DEMONSTRATION OF A PROTOTYPE DUAL-RECYCLED
CAVITY-ENHANCED MICHELSON INTERFEROMETER FOR
GRAVITATIONAL WAVE DETECTION
By
THOMAS DELKER
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
2001
ACKNOWLEDGMENTS
This work would have not be possible without the advice and guidance of my
advisor, Associate Professor David Reitze. I also give many special thanks to Professor
David Tanner, who often acted as a replacement advisor because of the incredible
travel schedule resulting from LIGO involvement.
The most notable piece of advice that I was given during the final years of
completely this study was that a Ph.D. is not necessarily proof that you are intelligent.
Most people show this by the time they complete their first two years of graduate
school. A Ph.D. is instead a badge of stubbornness and persistence. That advice
echoed through my head as more and more challenges appeared along the road to
finishing this dissertation. I believe that I have shown that I am stubborn enough.
The work in this thesis drew on many people's expertise. Dr. Guido Mueller is
largely responsible for the conception of the locking scheme. He also took me down
the first steps in the lab and continually helped with interpreting of results. Dr.
Gerhard Heinzel contributed greatly to this work through his electronic designs and
his electrical circuit simulation program, LISO. Andreas Freise provided an invaluable
tool in FINESSE, which I used on practically a daily basis. Lively discussions at the
LIGO Advance Interferometer Configurations meetings were very informative. For
that I must thank both the participants and the organizer, Dr. Ken Strain. The
thought that other graduate students were also out there working on similar problems
and succeeding (namely Jim Mason and Daniel Shaddock) also was a great help.
The most important component of being able to finish this work was the love
and support of the people in my life: most notable Crystal Lewis, my hiking com-
panion and so much more. I cannot express how important Crystal has been to the
completion of this work, so I won't try. It was my parents who gave me the gift that
allowed me to finish: my stubbornness. I must also thank Kenny for introducing me
to the world of outdoor activities that kept me sane over the past few years. I also
thank Willie for showing me how to succeed in academia.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ............... . . . ii
LIST OF TABLES . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . viii
ABSTRACT . . . . . . . . . . . . . . ix
1 INTRODUCTION . . . . . . . . . . . . . 1
1.1 Gravitational W aves ........................... 1
1.2 Interferometer as a Detector ........... ........... 3
1.2.1 Shot N oise . . . . . . . . . . . . . . 9
1.2.2 Radiation Pressure Noise ................ .. .. . 16
1.2.3 Thermal Noise .................. ....... .. 16
1.3 Sources. .................. ............... .. 17
1.4 Motivation for Future Detectors ................ .. .. 20
2 THEORY . . . . . . . . . . . . . . 23
2.1 Light Fields .................. ............. .. 24
2.1.1 Generating Sidebands .................. ..... 24
2.1.2 Sideband Detection .................. ... .. 26
2.2 Locking M atrix .................. ........... .. 29
2.3 Simple Cavity .................. ............ 33
2.3.1 Light Fields .................. ......... . 33
2.3.2 Error Signals .................. ...... .. .. 40
2.4 M ichelson .................. .............. .. 44
2.4.1 Light Fields .................. ......... .. 44
2.4.2 Error Signals .................. ...... .. .. 46
2.5 Michelson with Arm Cavities .................. ..... 50
2.5.1 Light Fields .................. ......... .. 51
2.5.2 Error Signals .................. . .. 53
2.5.3 Frequency Response .................. .. . 61
2.6 Three-Mirror Coupled Cavity .................. ..... 62
2.6.1 Light Fields .................. ......... .. 62
2.6.2 Error Signals .................. . .. 63
2.7 Power-Recycled Cavity-Enhanced Michelson ..
2.7.1 Light Fields . . . . . . . . .
2.7.2 Error Signals .. ..............
2.8 Dual-Recycled Cavity-Enhanced Michelson .....
2.8.1 Light Fields . . . . . . . . .
2.8.2 Error Signals .. ..............
2.8.3 Frequency Response .. ..........
3 EXPERIMENT . . . ............
3.1 Design ......... . . . . .
3.1.1 Selection of Mirror Parameters . . . ..
3.1.2 Length and Frequency Considerations . .
3.1.3 Physical Layout and Components . . .
3.1.4 Feedback . . . . . . . . . .
3.2 Calculated Locking Matrix . . . . . . ..
3.2.1 Cavity-Enhanced Power-Recycled Michelson
3.2.2 Dual-Recycled Cavity-Enhanced Michelson .
3.3 Measurement of Losses and Resonances . . . .
3.3.1 Arm Cavities . . ............
3.3.2 Simnle Michelson ...............
3.3.3 Power-Recycled Interferometer .
3.3.4 Dual-Recycled Interferometer .
3.4 Measured Locking Matrix . . ...
3.4.1 Power Recycling . . ....
3.4.2 Dual Recycled . . . .....
3.5 Measured Frequency Response . . .
3.5.1 Low Frequency Response . . .
3.5.2 High Frequency Response . .
3.5.3 Detuned Frequency Response .
3.6 Lock Acquisition . . . .......
3.6.1 Initial Lock Acquisition . . .
3.6.2 Repeatable Lock Acquisitions and
4 CONCLUSION . . ..
4.1 Summary of Results . . ......
4.2 Future W ork . . . ..........
Lock
Stability
A GAUSSIAN MODES . .
A.1 Modal Decomposition.
A.2 Propagation of Gaussian
Modes via ABCD Matrices .
B ELECTRONICS . . . . . . . . . . .
REFERENCES . . . . . . . . . . . .
. . 155
BIOGRAPHICAL SKETCH .. . . . ..............
84
84
84
87
98
102
106
106
110
113
113
114
116
119
122
126
127
129
129
133
135
136
137
139
142
142
144
LIST OF TABLES
Locking matrix for Michelson with arm cavities .. .........
LIGO-like parameters for coupled cavity .. ............
Locking matrix for coupled cavity with in-phase scheme .......
Locking matrix for coupled cavity with quad scheme .........
Locking matrix for LIGO I configuration .. ............
Designed mirror specification........... . . . . .....
Phase shift from arm cavities.. . . . .....
Final length parameters..... . . . ......
PZT tube parameters.. . . . . . .....
Locking matrix for LIGO I configuration . . . . .
Locking matrix for LIGO II configuration . . . . . . ..
Measurement of arm cavity mirrors via FWHM and FSR technique
OSA measurements for LIGO I configuration . . . . . .
OSA measurements for LIGO I configuration . . . . . .
OSA measurements for blocked interferometer . . . . .
OSA measurements for signal-recycling configuration . . . .
OSA measurements for blocked interferometer . . . . .
Buildup of 31 MHz sideband . . . . . . .
Measured locking matrix for power-re.' 11 configuration . . .
Measured locking matrix for dual-recycled configuration . . . .
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
86
92
95
101
107
110
114
115
116
116
119
119
122
127
128
LIST OF FIGURES
1.1 A gravitational wave, h+, on a simple Michelson . . . . ... 6
1.2 Example of a delay line to increase detector sensitivity . . ... 10
1.3 Example of a cavity to increase detector sensitivity . . . .... 11
1.4 Power-recycled simple Michelson .................. .. . 12
1.5 Signal-recycled simple Michelson. .................. 13
1.6 Signal recycling with arm cavities ................ . 14
1.7 Shot-noise limited sensitivity ................ .... 15
1.8 Expected noise levels for LIGO II ............. .. . 21
2.1 Carrier with sidebands in frequency space . . . ..... 25
2.2 Feedback loop for a general system .............. .. .. 29
2.3 Fields in a simple cavity .................. ..... 33
2.4 Phase shift of light reflected from an over-coupled cavity ...... .. 36
2.5 Transmission of an impedance matched cavity ............ .. 37
2.6 Light fields with antiresonant sidebands in a cavity . . . .... 41
2.7 In-phase error signal for a cavity in reflection ............. ..43
2.8 Fields in a simple Michelson .................. .. 44
2.9 Transfer function of the Michelson ................ 47
2.10 Fields in Michelson with arm cavities ................. 50
2.11 Frequency response of LIGOI-like instrument ............. .61
2.12 Fields in a three-mirror coupled cavity ................ ..62
2.13 Error signal in reflection for the two degrees of freedom . . ... 66
2.14 Error signal at pick off for the two degrees of freedom . . . ... 66
2.15 Fields in a power-re' ivi1. Michelson ................. 68
2.16 Fields in a signal-recycled Michelson ................. .. 73
2.17 Frequency response of a signal-recycled Michelson with arm cavities 81
2.18 Signal-recycling cavity resonances for recycled and detuned case . 83
3.1 Definition of lengths .................. ...... .. .. 87
3.2 Electric fields in the reflected port ................ 89
3.3 Electric fields in the antisymmetric port ............... ..90
3.4 Phase shift cavities impart to sidebands ................ ..93
3.5 Sidebands in inline cavity .................. .... .. 96
3.6 Sidebands in perpendicular cavity ................ 96
3.7 Sidebands in power-recycling cavity .................. 97
3.8 Sidebands in the Michelson .................. .... .. 97
3.9 Optical components on table .................. .. 99
3.10 31 MHz feedback signals .................. .... 103
3.11 60 MHz feedback signals ............. ... ..... . 103
3.12 31 MHz in-phase signal used to lock ..... . . . . ... 109
3.13 60 MHz quadrature signal used to lock Lpr .............. .109
3.14 31 MHz quadrature signal used to lock 1_ .............. . .110
3.15 60 MHz quadrature signal used to lock Lpr .............. .112
3.16 31 MHz in-phase signal used to lock L.s ............... .112
3.17 OSA in reflection for (a)power-recycled and (b)signal-rc' i, . ... . 123
3.18 OSA at pick off for (a)power-recycled and (b)signal-recycled ..... 123
3.19 OSA at antisym. port for (a)power-recycled and (b)signal-recycled . 124
3.20 Method for measuring locking matrix in-loop . . . . ..... 124
3.21 Power-recycling sensitivity at low frequency ............. ..130
3.22 Signal-recycling sensitivity at low frequency ........... . .130
3.23 Signal-recycling gain at low frequency .... . . . . 131
3.24 Signal-recycling gain on a linear scale .... . . . . 132
3.25 Signal-recycling gain over power-recycling . . . . . 134
3.26 Signal-recycling gain over power-recycling for detuned . . . ... 136
3.27 DC power in the pick-off port .................. ..... 140
3.28 DC power in the reflected port .................. ..... 140
3.29 DC power in the antisymmetric port ................. ..140
3.30 DC power transmitted through the inline cavity . . . ..... 140
A.1 Gaussian mode in a cavity .................. .... 147
B.1 Tunable phase shifter for the local oscillator . . . . 155
B.2 High voltage driver for PZTs ................ ... . 156
B.3 Current driver for the galvometer ................ 157
B.4 PZT feedback loop with f-3/2 roll off ................. ..158
B.5 Feedback loop for the galvometers ................ 159
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
DEMONSTRATION OF A PROTOTYPE DUAL-RECYCLED
CAVITY-ENHANCED MICHELSON INTERFEROMETER FOR
GRAVITATIONAL WAVE DETECTION
By
Thomas Delker
May 2001
C(i iii, i'I: David H. Reitze
Major Department: Physics
The direct detection of gravitational radiation has long been the goal of a large
international collaboration of researchers. The first generation of interferometric grav-
itational wave detectors is currently being constructed around the world. These de-
tectors will most likely be operating at their designed sensitivities within the next few
years. The detectors are expected to give insight into the fundamental nature of the
universe as well as operate as observatories for previously unobserved astronomical
events. Unfortunately, the event rate that these detectors will be sensitive to is on the
order of one per year. A 10-fold increase in sensitivity results in a 1000-fold increase
in the event rate.
The first generation of the United State's detector, LIGO, is a power-recycled
Michelson interferometer with arm cavities. The next upgrade to this detector will
come in several forms, one of which is altering the topology. The addition of a signal-
recycling mirror to the detection port of the interferometer yields a sensitivity increase
of approximately one order of magnitude. The topology alteration also increases the
longitudinal degrees of freedom which must be controlled to 5.
This dissertation proposes a control scheme for the signal-recycled cavity-enhanced
Michelson interferometer with power recycling using frontal modulation. The control
scheme is developed through a series of increasingly complex detector topologies. The
control of all five degrees of freedom is discussed in detail. I present a measure of how
independent the control signals are from one another.
I also describes the experiment performed to demonstrate the control scheme.
The control scheme is simulated in full detail for the tabletop instrument. The results
of the simulation and the experiment are compared and show good agreement. A
detailed analysis of the operation of the tabletop interferometer is given including
the frequency response for two different operating points. The frequency response is
shown to be in good agreement with the theory. The results of the experiment are
discussed further and some possible future research in this area is proposed.
CHAPTER 1
INTRODUCTION
1.1 Gravitational Waves
In order to comprehend the universe on a fundamental level, the intricate details
of mass interaction must be understood. Newton's theory of gravitation is an excel-
lent approximation of masses reacting to other masses at slow speeds, provided the
gravitational potential is not too great. However, some anomalous problems existed,
such as the fact that Maxwell's equations were not invariant under Galilean's princi-
ples of relativity. Newton's Universal Theory of Gravitation agreed with physicists'
everyday encounters with the universe, and thus was thought to be a fundamental
law. As more and more anomalies were encountered, such as the Michelson-Morley
experiment in 1887, a new theory was developed, and special relativity was born. It
was investigated thoroughly by the interaction of electromagnetic waves with matter.
These waves also give us the only available view of the distant universe.
Just as Einstein's Theory of Special Relativity expanded on how mass travels
through space, in 1916 he also expanded on how mass affects space [1]. In his Theory
of General Relativity he laid the groundwork for a new explanation of gravity that
went far beyond the prevailing theory of gravity.
The success of experiments that test the predictions of General Relativity gives
us confidence in the theory's validity. Three classical tests demonstrated the short-
comings of Newtonian gravitation and firmly established General Relativity. In the
first, a finite shift of the perihelion precession of Mercury, predicted to be identically
zero in Newtonian gravity, was explained [2, p.282]. Second, Eddington's famous
experiment during the solar eclipse of 1919 detected a small deflection in the path
2
of a light ray from a distant star as it passed through the sun's gravitational field
[3]. Finally, the redshift as photons climb out of gravitational fields predicted by
General Relativity was observed by Pound and Rebka in 1960 [4]. Many other ex-
periments further verified General Relativity as the appropriate theory of gravity on
cosmological scales.
General Relativity has the additional prediction of the existence of gravitational
radiation. This gravitational radiation is the result of large masses interacting with
each other. Given a sensitive enough instrument, this radiation should be detectable.
Nonetheless, the detection of gravitational radiation has proven to be a difficult goal
and remains unseen almost ninety years after the formulation of General Relativ-
ity. Due to the very weak nature of gravity, gravitational waves continue to elude
physicists.
Some of our strongest evidence for the existence of gravitational radiation comes
from Taylor and Hulse, who made precise observations of the orbital motion of the
binary system PR 1913+16 [5, 6, 7]. One of the stars was a pulsar, providing a very
precise natural clock for the system. Over twenty years they mapped the decay of
the stars' orbit and showed that it very precisely matched what general relativity
predicted if the energy lost by the pair was radiated away by the emission of grav-
itational waves [8]. Their observations provided indirect evidence that gravitational
radiation existed and they were award the Nobel Prize for their work in 1993. There
is additional evidence from other areas of physics that further support the existence
of gravitational radiation [9]. Although this evidence is a massive triumph for general
relativity, it is no replacement for direct detection of gravitational radiation.
The Theory of General Relativity leads to very complex equations, many of
which are immensely difficult, if not impossible to solve. Assumptions are necessary to
make headway into the physics. This leads to a strong desire for confirmation that the
theory is correct, and that the approximations being made are valid. Just as quantum
mechanics was proven with and improved upon by observation of electromagnetic
radiation, detection of gravitational radiation would vastly improve our understanding
of general relativity.
More important however, a gravitational wave detector is more than simply
a confirmation of a theory. It will also be an observatory of the universe. When
astronomers first looked at the universe outside of the visible spectrum they discovered
objects never viewed before. Quasars and Pulsars were discovered. A-i in i .. ii, was
revolutionized, and with it, astrophysics. When the universe is viewed with an entirely
different medium, a revolution of understanding is a very strong possibility. The fact
that gravitational radiation is so hard to detect is also an advantage. Since it interacts
so weakly with matter it will allow observation of events that are otherwise obscured.
1.2 Interferometer as a Detector
In order to detect gravitational radiation, we must first understand how it in-
teracts with the world around us. This can be achieved by the following derivation
of how a gravitational wave affects light traveling to a distant free mass mirror and
back again.
Within the context of General Relativity, a space-time interval can be written
in general as
ds2 g~vdx'dxV, (1.1)
where g,, is the metric tensor that describes the curvature of space. Assuming weak
gravitational fields, the metric is essentially flat (\i 1il:, wski) space with a small added
perturbation, defined as
9gV,, = lpV + h1V )
(1.2)
where
-1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0100
0010
0001
(1.3)
and
I h,, I <1
(1.4)
The metric tensor g,, must satisfy Einstein's Field Equation, which in the weak
field approximation reduces to [2, p. 214]
( 2
(1.5)
1 02
c2 t2 O.
Since Einstein's Field Equations are gauge invariant, we can choose a transverse
traceless coordinate system as the gauge. For this gauge, waves propagating in the
z-direction provide a solution to Equation (1.5) when they have the form
0
0
h, =
0
0
0 0 0
hil h12 0
h12 -hll 0
0 0 0
(1.6)
where we define
hll = h+(z ct)
(1.7)
(1.8)
h12 =h (Z ct).
5
where h+ and h, represent two orthogonal polarization states of the wave and the
wave is traveling along the z-axis.
It is easy to see how light can be used to detect this bending of space. Light
alv--v- follows a geodesic, which means ds2 = 0, and thus Equation (1.1) becomes
0 -(cdt)2 + (1 + h+)(dx)2 + (1 h+)(dy)2 + (dz)2. (1.9)
For light traveling along the x-axis, Equation (1.9) becomes
dx c
(1.10)
dt t/1+h +
Assuming that light travels a length I and back again, we can calculate the amount
of time it takes the light to return in the presence of slightly curved space by the
following
21 dx dx= dt' (1.11)
t 1
c i htt') + O(h)}dt'. (1.12)
Rearranging the equation for the amount of time that passes before the light returns
gives
21 1 t
t, -+ h+(t')dt', (1.13)
c 2 2t_21
where we have made the assumption that t, can be replaced by the unperturbed
round trip time, 2, in the integration limits.
Assuming that the gravitational radiation has a simple sinusoidal variation such
that h+(t') = ho cos (cgt'), and writing it as a phase change in the light, where the
light frequency is wo, Equation (1.13) becomes
60(t) ho cos ( )dt' ( 1.14)
2 J_21
ho osin (,t) -sin Lg t ]- } (1.15)
wo sin / /
ho cos t (1.16)
Based on Equation (1.9), it is not difficult to see that if we perform an equivalent
calculation for light that travels along the y-axis, we find that the phase shift is equal
in magnitude but opposite in sign to that found in Equation (1.16). This -ii--.- -1-
that a Michelson interferometer should be a suitable instrument for detecting the
phase shift that accompanies the passage of a gravitational wave since a Michelson
interferometer measures the phase difference of the light in one arm relative to the
other arm. As the gravitational wave propagates down the z-axis perpendicular to the
plane of the interferometer, the phase shift imparted by the wave will modulate the
light paths, hence the intensity of the light hitting the detector. Figure 1.1 shows the
effects of a gravitational wave on a simple Michelson as it propagates perpendicular
to the plane of the detector.
0 7T/2 7t 3T7/2
Figure 1.1: A gravitational wave, h+, on a simple Michelson
From Equation (1.16) several things can be learned. The first point to notice
is that in the limit 91 < 1, the signal scales with the length of the arms. A second
c,
point to note is that the higher the frequency of the light, the larger the phase shift.
A third point is that there is a maximum length, or storage time for light, in the arms
which is
< -. (1.17)
c 2
This becomes extremely important in more advanced designs. For a typical detection
frequency of 300 Hz, the length of the arms would have to exceed 250 km before the
signal would start to degrade. This is equivalent to the gravitational wave interacting
with the light path for a quarter period. Should the length of the arms be so long
that the light interacts with the gravitational wave for a half period, then light phase
would be shifted back to zero, thus showing no apparent effect from the gravitational
wave.
A very important conceptual notion needs to be explored here. The mirror in
the above example was taken to be free of all forces, and space was taken to be pliable.
This was inherent in our choice of a transverse-traceless gauge. In the laboratory, this
is counter-intuitive. The more familiar gauge is one in which all points of space are
taken to be fixed. After all, what object is ever truly free of all forces? But the results
of the previous calculation can't depend on the gauge that is chosen. This would be
fundamentally contradictory to physics. The world does not depend on how we view
it. How do we reconcile the apparent contradiction?
In a gauge where distances are taken to be separated by infinitely rigid rods,
then the free mass in the above example must move in that coordinate system in
order to achieve the same results. This means that the gravitational wave can be
described as a force. The force on a freely falling object in this coordinate system is
1 a2 h11
F = m2 = -ml (1.18)
2 Ot2
If we assume the sinusoidal nature of the gravitational wave, Equation (1.18) becomes
1
F = -mlhou0 cos (wt) (1.19)
The mirror will be moving with a sinusoidal motion that depends on the path length
that the light traveled. This becomes important when we talk about how to generate
synthetic gravitational waves in the experiment.
The description of this gauge is even less intuitive than the previous explana-
tion, since it is hard to understand how the position of the mirror relative to the
measurement point can affect the force that the mirror experiences. However, both
explanations give completely consistent results that are expected from the experiment.
The very precise nature of interferometers makes them excellent choices for mea-
suring small length changes. In fact interferometers can measure length differences
in two arms that are less than the diameter of an atom. Conceptually this can be
difficult to understand. The atom itself is a moving object, and the mirror is made
up of all these objects with uncertain positions. However, the nature of coherent
light sampling all these moving objects together causes this effect to average out, and
high precision interferometry can be achieved. Phase sensitivities comparable to what
needs to be accomplished to detect gravity waves have already been demonstrated at
higher detection frequencies [10].
Unfortunately, interferometry, like any measurement technique, is plagued by
noise. Noise comes in two different forms. The first is fundamental noise associated
with the nature of light. No amount of improvement of materials, isolation, or other
technological advances will reduce this noise. Only by manipulating the light can
advances be made in driving this noise down. The two noise terms here are shot
noise and radiation pressure noise. The second noise term is associated with the
environment in which the interferometer operates. These noise terms are thermal
noise, seismic noise, and many more not mentioned here.
1.2.1 Shot Noise
Shot noise is the result of the random nature of the universe. Heisenberg's
uncertainty relation guarantees that not all the photons that arrive at the beam
splitter and go toward the antisymmetric port, defined by the port that the laser
light does not enter, are ones that were created by the gravitational wave. Some were
shifted due to the uncertain phase information intrinsic in the photon. The quantum
mechanical result for this error is stated as
1
AnA > -, (1.20)
2
where n is the number of photons and Q is the phase of the photons.
An equivalent derivation of the noise uses the classical description of shot noise.
In this case we can assume Poisson statistics for the photons. A detector at the
antisymmetric port would record the number of photons incident at a given time
interval, r. For many measurements the distribution of these measurements would
be centered at n with a variance (An)2 = n or
An (1.21)
For this simple calculation we assume that all the photons detected are the result
of the signal. The power detected is
P =h0on, (1.22)
where h is Planck's constant divided by 27. We can write down the relative error
AP 1
(1.23)
P
where P is the average power in the signal and n is the average number of photons
detected in a time interval r. This immediately shows that the more photons in the
signal, the larger the signal to shot noise ratio. Often we will discuss increasing the
signal at the antisymmetric port in order to increase sensitivity. This is equivalent to
stating that we are lowering the sensitivity to shot noise.
Higher sensitivity can be achieved in several v--=i-. Equation (1.16) tells us that
the signal is proportional to the length of the arms. There is a physical limit on how
long a straight path can be made on earth. A simple way to increase this length is
to fold the arms. This can be accomplished by one of two v---.
Figure 1.2: Example of a d,1 iv line to increase detector sensitivity
1.2.1.1 Delay Lines
The first way is to separate the beams on each reflection. This is shown schemat-
ically in Figure 1.2. In order to maximize the folding, the input beam is injected in
such a way that at each point where the beam reflects off the mirror the light creates a
circle around the mirror with spots at discreet intervals. This way of folding the arms
has the problem that light scattered out of the beam path can enter another beam
path leading to an additional noise term. This scattering becomes critical when the
beams start overlapping, so mirrors have to be big enough to accommodate folding
equal to about 250 km without beam overlap. This is very difficult to achieve [11,
p.86-90].
r2, t2 Einb r, t E
Ewlans 1' 1 r1
I inb *E
1 EL
+ E +
-mo
Figure 1.3: Example of a cavity to increase detector sensitivity
1.2.1.2 Fabry-Perot Cavities
A second method for beam folding is to use cavities in the arms. Figure 1.3 shows
the fields in a cavity. This is such a fundamental building block of a gravitational wave
detectors that we look deeply at the detailed behavior of this object in C'! Ipter 2. For
now the most relevant thing to mention is that the light in the cavity has an average
storage time that is equivalent to an increased arm length. That length increase is
related to the reflectivities of the input and end mirrors. An additional problem is
created by overlapping the beams. The arm cavities must be controlled such that
the light builds up power in the cavity. Although this may seem like an unnecessary
complication, the additional shot noise sensitivity that this achieves is well worth
it. In fact, controlling a single cavity is a simple affair using the Pound-Drever-Hall
technique [12].
1.2.1.3 Power Recycling
Another way to increase the amount of signal is to increase the number of photons
that are present to be phase shifted. This can be accomplished by increasing the
input power. There is a technological limit to the amount of power that can be
delivered to the input of an interferometer which will be quiet enough not to affect
the gravitational wave signal. That limit currently seems to be about 10 watts, and
is hoped to be around 100 watts for the next generation of detectors.
However there is another way to increase the power that is incident on the beam
splitter. An interferometer that is operating on a dark fringe in the antisymmetric
r,, tp
i, ti rpr, tp
rbs, tbs Eo
antisym
Figure 1.4: Power-recycled simple Michelson
port, where no light exits out the antisymmetric port, will reflect all its power back
toward the laser. If a mirror is added at the input port, that power can build up
inside the cavity formed by the end mirrors and the power-recycling mirror [13]. The
fundamental limit here is the amount of losses in the cavity.
We were very careful to call an increase of power in the arms an effective path
lengthening, not a power increase. A fundamental difference is determined by the
nature of the gravity wave. Since the arm cavities are essentially equivalent to in-
creasing the length of the simple Michelson arms, there is a frequency response of the
arm cavities to the gravity-wave signal that will roll off as the signal frequency ap-
proaches the storage time of the arm cavity. The roll off is shown in Figure 1.7 by the
curve with no signal recycling. There is no frequency response associated with power
recycling. The reason that we use arm cavities at all, and do not just remove them in
favor of more power, is because of the complication of the beam splitter. High power
at the beam splitter leads to thermal lensing and high absorption, which significantly
affects the performance of the interferometer, and limits the power recycling factor
that can be achieved.
r,, t
ri, ti
; Erefl
S rbs tbs Eo
rsr, tsr
E +
antisym
Figure 1.5: Signal-recycled simple Michelson
1.2.1.4 Signal Recycling
There is one last cavity that can be formed in order to increase the signal. If a
mirror is introduced to the dark port any signal that is sent out of that port will then
reflect off that mirror and build up in the cavity formed by it and the end mirrors of
the arms. This configuration is known as Signal Recycling and was first proposed by
Meers [14] and demonstrated by Strain [15].
When the signal-recycling mirror is placed in a position so that it builds up the
carrier in the cavity formed by it and the Michelson end mirrors, then the signal
is built up in such a way that shot noise sensitivity is reduced at the expense of
bandwidth. The signal-recycling mirror increases the storage time of the gravitational
wave signal, and when that storage time exceeds a half cycle of the gravitational wave,
then the signal degrades.
r2p, t2p
r1p, tip1b
r2i, t2i li tli
[ 1 Erefl
rbs, tbs E
r, s, t
r tsr5 sr
Figure 1.6: Signal recycling with arm cavities
1.2.1.5 Resonant Sideband Extraction
There is an alternative operating point for the signal-recycling mirror when one
includes arm cavities. When the mirror is placed in a position such that the light is
antiresonant in the cavity formed by it and the arm cavities, then the signal actually
gains bandwidth at the expense of shot noise compared with the equivalent system
on resonance for the carrier. If the arm cavity storage time is then increased, the shot
noise level increases, and the detector has a higher sensitivity at the desired detection
frequency than an equivalent signal recycling instrument. This is known as Resonant
Sideband Extraction (RSE) [16]. The signal on the light created by the gravitational
wave sees an effective lower reflectivity in the end mirrors of the arm cavities than the
carrier sees, and it is extracted by the signal extraction mirror. Without arm cavities
this effect does not occur.
There is of course a broad range of intermediate states, where the signal-recycling
mirror is neither in the recycled case or the RSE case, but rather in between. This is
known as detuned signal recycling.
-- Broadband Signal Recycling
100000- Detuned 10 Degrees From Broadband
-....... Detuned 87.4 Degrees From Broadband (LIGO II)
----- Resonant Sideband Extraction
10000 No Signal Recycling
S1000
100
10 .-----.-.---.... .. ---...-.-
o 10
0.1-
0.01 0.1 1 10 100 1000 10000
Frequency (Hz)
Figure 1.7: Shot-noise limited sensitivity
Figure (1.7) shows the shot noise limit for different positions of the signal-
recycling mirror. The parameters of the interferometer are those of the initial design
for the United States dual-re' i, .1 gravitational wave detector, LIGO II. The effects
of tuning are very clear. As the signal-recycling mirror is tuned from broadband to
RSE, the signal has a peak sensitivity at a specific frequency at the cost of bandwidth.
The RSE configuration has the highest bandwidth with the lowest sensitivity for fixed
arm cavities.
The detector was designed to operate at a specific tuning. If it was desirable
to operate in RSE mode, for example, then the interferometer would be redesigned
for peak performance. In this case, the arm cavity finesses would be increased to
increase the sensitivity and to lower the bandwidth. It is important to realize that
signal recycling will work with a simple Michelson, but RSE needs cavities in the
arms. Without arm cavities the signal simply doesn't build up in the signal-recycling
cavity, the shot noise becomes larger, and no advantage is gained.
1.2.2 Radiation Pressure Noise
Another noise that is quantum mechanical in nature is radiation pressure noise
[17]. This noise term is created by the discreet arrival of photons at the mirror surface.
This noise term is proportional to the square root of the power in the arms over the
mirror mass. This doesn't bode well for increasing the power on the end mirror, but
there is an additional component to this noise that has been ignored. The frequency
response of the radiation pressure noise falls off at one over the frequency squared.
At low frequencies this dominates the fundamental noise spectrum, but at higher
frequencies shot noise is dominant.
1.2.3 Thermal Noise
Thermal noise [18] mostly comes in the problematic form of thermal motion.
That is, any object that is used to hold the mirror, and the mirror itself, has some
thermal energy that causes the mirror to move, creating noise in the detection signal.
There are several tricks to pl i, here.
The first trick is to suspend the mirror from a wire [19]. This concentrates all
the thermal motion into very distinct frequencies that are defined by the pendulum
mode and the "violin" modes of the wire-mass system, with a spectral distribution
controlled by the Q value, or quality factor. The Q value depends on the material of
the wire, material of the mirror, and the manner in which the wire is attached to the
mirror and the supporting structure. These parameters can be fine-tuned so that the
resonances are far enough away from frequencies of interest, and so that the Q value
is as high as possible.
The thermal vibration in the mirror substrate itself is slightly more difficult to
manipulate. Once cylindrical mirrors have been chosen as the obvious shape, the
free parameters are the height-to-diameter ratio and the material of the substrate.
Picking a material with a good Q is extremely important, but nature has been kind
here, and fused silica works well for the first generation of detectors [18].
Another form of thermal noise is Braginsky Noise, or thermoelastic noise. This is
created by the fact that the mirror substrate absorbs quanta of energy from the light.
This leads to a nonuniform thermal distribution in the surface of the mirror, creating
small scattering centers for the light. This noise level is expected to be below other
noise levels for the first generation of detectors, but causes serious problems when the
power levels increase inside the interferometer in advanced detectors [20].
1.3 Sources
The most common source of gravitational radiation is from two objects orbiting
around each other. In order to achieve some sense of how large the gravitational wave
signal is, it is useful to calculate the strain for two objects orbiting each other in the
far field limit.
Making direct analogies with electromagnetic radiation we arrive at an equation
for the gravitational radiation. There will be no monopole radiation since it doesn't
exist in electromagnetic radiation. This is the result of the conservation of energy.
There will be no dipole radiation because there is only one "(*
*
The quadrupole moment [11, p.30] can be written as
/1, = dV (, -t 6r2) p (r) (1.24)
Writing the gravitational equivalent of quadrupole radiation gives
2G
h Rc = R4 /. (1.25)
Equation (1.25) can actually be derived more rigorously from Equation (1.5). [2,
p.226-233]
For this simple example, we assume each object has a mass M, that they are
separated by 2ro, and that they have a constant angular velocity w. If the z direction
is perpendicular to the plane of orbit, the quadrupole moments are
I, = 2Mr cos2 (t) (1.26)
I 2Mr2 cos2 (Lt)- (1.27)
and
Iy = Iy, = 2Mr2 cos (uw) sin (wt) (1.28)
Assuming the observation happens directly along the z-axis at a distance R then
Equation (1.25) becomes
32x22G
h, = -hy, = Mr cos (2wt) (1.29)
Rc4 0
and
-322G 2 sin (2) (.30)
he,= Ae = R MrYf2 sin (2wt). (1.30)
RC4
Using classical gravity to give the orbital angular velocity as
GM
2 GM (1.31)
4r3'
and defining the Schwarzschild radius for a point mass as
rs = 2 (1.32)
we can write the amplitude of the gravitational wave as
Ihl rss (1.33)
rR
Assume that the two orbiting objects are a binary neutron star system. The
mass of neutron stars is 1.4 times our sun, or 3 x 1030 kg. If the stars are almost
,uii' lir,.- then ro = 20 km, and w ; 400 Hz. Putting the binary stars in the Virgo
System means R w 4.5 x 1023 m. This gives
Ihl I 1 x 10-21. (1.34)
This is a very impressively low strain, but unfortunately does not tell the whole story.
An important question to ask is how many events like this are detectable by a
realistic earthbound detector. Binary coalescence actually will come in three forms,
Neutron Star-Neutron Star, Neutron Star-Black Hole, and Black Hole-Black Hole.
The coalescence will have to be in its final stages in order to emit the strongest
waves possible, since that is the point at which the masses are closest. Trying to
calculate the number of such entities that will be detectable by the first generation of
gravitational wave detectors is extremely difficult, and has huge errors, ranging over
several decades [21].
Using observations of how many neutron star binaries are in our galaxy, the best
guess at this time is that an event will be detectable every 10-6 to 10-7 times per
year that originates in our galaxy [22, 23, 24]. Making assumptions that they are
distributed evenly over all space, then a detector with 10-23 strain sensitivity will be
able to detect 1 event every 1000 to 10000 years. However, if the strain sensitivity
is improved by one order of magnitude, the number of events goes up by 103, since
Equation (1.33) is inversely proportional to distance from earth, but the number of
events goes up as the distance from earth to the third power. That means about
one event per year will be detectable. The lowest estimate is around one event every
10 years. Any improvement to the detector significantly increases the detection rate
[21].
Fortunately neutron star binaries are not the only source of gravitational waves.
They are the best understood, and most predictable. Other possible sources of grav-
itational waves are non-axial symmetric pulsars, non-axial symmetric supernovas,
coalescing black holes, and a stochastic background that is still present from the Big
Bang. There is also the important fact that there are many things about strong field
gravity that is not fully understood. This leaves open the possibility of strong sources
that we haven't yet recognized. One thing is very clear though. In order to detect
gravitational waves every possible avenue for reducing noise and increasing strain
sensitivity must be explored.
1.4 Motivation for Future Detectors
There is currently a huge effort in the experimental gravity wave world commu-
nity to build detectors that will detect gravitational waves. There are six detectors
that will come on line in the next 10 years. An array of detectors is important in
order to correlate the data since the signal to noise ratio will not be very large. The
United States project, LIGO, will be the one most frequently discussed in this work.
It is a power-recycled Michelson with 4-km long arm cavities [25].
Sapphire test masses
1 0 - :::: ::: : ::::: :::: :: :: :: : ::: : : ::::: : -: :::: : '': -:
......... .. .... : ,... .i i .. .. I . th e rm a l ........ ....... ... ..
....... Shot noise
Int. thermal
Susp. thermal
S. . -- Radiation pressure
10-22 ::::, :- Total noise
..-23
10-24
101 102 103
flHz
Figure 1.8: Expected noise levels for LIGO II
The first generation of LIGO has been designed to give a strain sensitivity of
S10-22/Hz1/2. The more favorable estimates of event rates that this detector will
be able to sense is not better than a few per year, and most likely no events will be
detectable. In order to guarantee that the detector will see gravitational radiation
at a rate of better than one per year the sensitivity must be increased by at least a
factor of 10. In order to do that all the noise sources mentioned previously must be
improved upon.
Figure (1.8) shows the expected noise in LIGO II for sapphire test masses [26].
Every noise source contributes significantly to the total noise curve. The most critical
are internal thermal noise of the test masses themselves and the shot noise of the
detector.
The next generation of gravitational wave detectors will require all the techniques
to reduce shot noise discussed in Section 1.2.1. In order for the detector to work a
scheme for locking all the mirrors so that the light is resonant in the appropriate
places needs to be devised. That is the main thrust of this work.
Chapter 2 will analyze the sensitivity of different interferometer configurations.
A detailed transfer function for different configurations will be derived. Control
schemes for the different configurations will be discussed and when possible, calcu-
lated. A method for comparing different control schemes for the same configuration
will be presented. A detailed explanation for how the experimental dual-recycled
tabletop instrument was controlled will be presented. The frequency response of a
signal-recycled instrument will be discussed in more detail.
Chapter 3 will discuss the prototype dual-recycled cavity-enhanced interferom-
eter that was constructed on a tabletop. It will explain in detail how the parameters
were decided on. It will give a detailed explanation for the error signals expected
from the instrument. The instrument will be characterized through measurements
of losses and the amount of power buildup in cavities. The performance of the error
signals will be assessed with a measurement of the locking matrix. The low frequency
sensitivity of the instrument will be shown. The measurement of the high frequency
response of the dual-recycled and slightly detuned instrument will be presented and
compared to the predictions.
Chapter 4 will summarize the work done. It will draw some final conclusions.
Possible future work will be discussed.
CHAPTER 2
THEORY
The analysis and control of a complex interferometer requires some general the-
ory along with detailed calculations. This chapter will discuss the input light field,
the generation of additional frequency components on that light field and detection of
the fields. It will then explain how those detected light fields can be used to control
the interferometer. The next step will be a careful analysis of the different topologies
for the instrument, including a section on the control signals for each topology.
There are two regimes in which an optical configuration can be analyzed. The
first is its response to time dependent perturbations. This analysis ignores the storage
times of complex cavities, but it is very useful in developing a locking scheme for the
interferometer. It will also explain the slow time response of the interferometer to
gravitational radiation, where the signal frequency is much lower than the storage
time of any of the cavities. The second regime is the response to time dependent
perturbations. This frequency analysis is fundamental to how the interferometer will
respond to the gravity wave in general.
In order to understand how signals are generated in complicated multi-cavity
interferometers, it is useful to build up a model for the static case using subsections.
This will allow us eventually to express a full transfer function of the complete inter-
ferometer, which will depend on the frequency of the light. Although there are pro-
grams which allow modeling of different interferometer configurations [27], one needs
to go through this exercise in order to develop and understand a locking scheme.
The frequency analysis will be accomplished largely through the modeling program
Finesse.
2.1 Light Fields
In most cases the light field going into a complex interferometer has some fre-
quency shifted components. The advantages of this will be explored more thoroughly
when we discuss locking the cavities and the Michelson interferometer to the center
frequency, or carrier. The frequency shifted fields come in two flavors, either single
sideband or paired sideband. Since our experiment uses only paired sidebands, that
is the situation that we consider in detail.
2.1.1 Generating Sidebands
In order to generate a pair of sidebands the carrier passes through an electro-
optic modulator (EOM) [28]. The modulator is driven with a sine wave typically in
the RF frequency region which modulates the index of refraction through the electro-
optic effect. This results in an effective path length change that oscillates at the drive
frequency.
The output field is then
Epr = Eoe-i(wot+msin nt), (2.1)
where Wo is the light frequency, and Q is the frequency at which the EOM is driven.
m is related to the amplitude of the driving frequency. If we assume m is much less
than one, as would normally be the case, then Equation (2.1) becomes
Epm Eoe-iot (1 im sin Qt) (2.2)
SE e-wo + -i(wo+Q)t m i(wo-n)t (2.3)
Figure (2.1) shows how the electric field looks in a frequency domain. There is
a phase relationship between the carrier and the sidebands that changes with time.
Eo
E,
E-1
EB,
Figure 2.1: Carrier with sidebands in frequency space
The sidebands rotate in the complex plane in opposite directions with respect to the
carrier at frequency f. For a good discussion of this, see Mizuno's thesis [29, p.20-28].
At this point it is important to note that light reflecting off a mirror oscillating
in the direction of the incident light z at a given frequency has the exact same effect
on the light field. It takes some portion off the carrier and frequency shifts it, such
that it creates a pair of sidebands. This is the same effect as doppler shifting. As the
wavefronts reflect off a mirror moving opposite the direction of incidence, the reflected
wave fronts are closer together than the original wavefronts, shifting the frequency
up. As the light reflects off a mirror moving .1-.v li, from the incident light it shifts the
reflected light's frequency down. For a mirror moving sinusoidally the reflected light
has the form of Equation (2.1)
It follows that when a gravity wave interacts with an arm of the interferometer,
given Equation (1.19), the effect the gravitational wave has on the light field in the
instrument can be described as a pair of frequency shifted sidebands. This realization
becomes very useful in synthesizing signals. Rather than trying to move a mirror to
simulate a gravitational wave, light can be added to the system through the end
mirror of one of the arms. If the added light is offset from the carrier by a frequency
ug then it simulates one half of the pair of sidebands that would be created by the
interaction of a gravitational wave of frequency ug with the detector.
2.1.2 Sideband Detection
Once this field has traveled through an optical device, be it either a cavity or
complex interferometer, the sideband amplitudes and phases have changed due to
the frequency dependent transfer function, T (k). We can write an electric field with
paired sidebands after it has interacted with an arbitrary transfer function as
E = [ (ko) + T (k +) e- T(k_) e] Eoe-iw (2.4)
2 2
where k = ko kn, k = ", and Eo is the amplitude of the carrier before interacting
with the system.
A photodetector will only detect the power of the total field, or
P EE*, (2.5)
which leads to
EE* =D.C. + -{T (ko) T* (k+) + T* (ko) T (+) (2.6)
2
T (ko) T* (k_) T* (ko)T (k_)} IEo 2 cos Qt
+ {T (ko) T* (k) T* (ko)T (k)
2T(o (_)- ()(_)} t (2.7)
+ T (ko) T* (k-) T* (ko) T (k-) I} I Eo 2 sin Qt (2.7)
Realizing that a + a* = 2R {a} and a a* = 2S {a} allows us to write
P =D.C. + mR {T (ko) [T (k+) T (k_)]*} IE 2 cos Qt (2.8)
ms {T (ko) [T (k+) + T (k_)]*} Eo 12 sin t (2.9)
Taking the photocurrent and using heterodyne detection [30, p.885-902], the
signal is beat against a sine wave with the appropriate demodulation phase, Q, such
that
2,
s= P cos (Qt + ) dt. (2.10)
Choosing the appropriate phase, Q, picks up a factor of 1 from the integration
and yields the final results of
Sinpase 72 {T (ko) [T (k+) + T (k_)]*} IEo 2 (2.11)
2
Squadrature {T (ko) [T (k+) T (ko)]*} IEol2 (2.12)
2
These two signals, the in-phase and the quadrature, can be generated by two
different processes. The in-phase component will be generated when there is a phase
shift between the sidebands and the carrier. Assuming that the sideband phase does
not change as the system departs from its operating point, but that the carrier receives
some phase shift, then it is the beat of the carrier with the sidebands acting as a local
oscillator that would create a signal which gives information about how the system
phase shifted the carrier. This signal could be used to control the system so that the
phase shift imparted to the carrier does not change with time. This signal is known
as an error signal. The in-phase signal will be the sine of the phase difference between
the paired sidebands and the carrier, which is a good error signal since it is zero when
the phases are the same, and has opposite signs depending on whether the phase shift
is positive or negative.
A quadrature signal will be created when there is an imbalance in the paired
sideband amplitudes. Assuming that the carrier and sideband phases do not change
as the system departs from its operating point, but that the amplitude of the two
sidebands changes antisymmetricly, then an error signal is created in this component.
Again, the error signal is zero at the operating point and changes sign depending on
which direction it has departed.
There are other v--,v in which the error signals can be created. For example, a
change in the phase of the sidebands will show up in the quadrature signal. Often the
transfer function of the system acting on the input light field will create a signal that
appears in both quadratures. If, for example, the transfer function depends on two
degrees of freedom, it is desirable to have signals to sense these degrees of freedom
that are independent from each other. The input light field and the free parameters in
the transfer function are usually chosen such that the degree of freedom that creates
the signal is as pure in one component as possible. This leaves the other component
of the signal to be used for the other degree of freedom, and the two processes then
create error signals that are independent from one another.
There may be some confusion here as to why the in-phase signal is obtained
when mixing down with sin Qt and the quadrature signal is obtained when mixing
down with cos Qt. Typically the situation would be the reverse, since the sine is the
imaginary part and the cosine the real part of the complex optical field. This should
not be a concern since it depends on the phase used to demodulate the signal. If
one started the derivation with sidebands being generated with a cosine modulation
rather than the sine that was chosen here, then the final demodulation phases would
have been shifted by 7 and you would have the standard concept of in-phase and
quadrature. The important thing to realize is that there are two signals that are
orthogonal to each other.
2.2 Locking Matrix
Ultimately the goal of the interferometer is to obtain the highest sensitivity
to the gravitational wave. To accomplish this, complex optical configurations are
used with many degrees of freedom. These degrees of freedom must be sufficiently
controlled such that they don't introduce noise into the gravity-wave channel.
Optimally a change of one of the degrees of freedom from the operating point
would not affect the other degrees of freedom. However, that is usually not the case
in complex optical configurations. Take the example of a power-recycled Michelson
with arm cavities. A change in the length of the arm cavities results in a phase shift of
the carrier coming from the arms. This phase shift causes the power-recycling cavity
to no longer have a buildup inside it, since the incoming light is no longer perfectly
constructively interfering with the light in the cavity. The power recycling factor is
reduced, thus reducing the sensitivity to the gravitational wave signal.
AL- M S
Figure 2.2 Feedback loop for a general system
Figure 2.2: Feedback loop for a general system
There is a general way to discuss how independent the signals used to control
the degrees of freedom are and how to decide on a control scheme. Figure 2.2 shows
a general system with feedback. The matrix M is the optical system, and in general
not diagonal. The matrix G is the gain matrix, and for simplicity sake is usually
diagonal. The vector S are the error signals that we use to feed back to the degrees
of freedom. The vector AL is the disturbance of each of the degrees of freedom. We
can now write down the error signals in general as
S MAL + GM S. (2.13)
Solving this equation for the signal vector we get
M
S M AL. (2.14)
1- GM
The situation that requires analysis is two degrees of freedom, where AL =
{AL1, AL2} and M is a 2 by 2 matrix. We will use this as a basic building block for
much more complicated situations. In this case
G1 0
G (2.15)
0 G2
The locking matrix is created from the equations that govern how S1 and S2
depend on the degrees of freedom. Writing down the error signals as
as1 as1
= AL1 + AL2 (2.16)
aL1 aL2
as2 aS2
2 = AL + AL2 (2.17)
aL1 aL2
The locking matrix is the Jacobian, defined as
as1 as1
tL1 tL2
M (2.18)
aS2 aS2
9L1 tL2
It is now useful to solve for the two signals in order to see how one signal depends
on the two degrees of freedom. After some linear algebra the results are
(-G2det M + AL + a AL2
Sa (2.19)
1 -&, aSG2- GG2detM
-G1 det M + AL2 + JAL1
S2 (2.20)
1 1 G 1G2 -GG2det M
There are a few cases to consider. The first case is when the two degrees of
freedom truly are independent from each other. That is equivalent to assuming that
M is diagonal. We can also assume G1,2 > 1 for the steady state solution, since
we want enough feedback to hold the resonant conditions for the light. Then these
equations become
9S1 ALl
S = AL L 1 (2.21)
as ALI
aCLl
2 aL2 AL2 --2 (2.22)
1 9Sz rAL
1 aG2 G2
The second case to consider is when the off-diagonal terms in M start to get
large compared to the diagonal terms. In this case the signals start to get mixed.
Without taking special steps with the gain distributions, changes in one degree of
freedom appear in the error signal for the other degree of freedom causing the system
to try to compensate for this signal. This is a bad situation, and in general we want
Si to be dependent only on ALi
Taking Equation (2.20), we realize that for S2 to be dominated by AL2 then the
condition
as2 aS2
G1 det M 2 > (2.23)
OL2 OLi
must be true.
Investigating the case where the degree of freedom L1 dominates signal S2, then
a2- >> 2. We can now write a simple expression for G1.
aS2
GI det (2.24)
det M
G1 > os, s (2.25)
0L1 &S2 8%2
OL1 OL1
OS2 OS1
If L2= IL2 corresponding to the det M = 0, the signals are linearly dependent.
OL1 OL1
We cannot lock this system since there is no unique locking point.
As long as the system is not very linearly dependent, the gain does not need to
be abnormally high. As the system becomes more and more linearly dependent, then
det M 0 and in order to achieve lock Gi oo.
There is an absolute measure for how orthogonal the signals are to one another,
given by the cross product of the two signals. The angle between the two signals is
defined as
Six S
sina -- (2.26)
S1 S2
Looking at equations (2.16) and (2.17) we can see that the cross product of the two
signals is
aS1 aS2 as1 aS2
sina 2 (2.27)
as, ++ a S s2 2s2
Ay 1 aL2 aL1 aL2
which we recognize as
det M
sin a (2.28)
sSi S as1 as2 I S2
y aL1 L2 yL1 a iL2
The smaller a is, the more linearly dependent the two signals are. This gives us an
absolute measure with which to compare locking schemes.
2.3 Simple Cavity
The simple cavity is the most fundamental improvement for an interferometric
gravitational wave detector. It is very useful to explore the properties of a simple
cavity completely. Since a suspended cavity cannot detect gravitational radiation the
frequency analysis will be saved for when it is applicable in a detector.
r1'
Eansr2, t2 Einb r, t]
+" L + E"
Eino
Figure 2.3: Fields in a simple cavity
2.3.1 Light Fields
Figure (2.3) shows the notation used. We adopt a few conventions here that
are not necessarily universal. Light reflected off the coated side of a mirror gets
a minus sign. Light reflected off the substrate side of the mirror gets a plus sign.
All transmitted light is real and positive and the substrates have zero thickness.
As for notation, all fields traveling away from a mirror have the subscript o and
all fields traveling to the mirror have the subscript b. The mirror reflectivities and
transmissivities, ri and ti, are for field amplitudes. For power quantities they are
related by Ri =|ri2 and T = It 2.
Writing all the equations for the fields gives
Eino = t1E0 r1Einb (2.29)
Einb -r2-2ikLEEno (2.30)
Etrans = t2-ikLEino (2.31)
refl t= rIE + tlEinb. (2.32)
Continuing on in painful detail, we substitute Equation (2.29) into Equation (2.30)
and solve for Einb
Enb = -2-2ikL (tiE rTEib) (2.33)
Eib (1 r1r2e-2ikL) -r2 -2ikLE (2.34)
r2t1e-2ikL
Einb 2= Eo (2.35)
1 rlr2e-2ikLe
Solving for the other fields gives :
Eino --2ikL E (2.36)
1 rr2e2
tlt26-ikL
Etrans = r -2ikL E0 (2.37)
1 rr2e 2kf
ri (2 + 2) e-2ikL
Ef -2kL Eo. (2.38)
1 rir26
It is useful to define a complex reflectivity of the cavity such that
Erefl = -rFPEo. (2.39)
Assuming that there are no losses in the coating (generally a good assumption), so
that r2 + t 1, we can write
-r1 + r26-2ikL
rFpp 1 22ikL (2.40)
1 rlr2C-2ikL
There is a lot to be learned from these equations. If we assume that the cavities
are resonant for the carrier, such that L ', we can define the buildup inside the
cavity as
Eio 1
io > 1. (2.41)
tlEo 1 r1r2
This number is ahv-l- greater than or equal to 1 since the limit where r2 -- 0 implies
that the field inside is simply the light transmitted through the first mirror. As r2
gets larger than 0 the amplitude of electric field in the cavity increases. This changes
if the light is not on resonance.
The light reflected from the cavity is
Erefl r1 72
(2.42)
Eo 1 r1r2
The numerator is negative if r2 > r1, and reflected light is shifted by 180 degrees in
phase from the incoming light. This condition is known as an over-coupled cavity. If
r2 < r7 then the reflected light has the same phase as the incoming light, which is
known as an under-coupled cavity. If r2 r1, then there is no light reflected from
the cavity, and it is impedance matched, or critically coupled. If there are no losses,
then the transmitted power of the impedance matched cavity is equal to the input
power. Losses in the cavity can often be combined to be included into the r2 term,
such that r2 2 r2 1 sf and t2 is unchanged. This approximation concentrates
all the losses in the second mirror, and does not affect any of the equations used thus
far.
It is also useful to calculate the phase shift that the light receives from a cavity.
From Equation (2.42) we calculate
ri (1 + R2)- 2 + R1) cos (2kL)
tan r2 sin(2L)
T2 (1 1) Sin (2kL)
(2.43)
Figure 2.4 shows the phase shift that the reflected light receives from an over-
coupled cavity. When the light is resonant with the cavity the reflected light is shifted
by 180 degrees. When the light is antiresonant the phase shift is zero or 27. This will
be a useful fact for deciding where to place sidebands with respect to a cavity's free
spectral range in order to get the most useful locking signals.
2
Ca
0)1
Ca
04--
-FSR/2
0
Frequency (MHz)
FSR/2
Figure 2.4: Phase shift of light reflected from an over-coupled cavity
There are two quantities that are often used to define a cavity. They are related
to the length of the cavity and the reflectivities of the mirrors. The free spectral
FSR
-c/2L 0 c/2L
Input Light Frequency
Figure 2.5: Transmission of an impedance matched cavity
range of a cavity is defined as
FSR
(2.44)
('!I ,:1i5; the cavity's length on the order of wavelengths causes the cavity to go
through resonances every free spectral range. The resonances occur at a length change
equal to a half wavelength of the input light. If there are sidebands on the light, and
they are at a frequency that equals the FSR then they will also be resonant in the
cavity when the carrier is resonant.
The finesse of the cavity is defined as
F= -
1 rr2
(2.45)
This quantity is a measure of how much light is built up in the cavity.
Figure (2.5) shows the transmitted field of a cavity as it is scanned over a wave-
length. The physical parameters that can be measured are the FSR and the full
width half maximum (FWHM). The FWHM is related to the Finesse and the FSR
[31, p.408-436] by
FSR
FWHM F
F
(2.46)
One can determine the reflectivities of the mirrors of a cavity by measuring the
FWHM, the FSR, and the losses of the cavity. By measuring the FWHM and the
FSR the finesse can be calculated from Equation (2.46). Solving Equation (2.45) for
the square root of the reflectivities gives
A r-I 2
-Tr /r2 + 4F2
2F
(2.47)
(2.48)
A2
F1
Defining the amplitude loss as
IErefl Sloss IEol ,
we can write Equation (2.42) as
losefl
Sloss P0
V =P
and solving for r1 gives
loss (1 A2) V lo)2 (1 A2)2 + 4A2
2
and
(2.49)
(2.50)
(2.51)
Note that s/2 is a direct measurement of the total power loss in the cavity. This
is accomplished by measuring the power reflected from the cavity on resonance and
dividing it by the power reflected when the cavity is off-resonant.
Losses in a cavity are rarely negligible. Assuming 1/8th of a percent losses on
each bounce off a mirror, which seems to be pretty realistic for the mirrors in this
experiment, a cavity with a finesse of 60 has total losses of almost 10' ,
If the light isn't on resonance then the analysis needs to be redone. This will be
useful to calculate the length of a cavity when the RF sideband is sitting somewhere
near the FWHM of the cavity. We can calculate the length of the detuning from the
measurement of the power buildup inside the cavity.
1E22 2
I -i--12 2- B (2.52)
T1 Eol2 r1r2eL-2ikL
gives
1 + RIR2 1
cos (2L) = 2 (2.53)
2rir2
Here we've used the familiar r = Ri. In order to calculate the effective reflectivity of
the end mirror we need to use the reflected field. We'll divide the incoming field by
the reflectivity of the first mirror, since normally the fields measured from a cavity
are the reflected field when the cavity is locked, and the reflected field when a beam
block as been inserted into the cavity. Solving these equations
|Erep|2 1 1 26-2ikL 2
I E2 1 1 rr- 2e-2kL C, (2.54)
Eo12 1 rlr2e-2 ikL '
RB R
C 1 r26-2ikL 2 (2.55)
B R,
gives
1 RI( 1) 1
R2 + R B B (2.56)
1 R1
It will be useful to calculate the reflectivity of a cavity for small changes in
length. This will not only be used for calculating how the gravity-wave signal shows
up in the light fields of the cavity, but also will be used to generate error signals for
the cavity.
For a change in length of the cavity L -- L 61, Equation (2.40) can be written
as
-rl + r26-2ik(Ll)
rFp 1 rlr2e-2ik(Ll) (2.57)
Assuming that the arm cavity is on resonance for the light, Equation (2.57) becomes
rFp = _r 2ik(- ) (2.58)
2.3.2 Error Signals
There are two common processes that generate an error signal [12]. We showed
that signals can be generated either by the change of the phase of the carrier with
respect to the sidebands, resulting in an in-phase signal, or by an amplitude difference
in the paired sidebands, resulting in a quadrature signal.
An in-phase signal is created when the carrier receives a phase shift relative to
the sidebands from a slight change in length of the cavity, but the amplitudes of the
sidebands stay that same. The obvious place to put the sidebands would be at the
antiresonant points of the cavity, which is half the free spectral range. At that point
the sidebands are very insensitive to length changes of the cavity. Figure 2.6 shows
1.0
0.8
U5 0.6
Eo
0.4-
9Lq
0.2
-FSR 0 FSR
Frequency
Figure 2.6: Light fields with antiresonant sidebands in a cavity
where the electric fields lie with respect to the cavity resonances in frequency space.
From figure 2.4 we can see that the sidebands do not receive any phase shift when
they are placed at half the free spectral range.
Referring to Equation (2.38) and using the fact that the sidebands are antires-
onant, we can write the reflected sideband fields as
rl + r2
E-1 refl = 12 E- input (2.59)
1 + rir2
The electric fields reflected off the cavity can now be expressed as
E = rl r2-2iko + 72 -it 1 +r2 Ci Eoe-2iwot (2.60)
1 r7r2e-2iko06 2 1 + r1r2 2 1 + rr2
Comparing this field to equations (2.11) and (2.12), we can calculate the error
signal. Since the sideband transfer function amplitudes are equal, as expected, then
the quadrature component is zero. The sideband transfer functions are also real as
written, and so we just need to calculate the imaginary part of the reflected carrier.
S{Eo re{(rl- (2.61)
1 + RIR2 2r1r2 cos (2ko61)
r2 ( R1) sin (2ko61)
1 + RIR2 2r1r2 cos (2ko61)
Putting this into Equation (2.11) gives
r + r2 r2( R1) sin 2kol6
Sin chase = -m (2.63)
h 1 + r1r2 1 + R1R2 2r1r2 cos 2ko61
In hindsight, the general shape of Equation (2.63) would have been easy to
guess. The carrier picks up an extra phase shift from movement of the back mirror
of the cavity that the sideband doesn't see since it is reflected directly off the front
mirror. The phase shift would be 2ko6l. Equation (2.11) shows that the overall error
signal would go like the sine of the phase shift. Since that equation also -o- that the
amplitude of the error signal corresponds to the amplitude of the carrier, we would
also predict the denominator. The amplitudes of the sidebands are also what would
be expected.
Figure (2.7) shows what the in-phase error signal looks like when the sidebands
have been placed at the antiresonances of the cavity. The slope at the center of the
error signal is related to the finesse of the cavity. The features at half the free spectral
range is the signal created by the sidebands becoming resonant in the cavity. The
locking range of the signal is defined as when the signal changes sign. In this case the
locking range is A/4. For more complicated systems the locking range can become
small, and it is then a concern since the system must be very close to the operating
point in order to generate the correct locking signal.
1.0
0.5
I 0.0-
0 -0.5
-1.0
-/4 0 1/4
Scanning Cavity Length
Figure 2.7: In-phase error signal for a cavity in reflection
The quadrature error signal can not be produced without also producing the
in-phase signal. There is no frequency for the sidebands such that a change in length
of the cavity results in a change of amplitude of the sidebands but not the phase of
the sidebands relative to the carrier.
An important point here is the concept of a local oscillator. For the two error
signal quadratures, the carrier and the sidebands shift in their role of local oscillator.
In the case where a signal is produced by a phase shift of the carrier, the sidebands
are the local oscillator, providing a constant phase reference. There must be sideband
present in order to detect an error signal. In the case where the error signal is created
by an imbalance in the amplitudes of the sidebands the carrier is 1p giving the role of
the local oscillator. This means that if a cavity is impedance matched for the carrier,
then there is an in-phase locking signal in reflection, but there is no quadrature signal.
If one can take the signal from inside the cavity, then both error signals are present.
2.4 Michelson
The simple Michelson is the fundamental building block for a gravitational wave
detector. The Michelson's control is important to investigate in order to understand
how to accomplish that control in a more complicated configuration. The frequency
response is a simple cosine function as would be expected in any Michelson, and does
not deserve any special attention.
+
r, t Epot Epb
Eib refl
EE0
+ E B +t
rbs, tbs
1 i
antisym
Figure 2.8: Fields in a simple Michelson
2.4.1 Light Fields
The sensitivity of a simple Michelson to a differential change in the arm lengths
is a relatively easy calculation. A few conventions need to be chosen. As in the case
of the simple cavity all reflectivities and transmissivities of simple mirrors are real
and positive. Light reflected off the coated side of the optic in air will get a minus
sign, light reflected off the coated side in the substrate will have a plus sign. This
applies to the beam splitter as well.
Figure (2.8) shows the notation that will be used. The fields are all drawn as
they interact with the beam splitter. A quick explanation of the subscripts is useful.
Fields and optics in the arm that are in line with the incoming beam get a subscript i.
Fields and optics that are perpendicular get a subscript p. Fields traveling away from
the optic get a subscript o and fields traveling back toward the optic get a subscript
b.
tbsEO
re- 2ikliEio
(2.64)
(2.65)
ritbse-2ikliEo
rbsEo
rpe-2ikl Epo
r p e p
These equations give us
antisymmetric port:
Eantisym
T (k)antisym
(2.66)
(2.67)
rp bs -2iklpEo
a solution for the transfer function of the light to the
rbsEib + tbsEpb
rbstbs (rb p-2ikl ie- 2ikl')
M Ip+li k je- Pi
rbtbse -2ik pe -2ik
tMI (k)
(2.68)
(2.69)
(2.70)
(2.71)
rieZiz ) "
If we assume that the lengths of the two arms are unchanged except for a small
differential movement such that
li liJ + 6l,
Ip l= 61
(2.72)
and that the end mirrors have the same reflectivity, r2 then (2.70) becomes
T (k)antisym = 2irbstbs2 sin r2k (1- 1
247"
6)] g2"(1
(2.73)
We can see that the choice of how we define reflections and transmissions at surfaces
has created an antisymmetric port that is dark for equal length arms. It is also useful
to write down the transfer function for the reflected port
S s2 -2ik T rit2 2ikC) -2ik'
-rMI (k).
(2.74)
(2.75)
2.4.2 Error Signals
We have already derived the transfer functions of the Michelson interferometer.
It makes sense to lock the Michelson at the antisymmetric port using an in-phase
signal. In order to do this we need to set the sideband at
SRESMI
R=
RESMI -
2 (lp li)
(2.76)
(2.77)
This ensures that the sidebands are bright in the antisymmetric port.
T (k),,f
where
1.0
0
0.5
0.0
S-0.5- E E
E b sb
-1.0
-4 0 1/4
Scanning Cavity Length
Figure 2.9: Transfer function of the Michelson
Figure (2.9) shows where the fields lie with respect to the carrier. The two
sidebands in fact do have the same sign at the antisymmetric port. This is the
result of having opposite signs at the input of the Michelson, and the fact that the
transfer function of the Michelson to the antisymmetric port flips the sign of the
upper sideband, but not the lower sideband, as is clear from the figure. The carrier
has a sign opposite that of the sidebands.
It is useful to solve this situation analytically. Writing down the field for the
carrier at the antisymmetric port using a 50/50 beam splitter and full reflector end
mirrors we get
Eo ,tis,, i sin (2kol)e--'" Eo (2.78)
The sidebands take a little more work to get. From Equation (2.70) we get
E = -ie-2k sin 2kP i (2.79)
It~~~~~~~ isueu)osleti iuto nltial.Wiigdw h il o h
Substituting lp = lp 61, li = li + 61 and realizing that setting the carrier to be dark
at this port, and the sidebands to be bright gives the conditions 2k-O2- n7 and
2k-P2 (n + 1) yields
E = Ie-2ik Ip 2k6
E ie-2ik 2 cos 2k161.
(2.80)
Writing
cos 2 (ko kQ) 61 = cos (2ko61) cos (2kQ1l) F sin (2ko61) sin (2kQl) ,
(2.81)
and taking 61 << ka, which is true since typically we are locking to a fraction of the
carrier wavelength, and the sidebands are RF, gives
cos 2 (ko ka) 61 cos 2ko61
(2.82)
Finally we can write the sideband fields at the antisymmetric port as
e-2ikplico2ko
E =-Fie- 2 cos 2ko61
(2.83)
The electric field at the antisymmetric port is
E
E0 input
D.C. + e-'
.m 2ik
sin (2kol) i-e cos (2ko6) eit
2
I M-2ik- 1+ cos (2ko6l) et.
2
(2.84)
When calculating the error signal the overall phase shift of the carrier is lost, and we
get
EE* 2m sin (2ko61) cos (2ko,6) cos 2ko + f) Eoincident2 (2.85)
Demodulating this signal with cos 2kA + Qt) gives us the locking signal
Squad = -msin (2ko6l) cos (2;o6l) |Eo incident 2 (2.86)
The demodulation phase 2ko-^- can be further simplified. Using the fact that
Q = gives
demod P + I (2.87)
2 1P Ii
We have called this signal a quadrature signal because the signal came from
Equation (2.12). However, it is created by an in-phase process. The signal arises
from a phase shift in the carrier with respect to the sidebands. As the Michelson
drifts from the locking point the sideband amplitudes stay the same relative to each
other. It is the phase of the carrier that changes with respect to the sidebands. The
sidebands themselves are transmitted with opposite signs from their original sign,
resulting in a signal that is created by an in-phase process but showing up in the
quadrature component. The demodulation phase is picked to optimize the size of the
signal. We will hold off plotting this error signal until we discuss the Michelson with
arm cavities, since the error signals are essentially the same.
As was the case for the cavity error signal, it would have been relatively easy to
predict Equation (2.86). The sine dependence is a direct result of the transfer function
of the Michelson on the carrier. The cosine dependence could also be guessed at. Since
the sideband is bright in the antisymmetric port, and we can expect length changes
on the order of the carrier wavelength to change the amount of sideband present,
the cosine function is the obvious result. The demodulation phase is the phase the
sidebands experience that the carrier does not.
Since there is a very clear symmetry between the antisymmetric port and the
reflected port, there is a second possible port for the locking signal. Since Equation
(2.86) does not have any dependence of the sidebands, it is easy to argue that the
result is the same with a minus sign from the fact that the sign of the sidebands are
different.
The common movement of the arms has been completely ignored. In fact, com-
mon movement of the end mirrors in a simple Michelson will not change the locking
signal nor the sensitivity. Since the only quantity that depends on the common mode
of the Michelson is the demodulation phase which is determined experimentally, the
common mode of the Michelson can be ignored.
r2p, 2p' t2
Lp
rip, tip +
T2i, t2i rli, ti Epo Epb
T Eib Ere
_10
EO
+ + Eio + -
Sbs tbs
antisym
Figure 2.10: Fields in Michelson with arm cavities
2.5 Michelson with Arm Cavities
This improvement allows the physical length from the beam splitter to the end
mirrors in the arms to be shortened while increasing the sensitivity. The control
becomes more complicated, but can still be calculated analytically. The frequency
response is more complicated due to the nature of the arm cavities.
2.5.1 Light Fields
The transfer function for a Michelson with arm cavities follows very easily from
previous results. Including arm cavities into the Michelson means substituting the
reflectivity of the Fabry-Perot cavities for the end mirrors into Equation (2.70) and
gives
., I+l, / Ipl-i ., lv-Ipi
T (k)antisym = bstbs -2ik (rFPp-2ik _2 i r 2e ) (2.88)
For the carrier we can simplify this equation. Using Equation (2.58) for the
reflectivities of the arm cavities, and assuming that the carrier is dark at the an-
tisymmetric port, and that the cavities in both arms are the same, a differential
movement of the end mirrors of the cavity gives
21rbs bs2 (1- 1) sin (2k61) __,,
E0 antisym =1 + R 2 2 cos e o. (2.89)
1 + RIR2 2rlr2 cos 2ko61
The difference between Equation (2.73) and Equation (2.89) is the term
EO antisym w/ arms 1 R1
(2.90)
Eo antisym w/o arms 1 + RIR2 2rlr2 cos 2ko61
Assuming the length change in the arm cavities is much less than the wavelength of
the light, then Equation (2.90) becomes
Eo antisym w/ arms 1 rf
)2 (2.91)
Eo antisym w/o arms (1 rlr2)2
Assuming the cavity finesse has already been optimized for frequency response,
then we can derive how rl and r2 should be related. From Equation (2.45) it is
evident that a fixed finesse value is equivalent to the product rlr2 being fixed. We
immediately recognize the denominator in Equation (2.41) as the power buildup in
the arms, a fixed value. This then leaves the numerator as the only free parameter.
It is very clear that ri should be as small as possible to give the largest response to
the gravity wave. It then follows that r2 needs to be as large as possible. Without
optical gain (which is not practical) we are limited to r2 being less than or equal to
1.
If the end mirrors are highly reflective then Equation (2.91) becomes
EO antisym w/ arms 1 + rl (2
(2.92)
Eo antisym w/o arms 1 r1
For a high Finesse over-coupled cavity the numerator is approximately 2 and the
denominator is the buildup of light in the cavity.
For the case where the cavity is impedance matched r2 rl, Equation (2.91)
becomes
Eo antisym w/ arms 1 (2
." (2.93)
E0 antisym w/o arms 1 r"
This is essentially half of the response to the over-coupled cavity. The impedance
matched cavity transmits half of the signal through the cavity, increasing the shot
noise.
Since there is a maximum amount of time the light can spend in the arm before
the gravity-wave signal starts degrading, and using the result that r2 should be as
large as possible, this sets a limit on rl such that
w9L rl 7r
< -. (2.94)
c 1- r 2
For the error signals we will need the light reflected from the Michelson. From
Equation (2.75) we can write
/ ),, I -i Ip + Ii IW li
T (k) = e-2i +FP t 2ik )-2i (2.95)
2.5.2 Error Signals
Locking this configuration requires a combination of techniques from the previ-
ous examples. The combination of Michelson and arm cavities do add a significant
complication.
It's useful to define the degrees of freedom that we want to lock. There are three
degrees of freedom in this system: the two cavities and the Michelson. The places
to get error signals are either the reflected port or the antisymmetric port. The
placement of detectors affects our choice of the degrees of freedom. A common length
increase in the arm cavities appears in the reflected port. Differential length changes
of the arm cavities will appear in the antisymmetric port. Common movement of the
short Michelson arms is unimportant. Differential movement of the short Michelson
arms shows up in both the reflected and antisymmetric port.
If we use the same locking scheme as was used for the short Michelson, the
sidebands are completely bright in the antisymmetric port. The differential Fabry-
Perot locking signal shows up strongly. The differential Michelson locking signal is
also present there, and is in fact in the same quadrature as the differential Fabry-
Perot error signal since they are both in-phase signals, relying on the phase shift of
the carrier with respect to the sidebands to carry all the information necessary to
determine the offset. The error signal for the differential Michelson is much weaker
than the cavities.
In the reflected port there are no locking signals at all when the sideband is bright
in the antisymmetric port. This is caused by the fact that there are no sidebands
present in the reflected port. The sidebands act as a local oscillator for the carrier,
just as a reference sine wave is a local oscillator used to mix down the photodiode
signal in order to retrieve the error signal. The sideband needs to be present in order
to act as a local oscillator to detect the phase shift between the sidebands and the
carrier.
There are two solutions to this dilemma. The first is to use a second pair of
sidebands that are bright in the reflected port. Although this solution is easy to
comprehend, it is not very elegant.
Another solution manipulates where the sideband lies with respect to the free
spectral range of the Michelson. To understand what is happening here it is useful
to write down the fields in the reflected port.
2.5.2.1 Common Mode Fabry-Perot
Since it is the common mode that we are interested in we substitute Lp
L + 61 and Li -- L + 61, where we have assumed the arm cavities to have the same
macroscopic length L, into Equation (2.95) and we get
r1 r2e-2ik(L+61) ( / 2ik+~1
T (k) = cr2ik(L+) cos 2k e-2) (2.96)
1- rr2e-2ik(L+l) 2 /
Specifically writing down the transfer function for the sidebands, putting them exactly
at the antiresonances of the cavities, and assuming that r2 t 1 gives
T (k) = cos (2kL e-2i ) (2.97)
Continuing, we expand out the cosine term to get
cos (2kL P -i cos (2koP cos (2klP T
sin (2ko0 2 I) sin (2k (2.98)
Once again using the fact that the carrier is completely reflected, that is r2 = 1, then
2ko0 = nr and the transfer function for the sidebands becomes
T(Vk) cos (2k1 e-2i ). (2.99)
Writing down the electric field in the reflected port we get
Er-fl C 2ik, 1 r2-2-iko6l IP _i(i 2k,, +QiLt)
E e o --{ + cos 2ka 2
EO t rer26-2ikol + c m 22
2 2
cos (2k 2 i ) j2ks i 2 Q)}
We can now use equations (2.11) and (2.12) to get the error signal if we demod-
ulate the signal with the correct phase. It is immediately evident that the sideband
amplitudes do not change as the arm cavities common mode length is dithered. This
is as expected, and there is no quadrature signal. The sideband transfer function is
completely real. All that is needed is the imaginary part of the carrier transfer func-
tion, which was calculated in Equation (2.62). We can now write down the common
mode Fabry-Perot error signal.
r2 (1 R) sin (2ko61) os I 1
Sinphase = -m cos 2ka (2.101
1 + R1R2 2rlr2 cos (2ko61) 2
where the demodulation phase is
Pdemod 2k +. (2.102)
2
The common mode error signal is exactly the same as the error signal reflected
directly from a cavity. This is as expected, since the Michelson should be essentially
invisible to the common mode of the cavities. At this point it is important to realize
that this exercise has not been in vain. This calculation must be done in order to
understand how the signal is coupled out of the interferometer. This will become even
more evident for the differential Fabry-Perot mode, where the error signal actually
changes which quadrature it is in.
There is an additional term in the error signal which comes from the amplitude
of the sidebands. If the Michelson is made completely bright in the symmetric port,
then this term goes to one. This would be an undesirable operating point, since there
would be no local oscillator in the antisymmetric port for the differential mode of the
arm cavities. Another difference in the equations is the result of assuming r2 is close
to unity. This simplified the equations significantly, and for a gravity-wave detector
this is the situation that is desired. However, the final formula can easily be modified
by including the term calculated from the simple cavity.
2.5.2.2 Differential Michelson
For the differential Michelson it is most useful to take the signal in the reflected
port. It will turn out that it will be orthogonal to the common mode Fabry-Perot
signal in this port. Writing down the transfer function again in the reflected port,
and using Equation (2.95), substituting lp lp 61 and 1i 1i + 61, and make the
two arm cavities the same we get
T (k) -rF (k) cos 2k 2 61 e (2.103)
Assuming that r2 m 1 and r, < 1 then for the carrier
rFp (ko) 1 (2.104)
and for the sideband
rFp = -1.
(2.105)
The transfer function for the carrier is then
T (ko) = cos (2ko6l) e-2 (2.106)
using the fact that the carrier is dark in the antisymmetric port.
For the sidebands we can show that
cos 2k ( 61 cos (2ko61) cos (2kQl1)
2 ) (2.107)
+ sin (2ko6l) sin (2kl6) .
This lets us write down the electric field.
Ef = e-2io cos (2ko6) + m {A + B} -2
2, (2.108)
m {A B }ei21 +Q
2
where A is the product of the sine terms in Equation (2.107) and B is the product of
the cosine terms.
Calculating the error signal, it is immediately evident that the in-phase signal is
zero because both the sideband and carrier transfer functions are real. The quadrature
signal is then
Squad = -mcos (2ko6l) sin (2ko6l) sin 2kQ (2.109)
where
Pdemrod 2ka l (2.110)
2
This is the same result as for the simple Michelson in the antisymmetric port
except for the additional phase shift and the slight change in the sideband amplitude.
Note that this is in the quadrature phase and the Fabry-Perot common mode was
in the in-phase, with the same demodulation phase. This puts these two signals
exactly ninety degrees out of phase, and thus completely orthogonal. This signal
is a pure quadrature signal because all of the in-phase signal is transmitted to the
antisymmetric port because of the nature of a Michelson.
2.5.2.3 Fabry-Perot Differential Mode
The transfer function of a differential length change of the two arm cavities to
the antisymmetric port for the carrier is
ir2 (1 R1) sin (2ko61) ,
T (ko) +RR2 r12 0 (2.111)
1 -+ R R2 2rlr2 COS 2ko61
For the sidebands it is
T (k) isin (2k e-2i2 (2.112)
Expanding the sine
/ /- li I /I/ /I I.
sin 2k = sin 2ko cos 2kQ + cos 2ko sin 2kQ (2.113)
2 2 2 2 2
sin 2kn (2.114)
2
using the fact that the carrier is dark in the antisymmetric port. This shows the
Michelson flipping the sign of one of the sidebands but not the other, as was seen in
the case of the simple Michelson.
Writing down the electric field in the antisymmetric port
S. _,- r2(1 R) sin(2ko61)
1 + RiR2 2rlr2 cos 2ko61
sin (2k e-i 2k) (2.115)
m ( l 2 ) i(2k,1P +Qt)
-- sm 2kei)
2 2
The transfer function of the sidebands is real and has opposite signs. This results
in the in-phase being zero. The quadrature error signal is
r2 (1- R) sin (2ko61) IP 1(2.
Squad = -m sin 2kan (2.116)
1 + R1R2 2rlr2 cos 2ko61 2
This is the same result as the common mode but it shows up in the quadrature
signal. The differential Michelson will also show up in this quadrature, and thus the
differential Fabry-Perot error signal is not orthogonal to the differential Michelson.
2.5.2.4 Locking Matrix
Our optical configuration is now complicated enough to introduce the locking
matrix. Although for this example the locking matrix is fairly well diagonalized it is
useful to use as an illustration of what a good locking matrix is. Again, the locking
matrix is a measure of how linearly independent our locking signals are to one another.
It is an extremely important item because it will decide how to design the locking
loops of the interferometer.
The locking matrix is defined as
Mn, D.O.F. (2.117)
OD.O.F.
The degrees of freedom are defined by the physical system. In the case of the Michel-
son with arm cavities the degrees of freedom are the common mode of the arm cav-
ities, L+, the differential mode of the arm cavities L_, and the differential mode of
the Michelson, l_.
S, is the signal that is being used. This signal depends on several variables. In
the current case
S, = S, {L_, L+, 1, Q, vdemod} (2.118)
The subscript n is the number of ports that have photodetectors multiplied by two,
the in-phase and quadrature component in each of those ports.
We are now prepared to write down the locking matrix for a Michelson with
arm cavities using the locking scheme previously discussed. Although there are four
locking signals, since there are only three degrees of freedoms to lock, one of the
locking signals will not be used. The three signals are picked in such a way that they
give us the most diagonal locking matrix.
The only signal that is not orthogonal is the in-phase antisymmetric port. As-
suming that the cavity end mirrors are highly reflective, we calculate
OSantisym I 1- R P -- li
m -m sin 2kQ (2.119)
BL (1- ri)2 2
and
-satm sin 2kn- (2.120)
01_ 2
Writing the locking matrix where each signal has been normalized by the degree
of freedom that is has been designed to detect yields Table 2.1.
The off-diagonal term in the antisymmetric port is the differential Michelson
contribution. We don't expect this to be a problem since the error signal for the
Fabry-Perot is much sharper then for the differential Michelson. In fact, as R1 1
Table 2.1: Locking matrix for Michelson with arm cavities
then this term goes to zero. Even at an intensity reflectivity for R1 of 91iI' this term
is almost a 100 times less than the differential Fabry-Perot signal.
L = 4000 m,F= 625
1 -------- L = 4000 m, F= 206
-- - -- - -- -------------- - --._ . _ _\
C)
0.1
0.01
1 10 100 1000
Frequency (Hz)
Figure 2.11: Frequency response of LIGOI-like instrument
2.5.3 Frequency Response
The frequency response of a Michelson with arm cavities is the first configuration
that we have encountered that has some structure to it. The response is governed
by the arm cavities. Since the cavity is locked to the wavelength of the carrier and
the gravity wave can be interpreted as frequency shifted light, then the amount of
L_ l_ L+
Antisym Q 1 (1"), 0
Refl Q 0 1 0
Refl I 0 0 1
gravity-wave signal that is built up in the cavity is dependent on the finesse and free
spectral range of the cavity.
Figure 2.11 shows the frequency response of a Michelson with arm cavities for two
different finesses, and with the same free spectral range. The low frequency response
is larger for the higher finesse, but the 3 dB point is lower. This is commensurate with
idea that as the gravity-wave signal frequency becomes higher than the linewidth of
the cavity the amount of gravitational wave signal is reduced.
r3t3 r2, t2 Erl, t1
E b E ref
+ + Em + Eo
Figure 2.12: Fields in a three-mirror coupled cavity
2.6 Three-Mirror Coupled Cavity
2.6.1 Light Fields
The next step to improve the sensitivity of the interferometer is to introduce a
coupled cavity. Before we include this in our instrument, it is useful to look at it as
a pure cavity. Figure 2.12 shows the fields. The front cavity is formed by M1 and
i.[. and the back cavity is formed by i_ and I[.. We can reduce the second cavity
into a single mirror with a complex reflectivity, rFp. We can write down the transfer
functions for the light fields reflected and inside the cavity
-T ( 2 73e-2ikL
r (k) 3r 2ik (2.121)
1 T3e26
1 rlrpe-2ikl
Ti, (k) 1 2ikl (2.123)
1 rrpe-i
(2.124)
2.6.2 Error Signals
Locking the three mirror coupled cavity is a much more challenging task than
previously undertaken. There are two philosophies for locking this type of system.
The first is to use the largest locking signal for the cavities, the in-phase signal.
Both cavities will create an in-phase signal, but it is possible to create a locking
matrix that is linearly independent out of the signals from the reflected port and
from a pick-off inside the first cavity.
Developing a locking scheme for the in-phase approach is relatively straight
forward. The sideband needs to be antiresonant in the back cavity, and resonant in
the front cavity. Resonance in the second cavity is at half the free spectral range
because the sidebands get a 180 degree phase shift upon reflection from the second
cavity when the carrier is on resonance. For a LIGO-like coupled cavity, we can use
the following parameters.
Table 2.2: LIGO-like parameters for coupled cavity
Parameter Value
R1 95'.
R2 97'
R3 1
L 4000m
I 6m
The FWHM, or line width, of the second cavity is very small. The free spectral
range is 37.5 kHz and the FWHM is 182 Hz. As long as we don't pick a frequency
that lies within the line width, it is effectively antiresonant. For the first cavity, a
12.5 MHz sideband is exactly resonant in it, and is sufficiently antiresonant in the
second cavity.
The locking matrix that we get in this case is shown in table 2.3. The error
signals have been normalized such that they are a vector row with length unity. That
Table 2.3: Locking matrix for coupled cavity with in-phase scheme
L I
Refl I 1 3.1 x 10-4
P.O. I 1 -1.0 x 10-2
allows us to immediately see what the determinant of it is. We can see that this
matrix is not linearly dependent, but also not very orthogonal. Calculating the angle
between the two vectors yields
a 10 mrads (2.125)
so they are fairly parallel. We will need a locking hierarchy such that G1 ~ 100G2.
This is very possible, and similar to a previous experiment [32].
A second way to lock this configuration is the use of the quadrature signal. By
using the signal created by an imbalance in the amplitudes of the sidebands, we get
a locking signal for one cavity that will be relatively independent of the other cavity.
It is still desirable to have the sidebands antiresonant in the second cavity. The
in-phase signal that this creates will be used to lock this degree of freedom. In order to
create an imbalance in the two sideband amplitudes, the sideband needs to resonate
to some degree in the cavity that it will lock. A sideband with a frequency of 12.614
MHz satisfies these requirements for the parameters in table 2.3.
The cost of using the quadrature signal is that the demodulation phase must be
picked to minimize the in-phase component from the second cavity in the signal that
locks the first cavity. The accuracy of the demodulation phase, and the noise related
to phase jitter, is something that must be analyzed, and unfortunately will only be
touched on in this work.
As we can see, using the quadrature signal to lock the first cavity has almost
completely decoupled the degrees of freedom. They are now almost completely or-
Table 2.4: Locking matrix for coupled cavity with quad scheme
L I
Refl I 1 5.2 x 10-3
P.O. Q 0 1
thogonal, and in fact if the demodulation phase is picked with perfect accuracy both
off-diagonal terms can be driven to zero. The amount that they can be decoupled
is related to the phase noise in the signal being used to demodulate the signal. In
this scheme the in-phase signal starts to become of the order of the quadrature signal
when the phase noise exceeds 1.7 mrads.
Figures 2.13 and 2.14 shows the actually error signals in each of the two ports.
As we can see, the signals are very orthogonal. In the reflected port some of the in-
phase signal created by the front cavity shows up. This coupling is unavoidable since
those two in-phase signals are essentially parallel in phase space. At the pick-off the
in-phase signal from the rear cavity has been essentially zeroed, and all that remains
is the quadrature signal created by the front cavity. Before we can claim this scheme
a complete success we have to realize that in the case of a full interferometer there is
the complication of the differential Michelson.
Another possible problem with using this scheme is the question of how to find
the correct phase. For the in-phase technique the phase is tuned to maximize the
signal. Since the signal has such a broad maximum, this is rather easy. In the case
of the quadrature signal, finding a zero requires being very close to it.
There are two possible v--v to solve this problem. The first is to use the in-
phase locking scheme with the detuned frequency. Once the instrument is locked up
the in-phase component can be zeroed using a second channel, and then switching to
that channel for lock. Lock acquisition after this has been achieved should be easier
-- Scanning L
0.010- ----- Scanning 1
0
P-4
0.005
0.000- ------ ------- -------------
S-0.005 -
-0.010
-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05
Mirror Detuning (Degrees)
Figure 2.13: Error signal in reflection for the two degrees of freedom
0.20-
- Scanning L
-----Scanning 1
0.00 ----------------
-0.05
-0.10-
-0.15-
-0.20
-4 -2 0 2 4
Mirror Detuning (Degrees)
Figure 2.14: Error signal at pick off for the two degrees of freedom
because the error signals are relatively independent of each other, and lock stability
should be better, since noise in one degree of freedom doesn't feed into the other.
A second way to acquire the demodulation phase is to misalign the first mirror,
and lock the second cavity with the in-phase signal. The demodulation phase for
the quadrature detector can then be tuned to zero the in-phase component. This
is not the final demodulation phase, but it does give an absolute measurement for
the demodulation phase. The misaligned case can be simulated, and from that the
final demodulation phase can be calculated. Tuning the quadrature signal to that
phase and realigning the cavity should be enough to get the two degrees of freedom
independent. Experimentally both of these techniques are used, and both work to
varying degrees of success.
Although the complication of tuning the demodulation phase seems large, the
same steps must be taken to separate the common mode signal from the differential
Michelson signal in the reflected port. The only simplification is that those two
signals are exactly ninety degrees out of phase, where as in this case there is no
simple relationship with the demodulation phase used for the pick-off signal and that
used for the reflected signal.
2.7 Power-Recycled Cavity-Enhanced Michelson
The power-recycled cavity-enhanced Michelson is the configuration that LIGO
will be using for its first generation of detectors. The control scheme is much more
complicated and analytical solutions are now more obscure due to the fact that not all
locking configurations involve sidebands either on or off resonance. When sidebands
sit on the edges of cavity resonances then expressions for the fields are nontrivial.
The frequency response is the same as without power recycling since the effect of
the power-recycling mirror can be explained completely as an increase in laser power
which clearly has no effect on the frequency response.
+
rFPP
rFPi o Epbr rp tpr
pr
SE prb E
Ii rbs, tbs E
E Epro
antisym
Figure 2.15: Fields in a power-re, 1.1 Michelson
2.7.1 Light Fields
Figure (2.15) shows the fields defined for a power-recycled interferometer. Using
previous results to reduce the work, if we substitute Equation (2.75) for r2 and r,,, tp
for rl, t1 in the cavity equations (2.35), (2.36), (2.38), this gives the equations for the
fields in the power-recycling cavity and reflected from the power-recycling mirror.
Eo = tpr 2 Eo (2.126)
1 rprrMie-C
--MIpri -2ikl"
Eprb = It 2ik Eo (2.127)
1 rMIrpre -2iklp
Spr- (MI (r + t 2ikl ,
Erefl = r 2r Eo. (2.128)
1 TprTMIC-2iklpr
For the antisymmetric port, we need to multiply Equation (2.70) by Eproe-iklpI
to get the transfer function to the dark port, giving us
T (k) tMItpr _-ikl. (2.129)
S()antisym 1 rprrMIe 2ikl
To include arm cavities is a simple matter of substituting rFP into tMI and rMi.
For a simple analysis on how the power-recycling mirror affects the sensitivity
we can use the simple Michelson. Assuming that the end mirrors are completely
reflective, that the beam splitter is nearly to 50/50, that the arm lengths obey (2.72),
and that lpr + 'P is resonant for the light, then for the carrier
rMI = -2ik cos (2iko61) e-2ik0 (2.130)
and
itpe C
Eantsym sin (2ko61) E. (2.131)
1 rpr
Comparing this to the same equation for the simple Michelson, Equation (2.78), we
have gained the factor
r > 1. (2.132)
1 rpr 1 rpr
This is the buildup of power in the cavity formed by the end mirrors of the Michelson
and the power-recycling mirror.
We already know how the arm cavity mirrors would have their reflectivities
determined by the ,in i, .-;i of the Michelson with arm cavities. R2 is as large as
possible, usually only transmitting around 10 parts per million. R1 is selected so that
the storage time of the arm cavities does not exceeds half the cycle of a gravity wave
at the frequency of interest. For LIGO I this means R1 is 97'.
The power recycling reflectivity is optimal when the power buildup in the cavity
is as large as possible. In order to pick a value for the reflectivity of the power-recycling
mirror we need to express the power build in the cavity in terms of the reflectivity
of the power-recycling mirror Rpr and the total losses in the interferometer for the
carrier, Lo. Then we need to take a derivative of this with respect to Rpr and set it
equal to zero. This yields the result that Rpr = Lo. The carrier is impedance matched
into the interferometer. For LIGO I Rpr is around 97'.
2.7.2 Error Signals
For this optical configuration there are four degrees of freedom. We still have
the same three as in the case without the power-recycling mirror, L_, L, 1_ but we
have picked up control of the power-recycling cavity, Lpr. The physical length of the
power-recycling cavity is lpr + 2
Once again the entire system can be locked with a single pair of sidebands if
the sidebands are picked to lie at the correct frequencies with respect to the different
cavity and Michelson free spectral ranges.
There is now a third port from which a locking signal can be obtained. Besides
the normal antisymmetric port and reflected port, we can now use a signal that is
reflected off the antireflection coated side of the beam splitter. This signal will create
a linearly independent signal from the reflected port.
The sidebands should again be antiresonant for the arm cavities. This will allow
us to essentially use the same locking signals for L_, L+, and 1_. Again, L_ is locked in
the dark port, and depends weakly on the differential Michelson signal. The reflected
port locks L+ and 1_, which are ninety degrees out of phase with each other.
In order to lock the coupled degrees of freedom, L+ and Lpr using the in-phase
technique discussed in Section 2.6, the sideband should be resonant in the power-
recycling cavity. Taking the signal at the pick-off we have a signal that is linearly
independent from the common mode of the Fabry-Perot, but is far from orthogonal.
Table 2.5 shows the locking matrix for this scheme using LIGO-like parameters
published by Martin Regehr [32, p.69]. We can see that the differential signals are lin-
early independent and fairly orthogonal. The coupling of the differential Fabry-Perot
in the signal used to lock the differential Michelson is weak because the differential
Michelson is taken in the reflected port, and very little differential Fabry-Perot shows
Table 2.5: Locking matrix for LIGO I configuration
L_ 1_ L+ Lpr
Antisym Q 1 7.4 x 10-3 0 0
Refl Q 4.5 x 10-3 .58 -4.8 x 10- -.82
Refl I 0 3.2 x 10-4 1 9.9 x 10-4
P.O. I 0 -1.6 x 10-5 1 -2.4 x 10-3
up in that port. The differential Michelson is very weak in the antisymmetric port
compared to the Fabry-Perot.
The reflected port quadrature signal is demodulated with a phase such that
the in-phase common mode signal is zeroed. Ninety degrees from this phase is the
maximum common mode Fabry-Perot signal because they are orthogonal signals. The
power recycling degree of freedom is starting to show up in the differential Michelson
because it is also on this channel. However, because one of these signals is common
mode and the other is a differential mode they will ahiv-- be linearly independent.
We are using an in-phase signal to lock both the common mode Fabry-Perot and
power recycling. We immediately know that they are linearly independent because
the sign of the product of the diagonal terms is different than the sign of the product
of the off-diagonal terms. Using our cross product to show how linearly independent
they are gives us
sin a+ = 3.4 x 10-3 (2.133)
As we can see the angle between the two signals is very small. As long as the angle
is large enough to separate them, then we can still lock the system. The question of
how large is large enough is determined by how much gain is needed to overcome the
coupling.
Since they are not completely orthogonal, a gain hierarchy needs to be estab-
lished in order to eliminate the coupling. From the calculation in Section 2.2 we know
how large the gain in the common mode Fabry-Perot needs to be in order to suppress
that degree of freedom in the signal used to lock power recycling.
We could also have used the quadrature signal to lock the power-recycling cavity
in order to separate the degrees of freedom. Since the analysis is similar to that done
for the full signal-recycled interferometer, we will put off the detailed discussion for
that section. We will note that for this scheme both the differential Michelson and
the power-recycling cavity have similar demodulation phases.
LIGO I has the possibility of using a different locking scheme. Although the
above locking scheme has proven to work well [32], there is not enough signals to
control the angular degrees of freedom of the mirrors. For that reason a second
sideband is used. This second sideband could give us a chance to decouple the common
mode Fabry-Perot and the power-recycling cavity even more. Making it antiresonant
in the power-recycling cavity, we can use it to control the common mode of the arm
cavities. Since it is completely antiresonant it is much more sensitive to L+. Putting
the original sideband on resonance with the power-recycling cavity and having some
leak out the dark port gives us the remaining locking signals.
2.8 Dual-Recycled Cavity-Enhanced Michelson
The final upgrade to an interferometer that this work will analyze is the use
of a signal-recycling mirror. The introduction of the signal-recycling mirror makes
the equations much more complex. The solutions for the light fields are not terrible
difficult to arrive at but, do require some extensive algebra. The frequency response
of such an instrument is rather complicated, and also deserves some attention.
2.8.1 Light Fields
The light fields in this instrument are much more intermixed than before. There
is now a second Michelson that has be introduced to the interferometer consisting
of the fields entering the beam splitter from the antisymmetric port. Any field that
P
p +
Blb E p pr r
li rbs, ts
+ Eio Epro Eo
rSO
BEsror srb
I i r s, t s r
Eantisym
Figure 2.16: Fields in a signal-recycled Michelson
goes towards the antisymmetric port will be reflected off the signal-recycling mirror
and will see the Michelson from the other side. Light fields that are bright in the
antisymmetric port will then enter the second Michelson and be bright in the reflected
port. A second cavity has also been added. For light that is dark in the antisymmetric
port, the cavity is formed by the signal-recycling mirror and the arm cavities. For light
that is bright in the antisymmetric port, the cavity is formed by the signal-recycling
mirror and the power-recycling mirror.
Figure (2.16) shows the fields inside a signal-re 1'. 1 Michelson. In order to
solve for the electric fields we want to solve the equations for the case without power
recycling. It's a simple matter to add power recycling in at the end. We will also call
the reflectivity of the arm cavities rp, ri in order to save some subscripts. We need to
keep in mind that they are the reflectivities of the Fabry-Perot cavities.
Writing down the equations
tbsEo + "bsEsrb
-r_ ie-2iklEi _
-rbsEo + tbsEsrb
_p e-2iklpEpo
tbsEpb + rbsEib
-r se-2ikl, Eo
I~~~~ ygO -
rie-2ikl (tbsEo + rbsEsrb)
-rpe-2ikl ( -rbEo + tbsEsrb)
-rre-2ikl (tbsEpb + rbsEib)
Unfortunately most of the work done in the previous sections will not really help for
this configuration due to the mixing created by the signal-recycling mirror.
In order to make the equations more manageable some abbreviations must be
used. The average length of the common mode Michelson and differential mode
Michelson can be defined as
12k + l
2
S= 2k -
2
c = 2klr.
(Psr = 2klsr-.
(2.140)
(2.141)
(2.142)
(2.143)
The fields reflected back from the beam splitter towards the power-recycling
mirror and in the antisymmetric port are
S { (rrpe- r r+2 ~~ r- [rt4 + 2rrpt r r ++ r} ]-+
rp -% bs' I 2.i1s4 I s pbsbs P I bsl-
1 + rtb2s rir) e-i(2y ++rPs() rPr t r, +( rt ) g(e-(+S+S+
-r (k) (2.144)
(2.134)
(2.135)
(2.136)
(2.137)
(2.138)
(2.139)
and
r(t_ +r i r1)t8 r(1 1 + I-
T 1 + rf rt (1 rpr,) e-i(2,p++sp,r) rs(ter 2 C-, + jei ) e e-++)
t' (k) (2.145)
Now using the normal cavity equations we can write the reflected, transmitted
and intra-power-recycling cavity transfer functions.
Tref (k) = M (k) -2il (2.146)
1 r1rIM (k) e-2iklpr
I t, P (k) p-iklpr
Tati (k) 1 (k) e-ikl (2.147)
1 rpr MI (k) e-2ikl,
Tp.*. (k) = 2il (2.148)
1 rperMI (k) e-Cp2i
We want to now calculate the sensitivity gain of this configuration over one
without signal recycling. For this we can assume that there are no arm cavities, that
the end mirrors are completely reflective, and that there is no power recycling. Then
the transfer function to the antisymmetric port for a differential movement in the
arms reduces to
2itsrrbstbs Soin 2k P 1 9 -2
T (k)antm L (2.149)
1 rsr cos L2k (P2 61 2ik (ls,+
For the carrier we assume that it is dark in the antisymmetric port and that it is
resonant in the signal-recycling cavity formed by the signal-recycling mirror and the
end mirrors of the arms. We also assume that the movement of the end mirrors is
very small compared to the wavelength. Then Equation (2.149) becomes
T (ko)2itsrFbstbs sin (2kol6) e-2ik
T ( ansym r(2.150)
1 rs,.
Comparing this to Equation (2.78) gives us
T (ko)sR tsr2
T (ko)MI rsr
This result is not a surprise and could have be written down immediately without
this detailed analysis. The gain is simply the buildup of the carrier in the cavity
formed by the signal-recycling mirror and the ends of the Michelson mirror. Since
the Michelson is held dark all differential movements go towards the antisymmetric
port, and then encounter the signal-recycling mirror. The signal-recycling mirror is
then controlled so that the differential signal is resonant in that cavity.
2.8.2 Error Signals
There are several possible v--,v- to lock this configuration. The approach that
was taken in this work was that the locking scheme should be relatively easy to
implement. This rules out using single sidebands, since generation of single sidebands
on a carrier is rather complex and involves frequency stabilizing two lasers. This is
not an insurmountable problem, but since there are other efforts ongoing to lock this
configuration that use single sidebands, it would have been a redundant effort [33, 34].
The next criterion was that it would lock the interferometer in a dual-recycled state,
where the carrier is built up in the signal-recycling cavity. The locking scheme was
not designed to operate in a detuned state. However, small detunings should be
possible if the signals are sufficiently independent. The third criterion was that it
would be locked using only frontal modulation. Injecting light from other ports for
locking would not be allowed. The reason for this is mainly the complex noise and
implementation issues for an external modulation locking scheme when applied to a
full scale gravitational wave detector.
There are five degrees of freedom in this configuration. There are several differ-
ent v-,v to express these degrees of freedom. For all cases the common mode and
differential mode of the arm cavities and the differential Michelson are the most con-
venient ones to use because of the properties of the Michelson. The power-recycling
cavity length is also a fairly usual choice for control. For the signal-recycling mir-
ror the most obvious degree of freedom is the power-recycling, signal-recycling cavity
length. This degree of freedom will be controlled by a sideband that enters in through
the power-recycling mirror, must reach the signal-recycling mirror, and must sense
its movement.
2.8.2.1 Common Modes
Control of the common modes is relatively straight forward, and similar to con-
trol schemes already discussed. We want to control these degrees of freedom with one
of pair of sidebands, Q2 (This sideband has the subscript 2 because in the final design
it has the higher frequency). In order to control the common modes, we are going to
have to deal with the coupling of the common mode of the Fabry-Perot cavities and
the power-recycling mirror.
For our scheme we want to use an in-phase signal for the locking of the arm
cavities. The sideband should be antiresonant in this case. To lock the power-
recycling cavity the sideband should sit in the edges of the resonances, somewhere
near the full width, half maximum. This will generate a quadrature signal for the
power-recycling cavity degree of freedom.
2.8.2.2 Signal Recycling
The signal-recycling mirror is controlled by an in-phase signal created by a side-
band that is resonant in the cavity formed by it and the power-recycling mirror. This
cavity is rather complex and samples many optics in the interferometer twice before
it makes a round trip in the cavity.
The key to sensing the signal-recycling mirror is to have the second pair of
sidebands, fi, bright in the antisymmetric port (Since it will turn out that this
pair of sidebands is lower in frequency than the other pair, it has the subscript 1).
When f1 is reflected back into the interferometer from the signal-recycling mirror it
sees the Michelson from the opposite side, and is then bright in the reflected port.
The signal-recycling cavity length is then adjusted so that it forms a cavity with the
power-recycling mirror and signal-recycling mirror.
Ideally it would only sense a cavity formed by the signal-recycling mirror and
the arm cavities. The limitation of frontal modulation that we imposed excludes this
because there is no way for a light field to be bright for the antisymmetric port when
coming from the reflected port, but dark for the reflected port when coming from the
antisymmetric port. The symmetry of the system forbids it. External modulation
would allow injecting a light field in the antisymmetric port that would sample only
the signal-recycling cavity and the arm cavities.
With this resonance condition on fR we get the signal-recycling mirror error
signal. A change in the position of the signal-recycling mirror results in a phase
shift of f with respect to the carrier. Again, the concept of a local oscillator is
important because the carrier acts as a local oscillator for the sidebands to sense this
degree of freedom. That means that there is no locking signal for this sideband in
the antisymmetric port because the carrier is not present there. The locking signal
must be taken either in reflection or at the pick off. Both these ports give the same
signal with slightly different shot noise.
The signal-recycling mirror control is very difficult to obtain. Since the signal-
recycling cavity will have a low finesse, the error signal simply will not be as strong
as the other degrees of freedom. The cavity it forms also samples every other degree
of freedom. This is unlike the other degrees of freedom, except for the differential
Michelson which rivals it in complexity, since other signals in general only have a
strong dependence on one other parameter.
We will find that the differential Michelson causes the most problems with the
signal-recycling mirror. It is important that we create a signal for the differential
Michelson that has a different phase from signal recycling. The amount that the
differential Michelson degree of freedom appears in the signal recycling error signal
can then be zeroed with the phase of the local oscillator for the mixer.
2.8.2.3 Differential Modes
Control of the differential Fabry-Perot degrees of freedom will again be similar
to previously discussed schemes. In this case the sideband will be fairly bright in the
antisymmetric port. This gives us a strong local oscillator for measuring the in-phase
signal created by the differential Fabry-Perot Cavities.
The difficult degree of freedom is the differential Michelson. For the case of a
simple Michelson with signal recycling, the locking signal could be taken in the dark
port. This gives us a strong in-phase error signal that is created by the differential
Michelson, since the sidebands act as the local oscillator, and the carrier gets the
phase shift. There is no locking signal contribution from the signal-recycling mirror
since there is no carrier in the dark port to act as the local oscillator.
For the case of arm cavities the dark port error signal will ahl--i be dominated
by the differential Fabry-Perot signal. That leaves the reflected port and the pick off
signal. For the differential Michelson and the signal recycling error signals there is
no difference in the pick-off and the reflected port, except for an overall amplitude
change. The locking matrix for these two signals in those two ports is completely
linearly dependent. This would not be the case for the signals in the antisymmetric
port and the pick off signal. The locking matrix for these two ports as a function of
the differential Michelson and signal-recycling mirror would be linearly independent
if error signals were present. Unfortunately there is no locking signal for the signal-
recycling mirror in the dark port. A differential Michelson signal created by an in-
phase mechanism would be completely overwhelmed by the differential Fabry-Perot
in the antisymmetric port.
There is another way to get a locking signal for the differential Michelson that
is linearly independent from the signal recycling signal. A quadrature signal could
be used. By slightly detuning the lengths of the Michelson we can put the sideband
so it is not completely dark in the antisymmetric port. As the differential Michelson
changes the amplitude of the two sidebands will change, and a quadrature signal will
be created.
In order to ensure that the differential Michelson and the signal-recycling mirrors
are linearly independent the sideband has to be completely resonant in the signal-
recycling cavity. There would then be essentially no quadrature signal created by
the signal-recycling mirror. By tuning the phase of the local oscillator the amount
that the signal recycling degree of freedom appears in the differential Michelson error
signal can be completely zeroed.
We can again get the differential Fabry-Perot modes in the antisymmetric port
and the differential Michelson signal will be obtained at the pick-off inside the power-
recycling cavity. This can be the same port that signal recycling is take in. There
is no advantage in taking one in reflection and the other at the pick off. Again,
the reflected port and the pick off are completely linearly dependent for the signal
recycling and differential Michelson error signals.
Note that the error signal to lock the differential Michelson is now created by
a completely different mechanism than what will be used to lock the LIGO I con-
figuration. Simulations using Finesse show that this locking technique does indeed
give linearly independent solutions. The full locking matrix will be discussed in the
experimental section were there are known parameters for calculating the locking
scheme.
81
Broadband Signal Recycling
Detuned 10 Degrees From Broadband
.----... Detuned 87.4 Degrees From Broadband (LIGO II)
------- Resonant Sideband Extraction
10- No Signal Recycling
I 1 -
0.01
S1E-3
o 1E-4
1E-5
0.01 0.1 1 10 100 1000 10000
Frequency (Hz)
Figure 2.17: Frequency response of a signal-recycled Michelson with arm cavities
2.8.3 Frequency Response
The frequency response of a cavity-enhanced signal-re. 1, ,1 Michelson has al-
ready been touched on in the introduction. It requires a little more explanation
though.
Figure 2.17 shows the frequency responses of a signal-re' v 1' 1 interferometer
for various tunings of the signal-recycling mirror. The parameters used are from the
LIGO II conceptual design, so that the frequency behavior is what will be evident
in the real instrument. They have been normalized for the instrument without any
signal-recycling mirror. There are several features in the responses that should be
understood.
One feature in every curve is that there is an overall high frequency cut off. The
source of this cut off is the cavities which the gravity-wave signal sees. The cut off
changes frequency as the tuning of the signal-recycling mirror changes.
For the dual-recycled case the mirror is in the position that the carrier is resonant
in the cavity formed by it and the arm cavities. The -3 dB point is lower than if there
is no signal recycling, because the storage time for the gravitational wave signal is
effectively longer, resulting in a lower frequency at which the phase of the signal
already in the cavity is 180 degrees out of phase with the new signal being generated.
It's at this point that the signal begins to roll off. The frequency response for this
cavity is the same as a Fabry-Perot cavity with a higher finesse.
For the resonant sideband extraction case the mirror is positioned such that the
carrier is exactly antiresonant in the cavity formed by the signal-recycling mirror and
the arm cavities. In this case the -3 dB point is actually higher in frequency than if
there were no signal-recycling mirror. The reason for this is that the signal-recycling
mirror results in a low storage time in the arm cavities. The three-mirror cavity for
the gravity-wave signal formed by the arm cavities and the signal-recycling mirror
can be described as a two-mirror cavity. The gravity-wave sideband enters through
the end mirror and then encounters a complex mirror formed by the signal-recycling
mirror and the input arm cavity mirror. This complex mirror has an overall lower
reflectivity than the input arm cavity mirror, and the signal is coupled out through
the signal-recycling mirror. The response is exactly like a Fabry-Perot cavity with a
lower finesse.
Every case in between these two extremes results in a cavity that has a peak
response for a certain frequency. Figure 2.18 explains pictorial why that is so. The
signal-recycling cavity is detuned in such a way that the frequency that is resonant
is no longer the same as the carrier. This results in a frequency response that is
peaked at a frequency that is a function of the detuning. There is some cost for
this. As is clear in the figure 2.18, only one sideband can be recycled at the detuning
frequency. This is why the peak with 10 degrees detuning is approximately half that
of the broadband case. As the detuning approaches the RSE case the effective finesse
6 -- No Detuning
------ Detuned
5- \
a 4
o I
3-
Lower
1 Gravity Wave Sid and Upper
GravityjWave Sideband
--- -----------
Frequency
Figure 2.18: Signal-recycling cavity resonances for recycled and detuned case
of the cavity is lowered because of the effect discussed in the RSE case. This explains
the lower peak in the case where the detuning is 87.4 degrees with respect to the 10
degree case.
We can also explain the terminology broadband and narrow band instrument.
Although there is no narrowing of the resonance peak, the fact that only one sideband
is recycled makes the detuned case 1v 1i row band". Both the RSE and dual-recycled
case are termed broadband since they recycle both signal sidebands equally.
CHAPTER 3
EXPERIMENT
This chapter will concern itself with the tabletop dual-recycled cavity-enhanced
Michelson interferometer built to demonstrate the locking scheme. Section 3.1 will
discuss the physical parameters and components used in the experiment. Section
3.2 will present the locking matrices resulting from simulations. Section 3.3 will
characterize the losses and resonances for the carrier and sidebands in the tabletop
instrument. Section 3.4 will present the measured locking matrix and compare it
to the simulation. Section 3.5 will show the experimental results of the sensitivity
measurements as a function of frequency and compare them to the theory.
3.1 Design
The first step in designing the tabletop interferometer, detailed in Section 2.8.2,
is creating a locking scheme. The next step is to decide on mirror reflectivities.
Once that is done, the length and sidebands are specified to match the resonance
conditions stipulated in the locking scheme. This section will focus on these two
steps. The design section will also show the physical layout of all the components
and discuss the key components used to control and monitor the interferometer.
3.1.1 Selection of Mirror Parameters
Our locking scheme requires that sidebands are not albv--i- resonant or antireso-
nant in a particular cavity. In order to calculate all the conditions that are necessary
for the sidebands, we have to take into account the phase shifts encountered by the
sidebands from cavities when they are not fully resonant. Also, in order to set the
sideband in the full width, half maximum of the power-recycling cavity (a stipulation
of our locking scheme) we must know the finesse of the cavities. These considerations
dictate that we select mirror reflectivities before sideband frequencies and cavity
lengths are chosen.
We begin by specifying the arm cavity mirrors. The losses that are incurred in
these cavities will constrain the power-recycling mirror reflectivity. The first condi-
tion on these optics is that the carrier should be strongly over-coupled in the entire
interferometer. In order to achieve this, we must estimate the mirror losses that we
will be using. We selected mirrors manufactured by CVI Lasers for cost and avail-
ability reasons. The CVI high reflector (HR) mirrors were found to have losses of
approximately 0.001;:'. from our measurements. This limited the cavity input cou-
pler to 91,' reflectivity in order to stay well within the stipulation that the overall
interferometer be over-coupled. The finesse of the arm cavities is then limited to ap-
proximately 60. An added advantage of going with lower finesse arm cavities is that
the cavities are much more stable since their linewidth is larger. More stable cavities
are definitely an advantage when dealing with such a complex instrument, and the
mirror parameters chosen still provide a solid test of principles.
The power-recycling mirror can now be specified. With the losses in the arm
cavities around I' it seems prudent to choose a power-recycling mirror of 1I' .
reflectivity to ensure over-coupling. In the end, the interferometer to be very close
to impedance matched because of additional losses in the power-recycling cavity.
Without losses the power recycling factor would be about 17.
The signal-recycling mirror should give a reasonable recycling factor. It should
also keep the power-recycling, signal-recycling cavity over-coupled for the sideband
that will be used to lock it. We selected a signal-recycling mirror of I,.'. reflectivity.
If the differential Fabry-Perot signal is taken from behind the signal-recycling mirror,
then we would have a signal gain of
G, -tsr 3. (3.1)
1 rsr
The mirror diameters are chosen according to several different considerations.
For the mirrors that are actuated directly, the mirror should weigh as little as possible
because of resonant frequency considerations. This is the case for the cavity end
mirrors and the signal-recycling mirror. The cavity end mirrors were chosen to be
0.5 inches, and the cavity input mirror were also 0.5 inches in diameter for symmetry
reasons. The signal-recycling mirror is 1 inch, but, as will be described later, this
degree of freedom has the lowest feedback gain, and the resonant frequency is not
a concern. The power-recycling mirror is not directly actuated. A 1 in mirror was
chosen. The beam splitter is at 45 degrees and it is easiest to use a 2 inch optic in
order to avoid any of the beams clipping on an edge.
Table 3.1: Designed mirror specification
Designed Measured Mirror Mirror
Parameter Reflectivity Reflectivity Curvature Diameter
Power Recycling '. 82. !' 2 m 1 in
Arm Cavity Input Coupler 91i' 911' 4 m 0.5 in
Arm Cavity End Mirror HR 99.>' 6 m 0.5 in
signal-recycling mirror 51', 86.1 4 m 1 in
Beam Splitter 50'. 50'. Flat 2 in
The actually values vary from the specified value by a few percent, which is as
close as the manufactures generally quotes them. The only optic that we needed to
make sure is very close to the actual value specified is the beam splitter. A beam
splitter that is not 50/50 would create many problems, and would not be a good test
of the locking scheme.
There is also the matter of picking the correct radius of curvature for the mirrors
so that it matches the gaussian mode. The details of this will not be gone through in
this work, but appendix A will describe how to do the calculation. The cavity lengths
must be known before one can calculate the modes.
L,
SEo.t ERefl
4 E
/ j antisym
Figure 3.1: Definition of lengths
3.1.2 Length and Frequency Considerations
Figure 3.1 shows the layout and length definitions on our tabletop interferometer.
The main difference between this layout and the typical signal recycling topology is
that the signal-recycling cavity has been turned, and the partially reflective optic is
the turning mirror. The turned signal-recycling cavity has the advantage that the
end mirror in the signal-recycling cavity can be blocked, but all the light fields are
still present on the photodetectors. With the turned signal-recycling cavity, blocking
the signal-recycling mirror allows the LIGO I configuration to be locked. Ideally the
interferometer will remained locked when the beam block is removed from in front of
the signal-recycling cavity, resulting in a very simple lock acquisition.
The first condition of the locking scheme which must be satisfied is that one on
the sidebands, Q2 is bright in the reflected port for the Michelson. The other sideband,
Q1, should be almost bright in the antisymmetric port. This has the unfortunately
consequence that they are multiples of each other since 2 is at the free spectral range
of the Michelson and QR is half the free spectral range of the Michelson. We will solve
this problem later.
The determination of the radio frequency (RF) of the sidebands is governed by
physical constraints. The wavelength of the RF sidebands must not be so long that
the cavity lengths can not fit on the table. This puts a lower limit on the frequencies
around 10 meters, or 30 MHz. In general, it is desirable to select the lowest possible
frequency since electronics become exponentially more difficult to find and build as
their frequency increases. The fact that the sidebands are multiples of each other
results in the other sideband being 60 MHz.
The next step is determining the .- i-vi ii. I1 ry in the Michelson arm lengths, i -lp.
We will put the 60 MHz sideband, Q2, bright in the reflected port. That immediately
sets the arm length mismatch. The sidebands will be bright in the reflected port
every free spectral range of the Michelson. In this case
c
FSRMI = ( 1 =60 MHz, (3.2)
2 (- lp)
or
li 1p = 2.5 m. (3.3)
We will find that 1p = .3 m and i = 2.8 m satisfy our requirements well.
The next length that we specify is the power recycling length, i,,. First we
calculate the length of the power-recycling cavity so that the 60 MHz sideband is
resonant in it. We then detune the length of the power-recycling cavity so that the
sideband is sitting somewhere within the full width, half maximum. This will give us
a strong quadrature signal to lock the power-recycling cavity.
In LIGO I, the sideband used to lock the power-recycling cavity is on resonance
for the power-recycling cavity when the frequency is at integer multiples plus a half of
the free spectral length. The reason for this is because the Michelson arm .-i.!liii_. I ry
is small, 0.58m, and the modulation frequency has a long wavelength, 24m. The
Michelson acts as a small amplitude loss mechanism for the sideband, but otherwise
does not disturb it. The power-recycling mirror and the arm cavities behave as a
normal three mirror coupled cavity. The carrier receives a r phase shift from the
cavity. In order for the carrier to be resonant in the power-recycling cavity the cavity
length must be shifted by j and the cavity now has resonances at FSR (n + '). Since
the sidebands do not get this r phase shift they need to be at an integer plus a half
free spectral range in order to be resonant.
1.0-
0
0.5- + 30MHz
H 0.0
S-05- 30MHz
-1.0
-3k/8 0 3R/8
Scanning Differential Michelson Length
Figure 3.2: Electric fields in the reflected port
In contrast we place the sideband used to lock the power-recycling cavity, Q2
completely in the next dark fringe. This gives an additional r phase shift to Q2 that
the carrier does not see. This is shown pictorially in Figure 3.2. The carrier sits in a
bright fringe. The 30 MHz sideband pair is transmitted fully to the dark port. The
60 MHz sideband pair sits at the next bright fringe, which has given each sideband
a sign flip from when it entered the Michelson. Since the carrier does not receive
this sign flip, but does receive a sign flip from the arm cavities, they are both again
resonant at integer multiples of the free spectral range.
We are now ready to specify the length of our power-recycling cavity.
c
= 60 MHz, (3.4)
2 + 'Pj+
or
lp, + = 2.5 m (3.5)
2
For our previous picks of lp and Ii, then lpr = 0.95 m.
1.0-
0.
0.5
0.0
r-e
S-0.5 30MHz + 30MHz
-1.0 .
-3/8 0 3/W8
Scanning Michelson Length
Figure 3.3: Electric fields in the antisymmetric port
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