Electrooptic Modulators
for Cavity Locking
Case Study
Physics 208, Electro-optics
Peter Beyersdorf
Document info
1
Class Outline
Overview of cavity locking
LIGO input spectrum
The role of modulation sidebands
Methods to generate various input spectra
Advanced LIGO modulators
through the STAIC website.[49]
Advanced LIGO Configuration
3.2
The RSE Interferometer
Figure 3.6 shows the RSE interferometer, along with the labels relevant to this section.
There are five longitudinal degrees of freedom in RSE. “Longitudinal” indicates that
ETM2
Φ+ = φ3 + φ4
Φ− = φ3 − φ4
φ+ = φ0 + (φ1 + φ2 )/2
φ− = (φ1 − φ2 )/2
φs = φ5 + (φ1 + φ2 )/2
φ4
ITM2
rac2
φ2
PRM
Laser
+ −
φ0
ITM1
+
rprm
−
φ1
φ5
PD1
PD2
rsem
ETM1
rbs r
ac1
+
−
PD3
SEM
φ3
Cavity Locking
Laser field must resonate in the cavities for
interferometer to operate as intended
External disturbances cause cavity length and laser
frequency to fluctuate, thus active sensing and control is
required to keep laser resonant in the cavities
Microwave technique for locking cavities developed by
Pound was adapted to optical cavities by Drever and
Hall, and used by Jan Hall for advances in laser
stabilization that won he and Ted Hansch the 2005
Nobel prize in Physics
Pound-Drever-Hall technique for cavity locking
(alternatively laser frequency stabilization) is widely
used
Conceptual Model
Consider a Fabry-Perot cavity illuminated by a laser of
frequency f, slightly detuned from the cavity resonance
by δf=f-fres
Ein
Er
Ec
Et
L
Treat the input mirror as having a reflectivity of r1>0
when seen from inside the cavity, and the end mirror as
having a reflectivity of r2>0 when seen from inside the
cavity.
The cavity reflectance is
t21 r2 e2ikL
rcav (f ) = −r1 −
1 − r1 r2 e2ikL
Conceptual Model
t21 r2 e2ikL
rcav (f ) = −r1 −
1 − r1 r2 e2ikL
Cavity Reflectance
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Modulation of the round trip phase
(by modulating laser frequency or
cavity length) at a frequency fm
produces a modulation on the
reflected power at fm if cavity is
off-resonance
Fig. 1. Transmission of a Fabry–Perot cavity vs frequency of the incident
Fig. 2. The reflected light intensity from a Fabry–Perot cavity as a function
of laser frequency, near resonance. If you modulate the laser frequency, you
Carrier-Sideband Picture
1
R(f)
0.5
-1.5
-1
-0.5
sideband
P(f)
f0-fm
0
0.5
f/ffsr
carrier
sideband
f0
f0+fm
1
1.5
A (phase) modulated beam has a carrier and sidebands spaced by
frequency the modulation frequency fm, as long as fm≠nffsr, the
modulation frequency is not an integer multiple of the cavity’s free
spectral range (ffsr=c/2L), the sidebands will reflect off the cavity with
high efficiency, while the carrier will “see” a much lower reflection
coefficient due ot the frequency dependence of the reflection coefficient
Carrier-Sideband Picture
2!
φr(f)
!
f/ffsr
-1.5
-1
-0.5
sideband
P(f)
f0-fm
0
0.5
carrier
sideband
f0
f0+fm
1
1.5
The phase shift upon reflection from the cavity is essentially 0 for the
sidebands, but the carrier has a phase shift that is a strong function of
the detuning of the cavity (i.e. the difference between the cavities
resonant frequency and the carrier frequency) when the carrier is near
resonance (within a linewidth Δf=ffsr/F). Near resonance it has a slope
of 2πF/ffsr, when measured as a function of frequency
Carrier-Sideband Picture
Phase shift of carrier transforms phase
modulation into amplitude modulation
negative phase shift
noitcnuf a sa ytivac toreP – yrbaF a morf ytisnetni thgil detcefler ehT .2 .giF
uoy ,ycneuqerf resal eht etaludom uoy fI .ecnanoser raen ,ycneuqerf resal fo
rewop detcefler eht woh yb no era uoy ecnanoser fo edis hcihw llet nac
no phase shift
tnedicni eht fo ycneuqerf svFig.
ytiv1.acTransmission
toreP – yrbaF of
a foa Fabry–Perot
noissimsnarTcavity
.1 .giFvs frequency of the incident
fo erutcurts eht ekam ot ,21light.
tuobaThis
,essecavity
nfi wohas
l ylraiafairly
f a sahlow
ytivfinesse,
ac sihTabout
.thgil12, to make the structure of
positive phase shift
Fig. 2. The reflected light intensity from a Fabry–Perot cavity as a function
of laser frequency, near resonance. If you modulate the laser frequency, you
can tell which side of resonance you are on by how the reflected power
changes.
the transmission lines easy to see.
compared with the local oscillator’s
resonance with the cavity, you can’t tell just by looking at
can think of a mixer as a device who
the reflected intensity whether the frequency needs to be inof its inputs, so this output will conta
creased or decreased to bring it back onto resonance. The
very low frequency" and twice the m
derivative of the reflected intensity, however, is antisymmetis the low frequency signal that we
ric about resonance. If we were to measure this derivative,
that is what will tell us the derivativ
we would have an error signal that we can use to lock the
sity. A low-pass filter on the output o
laser. Fortunately, this is not too hard to do: We can just vary
low frequency signal, which then go
the frequency a little bit and see how the reflected beam
plifier and into the tuning port on the
responds.
to the cavity.
Above resonance, the derivative of the reflected intensity
The Faraday isolator shown in Fi
with respect to laser frequency is positive. If we vary the
beam from getting back into the las
laser’s frequency sinusoidally over a small range, then the
This isolator is not necessary for u
reflected intensity will also vary sinusoidally, in phase with
Actuator
nique, but it is essential in a real s
the
variation
in
frequency.
!See
Fig.
2."
Pockels Cell
Cavity
Laser
small amount of reflected beam that
Below resonance,
(PC)this derivative is negative. Here the reisolator is usually enough to destabil
flected intensity will vary 180° out of phase from the frethe phase shifter is not essential in
quency. On resonance the reflected intensity is at a miniuseful in practice to compensate for
mum, and a small frequency variation will produce no
two signal paths. !In our example, it
change in the reflected intensity.
between the local oscillator and the
By comparing the variation in the reflected
intensity
with
Photodetector
This conceptual model is really on
the frequency variation we can tell which side of resonance
ering the laser frequency slowly. If y
we are on. Once we have a measure of the derivative of the
too fast, the light resonating inside
reflected intensity with respect to frequency, we can feed this
time to completely build up or settl
measurement back to the laser to hold it on resonance. The
will not follow the curve shown in
purpose of the Pound–Drever–Hall method is to do just this.
technique still works at higher modu
Figure 3 shows a basic setup. Here the frequency is moduboth the noise performance and ban
lated with a Pockels cell,20 driven by some local oscillator.
The reflected Local
beamOscillator
is picked off with
isolator
!aAmp typically improved. Before we addre
Mixeran optical
Servo
low pass
(LO) and a quarter-wave filter
that does apply to the high-frequency
polarizing beamsplitter
plate makes a
lish a quantitative model.
good isolator" and sent into a photodetector, whose output is
Experimental Schematic
Modulation of the laser frequency
is accomplished by a pockels cell
driven by a sinusoidal oscillator.
Any modulation at fm on the light
reflected from the cavity is
detected (by mixing with the local
PDH schematic for locking a cavity onto a laser
Figure 1: The basic layout for locking a cavity to a laser. Solid lines are optical paths, and dashed
oscillator) and used to provide
lines are signal paths. The signal going to the far mirror of the cavity controls its position.
feedback to the cavity (or laser). 2 A conceptual model
Fig. 1 shows a basic Pound-Drever-Hall setup. This arrangement is for locking a cavity to a laser, and
it is the setup you would use to measure the length noise in the cavity.1 You send the beam into the
cavity; a photodetector looks at the reflected beam; and its output goes to an actuator that controls the
length of the cavity. If you have set up the feedback correctly, the system will automatically adjust
the length of the cavity until the light is resonant and then hold it there. The feedback circuit will
compensate for any disturbance (within reason) that tries to bump the system out of resonance. If you
keep a record of how much force the feedback circuit supplies, you have a measurement of the noise
in the cavity.
Setting up the right kind of feedback is a little tricky. The system has to have some way of telling
which way it should push to bring the system back on resonance. It can’t tell just by looking at the
1
Locking the laser to the cavity follows essentially the same design, the only difference being that you would feed
back to the laser, rather than the cavity.
80
PDHAm.schematic
for
locking
a laser onto a cavity
J. Phys., Vol. 69, No.
1, January
2001
3
Fig. 3. T
cavity to
paths and
The sign
its freque
Quantitative Model
Recall that the phasor amplitude for phase modulated light can
be written as
!
Ein ≈ E0 J0 (m)e
iωt
!
+ iJ1 (m) e
i(ωt−Ωt)
+e
i(ωt+Ωt)
""
where m is the modulation depth, Ω=2πfm is the modulation
frequency and ω=2πf0 is the carrier frequency. The field
reflected from the cavity is
!
Er ≈ E0 r(ω)J0 (m)e
iωt
+ iJ1 (m) r(ω − Ω)e
which can be approximated by
!
E ≈ E0 −r0 J0 (m)e
!
i(ωt+2πF δf /ff sr )
i(ωt−Ωt)
!
+ iJ1 (m) e
+ r(ω + Ω)e
i(ωt−Ωt)
i(ωt+Ωt)
+e
i(ωt+Ωt)
""
""
when the carrier is near resonance, the sidebands are far offresonance and r1,r2≈1. Here r0 is the magnitude of the cavity
reflection on resonance and δf is the detuning of the laser from
the cavity
Quantitative Model
The relative intensity detected by a photodetector is proportional to the magnitude of
!
!
""
the field squared
i(ωt+2πF δf /ff sr )
i(ωt−Ωt)
i(ωt+Ωt)
E ≈ E −r J (m)e
+ iJ (m) e
+e
0
giving
I ∝ E∗E
!
0 0
1
= E02 r02 J02 (m) + 2J12 (m)
!
− ir0 J0 (m)J1 (m)
+
or
E∗E
E∗E
"
"
i(2πF δf /ff sr +Ωt)
e
−e
ei(2πF δf /ff sr −Ωt) − e−i(2πF δf /ff sr −Ωt)
%
&
+ J1 (m)2 e2iΩt + e−2iΩt
−i(2πF δf /ff sr +Ωt)
$
#
$
#
= DC terms + 2ω terms
!
"
2
+ 2E0 r0 J0 (m)J1 (m) sin (2πFδf /ff sr + Ωt) + sin (2πFδf /ff sr − Ωt)
= DC terms + 2ω terms
+ 4E02 r0 J0 (m)J1 (m) sin(2πFδf /ff sr ) cos(Ωt)
This is the amplitude of the modulated power at the modulation
frequency, near resonance (δf<<ffsr/F) this is proportional to δf
Demodulation
Laser
Pockels Cell
(PC)
Cavity
Consider a detected photocurrent
of the form
Actuator
Photodetector
Vlo
Vdet = V0 + VΩ cos(Ωt) + V2Ω cos(2Ωt)
Vdet
Vmix Vlpf
it gets “mixed” with a “local oscillator”
of the form Vlo=cos(Ωt) which is equivalent to multiplication
Local Oscillator
(LO)
Mixer
low pass Servo Amp
filter
Figure 1: The basic layout for locking a cavity to a laser. Solid lines are optical paths, and dashed
lines are signal paths. The signal going to the far mirror of the cavity controls its position.
Vmix = Vlo Vdet = V0 cos(Ωt) + VΩ cos2 (Ωt) + V2Ω cos(2Ωt) cos(Ωt)
2 A conceptual model
Fig. 1 shows a basic Pound-Drever-Hall setup. This arrangement is for locking a cavity to a laser, and
it is the setup you would use to measure the length noise in the cavity.1 You send the beam into the
cavity; a photodetector looks at the reflected beam; and its output goes to an actuator that controls the
length of the cavity. If you have set up the feedback correctly, the system will automatically adjust
the length of the cavity until the light is resonant and then hold it there. The feedback circuit will
compensate for any disturbance (within reason) that tries to bump the system out of resonance. If you
2
keep a record of how much force the feedback
circuit supplies, you have a measurement of the noise
1
0 0 0
in the cavity.
ΩSetting up the right kind of feedback is a little tricky. The system has to have some way of telling
which way it should push to bring the system back on resonance. It can’t tell just by looking at the
the mixed signal is then low pass filtered to get
Vlpf
1
≈ lim
T →∞ T
!
t
t−T
Vmix dτ =
1
V
2
1
This is 4E r J (m)J (m)δf
and provides a measure of the detuning of
the laser from the cavity (or vice versa)
Locking the laser to the cavity follows essentially the same design, the only difference being that you would feed
back to the laser, rather than the cavity.
3
Hall in practice
the carrier
near resonance
andmodulation
the modulation
When When
the carrier
is nearis resonance
and the
frequency
high enough
that
the sidebands
arewe
not,can
we can
frequency
is highisenough
that the
sidebands
are not,
assume that the sidebands are totally reflected, F( $ $!)
assume that the sidebands are totally reflected, F( $ $!)
The reflected power is just P ref! P 02! F( $ ) ! 2 , and we
&#1. Then the expression
The reflected power is just P ref! P 0 ! F( $ ) ! , and we
&#1. Then the expression
might expect it to vary over time as
might expect it to vary over time as
F " $ # F * " $ "! # #F * " $ # F " $ #! # &#i2 Im( F " $ # ) ,
F " $ # F * " $ "! # #F * " $ # F " $ #! # &#i2 Im( F " $ # ) , "4.1#
d P ref
d"P$ref# "
! % cos !t
P " $ "! % cos !t # & P ref
"4.1#
$ !t
% cos !t # & P ref" $ # "
! %d cos
P ref" $ "!ref
is
purely
imaginary.
In
this
regime,
the
cosine
term
in Eq.
d$
is
purely
imaginary.
In
this
regime,
the
cosine
term
in
Eq.
"3.3# is negligible, and our error signal becomes
2
"3.3#
is
negligible, and our error signal becomes
2 d!F!
"!P 0
! % cos !t.
& P ref" $d #! F
' !#2 !P c P s Im( F " $ # F * " $ "! # #F * " $ # F " $ #! # ) .
! %d $cos !t.
& P ref" $ # " P 0
' !#2 !P c P s Im( F " $ # F * " $ "! # #F * " $ # F " $ #! # ) .
d$
Figure 7 shows a plot of this error signal.
In the conceptual model, we dithered the frequency of the
Figure
7 Near
showsresonance
a plot of the
this reflected
error signal.
In the laser
conceptual
model,
we
dithered
the
frequency
of
the
power essentially vanishes,
adiabatically, slowly enough that the standing wave in2
Near
resonance
the
reflected
power
essentially
vanishes,
laser adiabatically,
slowly
the standingwith
wavethein-incident
terms to
first order in
since !2F( $ ) ! &0. We do want to retain
side the cavity
wasenough
alwaysthat
in equilibrium
!
F(
$
)
!
&0.
We
do
want
to
retain
terms
to
first
order
in
since
side the beam.
cavity We
wascan
always
in
equilibrium
with
the
incident
express this in the quantitative model by makF( $ ), however, to approximate the error signal,
beam. We
this In
in the
model by makF( $ ), however, to approximate the error signal,
ingcan
! express
very small.
this quantitative
regime the expression
P ref&2 P s #4 !P c P s Im( F " $ # ) sin !t" " 2! terms# .
ing ! very small. In this regime the expression
P ref&2 P s #4 !P c P s Im( F " $ # ) sin !t" " 2! terms# .
modulated beam, the instantaneous frequency is
d$ " t # ! d " $ t" % sin !t # ! $ "! % cos !t.
$ " t # ! " $ t"dt
% sin !t # ! $ "! % cos !t.
dt
Error Signal
The demodulated (and low pass filtered signal) is
a measure of the laser’s detuning from the
cavity resonance and is called an “error signal”
Fig. 7. The Pound–Drever–Hall error signal, ' /2!P c P s vs $ /* + fsr , when
Fig. 6. The Pound–Drever–Hall error signal, ' /2!P c P s vs $ /* + fsr , when
the modulation frequency is high. Here, the modulation frequency is about
Fig.
Pound–Drever–Hall
' /2!Prange,
$ /*a+cavity
c P s vswith
fsr , when
the modulation frequency is low. The modulation frequency is about half
a 7. The
20 linewidths:
roughly 4%error
of a signal,
free spectral
finesse of
Fig. 6. Thelinewidth:
Pound–Drever–Hall
#3 error signal, ' /2! P c P s vs $ /* + fsr , when
the
modulation
frequency
is
high.
Here,
the
modulation
frequency
is about
about 10 of a free spectral range, with a cavity finesse of 500.
500.
the modulation frequency is low. The modulation frequency is about half a
20 linewidths: roughly 4% of a free spectral range, with a cavity finesse of
linewidth: about 10#3 of a free spectral range, with a cavity finesse of 500.
500.
83
Am. J. Phys., Vol. 69, No. 1, January 2001
Eric D. Black
83
error signal as a function of
detuning (for small tuning range)
83
Am. J. Phys., Vol. 69, No. 1, January 2001
error signal as a function of
detuning (for large tuning range)
Eric D. Black
83
Negative Feedback
noise
δx
x1
system
-Gx1
G
Gx1
inv
In the steady state we must have
% % % δx-Gx1=x1
giving
δx
x1 =
1+G
meaning the noise is suppressed by a factor of 1+G. The
“noise” can be laser frequency or cavity length fluctuations
1
1
2
state of the interferometer as well as design and debug the experiment. It is available
where n1 is an integer, and ff srprc = c/(2lprc ). This argument applies to each of
through the STAIC website.[49]
the modulation frequencies, with a different integer, n2 for the second RF sideband.
Cavities
in
LIGO
The RSE Interferometer
Clearly the difference of the two frequencies will be some integer multiple of the free
3.2
spectral range.
It’s desired that one of the RF sideband frequencies resonates in the signal cavity,
while the other does not. The non-resonant RF sideband then acts as a local oscillator
Figure 3.6 shows the RSE interferometer,
along with the labels relevant to this section.
for the resonant RF sideband, thus providing a signal extraction
port for the φ s
of freedom.
An example
solution is shown
in Figurethat
3.10, where f2 = 3f1 .
There are five longitudinal degrees ofdegree
freedom
in RSE.
“Longitudinal”
indicates
ETM2
Φ+ = φ3 + φ4
Φ− = φ3 − φ4
φ+ = φ0 + (φ1 + φ2 )/2
φ− = (φ1 − φ2 )/2
φs = φ5 + (φ1 + φ2 )/2
−f2
Laser
+ −
rprm
−f1
f0
+f1
+f2
f
φ4
ITM2
rac2
PRM
For
Input spectrum
f
Power recycling cavity
spectral schematic
f
Signal cavity
spectral schematic
Figure 3.10: Power and signal cavity spectral schematic for broadband RSE. The RF
φ2 are related by f2 = 3f1 .
sidebands
φ0
ITM1
rbs r
ac1
+
ETM1
φ1RSE, some care must
φ3 be taken as to which frequency is resonant in the
broadband
−
signal
φ5 cavity. For example, in the f2 = 3f1 case, clearly if f1 were resonant in the
PD1
PD2
signal cavity, f2 would be resonant as well.
+
SEM
The− constraint
on which frequency is resonant in the signal cavity doesn’t usually
rsem
16
apply to a detuned RSE interferometer. Since both the frequency of one of the RF
PD3 and the frequency of the detuning must be resonant in the signal cavity,4
sidebands,
4
LIGO Input Spectrum
Various frequency components are necessary for
length and alignment sensing of the many
cavities and interferometers in the LIGO
detector
phase modulation sidebands for locking “power recycling
cavity” and Michelson interferometer (9 MHz)
phase modulation sidebands for locking the arms (180 MHz)
carrier
9 MHz sidebands
P(f)
f
180 MHZ sidebands
17
LIGO Modulators
Initial LIGO
LiNbO3 slabs with 10mm x 10mm clear aperture for Initial
LIGO (operates with up to 10W of power)
Transverse modulation
resonant circuit geometry
9MHz phase modulation sidebands for interferometer
sensing
Advanced LIGO
RTP (RbTiOPO4) (operates with up to 300W of
power)
9 MHz and 180 MHz PM sidebands for interferometer
Sensing
18
Role of Modulation Sidebands
Cavity Reflectance
t2
rcav (f ) = r −
1 − r2 ei2πf /ff sr
|r(f)|
-1.5
-1
-0.5
0
carrier
0.5
1
1.5
f/ffsr
9 MHz sidebands
P(f)
f
180 MHZ sidebands
Sideband frequencies are such that only one component is resonant in the cavity of interest
19
Role of Modulation Sidebands
Cavity Reflectance
t2
rcav (f ) = r −
1 − r2 ei2πf /ff sr
φr=arg(r(f))
2!
f/ffsr
-1.5
-1
-0.5
0
carrier
0.5
1
1.5
9 MHz sidebands
P(f)
f
180 MHZ sidebands
Component that resonates in the cavity acquires a phase shift upon reflection that is a
function of the cavities detuning.
20
Higher Order Sidebands
Higher order modulation harmonics
P(f)
f
P(f)
sidebands of sidebands
f
Demodulation gives the sum of all frequency
components spaced by flo. Higher order modulation
harmonics and sidebands-of-sidebands can produce
unwanted contributions
21
Unintended Sidebands
Higher order (for example 18 MHZ sidebands
from the 9 MHz modulator)sidebands are
problematic because the intermodulation they
produce can obscure intended signals
Solutions:
Low modulation depth
Sideband cancellation
Parallel modulation
22
Serial Modulation
For small modulation depth J1(m)≈m/2 and J2(m)≈m/24
so intermodulation depth is m1m2/48
9 MHz
Power spectrum after
9 MHz modulator
180 MHz
Power spectrum after
9 MHz and 180 MHz
modulators
23
Harmonic Compensation
Through proper adjustment of amplitude and
phase of 189 and 171 MHz signals, the
intermodulation harmonics can be cancelled
9 MHz
Power spectrum after
9 MHz and 180 MHz
modulators
180 MHz
-
x
=
Modulation spectrum
from third modulator
Power spectrum after
9 MHz and 180 MHz
modulators
Parallel Modulation
A Mach-Zehnder interferometer can
be used to combine modulation
sidebands from two independent
modulators
Drawbacks include reduction in
effective modulatin depth by a
factor of 2 due to sidebands
lost to the unused port and
increased complexity of maintaining
Mach-Zehnder interference
condition
180 MHz
9 MHz
25
Single Sideband Generation
Some control and readout schemes require a
“Single sideband” rather than a pair of phase
modulated sidebands or amplitude modulated
sidebands
Example: mapping out the frequency response
of a detuned interferometer
26
Single Sideband Generation
Phase modulation produces a pair of sidebands at
the modulation frequency that are each in phase
with the carrier (at some instant in time)
Amplitude modulation produces a pair of
sidebands a the modulation frequency with one
in phase with the carrier while the other is π
out-of-phase with the carrier.
Combining amplitude and phase modulation at the
same frequency, one sideband in a pair can be
enhanced while the other is suppressed.
Sub-Carrier Generation
Single Sideband Production
An alternative way
to generate a single
sideband is to phase
lock two lasers with
a given frequency
offset. This had
been proposed for
Advanced LIGO
28
Advanced LIGO modulators
Three sets of modulation electrodes on one
crystal
29
Modulator Crystals
To avoid interference effects from reflections off
the modulator faces, the crystal has a 2x2.8°
wedge
The wedge leads to polarization dependent
transmission angle, and thus the modulator also
acts like a polarizer
30
Resonant Circuit
The resonant circuit is designed to have 50Ω
impedance at the resonant frequency, but high
impedance at DC and low frequencies.
186
APPENDIX B. ELECTRONICS
Figure B.10: Alternative resonant transformer circuit with a capacitive voltage divider.
the example, the capacitor C needs to have about 1.6 nF. It behaves differently from
the previously discussed circuits in so far as there are now two resonances close to
each other (a parallel resonance and a series resonance). Figure B.11 shows the input
impedance of this circuit. The presence of the two resonances makes the adjustment
more difficult. On the other hand, it may be an advantage that at low frequencies the
input impedance is very high (instead of a short circuit as in Figure B.9).
Impedance [!]
100
10
1
9.4
9.6
9.8
10
10.2
Frequency [MHz]
10.4
10.6
Figure B.11: Input impedance of the alternative resonant transformer circuit with a capacitive
voltage divider.
31
References
Guido Mueller, David Reitze, Haisheng Rong, David Tanner, Sany Yoshida,
and Jordan Camp “Reference Design Document for the Advanced LIGO
Input Optics”, LIGO internal document T010002-00 (2000)
James Mason, “Signal Extraction and Optical Design for an Advanced
Gravitational Wave Interferometer” Ph.D. Thesis, CIT (2001)
M Gray, D Shaddock, C Mow-Lowry, D. McClelland “Tunable Power
Recycled RSE Michelson for LIGO II”, Internal LIGO document
G000227-00-D
G. Heinzel “Advanced optical techniques for laser-interferometric
gravitational-wave detectors” Ph.D. Thesis MPQ (1999)OSA
E Black “An introduction to Pound – Drever – Hall laser frequency
stabilization”, Am. J. Phys. 69 (2001)