑 = ∫ 𝜈 𝑔(𝜈)+∞
𝜙𝑖ℎ
𝑑𝜈 (B-8)
124
𝐶𝑖𝑑 = ∫ 𝜈2𝑔(𝜈)+∞
𝜙𝑖ℎ
𝑑𝜈 (B-9)
Equation B-1 becomes;
��𝑖𝑗 = 𝛼𝑖 ∙ 𝑎𝑖 ∙ 𝑄𝑌𝑖(𝜈) ∙ 𝐼𝑖.𝑒2∆𝑉𝑖𝑗
2𝐴𝑖 + (2𝑒𝜙𝑖𝐴𝑖 − 2𝑒ℎ𝐵𝑖)∆𝑉𝑖𝑗 + ℎ2𝐶𝑖 − 2ℎ𝜙𝑖𝐵𝑖 + 𝜙𝑖2𝐴𝑖
ℎ2𝐶𝑖𝑑 − 2ℎ𝜙𝑖𝐵𝑖𝑑 + 𝜙𝑖2𝐴𝑖𝑑
(B-10)
For electrons extracted from the j-th surface;
��𝑗𝑖 = 𝛼𝑗 ∙ 𝑎𝑗 ∙ 𝑄𝑌𝑗(𝜈) ∙ 𝐼𝑗 ∙
∫ 𝑔𝑗(𝜈) 𝑑𝜈+∞
−𝑒∆𝑉𝑖𝑗+𝜙𝑗
ℎ
∫ − 𝑒
𝐾𝑏𝑇 (−∆𝑉𝑖𝑗 − 𝑉𝑚)∆𝑉𝑖𝑗
−(ℎ𝜈−𝜙𝑗)
𝑒
𝑑∆𝑉𝑖𝑗
∫ 𝑔𝑗(𝜈) 𝑑𝜈+∞
𝜑𝑗
ℎ∫ −
𝑒𝐾𝑏𝑇 (−∆𝑉𝑖𝑗 − 𝑉𝑚)
ℎ𝜈−𝜙𝑗
𝑒0
𝑑∆𝑉𝑖𝑗
(B-11)
Numerator of the integral ratio in Equation B-11;
1
2𝐾𝑏𝑇𝑒 (ℎ2 ∫ 𝜈2𝑔(𝜈) 𝑑𝜈
+∞𝑒∆𝑉𝑖𝑗+𝜙𝑗
ℎ
− 2ℎ𝜙𝑗 ∫ 𝜈 𝑔(𝜈) 𝑑𝜈+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑗
ℎ
+ 𝜙𝑗2
∫ 𝑔(𝜈) 𝑑𝜈+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑗
ℎ
+
𝑒2∆𝑉𝑖𝑗2
∫ 𝑔(𝜈) 𝑑𝜈+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑗
ℎ
+
2𝑒∆𝑉𝑖𝑗ℎ ∫ 𝜈 𝑔(𝜈) 𝑑𝜈 − 2𝑒∆𝑉𝑖𝑗𝜙𝑗+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑗
ℎ
∫ 𝑔(𝜈) 𝑑𝜈) +∞
𝑒∆𝑉𝑖𝑗+𝜙𝑗
ℎ
(B-12)
Denominator of Equation the integral ratio in B-11;
1
2𝐾𝑏𝑇𝑒 (ℎ2 ∫ 𝜈2𝑔(𝜈) 𝑑𝜈
+∞𝜙𝑗
ℎ
− 2ℎ𝜙𝑗 ∫ 𝜈 𝑔(𝜈) 𝑑𝜈+∞
𝜙𝑗
ℎ
+ 𝜙𝑗2
∫ 𝑔(𝜈) 𝑑𝜈+∞
𝜙𝑗
ℎ
(B-13)
Define Aj, Bj and Cj and Ajd, Bjd and Cjd for Equation B-12 and Equation B-13 by
replacing the indexes accordingly to get;
125
��𝑗𝑖 = 𝛼𝑗 ∙ 𝑎𝑗 ∙ 𝑄𝑌𝑗(𝜈)
∙ 𝐼𝑗
𝑒2∆𝑉𝑖𝑗2𝐴𝑗 − (2𝑒𝜙𝑗 𝐴𝑗 − 2𝑒ℎ𝐵𝑗 )∆𝑉𝑖𝑗 + ℎ2𝐶𝑗 − 2ℎ𝜙𝑗𝐵𝑗 + 𝜙𝑗
2 𝐴𝑗
ℎ2𝐶𝑗𝑑 − 2ℎ𝜙𝑗𝐵𝑗𝑑 + 𝜙𝑗2𝐴𝑗𝑑
(B-14)
To fully evaluate the defined expressions for A, B and C; consider that the
function g(ν), follows a normal distribution of the form;
𝑔(𝜈) =1
𝜎√2𝜋𝑒
− (𝜈− 𝜈)2
2𝜎2 (B-15)
Where ν and 𝜎 are the mean and standard deviation of the distribution
respectively. By making the following simple conversions;
If 𝕏= a+bx ≡ 𝜈− 𝜈
𝜎 , x ≡ 𝜈, a ≡
−𝜈
𝜎, b ≡
1
𝜎 , x =
𝕏−𝑎
𝑏 (B-16)
And using the list of integrals of Gaussian functions as follows;
𝜙(𝕏) = 1
√2𝜋𝑒−
(𝕏)2
2 (B-17)
∫ 𝜙 (𝕏)𝑑𝕏 = 𝜱(𝕏) = 1
2 (1 + 𝑒𝑟𝑓 (
𝕏
√2)) + Constant (B-18)
∫ 𝕏 𝜙 (𝕏) 𝑑𝕏 = −𝜙(𝕏) + Constant (B-19)
∫ 𝕏2 𝜙(𝕏) 𝑑𝕏) = 𝜱(𝕏) − 𝕏 𝜙(𝕏) + Constant (B-20)
Using the defined relationships in Equation B-16 and Equation B-17 to rewrite
Equation B-15, Equation B-19 and Equation B-20 respectively as;
𝑔 (𝑥) = 1
𝜎𝜙(𝕏) (B-21)
∫ 𝑥 𝜙(𝕏) d𝕏 = ∫𝕏−𝑎
𝑏 𝜙(𝕏) d𝕏 =
1
𝑏 ∫(𝕏 − 𝑎) 𝜙(𝕏) d𝕏 =
1
𝑏 (∫ 𝕏 𝜙(𝕏) d𝕏 – a ∫ 𝜙(𝕏) d𝕏)
= 1
𝑏(−𝜙(𝕏) −
𝑎
2 (1 + 𝑒𝑟𝑓 (
𝕏
√2))) (B-22)
And
126
∫ 𝑥2 𝜙(𝕏) d𝕏) = ∫(𝕏−𝑎
𝑏)2 𝜙(𝕏) d𝕏 =
1
𝑏2 ∫(𝕏2 − 2𝕏𝑎 + 𝑎2) 𝜙(𝕏)𝑑𝕏
= 1
𝑏2 (𝜱(𝕏) − 𝕏 𝜙(𝕏) + 2𝑎 𝜙(𝕏) + 𝑎2 𝜱(𝕏)) (B-23)
So that Ai, Bi and Ci can be written as follows for surface i;
𝐴𝑖 = ∫ 𝑔(𝜈) +∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
𝑑𝜈 =1
2 (1 + 𝑒𝑟𝑓 (
𝕏
√2))|
+∞𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
(B-24)
𝐵𝑖 = ∫ 𝜈 𝑔(𝜈)+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
𝑑𝜈 = −1
𝑏 (𝜎𝑔(𝑥) +
𝑒
2 (1 + erf (
𝕏
√2)))|
+∞𝑒∆𝑉𝑖𝑗 + 𝜙𝑖
ℎ
(B-25)
𝐶𝑖 = ∫ 𝜈2𝑔(𝜈)+∞
𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
𝑑𝜈 =1
𝑏2 ((1 + 𝑒2)1
2(1 + 𝑒𝑟𝑓 (
𝕏
√2)) − (𝕏 − 2𝑒)𝜎𝑔(𝑥))|
+∞𝑒∆𝑉𝑖𝑗+𝜙𝑖
ℎ
(B-26)
Likewise, Aid, Bid 𝑎𝑛𝑑 Cid can be written also for surface i as;
𝐴𝑖𝑑 = ∫ 𝑔(𝜈) +∞
𝜙𝑖ℎ
𝑑𝜈 =
1
2 (1 + 𝑒𝑟𝑓 (
𝕏
√2))|
+∞𝜙𝑖ℎ
(B-27)
𝐵𝑖𝑑 = ∫ 𝜈 𝑔(𝜈)+∞
𝜙𝑖ℎ
𝑑𝜈 = −
1
𝑏 (𝜎𝑔(𝑥) +
𝑒
2 (1 + 𝑒𝑟𝑓 (
𝕏
√2))|
+∞𝜙𝑖ℎ
(B-28)
𝐶𝑖𝑑 = ∫ 𝜈2𝑔(𝜈)+∞
𝜙𝑖ℎ
𝑑𝜈 =
1
𝑏2 ((1 + 𝑒2)1
2(1 + 𝑒𝑟𝑓 (
𝕏
√2)) − (𝕏 − 2𝑒)𝜎𝑔(𝑥))|
+∞𝜙𝑖ℎ
(B-29)
Use −𝑒∆𝑉+𝜙𝑗
ℎ in the limits of 𝐴𝑗, 𝐵𝑗 and 𝐶𝑗 for electrons going in the reverse
direction (��𝑗𝑖).
127
APPENDIX C
CHARGE MEASUREMENT EQUATIONS
The electrostatic force acting on the TM along the x axis, given by 𝐹𝑥 is
𝐹𝑥 = 1
2∑
∂C𝑒
∂𝑥𝑒 (𝑉𝑇𝑀-𝑉𝑒)2
(C-1)
𝐹𝑥 = electrostatic force acting on the TM along the X axis
𝑉𝑇𝑀 = Test Mass (TM) potential
𝑉𝑒 = e-th surface voltage, e = 1, 2, …
𝐶𝑒 = e-th surface capacitance, e = 1, 2, …
𝑉𝑇𝑀 = 𝑞
𝐶𝑇 +
1
𝐶𝑇∑ 𝐶𝑒𝑉𝑒𝑒 (C-2)
A = Amplitude of sinusoidal voltage applied to TM
𝐶𝑇 = total capacitance of the TM toward surrounding surfaces
q = net charge
For charge measurement, set,
𝑉1 = 𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡) and 𝑉2 = − 𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡)
𝑉𝑇𝑀 = 𝑞
𝐶𝑇+
1
𝐶𝑇 (𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡)(𝐶1 − 𝐶2)) (C-3)
Substituting Equation C-3 in Equation C-1,
𝐹𝑥 = 1
2 𝜕𝐶1
𝜕𝑥 (
𝑞+ 𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡)(𝐶1−𝐶2)
𝐶𝑇− 𝐴 𝑐𝑜𝑠(2𝜋𝑓𝑡))2 +
1
2
𝜕𝐶2
𝜕𝑥 (
𝑞+ 𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡)(𝐶1−𝐶2)
𝐶𝑇 +
𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡))2 (C-4)
Therefore,
𝐹𝑥 =1
2
𝜕𝐶1
𝜕𝑥(
𝑞
𝐶𝑇 − 𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡)) 2 +
1
2 𝜕𝐶2
𝜕𝑥(
𝑞
𝐶𝑇 + 𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡)) 2 (C-5)
If 𝑋+ and 𝑋− indicate the movement of the TM in the positive and negative X
direction respectively with corresponding capacitances of 𝐶𝑋+ and 𝐶𝑋−
towards an
128
opposite surface with area A, then for a TM displaced from the center position by
𝑥 as described by Giacomo Ciani (Diss. Univ. of Trento, 2008)
𝐶𝑋±=
𝜖0𝐴
𝑑∓𝑥 (C-6)
𝜕𝐶𝑋+
𝜕𝑥|
𝑥=0=
𝐶𝑋+
𝑑 (C-7)
𝜕𝐶𝑋−
𝜕𝑥|
𝑥=0=
−𝐶𝑋−
𝑑 (C-8)
If 𝐶1 = 𝐶𝑋− and 𝐶2 = 𝐶𝑋+
and 𝐶1 = 𝐶2, then ∂C2
∂x=
C
d and
∂C1
∂x=
−C
d.
Substituting for 𝐶1 and 𝐶2 in Equation C-5 gives,
𝐹𝑥 = 2𝐶𝑞𝐴 𝑐𝑜𝑠 (2𝜋𝑓𝑡)
𝑑𝐶𝑇 (C-9)
A is the amplitude of sinusoidal voltage applied to the electrodes and q is the net
charge on the TM. For small rotations, the equation of motion of the pendulum is
given by the second order ODE,
�� = −𝜔02𝜑 +
𝐿
𝐼𝐹𝑥 (C-10)
I is the moment of inertia (1.63 × 10−2 kgm2) while L is the arm length (0.22m)
Homogenous solution:
�� + 𝜔02𝜑 = 0 (C-11)
𝜑 = 𝐴 𝑐𝑜𝑠 (𝜔0𝑡) + 𝐵 𝑠𝑖𝑛 (𝜔0𝑡) (C-12)
�� = −𝐴 𝜔0𝑠𝑖𝑛 (𝜔0𝑡) + 𝐵 𝜔0𝑐𝑜𝑠 (𝜔0𝑡) (C-13)
�� = −𝐴 𝜔02𝑐𝑜𝑠 (𝜔0𝑡) − 𝐵 𝜔0
2𝑠𝑖𝑛 (𝜔0𝑡) (C-14)
Substituting Equation C-14 and Equation C-12 into Equation C-11
−𝐴 𝜔𝑜2𝑐𝑜𝑠 − 𝐵 𝜔0
2𝑠𝑖𝑛 (𝜔0𝑡) + 𝜔02(𝐴 𝑐𝑜𝑠 (𝜔0𝑡) + 𝐵 𝑠𝑖𝑛 (𝜔0𝑡)) = 0 (C-15)
So that,
𝜑ℎ = 𝐴 𝑐𝑜𝑠 (𝜔0𝑡) + 𝐵 𝑠𝑖𝑛 (𝜔0𝑡) (C-16)
129
The pendulum is subjected to an external periodic force so the particular solution
is the solution of interest in charge measurements. For the particular solution;
Let;
𝜑𝑝 = 𝐸 𝑐𝑜𝑠 (2𝜋𝑓𝑡) + 𝐹 𝑠𝑖𝑛 (2𝜋𝑓𝑡) (C-17)
��𝑝 = − 2𝜋𝑓 𝐸 𝑠𝑖𝑛 (2𝜋𝑓𝑡) + 2𝜋𝑓 𝐹 𝑐𝑜𝑠 (2𝜋𝑓𝑡) (C-18)
��𝑝 = −(2𝜋𝑓)2 𝐸 𝑐𝑜𝑠 (2𝜋𝑓𝑡) − (2𝜋𝑓)2 𝐹 𝑠𝑖𝑛 (2𝜋𝑓𝑡) (C-19)
��𝑝 + 𝜔02𝜑𝑝 = 𝐷 𝑐𝑜𝑠(2𝜋𝑓𝑡) (C-20)
Substitute Equation C-17 and Equation C-19 into Equation C-20
𝐸(𝜔02𝑐𝑜𝑠 (2𝜋𝑓𝑡) − (2𝜋𝑓)2 𝑐𝑜𝑠 (2𝜋𝑓𝑡)) + 𝐹 (𝜔0
2𝑠𝑖𝑛 (2𝜋𝑓𝑡)
− (2𝜋𝑓)2 𝑠𝑖𝑛 (2𝜋𝑓𝑡))
= 𝐷 𝑐𝑜𝑠(2𝜋𝑓𝑡) (C-21)
Comparing LHS and RHS
𝐹 = 0 (C-22)
𝐸 𝑐𝑜𝑠 (2𝜋𝑓𝑡) (𝜔02 − (2𝜋𝑓)2) = 𝐷 𝑐𝑜𝑠(2𝜋𝑓𝑡) (C-23)
𝐸 = (𝐷
𝜔02− (2𝜋𝑓)2
) (C-24)
From Equation C-10 and Equation C-20,
𝐷 𝑐𝑜𝑠(2𝜋𝑓𝑡) = 𝐿
𝐼𝐹𝑥 (C-25)
𝐷 = 2𝐶𝑞𝐴 𝐿
𝑑𝐶𝑇𝐼 (C-26)
𝐸 =
2𝐶𝑞𝐴 𝐿
𝑑𝐶𝑇𝐼
𝜔02− (2𝜋𝑓)2 (C-27)
Find θp from Equation C-17
130
𝜑𝑝 =
2𝐶𝑞𝐴 𝐿
𝑑𝐶𝑇𝐼
𝜔02− (2𝜋𝑓)2 𝑐𝑜𝑠 (2𝜋𝑓𝑡) (C-28)
The charge q, can be found from the amplitude of 𝜑p as;
𝑞 =𝜑𝑝𝐶𝑇𝐼 (𝜔0
2− (2𝜋𝑓)2)
2𝐶𝑑𝐴𝐿 (C-29)
131
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BIOGRAPHICAL SKETCH
Taiwo J. Olatunde was born in Lagos, Nigeria. She obtained her BS. in
mechanical engineering at the University of Lagos, Nigeria in 2010 and a master’s
degree in aerospace engineering at the University of Florida in 2013, She graduated
with a PhD in aerospace engineering at the University of Florida in 2018. Taiwo is
married to Yomi Olatunde and is a mother to two children, Mayowa Katherine and
Ayomide Joanne.