Analysis of spatial mode sensitivity of a gravitational wave
interferometer and a targeted search for gravitationalradiation from the Crab pulsar
byJoseph Betzwieser
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 2007
c Joseph Betzwieser, MMVII. All rights reserved.
!
The author hereby grants to MIT permission to reproduce and distribute publicly
paper and electronic copies of this thesis document in whole or in part.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
August 31, 2007

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nergis Mavalvala
Associate Professor of Physics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thomas J. Greytak
Associate Department Head for Education

2

Analysis of spatial mode sensitivity of a gravitational wave interferometerand a targeted search for gravitational radiation from the Crab pulsar
byJoseph Betzwieser
Submitted to the Department of Physics
on August 31, 2007, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Over the last several years the Laser Interferometer Gravitational Wave Observatory (LIGO)
has been making steady progress in improving the sensitivities of its three interferometers,
two in Hanford, Washington, and one in Livingston, Louisiana. These interferometers havereached their target design sensitivities and have since been collecting data in their fifthscience run for well over a year.
On the way to increasing the sensitivities of the interferometers, difficulties with increasing the input laser power, due to unexpectedly high optical absorption, required theinstallation of a thermal compensation system. We describe a frequency resolving wavefront sensor, called the phase camera, which was used on the interferometer to examine theheating effects and corrections of the thermal compensation system. The phase camera wasalso used to help understand an output mode cleaner which was temporarily installed on
the Hanford 4 km interferometer.
Data from the operational detectors was used to carry out two continuous gravitational
wave searches directed at isolated neutron stars. The first, targeted RX J1856.5-3754,
now known to be outside the LIGO detection band, was used as a test of a new multiinterferometer search code, and compared it to a well tested single interferometer searchcode and data analysis pipeline. The second search is a targeted search directed at the Crabpulsar, over a physically motivated parameter space, to complement existing narrow time
domain searches. The parameter space was chosen based on computational constraints,
expected final sensitivity, and possible frequency differences due to free precession and asimple two component model. An upper limit on the strain of gravitational radiation fromthe Crab pulsar of 1.6 × 10−24 was found with 95% confidence over a frequency band of
6×10−3 Hz centered on twice the Crab pulsar’s electromagnetic pulse frequency of 29.78 Hz.
At the edges of the parameter space, this search is approximately 105 times more sensitive
than the time domain searches. This is a preliminary result, presently under review by the
LIGO Scientific Collaboration.
Thesis Supervisor: Nergis Mavalvala
Title: Associate Professor of Physics

3

4

Acknowledgments
I have been lucky to be able to work with an incredible group of people doing what somepeople said was impossible ten or twenty years ago. During my time at MIT and at the
Hanford site I have met, worked with, and learned from too many people to thank properly.
However, I’d still like to try to thank a few by name.
To Luca, thanks for showing me the ropes at Hanford.
To Keita, thanks for your all insightful suggestions.
To Fred, thanks for all the words of wisdom.
To Dave, thanks for teaching me the ins and outs of optics.
To Gregg, thanks for teaching me how to make a ringdown measurement.
To Myron, thanks for showing me how to use a machine shop the right way.
To Paul, Josh, and Richard, thanks for sharing your electronics expertise.
To the whole Hanford crew, thanks for showing me how to put together, break, fix, andfinally run a LIGO interferometer.
To Mike, Greg, MAP, and the rest of the continuous wave search group, thanks forteaching me how to hear the universe.
To the group at MIT, thanks for sharing your experience and your friendship.
To Nergis, thanks for your unwavering support and guidance.
I’d also like to thank Yoon and the rest of my family for the support and love they haveshown me these many years.
- Joe October 3, 2007

5

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6

Contents
I

Introduction

15

1 Generating gravitational waves
1.1

1.2

17

General relativity and gravitational waves . . . . . . . . . . . . . . . . . . .

17

1.1.1

Gravitational wave effects on test masses . . . . . . . . . . . . . . .

18

1.1.2

Generation of gravitational waves . . . . . . . . . . . . . . . . . . . .

19

Continuous gravitational wave sources . . . . . . . . . . . . . . . . . . . . .

21

1.2.1

Potential source: RX J1856.5-3754 . . . . . . . . . . . . . . . . . . .

23

1.2.2

Potential source: the Crab pulsar . . . . . . . . . . . . . . . . . . . .

25

2 Laser interferometer gravitational wave detectors
2.1

2.2

II

Laser interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.1.1

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

27

2.1.2

Laser and mode cleaners . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.1.3

Fabry-Perot arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.1.4

Power recycling cavity . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.1.5

AS port . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.2.1

Thermal heating and thermal compensation . . . . . . . . . . . . . .

32

2.2.2

Mode and power mismatches and the AS I signal . . . . . . . . . . .

35

Spatial mode analysis

37

3 The phase camera: a frequency sensitive wavefront sensor
3.1

27

Mechanical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

39
39

3.2

Testing the phase camera and the output mode cleaner . . . . . . . . . . . .

41

3.2.1

Motivation for the output mode cleaner . . . . . . . . . . . . . . . .

41

3.2.2

Theory and modeling . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.2.3

Phase camera measurements and comparisons . . . . . . . . . . . . .

54

4 Improving and applying the phase camera
4.1

4.2

III

65

Phase camera with reference beam . . . . . . . . . . . . . . . . . . . . . . .

65

4.1.1

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

4.1.2

Demodulation board redesign . . . . . . . . . . . . . . . . . . . . . .

67

4.1.3

Fitting and understanding the data . . . . . . . . . . . . . . . . . . .

67

H1 in different heating states . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.2.1

74

Discussion of phase camera results . . . . . . . . . . . . . . . . . . .

Targeted continuous gravitational wave searches

83

5 Neutron stars, spindown models, and estimating upper limits of sources 85
5.1

Neutron stars and pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

5.2

Estimated deformations of neutron stars . . . . . . . . . . . . . . . . . . . .

87

5.3

Energy considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.4

Differences between GW and EM pulses . . . . . . . . . . . . . . . . . . . .

92

5.4.1

Free precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.4.2

Two component model . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5.5

Application to RX J1856.5-3754 . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.6

Application to the Crab pulsar . . . . . . . . . . . . . . . . . . . . . . . . .

102

6 Search methods

105

6.1

Search codes and constraints . . . . . . . . . . . . . . . . . . . . . . . . . .

105

6.2

Detector response function . . . . . . . . . . . . . . . . . . . . . . . . . . . .

106

6.3

Signal form in terms of Doppler and amplitude parameters . . . . . . . . .

110

6.4

The F statistic: a coherent search method . . . . . . . . . . . . . . . . . . .

112

6.5

Coherent vs incoherent templates and unknown parameters . . . . . . . . .

116

6.6

Exact grid spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120

6.7

Interferometer sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

8

6.8

Estimating limits of detection . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Searches and results
7.1

7.2

IV

122
125

RX J1856.5-3754 directed search . . . . . . . . . . . . . . . . . . . . . . . .

125

7.1.1

Parameter space chosen . . . . . . . . . . . . . . . . . . . . . . . . .

125

7.1.2

Search results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

7.1.3

Version 1 coincidence step . . . . . . . . . . . . . . . . . . . . . . . .

128

7.1.4

Upper limit injections . . . . . . . . . . . . . . . . . . . . . . . . . .

129

7.1.5

Final comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

Crab wide parameter search . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

7.2.1

Chosen search parameter space and results . . . . . . . . . . . . . .

133

7.2.2

Upper limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

7.2.3

Checks on the search result and upper limits . . . . . . . . . . . . .

138

7.2.4

Constraining energy loss in the Crab pulsar . . . . . . . . . . . . . .

156

Conclusion

160

Conclusion

161

Appendices163

A Table of acronyms

163

B Phasecamera demodulation and control board schematic

165

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10

List of Figures
1-1 A passing gravitational wave’s effects . . . . . . . . . . . . . . . . . . . . . .

20

1-2 Neutron star with a non-axisymmetric bump . . . . . . . . . . . . . . . . .

23

2-1 LIGO interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2-2 TCS system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3-1 Phase camera and galvo control loop . . . . . . . . . . . . . . . . . . . . . .

40

3-2 OMC in relation to the interferometer . . . . . . . . . . . . . . . . . . . . .

48

3-3 Modal decomposition of OMC transmission . . . . . . . . . . . . . . . . . .

51

3-4 Phase map generated from OMC model . . . . . . . . . . . . . . . . . . . .

53

3-5 Michelson bright fringe on transmission of the OMC . . . . . . . . . . . . .

55

3-6 Direct reflection off ITMY on transmission of OMC . . . . . . . . . . . . . .

56

3-7 Michelson bright fringe directly to phase camera . . . . . . . . . . . . . . .

58

3-8 Direct reflection off ITMX directly to phase camera . . . . . . . . . . . . . .

59

3-9 Qphase fit of Michelson bright fringe data . . . . . . . . . . . . . . . . . . .

61

4-1 Carrier, sidebands and reference beam . . . . . . . . . . . . . . . . . . . . .

66

4-2 Reference beam setup for the phase camera . . . . . . . . . . . . . . . . . .

70

4-3 Carrier signal: horizontal and vertical profiles . . . . . . . . . . . . . . . . .

71

4-4 Fit of carrier on sideband signal . . . . . . . . . . . . . . . . . . . . . . . . .

72

4-5 “Reversed TCS” state images . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4-6 “No TCS” state images . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4-7 “Tuned TCS” state images . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4-8 “Power up” state images . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5-1 Diagram of angles for precession . . . . . . . . . . . . . . . . . . . . . . . .

93

11

5-2 Precession for small λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

5-3 Precession for small θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

5-4 Ellipticity as a function of current frequency . . . . . . . . . . . . . . . . . .

101

5-5 1st Spin down as function of current frequency . . . . . . . . . . . . . . . .

102

6-1 Reference angles for definition of interferometer location . . . . . . . . . . .

107

7-1 RX J1856.5-3754: version 1 largest 2F values for H1 . . . . . . . . . . . . .

127

7-2 RX J1856.5-3754: version 1 largest 2F values for L1 . . . . . . . . . . . . .

127

7-3 RX J1856.5-3754: version 2 largest 2F values . . . . . . . . . . . . . . . . .

128

7-4 RX J1856.5-3754: coincident largest 2F values . . . . . . . . . . . . . . . .

129

7-5 RX J1856.5-3754: upper limits . . . . . . . . . . . . . . . . . . . . . . . . .

131

7-6 RX J1856.5-3754: ratio of V1/V2 upper limits . . . . . . . . . . . . . . . .

132

7-7 Sensitivity scaling as a function of templates. . . . . . . . . . . . . . . . .

135

7-8 Confidence vs injection runs . . . . . . . . . . . . . . . . . . . . . . . . . . .

138

7-9 Strain vs SFT for all 3 interferometers . . . . . . . . . . . . . . . . . . . . .

140

7-10 Estimated strain vs frequency for all 3 interferometers . . . . . . . . . . . .

142

7-11 Gaussian noise strain vs SFT . . . . . . . . . . . . . . . . . . . . . . . . . .

143

7-12 Estimated Gaussian noise strain vs frequency . . . . . . . . . . . . . . . . .

143

7-13 Expected versus actual number of templates for all 3 interferometers . . . .

145

7-14 Expected versus actual number of templates for a Gaussian noise data set .

146

7-15 L1: weighted noise from 59 to 60 Hz . . . . . . . . . . . . . . . . . . . . . .

148

7-16 H1: weighted noise from 59 to 60 Hz . . . . . . . . . . . . . . . . . . . . . .

148

7-17 Distribution of templates with 2F greater than 25 for all 3 interferometers .

150

7-18 Distribution of templates with 2F greater than 25 for Gaussian noise data .

151

7-19 Response of the Hanford detector to varying ψ and i . . . . . . . . . . . . .

154

7-20 Response of the Livingston detector to varying ψ and i . . . . . . . . . . . .

154

7-21 Confidence versus injected signal strength . . . . . . . . . . . . . . . . . . .

155

7-22 Ellipticity versus moment of inertia . . . . . . . . . . . . . . . . . . . . . . .

157

7-23 Ratio of GW energy loss to total spin down energy loss . . . . . . . . . . .

158

12

List of Tables
3.1

OMC parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.2

Astigmatic beam decomposition . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.3

Fit and model of OMC scan . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.4

Amplitude modulation comparisons . . . . . . . . . . . . . . . . . . . . . . .

60

3.5

Model and fit comparisons for both the Michelson bright fringe and directreflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.1

NSPOB and sideband power ratios . . . . . . . . . . . . . . . . . . . . . . .

75

4.2

Heating states and phase camera power ratios . . . . . . . . . . . . . . . . .

76

7.1

Crab parameters at the start of S5 . . . . . . . . . . . . . . . . . . . . . . .

136

7.2

Positions of Hanford and Livingston detectors . . . . . . . . . . . . . . . . .

153

A.1 Acronym definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

13

14

Part I

Introduction

15

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16

Chapter 1

Generating gravitational waves
This chapter describes gravitational waves and considers their generation from isolated
neutron stars.

1.1

General relativity and gravitational waves

From the simple experimental observation of the equivalence of inertial and gravitational
masses and the desire to incorporate the principle of relativity into a theory of gravity
was developed one of the most elegant and successful theories of physics, that of Einstein’s
General Relativity [20]. It tells us that masses move along geodesics and that the curvatureof space-time along which mass moves generates the perceived gravitational force. In turnspace-time curvature is generated by the presence of mass and energy. This fact leads
General Relativity to predict the propagation of gravitational waves on space-time [40].
Consider the curvature encoded in the space-time metric gµν whereds2 = gµν dxµ dxν .

(1.1)

Here ds is the space-time interval or proper time of an observer who is at rest with thecoordinate system, dxµ and dxν are the indexed coordinates of space-time and the Einsteinsumming convention is assumed. From this most general case, let us consider the weak-field
limit where space can be considered mostly flat with a small perturbation on top of it,which is a reasonable approximation to human scale laboratories on Earth. In this case themetric gµν can be written as17gµν = ηµν + hµν(1.2)

where hµν is the perturbation (and thus holds the gravitational wave information) and ηµνis the Minkoski metric of flat space-time equal to
−1 0 0 0 0 1 0 0 .

ηµν = 
 0 0 1 0

0 0 0 1

(1.3)

In order for a gravitational wave to propagate, it must satisfy the wave equation!hµν =

'

∂
$ − 2
∂t
2

(

hµν = 0 .

(1.4)

We neglect the static solutions to the wave equation, as we are interested in radiation whichchanges with time as opposed to static Newtonian potentials. We are free to choose thecoordinate system in which to view the perturbation (gauge choice) and the one in which
the math is most clear is the transverse traceless gauge, in which freely falling masses movealong coordinate lines. In the transverse traceless gauge with the weak-field assumption,
the solution of a wave propagating in the x3 direction can be written as [40]


0

0

0



0 h + h ×hµν (x, t) = 0 h× −h+

0 0
0


0


)
*
+,
0
 cos 2πfGW x3 − t

c
0

0

(1.5)

where h+ and h× are the two possible polarizations of the wave and fGW is the frequencyof the gravitational wave.

1.1.1

Gravitational wave effects on test masses

Consider a h+ polarized gravitational wave acting on a pair of test masses at rest relative
to each other. Place both a distance L from the arbitrary origin of their common Lorentz
frame, one along the x1 axis and the other along the x2 axis. Apply the gµν metric to findthe space-time interval ds between the origin and each of the masses as a function of time.
18

As a h+ polarized gravitational wave propagates in the x3 direction, the ds between theorigin and the first mass will vary sinusoidally and be 180◦ out of phase with the sinusoidal
variation of ds between the origin and the second mass.
One method to sample this change in ds is to propagate a laser beam from the origin toeach test mass and back. One can integrate along the laser’s path from the origin to eachtest mass to find a measurable quantity, the difference in the light’s phase between the tworound trip paths:
2π
%Φ =2
λ

-L .
0

2π
|gx2 x2 |dx2 − 2λ-L .
|gx1 x1 |dx1

(1.6)

0

where λ is the wavelength of the laser light and %Φ is the difference in total accumulatedround trip phase by the laser light along the two paths. If the laser light round trip time issmall compared to the gravitational wave period, then the maximum total difference caused
by a h+ wave can be approximated by%Φ & 2h+

2πL
.
λ

(1.7)

One can do the same calculations with the h× polarization or any linear combination of the
two.

1.1.2

Generation of gravitational waves

To gain insight into the generation of gravitational waves, we look to electromagnetic theory
as a guide. Electromagnetic waves are governed by the same wave equation except that theelectromagnetic field E replaces the perturbation hµν . If we restrict ourselves to waves
whose wavelength is much larger than the source we can use a multipole expansion. In thesame way acceleration of charge is necessary for electromagnetic waves, acceleration of mass(and thus second derivatives of the moments) are necessary for gravitational waves. The
monopole of the expansion for both electromagnetic waves and gravitational waves does notradiate due to the conservation of charge on the one hand and the conservation of massand energy on the other. We can move on to consider the gravitational equivalent to theelectric dipole [40]19Figure 1-1: Effects of a passing gravitational wave on free masses and a laser interferometer.

dg =

-

ρ (r) rdx1 dx2 dx3 .

(1.8)

However, it is easy to see that the second derivative of dg will be zero because of theconservation of momentum (mẋ = constant). The next higher moments are equivalent
to the magnetic dipole and the electric quadrupole. The gravitational equivalent to themagnetic dipole is

µg =

-

ρ (r) r × v (r) dx1 dx2 dx3 .

(1.9)

Again in this case there is no radiation, this time due to the conservation of angularmomentum (mẋ × r = constant). Finally, with the quadrupole moment we lack any moreconservation laws to apply and thus have the first term in the expansion that can generateradiation. The typically quoted reduced gravitational quadrupole moment is

Iµν =

-

1
ρ (r) dxµ dxν − δµν r2 dx1 dx2 dx3
320(1.10)

Thus in direct analogy to electromagnetism’s potential [35]

Aµ =

d˙µ
R

(1.11)

where dµ is a charge dipole moment and R is the distance from the dipole to the point
where the potential is measured at, we can write

hµν =

2¨
Iµν .
R

(1.12)

Note that we are working in units where the gravitational constant, G, and the speed oflight, c, are set to 1. The factor of 2 comes from the correct tensor calculations which arepresented in detail in [40].

1.2Continuous gravitational wave sources

For the majority of this thesis we will focus on one type of source of changing quadrupole
moment, a non-axisymmetric isolated neutron star. One can imagine such a star with abump on its surface or an equatorial bulge producing an eccentricity as shown in Figure
1-2. Consider a non-axisymmetric neutron star with the principle moments I11 , I22 , I33 .
Let us assume it is rotating about the x3 axis with frequency fROT . In this case the second
derivative of the quadrupole moment tensor Iµν is [56]I¨µν =

0
0
0


2
2
0 −16π 2 fROT
(I11 − I22 ) cos (4πfROT t) −32π 2 fROT
(I11 − I22 ) sin (4πfROT t)22
0 −32π 2 fROT
(I11 − I22 ) sin (4πfROT t) 16π 2 fROT
(I11 − I22 ) cos (4πfROT t)

0
0
0


0


0
 .

0

0

(1.13)

I11 −I22 can be replaced in the above with )I33 by using the definition of equatorial ellipticity)=

I11 − I22.
I33
21

(1.14)

.
Using this I¨µν in Eq. (1.12) and choosing to observe the generated wave from a newcoordinate system with the x3 axis oriented at an angle i away from the axis of rotation weget [56]

hµν =


0


/
0


2 (i)
2 )I


0
−
cos
(4πf
t)
1
+
cos
−2
sin
(4πf
t)
cos
(i)
0
32π 2 frot
ROT
ROT
33
 .

×
/
0

R
2
0
−2 sin (4πfROT t) cos (i)cos (4πfROT t) 1 + cos (i) 0


0
0
0
0


0

0

0

(1.15)

This can be compared to Eq. (1.5) which allows one to read off h+ and h× and also notethat fGW = 2fROT .
From the above Eq. (1.15) it is easy to determine what properties a detectable continuous wave source would have. First of all, it must have some eccentricity. However there
are theoretical estimates of the maximum possible eccentricities supportable by neutronstars. These estimates are based off of possible equations of state for neutron stars, which
are equations which define the relationship between the temperature, pressure and density
within the star. By surveying the more plausible equations of state one finds a maximum
ellipticity of ∼ 10−6 [51] can be supported. Exotic equations of states for things such as

hybrid or quark stars, where the core is partially or completely made up of quark matter,
can have maximum ellipticities a factor of 100 larger. However, it should be noted thatsuch equations of state are considered by many astrophysicists to be speculative, and so
the ellipticities we infer from them may not be quite as plausible [42]. A more in-depth
discussion of ellipticities of neutron stars is covered in Section 5.2.
Secondly, the source needs to be rotating rapidly, due to the f 2 dependence of thestrain. It also needs to be within the LIGO detection band, which has a low end around
50 Hz determined by seismic noise. Known pulsars have a maximum spin periods of a few
milliseconds, yielding maximum gravitational wave frequencies of about 1 kHz. In the caseof an isolated neutron star this means it needs to be young and not have radiated away
22

too much of its angular momentum. Thirdly, an ideal source should be as close as possible
since the strain at a detector is inversely proportional to the distance to the source.
By considering the known properties of neutron stars we can now make a quick estimateof the maximum strength of gravitational waves reaching detectors on Earth. Typical radii
are of order 10 km and typical masses are around 1.4 M$ [40]. This leads to a typicalmoment of inertia 1045 g cm2 assuming a rigid rotator. Using 1 kpc as a distance standard
and simply note any change in distance just produces an inverse change in the detectstrain. The above equations can be converted from natural units to experimentalist units
by including correct factors of G and c. Using the typical maximum values for frequency
and ellipticity yields a GW amplitude of the order h ∼ 10−24 (1 kpc/R)()/10−6 )(f/500Hz)2

detected at Earth from a source 1 kpc away.

Figure 1-2: Diagram of a non-axisymmetric neutron star with a bump. The angle i is theinclination angle between the axis of rotation and the direction to the detector.

Next we consider two isolated neutron stars that could make good candidates for detection by LIGO.

1.2.1

Potential source: RX J1856.5-3754

A potential candidate that meets several of the previously listed criteria is the isolated
neutron star RX J1856.5-3756. RX J1856.5-3754 is the closest known neutron star, at a
distance of ∼ 120 pcs [31] [53], as determined by parallax measurements. While it is seenin the optical by the Hubble space telescope, it was actually discovered in the X-ray by
the Rosat All Sky Survey in 1996 [54]. It does not emit detectably at radio wavelengths,
23and thus was not in the original set of pulsar candidates being examined by LIGO. It
has a remarkably featureless x-ray spectrum which is well fit to a black body spectrum of
temperature 63.5 eV, although there is an excess in the UV-optical region of the spectrum
by about a factor of 7 from just the x-ray data fit [13]. A naive black body fit to the x-ray
data suggests that the star is of the order 4km in radius while a similarly naive black body
fit to the optical data leads to a star of radius of 14km [13]. The extreme mismatch between
these two fits indicates that one must model the effects of the neutron star’s atmosphere tounderstand the electromagnetic emission. At the time of considering RX J1856.5-3754 as asource Chandra and XMM observations had not been able to detect any periodic variationin the x-ray emission, having placed an upper limit of 1.5% on the pulsing fraction of x-rays
relative to the total x-ray flux [12].
A lack of spin period to focus on requires a blind search, although there are good
estimates of its age allowing us to limit the parameter space of the search, when coupledwith spin down energy calculations. Since the star is losing rotational energy as it radiatesaway gravitational radiation, an equation relating the current rotation frequency and therate of change of that rotation frequency (the spin down rate) can be written. If sufficient
time has passed since the birth of the neutron star, such that the current rotation frequency
is small compared to the rotation frequency at birth, the ratio between the current rotation
frequency and the current spin down rate will become a function only of the age of the star.
If the age is known then at any given frequency, spin downs only up to a certain maximum
need be considered, limiting the spin down parameter space necessary to be searched. Thisis covered in more depth in Section 5.3.
Age estimates for RX J1856.5-3754 come from considering its temperature and how
rapidly it cools, the lack of a visible super nova remnant, and possible associations withother stars from a common starting point. Its high temperature and x-ray brightness
implies that RX J1856.5-3754 must be around 106 years old or younger, since the star has
not lived long enough to cool too much by thermal emission. On the other hand, there is novisible supernova remnant, which implies an age greater than about 105 years, the length oftime most supernova remnants remain visible. RX J1856.5-3754’s proper motion of ∼ 0.33

mas yr−1 suggests that ∼ 5 × 105 years is correct. Tracing RX J1856’s trajectory back intime finds it crossing the trajectory of run-away star ζ Oph 5 × 105 years in the past [52].

They both may have originated in the Upper Sco OB association, where their paths meet,24

before the supernova of the progenitor of RX J1856.5-3754 kicked them out.
While carrying out a gravitational wave search directed at RX J1856.5-3754 it was
found to have a spin period below the LIGO detection band by X-ray astronomers. It hasa period of 7 seconds with only a variation in the X-rays of 1.5%, the smallest ever seen in
an isolated X-ray pulsar [50]. However the methodology and resulting test of the code andsearch pipeline performed using it as a target are still of use and thus it is presented here
and in Chapter 7.

1.2.2

Potential source: the Crab pulsar

The Crab pulsar and its surrounding nebula are remnants of a supernova that occurredin 1054 AD approximately 2 kpc distant [17]. Its been well studied over a broad band ofelectromagnetic frequencies from radio to gamma rays [38]. It produces very regular pulses
across the electromagnetic spectrum whose frequency and time evolution are known to highaccuracy. These electromagnetic wave pulses provide insight into the rotation rate and spin
down rate of the Crab pulsar and also act as a starting point from which to narrow down
directed searches for gravitational waves from the Crab pulsar.
The Crab has a known radio pulse frequency of ∼ 30 Hz which implies the rotation

frequency is close to that. It also known to be spinning down at the rate of ∼ 3 × 10−10

Hz/s [38] [28]. This information lets us calculate a naive classical upper limit on the strain
at an earth based detector of 1.4 × 10−24 and upper limit of 7.5 × 10−4 on the ellipticityof the Crab pulsar [43]. Thus the Crab is a good potential source because of its extreme
youth and rate of spin down, despite its distance. These radio observations also let us
narrow the parameter space down, but there is still some uncertainty left in how closely thegravitational radiation period matches the radio pulse period [56].

25

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26

Chapter 2

Laser interferometer gravitational
wave detectors
This chapter will summarize the basics of a laser interferometer detector. It will then discuss
heating problems with certain optics which arose while trying to reach the designed input
laser power and how these problems were addressed with a thermal compensation system.
An in depth examination of the laser light inside the interferometer from before and after
this compensation system was used can be found in Chapter 4.

2.1

Laser interferometers

2.1.1

Introduction

One of the simplest and most sensitive methods for measuring the difference in length along
two paths is a Michelson interferometer. Figure 2-1 shows the full Michelson interferometer
combined with the two Fabry-Perot arms, discussed in Section 2.1.3 and a power recycling
cavity, discussed in Section 2.1.4. This setup is the heart of all three LIGO detectors.
These interferometers have been undergoing construction and commissioning for the past
decade and recently have gone through several periods of designated data collection. These
periods, during which the interferometer is left to simply run without constant upgrades
and commissioning, are refered to as science runs. The most recent science run, the 5th sofar, started in November 2005 and was the first science run in which the interferometers
were operating at their designed sensitivity level.

27

At its most basic, a LIGO interferometer consists of a laser light source which is thendirected through a 50/50 beamsplitter. The transmitted and reflected beams travel along
perpendicular paths until reaching two end test masses (ETMs), which are highly reflective
mirrors. The reflected light from each mirror is then recombined at the beamsplitter. The
recombined electric field at the anti-symmetric (AS) port depends on the difference in the
optical paths of the two beams. The electric field at the AS port for this simple configuration
can be written as),
EAS = Ein rex tbs rbs ei2φx − rey tbs rbs ei2φy(2.1)

where Ein is the electric field incident on the beamsplitter from the laser source, φx andφy are the phases accumulated by the light in a one way trip down each arm, tbs and rbs
are the field transmission and reflection of the beamsplitter, and rex and rey are the fieldreflection of the end mirrors [3]. In reference to the earlier calculations in Section 1.1.1, the%Φ of Eq. (1.6) would be equal to 2(φy − φx ).

2.1.2

Laser and mode cleaners

The main laser is a 10 Watt Nd:YAG laser from Lightwave, which operates at a wavelengthof 1064 nm. It is both frequency and intensity stabilized. The laser output is passed
through a pre-mode cleaner (PMC). This is a 21 cm long triangular cavity designed to filter
the spatial mode and intensity noise above about 1 MHz. It then passes through the modecleaner, another triangular cavity, with a 24.492 meter round trip. This cavity spatiallyfilters the light, only allowing a single mode to pass, while higher order modes pick up
additional phase and fall out of resonance. It also acts as a stable angular reference, as
misalignments become translated to higher order modes within the cavity and also fall outof resonance [3].

2.1.3

Fabry-Perot arms

To increase the sensitivity of the basic Michelson interferometer, one would like to increase
L, the length of the arms, up until the length is comparable to the wavelength of thegravitational waves, roughly 300 to 6000 km. At that point the approximations leading
up to Eq. (1.7), which shows the signal grows linearly with L, break down. Effectively,

28

Figure 2-1: A diagram of a LIGO interferometer (taken from [3]). The purple light represents the input laser light. The blue line represents the resonant sidebands in the power
recycling cavity while the red line represents the carrier signal resonant in the recycling
cavity and the Fabry-Perot arms, as discussed in Sections 2.1.3,2.1.4, 2.1.5.

the light is picks up phase shifts from multiple cycles of the wave which cancel each other,preventing any further gain in sensitivity. There are technical problems with building thearms even 300 km long. One way to effectively make the arms longer is to have the laser
light make multiple trips down the same arm. As long as the time spent by the light inthe cavity is small compared to the period of the gravitational waves, this will significantly
increase the sensitivity of the instrument. However for changes occurring faster than thisstorage time, the resulting signal will be attenuated relative to the lower frequencies in thesame configuration.
By turning the Michelson interferometer arms into resonating Fabry-Perot cavities, one
can increase the number of round trips the laser light makes. This adds technical complexity
in the form of an additional set of partially transmitting input test masses (ITMs) and the29necessary feedback control systems to keep the Fabry-Perot cavities on resonance.

2.1.4

Power recycling cavity

In the case of the Michelson interferometer, the light interferes destructively in the directionof the AS port and constructively in the direction of the incoming laser light. Instead ofjust dumping the outgoing power, one can construct a cavity to take the outgoing light andrecycle it back into the interferometer. By adding a partially transmitting recycling mirror
between the laser and the beamsplitter a cavity can be made which recycles the power backinto the interferometer. This cavity also requires additional feedback control systems toremain on resonance and generally provides a factor of 50 gain to the carrier, and a factorof 26 gain to the sidebands when the input test masses are thermally tuned [5]. The recycling
mirror transmission was chosen to match the total round trip losses by the carrier in therest of the interferometer so as to couple all the carrier light into the interferometer [3].
The reason for using all available power is because of shot noise. The interferometer
operates on a dark fringe and thus very few photons carry the gravitational wave signal
to the AS port. Shot noise is due to the quantum fluctuations in the number of photons
reaching the detector. Shot noise scales as sqrtN , where N is the number of photons.
However, the gravitational wave signal scales with the power, and therefore the number ofphotons, and so the signal to shot noise ratio scales as sqrtN [3]. By increasing the total
power we increase the interferometer sensitivity at the frequencies which are limited by shotnoise.

2.1.5

AS portThe electric field at the AS port is detected with a photodiode, which produces a photoelectric current proportional to the average photon flux, or power, on the detector. For asimplified calculation of the power at the AS port we assume the end mirrors have identicalreflectivity and are equal to 1 (i.e. rex = rey = re = 1) and that the beamsplitter is aperfect 50/50 (i.e. rbs × tbs = 1/2). Using Eq. (2.1) we can then write the AS port poweras

PAS = E ∗ E = 4|Ein |2 (re tbs rbs )2 sin(%φ)2 = |Ein |2 sin(%φ)2

30

(2.2)

where we define %φ = φy − φx [3]. To produce signals that are at radio frequencies on thephotodiodes, the light entering the interferometer is first phase modulated to produce several
sets of sidebands. These sidebands are used in a Pound-Drever-Hall locking scheme [18] (seealso [23]), to keep the different cavities on resonance (or locked). One set of sidebands isreflected by the mode cleaner, one set is reflected by the power recycling mirror, and the last
set is reflected by the input test masses, and resonate in the recycling cavity. This last set ofsidebands which resonates in the power recycling cavity is used to beat against the carrier
and produce the final gravitational wave signal channel at the AS port. An asymmetry of
0.356 m in the lengths between the BS and the two ITMs, called the Schnupp asymmetry,causes the sidebands to come out the AS port. Nominally, the recycling cavity should bealmost critically coupled for the sidebands such that almost all the sideband light should
end up at the AS port.
These sidebands are produced by using a Pockels cell to phase modulate the input light.
The laser light modulated once for one set of sidebands can be written as

Ein = E0 eiΓ cos(Ωm t)∼= E0 (J0 (Γ) + iJ1 (Γ)e+iΩm t + iJ1 (Γ)e−iΩm t(2.3)− J2 (Γ)e+i2Ωm T − J2 (Γ)e−i2Ωm t )(2.4)
where Γ is the modulation depth in radians, Ωm is the modulation angular frequency, Jn isthe nth order Bessel function of the first kind, and E0 is the unmodulated field of the laser.
In the case of the sidebands resonant in the power recycling cavity, Ωm ≈ 2π × 24.5 MHzand Γ ≈ 0.4. The output of the photodiode is demodulated at the modulation frequency

used to run the Pockels cell. The In-phase (cosine) and Quadrature-phase (sine) of the AS
photodiode are separately recorded.

31

2.2

Problems

2.2.1

Thermal heating and thermal compensation

The ITMs were designed with the concept that a certain amount of absorption would betaking place leading to a curvature change due to thermal lensing, effectively the changing
index of refraction due to increasing temperature, when operating in a high input power (6Watt) state. The power gain of 50 in the recycling cavity and gain of 130 in the arms meansroughly 300 W and 10 kW are circulating in the recycling cavity and arms, respectively, inthis high input power state. The expected absorption was about 1 ppm in the HR coating
and about 4-5 ppm/cm in the substrate. Through the initial science runs, during which the
LIGO interferometers were left alone to take data as opposed to being worked on directly
to improve sensitivity, the LIGO interferometers were operating below this expected highpower state and consequently had a slightly wrong radius of curvature for the ITMs andETMs.
However, during attempts to reach high power on the H1 interferometer, it became clearthat optics were absorbing far more then expected. At their nominal radius of curvature
at 6 Watts of input power, the gain in power in the sidebands is theoretically expected tobe Gsb = 30 [5]. However, a maximum gain of Gsb = 26.5 was measured at 1.8 Watts oflaser power into the mode cleaner, a much lower power than expected. At higher power,
the sideband power gain reduced along with the sidebands decreasing in spatial size. Thissuggested that one or more optics of the power recycling cavity was absorbing too much
power.
Using spot size measurements on H1 at different IFO ports and at different heating states[41], we were able to determine that ITMX was absorbing the most power, 34 ± 4 mWatt

per Watt of power into the mode cleaner, although ITMY was only a factor 2.6 less thanthat. These in vacuum measurements were unable to distinguish between bulk substrateabsorption or coating absorption. However, since the power hitting the first coating layer on
the high reflectivity side of the ITM, which faces the Fabry-Perot arm cavity, is roughly 140
times the incident power, and the higher probability of surface contaminants, the absorption
is most likely due to the surface rather than the bulk. If one assumed all the absorption
were in the surface, then the ITMX coating would be absorbing 15±1.8 ppm and the ITMY
coating would be absorbing 5.6 ± 0.7 ppm. This is compared to the specification of 1 ppm.
32

One hypothesis is that contaminants were introduced to the surface during assembly andinstallation and resulted in this large difference from specification to the actual absorption.
However, we don’t have a definitive cause of the excess absorption. Based on these findings,
ITMX was replaced with a spare in June of 2005. Since the ITMs still had absorption
values above the design specifications, a system was necessary to compensate for the thermal
heating effects of the main laser.
A thermal compensation system (TCS) was originally studied by Ryan Lawrence [36]and was designed to apply heating to the test masses as needed to correct the curvature.
This eventually led to the full system installed on the LIGO interferometers [48] [6]. The
final systems installed for each ITM, shown in Figure 2-2, consist of a 10 Watt CO2 laser
operating at 10.6 µm wavelength that illuminates one of two interchangable mask patterns.
The laser light passes through a telescope with magnification of 26.5 and a ZnSe viewport
before reaching the high reflectivity side of the ITM (the side facing the arm cavity) of the
ITM. The 10.6 µm laser light is almost completely absorbed by the fused silica substrate of
the ITM.

33

Figure 2-2: A schematic of the Thermal Compensation System (TCS). This was installed
on both ITMs on all 3 LIGO interferometers. It is made up of a 10.6 µm CO2 laser, a set ofinterchangable masks, and a series of projection optics that produce an image of the maskon the optic.

34

The two masks are an annulus pattern and a hole. The annulus is used to compensate
for too much heat being deposited by the main LIGO laser light. The hole in the center
is used to correct for insufficient heat to reach the desired curvature. These masks are
installed in the Fourier plane of the projection system. A Bessel mask further downstream
clips the higher order maxima of the Airy diffraction pattern, leaving only the central lobeof the Airy disk. A polarizer on a rotation stage is used to adjust the power.
This system was successfully able to correct for the high power heating effects andallowed the LIGO interferometers to reach operation at 6 Watt input laser power state. To
control the long time dependence of the TCS on the radius of curvature, a servo system wasdevised and implemented. The full details of this system can be found in Stefan Ballmer’s
thesis [5].

2.2.2Mode and power mismatches and the AS I signal

The carrier light resonates mostly in the 4 km long arms of the interferometer, and thus
their spatial mode is mostly dependant on the state of the arm cavities. The sidebands only
resonate in the power recycling cavity and thus the sideband structure is determined by
the state of the power recycling cavity. The deformation and thermal lensing of the ITMs,
due to their excess absorption, affects primarily the power recycling cavity. This causes thetwo sidebands’ power and spatial profiles to become mismatched with the carrier and eachother. Since the gravitational wave signal is detected by the beat between the carrier andsidebands, poor matching will lead to excess noise.
To illustrate what poor matching can do, we write the incident light field on the AS
photodiode asΨ = Ac e−iωc t + iA+ e−i(ωc +Ωm )t + iA− e−i(ωc −Ωm )t(2.5)

where Ωm is the sideband modulation frequency, ωc is the carrier frequency, and Ac , A+ , A−
contains the modal content and the relative magnitudes of the carrier, upper sideband, andlower sideband respectively. When the photodiode signal is demodulated at frequencyΩm , two signals are produced, the In-phase signal and the Quadrature-phase signal. Since
these are recorded by the AS port photodiode, these are labeled AS I and AS Q. Theinterferometer is designed such that gravitational wave signals are only in AS Q and the35

AS I signal is zero.
We can write the power in each demodulated signal as

PAS I =

-

2π
Ωm

0

ΨΨ∗ cos(Ωm t) = (Ac A∗sb+ + A∗c Asb− ) − (A∗c Asb+ + Ac A∗sb− )(2.6)ΨΨ∗ sin(Ωm t) = (Ac A∗sb+ + A∗c Asb− ) + (A∗c Asb+ + Ac A∗sb− )

(2.7)

and

PAS Q =

-

2π
Ωm

0

where PAS I and PAS Q are the powers in the AS I and AS Q signals.
Note that when the sidebands have identical amplitude and modal content, namely
Asb+ = Asb− , PAS I goes to zero, leaving only AS Q with signal. However, if the overallmagnitude is different, effectively amplitude modulation of the sidebands, PAS I no longer
identically cancels. In addition if the modal content of the sidebands are not identical, andthere are higher order modes present to beat with in the carrier, PAS I will again not cancel.
The higher order modes in the carrier and sidebands will also add noise directly to the
AS Q signal as well.
If the mismatches between the carrier and sidebands are severe enough, the excess
AS I signal can cause saturation of the RF electronics and mixers, resulting in loss of any
gravitational wave signal [3]. There is an electronic system in place to provide feed-backthat cancels out RF photo current on the diode caused by the AS I signal. This system
can correct up to 10 mApk photo current per diode. The control signal for this system isrecorded in the channel AS I Corr. However in high power configurations this correction
system proved to be insufficient [5]. The excess AS I problem was eventually solved withthe thermal compensation system running a correction servo designed to null the AS I Corrsignal. It is important to examine the power and modal nature of the individual componentsof the light in the recycling cavity with and without TCS correction, to ensure that it iscorrecting properly. This examination is accomplished through the use of the phase camera,a system whose basics and initial test case are described in Chapter 3. The examination of
the thermal states and TCS correction of the interferometer is covered in Chapter 4.

36

Part II

Spatial mode analysis

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38

Chapter 3

The phase camera: a frequency
sensitive wavefront sensor
The “phase camera” is a high-resolution wavefront sensor that measures the complete spatial
profile of any frequency component of a laser field containing multiple frequencies. This
technique is useful as a probe of the spatial overlap of the carrier field with each sideband
component exiting the output port of a gravitational wave interferometer. It was first
demonstrated to work at MIT [25] and then brought to the LIGO sites.
This chapter describes the functioning of the phase camera at a basic level, where it
detects only the beat signal between the carrier and sidebands. At the Hanford site, thephase camera was first successfully used to test an output mode cleaner and confirm that
it was working as designed. That test is presented here to show the phase camera working
in a relatively simple to understand case on the actual interferometer. The addition of an
independant reference beam, described in Section 4.1.1, was necessary for the more complex
study of individual sideband and carrier components in different heating states describedin Chapter 4.

3.1

Mechanical details

The phase camera is the combination of a position scanning galvanometer and a New Focus
1811A RF photodiode. The galvanometer moves two mirrors in conjuction such that theincoming light reflects off both in a spiral pattern and onto the photodiode at a rate ofabout one full spiral scan every half second. The two output channels of the photodiode,39

DC power and RF power, are then sampled at a rate of up to 4000 points per full scan. The
RF sampled points are then demodulated and passed to a computer, along with the position
read back of the galvanometer as shown in Figure 3-1. The demodulation can be done at afrequency of 24.5 MHz, the difference between the carrier and sideband frequencies, and thus
looks at the beat between the carrier and sidebands of the incoming light. A reference beam
can be combined with the incoming light to produce beats with the carrier and sidebands
individually, allowing their individual demodulation, and which is described in Section 4.1

Figure 3-1: Phase camera control setup. The galvanometers move a pair of mirrors, directing
the light in onto an 1811A RF photodiode. The RF signal from the photodiode, produced
when the carrier and RF sidebands beat at 24.5 MHz, is demodulated, and then sent to acomputer, along with the DC signal, where the signals are recorded. The computer sends
the signal for the next position to a digital to analog converter board, which converts thesignal into a control voltage which moves the mirrors to the next point in the spiral pattern.

40

3.2

Testing the phase camera and the output mode cleaner

The phase camera was used to help test and understand an output mode cleaner (OMC) on
the Hanford 4km interferometer [10]. This also presented an opportunity to test the phasecamera itself, to ensure that it was operating as expected.

3.2.1

Motivation for the output mode cleaner

Laser interferometers have problems associated with imperfect interference at the AS port,
due to poor beam quality. Excess carrier light in higher order modes will produces excess
shot noise while not contributing to the signal. Imperfect matching in the reflectivity of thetwo arms will produce a static, TEM00 carrier field at the AS port, which when it beatswith the sideband light will produce noise in the signal channel. Any sideband field which
does not spatially overlap with the carrier does not contribute to increasing the opticalgain, but rather produces many detrimental effects like shot noise, acoustic sensitivity andso forth [3].
A potential solution to poor output beam quality is to place an output mode cleaner
in the path of the AS port light before it reaches the AS photodiodes. By using a cavity
with sufficiently large line width, both the RF sidebands and carrier TEM00 modes could betransmitted on the same resonance, while passively rejecting the higher order spatial modes.
The more recently adopted solution is for future upgrades to the LIGO interferometers is
to use only carrier light, without RF sidebands, a so called DC readout.
Experimental setup
The OMC was placed on the AS port of the Hanford 4k Interferometer as shown in Figure3-2. This OMC was a solid spacer triangular cavity with a piezo-electric transducer (PZT),a device which converts electrical field into length changes, attached to one mirror for
length control. It was designed to pass the TEM00 carrier and sidebands through the same
resonance, while rejecting all the higher order modes. The cavity was held on resonance byapplying a small dither signal to the OMC length at 40 kHz, and using a lock-in amplifier
to generate an error signal from the transmitted light. A few useful parameters of the OMC
are shown later in Table 3.1.

41

Incoming light
We have modeled and taken data for the OMC in the two simplifying cases of the laserreflected directly off an ITM onto the AS port, and of the interferometer being held on aMichelson bright fringe with the RM and ETMs misaligned. In these cases we assume thatthe modal content of the incoming carrier and sideband is essentially the same. However,a beam scan of the light on the table showed the light to be astigmatic, complicating thesituation. The light had a first beam waist of 137µm along an axis rotated 25◦ counterclockwise relative to the OMC horizontal axis. The second beam waist of 107µm located 6.6 cm
after the first waist was along an axis rotated 25◦ degrees relative to the OMC vertical axis.
Initially, the OMC was mode matched to only the horizontal waist, treating that as the
only waist. However, during the testing process it was moved such that the transmission
through the OMC was maximized. All data present here is from this later situation.
Phase camera
In this experiment a 50/50 beamsplitter was placed at the output of the OMC, redirecting
half of the transmitted light to the phase camera. On the path to the phase camera, shownin Figure 3-2, two lenses were placed so as to focus the light to the proper size to bescanned. This means the phase camera photodiode was approximately 0.45 m from a waist
size of 150µm, resulting in a Gouy phase of nearly π/2 radians. The Gouy phase is thephase difference between a Gaussian beam and a plane wave of the same optical frequency,
acquired by the Gaussian beam as it propagates through a focus. This effective phase shift
is different for higher order modes,

3.2.2

Theory and modelingIn the following sections a model for the astigmatic light from the interferometer reaching
the OMC, and then passing through the OMC, will be developed. This requires defining
a basis of modes affected by astigmatism, then transforming it into the OMC basis. By
comparing the relative power in the modes actually measured at the output of the OMCwith what we can determine if the OMC was built properly and performing as we expect.

42

Astigmatic beam
The Hermite Gaussian basis provides a complete set of solution for the fields which canpropagate inside or outside of the cavity [33]. These solutions for the electromagnetic field,
Ψ, which have been normalized for power, have the form

1

Ψmn =

2
1
Hm
(m+n)
w(z)
2
m!n!π

2√

2x
w(z)

3

2.
3
(2)y
Hn
w(z)

−1
2
2
2
2
2
2
× e(x +y )/w (z) e−ik(x +y )/(2R(z)) eikz ei(m+n+1) tan [(zλ)/(πw0 )](3.1)
where m and n are TEM mode numbers, x and y are the distance from the center of thebeam perpendicular to the direction of propagation, z is the distance from the waist in the
direction of propagation, w(z) is the radius of the beam at z, w0 is the radius at the waist,
R(z) is the radius of curvature of the beam front at z, λ is the wavelength of light, k is thewave number (2π/λ), and Hm is the Hermite polynomial of order m.
This can be generalized for the incoming light by breaking up the symmetry between
the x and y axes and allowing them to have different waists and waist positions. In thecase of a Michelson locked on a bright fringe and also a direct reflection of the laser light
off an ITM, the light should be a pure TEM00 , distorted only by the astigmatism. Thus,
the incoming light can be written as
4
4
2
1
1
Ψastigmatic (x& , y & , z & ) =(m+n)
2
m!n!π wx (zx ) wy (zy )
2 √
3
2 √
3
2x
2x
× Hm
Hn
wx (zx )wy (zy )
1

2

(3.2)

2

× e−[x/wx (zx )] −[y/wy (zy )] eik[(zx +zy )/2]

−1
2
−1
2
× ei[m+(1/2)] tan [(zx λ)/(πw0x )] ei[n+(1/2)) tan [(zy λ)/(πw0y )]where zx and zy are the distances from the x aligned waist w0x , and the y aligned waist
w0y , respectively.

43

The OMC basis
The OMC cavity imposes several boundary conditions on the general Hermite Gaussian
solutions of Eq. (3.1). First, for resonance to occur, after each full round trip the phase andamplitude must match. Second, at the curved back mirror the wave front of the beam mustmatch that of the cavity (neglecting the small incidence angle). Also note that, because ofsymmetry, the waist of the beam inside the cavity must occur midway between the two flat
mirrors and equidistant from the curved back mirror. These conditions imply

w04 =
and

' (2
λ
L(R0 − L)πc1= (q + 1) + (m + n + 1) cos−1
λν0
π

21

(3.3)

L
1−
R0

3

1
+ Mod2 (m)
2

(3.4)

where L is half the round trip length, R0 is the radius of curvature of the back mirror, Mod2
is modulo base 2, and ν0 is the free spectral range (FSR). The FSR is equal to c/2L, wherec is the speed of light, and is simply the distance in frequency space between successive
transmission peaks of a given spatial mode. These transmission peaks are numbered by theq parameter, since q+1 is the number of half wavelengths inside the cavity. The modulo base
2 term comes from the fact that the cavity is triangular, and that for odd mode numbers in
the direction parallel to the mode cleaner plane there is an additional π phase shift in theround trip )due to an odd number of reflections in the horizontal plane). Table 3.1 provides
the values for these and several other related parameters of the OMC.
Table 3.1: OMC parameters
Cavity Finesse
Half of Round Trip Path (L)End Mirror Radius of Curvature (R0 )Free Spectral Range (ν0 )Beam waist inside the OMC (w0 )Wavelength of Carrier light (λ)

3050.75 mm
75 mm
2.95 GHz109µm
1064 nm

The important part of Eq. (3.4) is that for a given frequency of light, each (m + n)needs a different length of cavity to resonate, due to the Gouy phase shift. This means the
OMC will decompose the incoming light into the Hermite Gaussian basis specified by its
44

geometrical parameters, and only let those modes whose (m + n) satisfy Eq. (3.4) to pass,
generally the carrier TEM00 mode. All others will be reflected back from the OMC. However,
because the mirrors of the OMC do not have perfect reflectivity, the range of frequencies
which can pass is broadened, allowing most of the sideband TEM00 modes to pass. Thiscan be seen by looking at the transmitted light of the OMC, considering the reflection andtransmission at each mirror. Assuming no losses in the cavity and no transmission on theend mirror, the complex amplitude of the transmitted light will beΨtransmitted =t1 t2 Ψincident iφe
1 − r1 r2

where
φ = 2kL − 2(m + n + 1) cos

−1

21

L
1−
Ro

3

(3.5)

− πMod2 (m),(3.6)

t1 = t2 represent the transmission of the mirrors, r1 = r2 represent the reflectivity of themirrors, and assuming t2i + ri2 = 1, for i = 1, 2. The k represents the wavenumber of thelight and varies due to the 24.5 MHz frequency difference between the carrier and sidebands,
resulting in a different transmission ratio for the carrier and sidebands of the same spatial
mode.
Decomposing into the OMC basis
In order to understand how the astigmatic beam of Eq. (3.2) passes through the OMC, weneed to decompose it into the Hermite Gaussian basis of the OMC, Eq. (3.1). This was donein two ways, both of which resulted in the same decomposition. The first decomposition
was done by taking the inner product between the astigmatic beam and a particular OMC
mode, essentially the inner production between Eqs. (3.1) and (3.2). This inner product,- ∞Ψmn Ψastigmatic = amn eiθ

(3.7)

−∞

yields the amplitude and phase of the TEMmn mode present in the beam.
The other method of decomposition started with creating a set of theoretical data with
Eq. (3.2), with the amplitude of the astigmatic beam stored at a series of x and y coordinates. The simple model of45

a00 eiφ00 Ψ00 + a01 eiφ01 Ψ01 + a10 eiφ10 Ψ10 + a02 eiφ02 Ψ02 + a11 eiφ11 Ψ11 + a20 eiφ20 Ψ20 , (3.8)
where Ψmn is the mnth OMC mode, was then fit to this artificial data set. The amplitude
and phase of each of the modes (defined by Eq. (3.1)) is then left as a free parameter of themodel. After fitting, we normalize the model by scaling so that the TEM00 mode has anamplitude of one and a phase of zero. This second method is very similar to the one used
to analyze the phase camera data. The results of the two decompositions are presented in
the first two columns of Table 3.2.2. The matches between modes from the two methods
are good, within ±0.02 in amplitude and ±11◦ in phase.

There are two additional decompositions which can provide insight, that of an astigmatic

beam without the 25◦ of rotation, where the two waists axes are aligned with the OMC axes,
and that of an astigmatic beam misaligned in vertical displacement coming into the OMC. In
the first case, the TEM11 mode completely disappears, leaving only the expected TEM02 andTEM20 modes due to the mode mismatch. In the second case a vertical displacement of one
third the waist size of the OMC was somewhat arbitrarily chosen because of measurements
indicating alignment drift of this size on short time scales. This decomposition gives afeel for what deviations to expect between the OMC model and the phase camera data.
The differences are mostly on the order of 15% for modes that are present in the perfectalignment and the inclusion of several misalignment modes, such as the TEM01 . Table 3.2.2
summarizes all of these decompositions.
Lastly, to calculate the effect of the astigmatic beam on transmission of the OMC, one
simply needs to apply Eq. (3.5) to each mode for both the carrier and sidebands, then add
all the modes back together for the carrier or sideband.
Decomposition versus Michelson bright fringe OMC scans
One straightforward experimental test of the astigmatic decomposition is to tune the OMC’slength such that modes other than the TEM00 become resonant and note the power of thetransmitted light. In practice this is done by applying a slowly increasing voltage to the
OMC’s PZT to change the length of the OMC while continuously recording the power of46

the light on transmission. By calculating the transverse mode spacing (TMS) from1
TMS = arccos
π

21

L
1−
R0

3

(3.9)

we can easily match peaks with sets of degenerate modes, since a TEMmn mode will beshifted in frequency from the TEM00 mode by (m+n)×TMS in units of free spectral range.
For the OMC, the TMS is 0.308 based off of the parameters from Table 3.1. To compare
the astigmatic decomposition to the data taken in this way during a bright Michelson lock,
we need to apply the OMC response function, Eq. (3.5), to the mode amplitude from
Table 3.2.2 while letting L vary, which lets us go through a free spectral range of resonant
frequencies. Then we take the power in each mode, sum them all, and plot as a function offree spectral range. The final formula used is thus/0
carriercarrier
A a200 |T00(dL)|2 + a201 |T01(dL)|2 + ...

(3.10)

where amn and θmn are amplitude and phase of mode mn from Table 3.2.2, Tmn (dL) is the
response of the OMC for mode mn for a given length change dL, and A is an overall scaleparameter. Note that we ignore the effects of the sidebands in this calculation since for thebright Michelson lock the DC power is dominated by the carrier. Figure 3-3 shows a plotof the scan data taken during a bright Michelson lock that is overlaid with Eq. (3.10) as afunction of dL. To produce amplitudes of the higher order modes that we can compare ourmodels to, we fit the scan data to
P ower(dL) =

m+n=4
5
m,n

carrier
Amn |Tmn(dL)|2(3.11)

where the amplitudes Amn are the free parameters of the fit. Amn are just the square of
amn . Due to degeneracy between modes, we cannot completely distinguish Amn terms, but
we can compare the sums of the degenerate modes. Table 3.2.2 provides a comparison ofthese sums from the model and OMC scan data, with all the values normalized to a TEM00
transmission of 1. Both Figure 3-3 and Table 3.2.2 show a very strong correspondence
between our model’s prediction and the actual experimental result, especially when we
consider that the model assumes perfect alignment and thus predicts no misalignment modes
such as TEM10 and TEM01 .
47

Figure 3-2: The triangular OMC cavity is show in relation to the full interferometer. It
is placed at the AS port to clean up the light reaching the AS photodiode. The red light
represents the carrier, while the blue light represents the sidebands. The phase camera wasplaced at pickoffs before and after the OMC to capture the state of the light. The phasecamera images are from when the interferometer was held on a Michelson bright fringe andare expanded in Figures 3-7 and 3-5. During this state, the RM and ETMs were misaligned.

48

Mode

00
01
10
02
11
20
03
12
21
30
04
13
22
31
40
05
14
23
32
41
50

Integration
MethodAmp. Phase
1
0
0.23
-67
0.26
80
0.21
31
0.07
-113
0.11
23
0.02
176
0.10
111
0.06
62
-

Fit
MethodAmp. Phase
1
0
0.24
-56
0.25
77
0.22
26
0.07
-113
0.11
21
0.01
177
0.10
103
0.06
51
-

Integration Method
Misaligned
Amp.
Phase
1
0
0.28
-155
0.13
25
0.22
-43
0.23
87
0.22
32
0.11
153
0.12
-63
0.05
-172
0.05
57
0.06
-86
0.09
43
0.003
-161
0.09
117
0.061
64
0.041
101
0.06
-120
0.02
46
0.04
-46
0.02
-181
0.02
88

Integration Method
Non-rotated
Amp.
Phase
1
0
0.28
-150
0.47
39
0.19
-136
0.02
38
0.15
45
-

Table 3.2: The mode column lists the mode mn number. All the amplitude columns givethe amplitude of the corresponding mode relative to the T EM00 which was fixed at 1. All
the phase columns give the phase difference from the T EM00 mode, which was fixed atphase of 0◦ . The integration method decomposition was done by taking the inner productbetween the astigmatic beam and a particular OMC mode, as described by Eq. (3.7). The
fit method amplitude and phase columns were done by creating a set of data then fit to asimple model described by Eq. (3.8). The misaligned integration method gives the mode
amplitudes and phases entering the OMC misaligned in the vertical direction by one thirdof a waist size. The non-rotated integration method gives the results for an astigmatic beam
without the 25◦ rotation, so that the waist axes are aligned with OMC axes.

49DegenerateMode
1 even1 odd
2 even2 odd
3 even3 odd
4 even
4 odd
5 even
5 odd

Modes
01
10
20,02
11
03,21
12,30
04,22,40
13,31
05,23,41
50,32,14

Data Fit(Power in modes)
0.13960.10400.06330.02400.02320.00660.02140.0128-Model(Power in modes)
0.10040.06860.00800.0206-

Misaligned Model(Power in modes)
0.07840.01690.09680.05290.01460.01690.00730.01620.00250.0056

Table 3.3: The degenerate mode column labels the resonances based on the order of themodes which pass through. The break in the degeneracy between modes of the same m + nvalue is due to the triangular nature of the cavity. For odd mode numbers in the direction
parallel to the mode cleaner plane there is an additional 0.5 FSR shift in resonance frequency.
The modes columns lists all the modes which resonated together in the OMC. The powerin modes is the sum of the power of all the resonant modes relative to the TEM00 , which isscaled to 1. The data fit column comes from fitting the scan data to Eq. (3.10). The modeland misaligned models come from calculating the power from the amplitudes in Table 3.2.2
and then summing the approriate modes together.

50

Figure 3-3: Plot of the OMC transmitted power versus detuning in units of FSR. The blackline is the actual data, the blue line is the model prediction and the red line is a fit to thedata. The fit was produced by fitting the data to equation Eq. (3.10).

51

What the phase camera should see
The phase camera sees the strength of the beat between the carrier and the sidebands,
by demodulating at 24.5 MHz. The sidebands are generated from the carrier via phasemodulation of the formi(ωt+Γ sin(Ωm t)) ∼

E0 e

*

= E0 J0 (Γ)eiωt

+ J1 (Γ)ei(ω+Ωm )t− J1 (Γ)ei(ω−Ωm )t+(3.12)

where E0 is the original carrier amplitude, ω is the angular frequency of the carrier, Ωm is thefrequency of the modulation, Γ is the modulation depth, and t is time. The approximationon the right hand side of Eq. (3.12) is simply an expansion, followed by collection ofterms into Bessel functions of 0th and 1st order. The first, second and third terms on theright hand side are the carrier amplitude, the upper sideband amplitude, and the lower
sideband amplitude, respectively. Since the carrier and sideband see the same surfaces for abright Michelson lock or a direct reflection off an ITM, and the sideband generation should
only produce a frequency and amplitude difference between the sideband and carrier, it isreasonable to assume that the modal content will be the same for both. In that case we
can write the individual field amplitude of the carrier and sidebands as),
carriercarrierΨ00 + a01 eiθ01 T01
Ψ01 + ...
Acarrier = E0 J0 (Γ) a00 eiθ00 T00
)
,
sb+
sb+
Asb+ = E0 J1 (Γ) a00 eiθ00 T00
Ψ00 + a01 eiθ01 T01
Ψ01 + ...
)
,
sb−
sb−
Asb− = −E0 J1 (Γ) a00 eiθ00 T00
Ψ00 + a01 eiθ01 T01
Ψ01 + ...

(3.13)(3.14)

carrier is the OMC response, obtained from Ψ
where T00
transmitted /Ψincident from Eq. (3.5)

for the appropriate mode, and the Ψmn are from Eq. (3.1) with the Gouy phases fixed tomatch that of the experimental setup. The total field of the light can then be written as
Acarrier eiωt + Asb+ ei(ω+Ωm )t + Asb− ei(ω−Ωm )t .

(3.15)

Demodulating the power in phase at an angular frequency Ωm yields the in-phase(Iphase), and 90◦ out of phase yields the quadrature-phase (Qphase) signals. These can be52

explicitly written as
2*(Acarrier A∗sb+ + A∗carrier Asb− ) = Qphase(3.16)

2+(Acarrier A∗sb+ + A∗carrier Asb− ) = Iphase(3.17)

andBy replacing the amplitudes amn and phases θmn with the appropriate model values
we can create an I or Qphase map which is directly comparable to the measured phasecamera phase maps. The phase map in Figure 3-4 is created using the parameters from theintegration method of astigmatic beam decomposition listed in Table 3.2.2, with an overall
demodulation phase rotation such that all the higher order mode “junk” is in one phase
but not the other.

Figure 3-4: Phase map generated from the OMC model, with the integration method parameters from Table 3.2.2. This shows spatial amplitude of the Qphase demodulation. The
demodulation phase rotation has been chosen such that all the higher order “junk” is in theIphase, leaving the TEM11 mode as the largest contributor to the Qphase signal.

53

3.2.3Phase camera measurements and comparisonsPhase camera data
The following images were taken by the phase camera in two different states. The first waswhen the OMC was locked with maximum light transmission while the interferometer washeld on a bright Michelson fringe. The second was when all the mirrors of the interferometer
were misaligned, except for ITMY and the BS, resulting in a direct reflection of light into
the AS port. The images are reproducible for a given state of the interferometer.

54

Figure 3-5: The above images from the phase camera were taken on transmission throughthe OMC when the interferometer was held on a Michelson bright fringe. The upper left
image shows the spatial amplitude of the DC signal, which should be made up mostly of theclean TEM00 carrier signal. The lower left image is the spatial amplitude of the Iphase of
the demodulated RF signal, demodulated at Ωm , 24.5 MHz. The fact that we see a TEM00
mode in the Iphase implies there is some amplitude modulation or mismatch between the
sidebands, since otherwise the beat the sidebands produce with the TEM00 carrier shouldcancel out. The lower right is the Qphase of the same signal. Our prediction for the Qphase
image from modeling the astigmatic beam passing through the OMC is Figure 3-4, mostly
the TEM00 mode
6 which we see. The upper right is the overall amplitude of the signal
calculated from(Iphase)2 + (Qphase)2 .

55

Figure 3-6: The above images from the phase camera were taken when only ITMY and the
BS were aligned, producing a direction reflection of light off ITMY, through the BS, andto the OMC. The signals from which the images are generated are described in Figure 3-5.
The factor of 4 reduction in overall power is due to the loss of 50% of the light at the beam
splitter because of the ITMX misalignment. This also means the electronics backgroundnoise stands out more in these images when compared to Figure 3-5.

56

The DC power images show fairly clean TEM00 modes, indicating the OMC is in facttransmitting mostly the mode we want. The factor of 4 drop in DC power intensity from
the Michelson bright fringe case to the direct reflection case is due to the loss of 50% of thelight at the beam splitter because of the ITMX misalignment. This also means the directreflection case is closer to the noise floor of the phase camera. The I and Qphases havebeen rotated to match the same overall phase rotation chosen for the model, so that one
phase is expected to be flat and the other will contain the higher order modes transmitted.
The fact that we see a clear TEM00 mode in one phase instead of a flat plane implies someamplitude modulation of the carrier and sidebands is being produced by or passing throughthe OMC. The TEM11 mode image in the Qphase is reassuring, as it looks similar to themodel prediction from Figure 3-4.
To determine if the TEM00 mode present in the Iphase is due to the OMC or present
on the light before reaching the OMC, we look at phase camera images when the OMC wasremoved. Figures 3-7 and 3-8 show the bright Michelson lock and an ITM direct reflection
respectively, without the OMC in the path. In the bright Michelson case the amplitude
modulation can be fully explained as being on the incoming light, since the ratio of the DC
power to the Iphase power is roughly the same or less for the OMC present and not present
cases as shown in Table 3.4. However, for the case of the direction reflection off an ITM,
the amplitude modulation seems to be greater after the OMC. However, in this case thereflection was off ITMX rather than ITMY. The ratio of the Iphase signal is down by much
more than the expected factor of 2 in amplitude when comparing between the Michelsonbright fringe and direct reflection cases, while the DC amplitude is down by the expectedfactor of 4. This indicates the sidebands in the ITMX direct reflection case were affectedby the reflection much differently from the ITMY case, and in such a way as to reduce theoverall amplitude of the beat between the carrier and sidebands.

57

Figure 3-7: The above images were taken when the interferometer was held on a Michelsonbright fringe and without the OMC in the light path. The signals from which the images
are generated are described in Figure 3-5. Without an OMC to clean the signal, we seesome combination of astigmatic modes. When compared to Figure 3-7, the overall powerin the DC image is roughly the same, but the beat signal increases, which shows that the
sidebands are more affected by the OMC then the carrier. This is expected since the OMC
was designed to pass the carrier, and the sidebands are slightly offset in frequency, reducing
their transmission.

58

Figure 3-8: The above images from the phase camera were taken when only ITMX and the
BS were aligned, producing a direction reflection of light off ITMX which was then directedto the phase camera. The signals from which the images are generated are described in
Figure 3-5. In comparison to Figures 3-7, 3-6, and 3-5, we see that the carrier on sideband
beat signal is lower than expected, while the DC light (representing mostly carrier) is
relatively unchanged, indicating a difference in the sidebands between the ITMY reflection
and the ITMX reflection.

59

Table 3.4: Amplitude modulation comparisonsMichelson Bright FringeWith OMC
Michelson Bright Fringe
Without OMC
Direct Reflection
With OMC
Direct Reflection
Without OMC

Data Set 1
Iphase/DC
2.28

Data Set 2
Iphase/DC
2.48

Data Set 3
Iphase/DC
2.45

3.7

3.68

3.75

7.03

3.9

4.97

3.12

3.18

3.2

Phase camera data and model comparison
To directly compare the models, we first need to set certain scale parameters. The easiest
method of getting these scale parameters is to fit the DC power data to AΨ00 Ψ∗00 (x +xoffset , y + yoffset , w), letting the overall amplitude A, the center position offsets xoffset andyoffset , and w the waist size vary as free parameters. Fixing the offsets and waist size inour model first makes the final stage of fitting to the model much easier. We then makea nonlinear fit of Eqs. (3.16) and (3.17) to the data, treating E02 J0 (Γ)J1 (Γ) as a single
amplitude free parameter A in addition to the a00 and θ00 free parameters. We use thevalues from Table 3.2.2 as our initial guesses, and restrict the parameter space to about±20% of those values. To determine the quality of the fit, we first determine the noise in

the data. By subtracting phase camera images from one another, and taking the standard
deviation of the resulting difference values gives us our σ. We then use this to calculate ourχ2 (normalized) merit functions with the values between 1 and 4, depending on how noisy
a particular image is, and whether it was the bright Michelson or direct reflection case.
However, the residuals show some distinct structure excess, perhaps some TEM00 mixed
with TEM20 in the case of the direct reflection fits. The two fits of the Qphase, one for thebright Michelson and one for the direct reflection in Figure 3.2.3 are typical of the sets.

60

Figure 3-9: Left: Qphase fit of a Michelson bright fringe data, as seen by the phase camera
on transmission of the OMC. Right: Qphase fit of a direct reflection off ITMY data, as
seen by the phase camera on transmission through the OMC. The respective top image isa plot of the fit including modes of up to order 4. The respective middle image is of thephase camera data for the Qphase. The background electronics noise stands out more in
the direct reflection images because of the lower overall power reaching the phasecamera.
61

Table 3.2.3 gives the parameter values for the two sets of measurements, along withthe model values from Table 3.2.2, normalized to a TEM00 amplitude of 1 and a phase of
0. This comparison provides several insights. The large TEM01 mode implies a significant
misalignment of the vertical axis between the incoming light and the OMC, on the order ofthe waist size or the divergence angle of the light at the OMC. The TEM02 , TEM11 , andTEM20 modes are consistent with being produced solely by the astigmatic nature of thelight. Ignoring the effects of a simple misalignment, the fact that the model matches thedata well implies the OMC is doing what it is expected to be doing. We get large amountsof higher order modes on transmission of the OMC mostly due to the known astigmatism,
and that other additional problems or effects are not needed to explain the OMC output.
After testing finished on the OMC it was removed and returned to GEO. However, there
are plans for a new OMC to eventually be added to the LIGO interferometers in a future
upgrade. The phase camera system was next used to examine the interferometer light inside
the recycling cavity in different heating states with and without the TCS in an attempt
to understand how the AS I signal was being generated and corrected. Those results arepresented in Chapter 3.

62

Mode
χ2
A
00
01
10
02
11
20
03
12
21
30
04
13
22
31
40

Model
Amp. Phase
1
0
0.23
-57
0.26
80
0.21
31
0.07
-113
0.11
23
0.02
176
0.10
111
0.06
62

Misaligned Model
Amp.
Phase
1
0
0.28
-155
0.13
25
0.22
-43
0.23
87
0.22
32
0.11
153
0.12
-63
0.05
-172
0.05
57
0.06
-86
0.09
43
0.003
-161
0.09
117
0.061
64

Mich BrightAmp. Phase
0.802
258
1
0
0.43
71
0.044
-15
0.30
-44
0.256
100
0.27
11
0.023
12
0.09
-70
0.06
-17
0.08
59
0.02
-90
0.127
129
0.03
191
0.10
131
0.04
82

Direct Reflection
Amp.
Phase
3.889
99
1
0
0.50
62
0.066
2
0.30
-72
0.20
100
0.27
11
0.03
12
0.04
62
0.06
-38
0.12
34
0.02
-90
0.11
175
0.03
191
0.08
91
0.04
82

Table 3.5: The χ2 row indicates the χ2 from the fit. The A factor indicates the factor
needed to rescale the TEM00 amplitude to 1. The amplitude columns indicate the relative
amplitude of the mode to the TEM00 and the phase columns indicate the difference in phasefrom the TEM00 , which was set to 0◦ .

63

THIS PAGE INTENTIONALLY LEFT BLANK

64

Chapter 4

Improving and applying the phasecameraThis chapter begins by describing the addition of an independent reference beam to thephase camera which is necessary for looking at the carrier and sideband components of thelight individually. The chapter then focuses on how this improved phase camera system
was used to investigate the heating problems of the LIGO 4k interferometer. The system
was able to show a consistent progression through the different heating states of the interferometer and what the TCS was correcting that reduced the excess AS I noise and allowed
the LIGO interferometers to operate at their designed operating power and sensitivity.

4.1

Phase camera with reference beam

4.1.1

Experimental setup

Interferometer light
Since the carrier resonates both in the arms and the power recycling cavity while the RF
sidebands only resonate in the power recycling cavity, they effectively “see” different partsof the interferometer. The Fabry-Perot cavities can be described by their finesse, which is
the FSR divided by the bandwith of the resonances and is inversely proportional to thelosses in the cavity, and their g factor, which is (1 − L/RITM )(1 − L/RETM ) where L is thelength of the cavity, RITM is the radius of curvature of the ITM, and RETM is the radiusof curvature of the ETM. In the same way that the OMC prevented higher order modes to65

propagate because of the Gouy phase shift (see Section 3.2.2), the Fabry-Perot arm cavities
suppress higher order modes. The cavities have high enough finesse, F = 219, combinedwith their g = 0.33, that the higher order modes are suppressed by a factor of up to 60
compared to a resonant TEM00 mode.
The almost flat input mirrors of the power recycling cavity support many modes, inwhich case the sidebands will tend to include higher order TEM modes. To understandbetter what is occurring inside the interferometer, it is helpful to be able to separate thecarrier light from the sideband light. One way to do this is to introduce a reference beamand measure the its beat note with the carrier and each RF sideband independantly.
Reference beam
A reference beam shifted off by 75 MHz from the main carrier frequency and then mixed
with the full interferometer light will produce a beat with the carrier at 75 MHz, a beatwith the upper sideband at 99.5 MHz and a beat with the lower sideband at 50.5 MHz, as
shown in Figure 4-1. For comparison, the carrier and sidebands beat at 24.5 MHz, and thesidebands beat with each other at 49 MHz.

Figure 4-1: This diagram shows the relative positions of the carrier, sidebands, and shifted
carrier reference beam in frequency. The carrier is 24.5 MHz from the sidebands and 75
MHz from the reference beam.

66

This reference light was generated by taking light directly from the output of the Lightwave laser and passing it through an acoustic optical modulator (AOM) driven at 75 MHz,as shown in Figure 4-2. This light was then coupled into a fiber optic cable which was run
from the laser table to the phase camera setup. A fiber to free space coupler was attachedto the fiber, and the resulting light was passed through a half-wave plate and a polarizing
beam cube so as to match the polarization of the incoming interferometer light. A beam
splitter was then used to combine the incoming interferometer light with the reference beam
before going on to the rest of the phase camera system.

4.1.2

Demodulation board redesign

It was quickly discovered that the overall phase of the reference beam varied on the order ofminutes, most likely due to thermal changes along the length of the fiber. In the time it took
to switch the demodulation signal to the demodulation board the signal phase could vary,making it much harder to piece together the individual structure. The solution was to takeall the data simultaneously, essentially recording the DC signal, the carrier and reference
beat, the upper sideband and reference beat, and the lower sideband and reference beat atonce. The images taken are simultaneously directly comparable, since they all have identicalreference beam effects.
A redesign of the control and demodulation board was required, as the initial design
only could demodulate and record one signal at time. The new board design can be foundin Appendix B. Essentially, the demodulation channels and computer recording channels
were increased by a factor of four, and the computer software was updated to handle thehigher data rate.
The boards were assembled and tested at Hanford and performed satisfactorily before
being installed in the phase camera system.

4.1.3

Fitting and understanding the data

The use of a reference beam creates several challenges in understanding the output of thedata. The demodulated output of each channel is a combination of the reference beamand the signal of interest (either carrier or individual sideband). The reference beam also
adds some noise. However, because of the new demodulation board where all the data istaken simultaneously, at each point recorded in the scan the reference beam must affect the67carrier and sidebands in exactly the same way. This provides us with enough information
to seperate the carrier and sidebands from the reference beam in software.
Since we record DC power of the light we know the total sum of the power of the four
signals. This suggests the following method for calculating the amplitude effects of thereference beam. We know that

PDC = A2cr + A2sb− + A2sb+ + A2ref(4.1)

where PDC is the recorded DC power level. By taking the sum of the squares of the Iphaseand Qphase for each demodulated signal we can calculate the total power in the IFO lighttimes the reference beam power, writing

Stotal ≡ A2cr A2ref + A2sb− A2ref + A2sb+ A2ref = A2ref (A2cr + A2sb− + A2sb+ )(4.2)

where the Stotal is just the sum of the signals. Multiplying Eq. (4.1) by A2ref and substituting
in our known sum of signals, we get the following equation

A4ref − A2ref PDC + Stotal = 0 .

(4.3)

This can be easily solved for A2ref which can then be in turn be used to find just Acr , Asb− ,and Asb+ . Figure 4-3 shows a horizontal and vertical cut of the power data before and after
this calculation was performed, effectively |Acr |2 and |Acr Aref |, to make it easier to see thedifference in the spatial power of the signal. The data was taken from a state with 6 Wattsof power input to the interferometer, and when the TCS was running. It also shows thecalculated reference beam power, |Aref |2 , which was then scaled up by a factor of 10 soas to fit in the plot. One problem with this method is when the electronics noise becomesan appreciable part of the total signal coming from the phase camera, the above equations
break down (and sometimes predict imaginary powers). In those regions of low signal tonoise (roughly one sigma in the electronics noise) the data was ignored, resulting in blanksections in those plots that use Eq. (4.3) to calculate the effects of the reference beam.
One test of the effectiveness of this method is to take the individual sideband and carrier
signals, and mix them in software in the same way they mix on the photodiode to produce
an image which can be directly compared to the recorded one omega signal. For example,

68the lower plot in Figure 4-4 shows a one omega signal in a thermally tuned state, such that
the AS I noise was minimized with the use of the TCS. To begin, we define a set of Iphaseand Qphase sideband signals as function of an overall rotation fitting parameter. For the
lower side band we can write

Isb−,rot (θsb− ) + iQsb−,rot (θsb− ) =(Isb− cos(θsb− ) + Qsb− sin(θsb− )Qsb− )+ i (Qsb− cos(θsb− ) − Isb− sin(θsb− ))(4.4)

where the rotation angle θsb− rotates the overall phase of the entire lower sideband data set.
The upper sideband is the same except with θsb+ as its rotation angle fitting parameter.
Lastly an overall amplitude factor a were used as a third fitting parameter in the leastsquares fit.
The full equation fit wasa[(Isb−,rot (θsb− ) + iQsb−,rot (θsb− ))(Icr − iQcr )+ (Isb+,rot (θsb+ ) − iQsb+,rot (θsb+ ))(Icr + iQcr ) + complex conjugate](4.5)

That least squares fit produced the top plot in Figure 4-4 which appears to be in good
agreement with the measured carrier on sideband signal in the lower plot.

69

Figure 4-2: This diagram shows the setup of the reference beam. A small portion of themain Lightwave laser was picked off and passed through an AOM driven at 75 MHz. This
was then coupled to an optical fiber and taken to the phase camera on a different table
where it was passed through a half-wave plate and a polarizing beam cube before beingcombined with the interferometer light on a beamsplitter.

70

Figure 4-3: The top plot shows a horizontal profile of phasecamera data from when theinterferometer had 6 Watts of power input and the TCS was on. The bottom plot shows avertical profile for the same case. The blue line is the power in the carrier only. The redline is the calculated reference beam only, scaled up by a factor of 10. The green line is thecarrier signal combined with the reference beam, effectively the measured 75 MHz signal
from the phase camera, and has been scaled to match in power. The x-axis is in millimeters
as measured at the position of the phase camera photodiode. By dividing out the reference
beam and associated noise from it, the carrier shape gets closer to the Gaussian profile one
expects of a TEM00 mode.

71

Figure 4-4: The top plot is a fit of the carrier mixed with sideband at 24.5 MHz createdfrom the individual sideband and carrier signals measured with the reference beam. The fit
was a least squares fit with three fitting parameters, specifically the phase rotation of theupper and lower sidebands, and the overall amplitude of the carrier on sideband signal. The
lower plot is the directly measured 24.5 MHz carrier on sideband signal without reference
beam. The color scale shows power in watts. The axes are in millimeters at the phasecamera photodiode.

72

The data analysis on a single phase camera scan that uses the reference beam proceeds asfollows. First the averaged dark noise is subtracted off from the data. The dark noise is justthe signal recorded from the phase camera when no light is incident on the photodetector.
Then the data is calibrated according to the photodetector response, including the effects
of a 3 dB roll off at 125 MHz. The demodulation board calibration is then taken into
account, which was measured before hand by using calibrated test signals in place of thephotodiode. Lastly, the reference beam amplitude effects are calculated and removed asindicated previously.

4.2

H1 in different heating states

By using the phase camera to look at the inteferometer light as the optics go from receiving
no heating, to optimal heating, and then to too much heating, we can understand what theeffects of heating are on the interferometer light and determine how it affects the AS I noise
and the final sensitivity of the instrument.
Data was taken in four different heating states by the phase camera in the month before
the S5 science run began. The data was taken from a pickoff inside the recycling cavity,
giving a view of both resonant sidebands and the power recycled carrier, as opposed to thedark port carrier which should mostly cancel itself. In each case the interferometer wasallowed to settle into the state for a half hour, roughly the thermal timescale of the ITMs.
The first, which we will refer to as the “No TCS” state, was with no thermal compensation at a low input power (only 1.8 watts incident on the interferometer). The second
state, “Tuned TCS”, was at a low input power but with thermal compensation system set
such that AS I signal was nulled. The third, “Reversed TCS”, was at a low input power butwith the thermal compensation system set to make the thermal problems worse, effectively
switching the direction of the correction effects and mimicking what the interferometer does
at high input power without correction. Lastly a set of data was taken while in the nominal
high power configuration (6 watts incident on the interferometer) with an early iterationof the thermal compensation servo running, which we will refer to as the “Power up with
TCS” state.
Table 4.2 lists some relevant recorded parameters of each state. The NSPOB channelrefers to a pickoff on the beamsplitter which measures the sideband on sideband signal at73

49 MHz. This signal is then normalized by the input power. The AS DC is refers to one of
the AS port’s photodiode DC currents, which is a measure of the total light incident on thephotodiode. AS I Corr is the Iphase AS port photodiode correction signal which is used toprevent saturation of the AS port photodiode.

4.2.1

Discussion of phase camera results

Table 4.2 lists ratios of the carrier to sideband power and the sideband to sideband forvarious states as measured by the phase camera. To make understanding of these easier, let
us assume for the moment that the modes are perfectly matched, and that the only differenceis the overall power of the carrier and the individual sidebands. With this assumption,
predictions can be made based on the recycling gains and the relative amplitudes of thecarrier and sidebands at the input to the interferometer.
In the cold state, the interferometer had known sideband recycling gain, Gsb , of roughly
13 in power. The highest Gsb achieved in a thermally tuned state was 26 [5]. With good
spatial overlap of the upper and lower sidebands, the NSPOB signal should be proportional
to Gsb . If we take into account the modulation depth Γ = 0.4 and a carrier gain Gcr of 50
we would predict a ratio of roughly 88 for the carrier power relative to the an individual
sideband’s power for the cold, untuned state. For a thermally tuned state with Gsb = 26,
we would expect the ratio of roughly 44.
The phase camera shows a mean ratio of 95 for the no TCS state and a ratio of 53 forthe good TCS state between the carrier and sideband power. Comparing the actual ratio ofof NSPOB of 395/220 = 1.8, and the ratio of the mean value of the power measured by the
sidebands in the two states of 95/53 = 1.8, we find good agreement. Similarly we expect
297/220 = 1.4 and 500/220 = 2.3 based on the NSPOB values in the Reversed TCS andPower up states relative to the No TCS state. The phase camera finds 95/49 = 1.9 and95/40 = 2.4 for the Reversed TCS and Power up states relative to the No TCS state. These
comparisons are summarized in Table 4.1. Here the the Reversed TCS state is in very poor
agreement with the NSPOB signal, which suggests poor spatial matching of the sidebands
might be the dominant effect.
As noted in Section 2.2.2, the AS I signal can be generated either by an amplitudedifference between the sidebands or higher order carrier modes beating with higher order
sideband modes. Table 4.2 lists the ratio of the power in the upper sideband to the ratio74

Table 4.1: NSPOB and sideband power ratios

NSPOB ratio(relative to No TCS case)
Ratio of Sideband Power(relative to No TCS case)

No TCS

Tuned TCS

Reversed TCS

1

1.8

1.4

Power upwith TCS
2.3

1

1.8

1.9

2.4

of the power in the lower sideband for the various states, which should be compared to the
AS I Corr signal on the same table. In the Tuned TCS case, where the AS I Corr signal has
effectively been nulled, the upper and lower sidebands have an average mismatch smaller
than the uncertainty of the overall measurement. Effectively, they have been balanced in
power successfully by the TCS system.
In the No TCS state the upper sideband is around 9% larger than the lower sideband,
which accounts for the non-zero AS I Corr signal. The Power up state also has a similar
amount of mismatch with, but in the opposite direction with more lower sideband thanupper sideband. In both cases, approximately 5% to 15% mismatch required the same
correction on the AS photodiode.
The Reversed TCS state interestingly has a very close match in overall power in the
sidebands, but required over twice the correction as the No TCS and Power up states. Thisimplies a spatial mismatch is dominating the AS I Corr signal. An example set of images
from the Reversed TCS state is present in Figure 4-5. If we compare with example images
from the other three states, shown in Figures 4-6, 4-7 and 4-8, we see the spatial form of the
sidebands are severely distorted, with a corresponding distortion in the carrier on sideband
24.5 MHz signal. Certainly in the No TCS and Tuned TCS states, the sidebands have amuch better spatial distribution matching the extent of the carrier.
The Reversed TCS case in Figure 4-5 clearly shows a mismatch between the spatial
extent of the sidebands and the carrier. This affects the carrier on sideband beat resulting
in only a small, non-symmetric portion of the carrier contributing to the AS photodiodesignal. The plot of the difference of the amplitudes between the two sidebands clearly showsa mismatch, which is large when compared to the other heating cases. So even though thetotal power in the sidebands are similar, the difference in spatial structure results in poor
beating and less NSPOB signal. There is a small difference in the position of the peaks as
75

Table 4.2: Heating states and phase camera power ratios

NSPOB (counts)
AS I Corr (counts)
Carrier/USB
Carrier/LSB
USB/LSBNo TCS

Tuned TCS

Reversed TCS

220
-200
90 ± 299 ± 41.09 ± 0.05395
052 ± 254 ± 2
1.03 ± 0.04

297
-450
50 ± 249 ± 30.98 ± 0.05

Power upwith TCS
500-200
42 ± 238 ± 20.9 ± 0.05

measured at the phase camera, roughly 10% of the waist size of the sidebands in this case,where the waist size is the radius where the amplitude of the sideband falls by 1/e.
The No TCS case in Figure 4-6 has the sidebands matching the carrier in spatial extent,
although their overall power and structure are different from each other. These differences
can lead to AS I signal when the sidebands are beat against the carrier. The sideband on
carrier signal, effectively seen at the AS port photodiode, is the same size as the carrier
signal, with a null band due to the difference in the curvature of the sidebands and carrier
signals on the phase camera photodiode. The asymmetry in the carrier on sideband signal
is due to the structure in the sidebands.
The Tuned TCS case in Figure 4-7 shares many similarities with the No TCS case,
except that the sidebands look more Gaussian, are better balanced in power and shape,
and the carrier on sideband signal is somewhat cleaner, with less overall power present.
Even though the central peak is slightly higher than in the carrier on sideband image thanin the No TCS case, there is less “junk” in the outer sections, reducing the overall power.
All these changes indicate less AS I signal should be seen by the AS photodiode and that
the AS I Corr signal required to balance it should decrease relative to the No TCS case,which was in fact the observed effect.
The Power up with TCS case in Figure 4-8 shows effects similar to the Reversed TCS
case, namely smaller spatial extent for the sidebands, although not to the same degree.
This is because the TCS servo, which was designed to null the AS I Corr signal, was not
fully tuned at the time this data was taken, and resulted in less compensation than was
required. The unusual shape of the carrier on sideband signal is related to the squashednature of the sidebands, although it is not precisely understood where the source of theasymmetry relative to the carrier is coming from. Due to time constraints before the start76

of LIGO’s 5th science run, data collection from different points of the interferometer were
never completed, limiting our ability to understand the source of these differences. One
possibility is uneven absorption and heating in the ITMs by the main interferometer laser.
Comparing the sidebands we see the lower sideband is spatially smaller than upper sideband,
but with nearly twice the peak power at its center. Such amplitude mismatches as noted
earlier are a clear source of AS I which needs to be corrected.
Taken together the images from the various heating states show a trend of shrinking
sideband spatial extent with increased heating of the ITMs, whether by the main interferometer laser, or the TCS system operating in such a way as to add heat. They also show
significant differences between the sidebands in spatial shape, especially in the high powerstate. However, they also show with a properly tuned TCS system, these differences in the
sidebands can be reduced, balancing their overall and peak power, producing a better match
between each other and the carrier, which corresponds with a drop in the necessary AS I
correction and allowing for higher power operation with the same AS port photodiodes andelectronics.

77

Figure 4-5: The above images are from the Reversed TCS case. The color scales are powerin watts. The axes are position in millimeters, measured at the position of the phase camera
photodiode. The top right plot shows the power from DC signal. The top left plot shows
the power in the upper sideband, demodulated at 50.5 MHz. The middle left plot shows
the lower sideband, demodulated at 99.5 MHz. The middle right plot shows the powerin the carrier, demodulated at 75 MHz. The lower left plots shows the subtraction of thelower sideband from the upper sideband. The lower right plot shows the carrier on sideband
beat, demodulated at 49 MHz. The upper sideband, lower sideband, sideband difference,and carrier images have had the effects of the reference beam removed.

78

Figure 4-6: The above images are from the No TCS case. The color scales are power inwatts. The axes are position in millimeters, measured at the position of the phase camera
photodiode. The top right plot shows the power from DC signal. The top left plot shows
the power in the upper sideband, demodulated at 50.5 MHz. The middle left plot shows
the lower sideband, demodulated at 99.5 MHz. The middle right plot shows the powerin the carrier, demodulated at 75 MHz. The lower left plots shows the subtraction of thelower sideband from the upper sideband. The lower right plot shows the carrier on sideband
beat, demodulated at 49 MHz. The upper sideband, lower sideband, sideband difference,and carrier images have had the effects of the reference beam removed.

79

Figure 4-7: The above images are from the Tuned TCS case. The color scales are power inwatts. The axes are position in millimeters, measured at the position of the phase camera
photodiode. The top right plot shows the power from DC signal. The top left plot shows
the power in the upper sideband, demodulated at 50.5 MHz. The middle left plot shows
the lower sideband, demodulated at 99.5 MHz. The middle right plot shows the powerin the carrier, demodulated at 75 MHz. The lower left plots shows the subtraction of thelower sideband from the upper sideband. The lower right plot shows the carrier on sideband
beat, demodulated at 49 MHz. The upper sideband, lower sideband, sideband difference,and carrier images have had the effects of the reference beam removed.

80

Figure 4-8: The above images are from the Power up with TCS case. The color scales are
power in watts. The axes are position in millimeters, measured at the position of the phasecamera photodiode. The top right plot shows the power from DC signal. The top left plotshows the power in the upper sideband, demodulated at 50.5 MHz. The middle left plotshows the lower sideband, demodulated at 99.5 MHz. The middle right plot shows thepower in the carrier, demodulated at 75 MHz. The lower left plots shows the subtractionof the lower sideband from the upper sideband. The lower right plot shows the carrier onsideband beat, demodulated at 49 MHz. The upper sideband, lower sideband, sideband
difference, and carrier images have had the effects of the reference beam removed.

81

82

Part III

Targeted continuous gravitational
wave searches

83

THIS PAGE INTENTIONALLY LEFT BLANK

84

Chapter 5

Neutron stars, spindown models,and estimating upper limits ofsourcesThis chapter describes some basics of neutron stars and discusses some basic models used
to estimate potential gravitational wave signals. When looking at sources with missing
information, such as neutron stars with unknown spin period and spin down, it is useful tomake estimates for the best possible cases to determine if its worth putting computation
time into the search. There are several simple but very informative calculations one can go
through in this regard.

5.1

Neutron stars and pulsars

Neutron stars are the end point of stellar evolution for intermediate mass stars, thosestarting with a mass of roughly 6 to 15 M$ [37]. Lighter stars will become white dwarfs
while more massive stars will end as black holes. As the nuclear fuel of a star becomes
exhausted, a core grows within an expanding outer shell, the total gravitational collapse of
the core being prevented by electron degeneracy pressure. If the mass of the core grows
greater than 1.4 M$ , the electron degeneracy pressure is no longer great enough to prevent
further collapse of the core. It begins to rapidly collapse until stopped by neutron degeneracypressure. However, by this time the collapse has released a large amount of gravitational85

potential energy in a few seconds, producing a supernova. Most of the original mass isblown off during the supernova, but the remainder of the star still has a mass on the orderof 1 M$ .
Neutron stars for which there are measured masses typically are found near 1.35 M$ .
The theoretical upper limit on the mass of a neutron star is somewhere around 3 M$ ,and is dependant which equations of state are used, which relate the density, pressure andtemperature inside the star. Using the more plausible equations of state the radius of atypical neutron star is ∼ 10 − 11 km. Modeling a neutron star as a sphere with uniform
density allows us to estimate the moment of inertia, I, as 1038 kg m2 . This moment ofinertia more of a scale rather than an exact number, because of the differences between
possible equations of state and other particular details of any given neutron star. Aftercomparing a range of equations of states, Cook, Shapiro and Teukolsky found a range of
0.5 × 1038 to 8 × 1038 kg m2 for the moment of inertia [15].

Pulsars are observed from the regular electromagnetic pulses that they emit. These

pulses are due to the magnetic field of the generating star, whose axis in general is notaligned with the rotation axis. As the star rotates, the magnetic field produces magnetic
dipole radiation, which would be highly beamed out from the magnetic poles and sweep
across space. If that beam intersects with a detector on Earth, it will be observed as a shortpulse, whose period is directly related to the period of rotation of the star. Periods of pulsars
vary from tens of seconds to milliseconds. The necessary rotation rates to produce pulses
with these periods means pulsars must be neutron stars, as other less compact stars can’t
rotate that fast and remain together. Observations of the power and frequency of pulsar
emission provides insight into the strength of the magnetic field necessary to produce them.
The estimated field strengths vary from 1012 Gauss down to 108 Gauss, depending on thepulsar. These dipole field strengths are consistent with the collapse of a normal star with apolar field of order 100 gauss and the flux being convserved in the collapsing stellar material.
The strongest polar fields of order 1012 Gauss occur in young pulsars, with spin periodsof order tens of milliseconds. The majority of pulsars have fields of order 1010 Gauss andspin periods of order seconds. There is a third population, refered to as millisecond pulsars
because of their spin periods of order 1 millisecond. These millisecond pulsars are the oldest
pulsars and have magnetic fields as low as 108 Gauss [24]. In general, the spin period of thepulsar pulses decrease with time (and thus decreasing frequency), with a measured rate of86changes in the period over the period, Ṗ /P , between 10−18 to 10−12 s−1 . However, in thecase of millisecond pulsars, they instead seem to have been spun up (and in some cases still
being spun up) by accretion from a low mass companion star. This is supported by the factthat well over half of millisecond pulsars are found in binaries. Millisecond pulsars appear
to be a smaller population of the primary body of pulsars, found through out the galaxy
and especially in globular clusters, with a longer lifespan but much lower re-birth rate [37].
There is an estimated active population of 105 to 106 pulsars in our galaxy, although
only about 1500 have been observed due to selection effects and strength of the signals [37].
Most of these observed pulsars are within a layer 1 kpc thick centered on the galactic plane
and within a radial distance of about 10 kpc of the center of the galaxy. Most have highvelocities, presumably from the initial supernova explosion which formed them, and are
moving away from the plane at around 200 km s−1 . This is consistent with an origin ofwithin 100 pc of the galactic plane and their lifetime of roughly 10 million years for theiractive stage. Using the current estimated numbers of pulsars, their lifetime of 10 million
years, and assuming a roughly constant birth rate would imply a population on the orderof 109 neutron stars (out of the 1011 total stars) in our galaxy [37].

5.2

Estimated deformations of neutron stars

It is important to consider the ellipticities of neutron stars as this plays directly into thestrength of any gravitational wave reaching us and influences heavily what we can detect.
Both the maximum and most likely ellipticities of neutron stars are highly uncertain [2].
An upper limit estimate on the maximal deformation supported by the rigidity of a neutron
star’s crust is [51])max ≈ 5 × 10−7

) σ ,
10−2

(5.1)

where σ is the breaking strain of the solid crust. The numerical coefficient is small mostly
because the shear modulus of the inner crust, where most of the crust’s mass is located,
is smaller than the pressure by a factor of 103 . Shear modulus is the shear stress over the
shear strain and as it goes to zero, the crust becomes a liquid which can’t support nonaxial deformations. Thus, the small shear modulus results in a small maximum ellipticity.
Equation (5.1) uses the fiducial breaking strain of 10−2 since that is approximately the best87

terrestrial alloys have. However, for a perfect crystal with no defects σ could be as high as
10−1 , or several orders of magnitude smaller for amorphous solids or a crystal with many
defects [32].
There have been suggestions that exotic and speculative star types could support larger
maximum ) values. The highest ellipticity model is that of a solid strange-quark star whichhas [42])max ≈ 4 × 10−4

) σ ,
.
10−2

(5.2)

The reason this maximum ) value is so high is due to the high shear modulus, which forsome strange star models can almost be as large as the pressure. Another speculative but
more robust model is the hybrid star. Such a star would consist of a normal neutron staroutside a solid core of mixed quark and baryon matter, extending from the center to justbelow the crust. Such hybrid star models have a maximum ellipticity of [42])max ≈ 9 × 10−6

) σ ,
.
10−2

(5.3)

However, the maximum supportable ellipticity is not the same as being able to achieve
the maximum ellipticity in practice. We can consider a few mechanisms that might produce
ellipticity in a neutron star. In the case of young pulsars, as they spin down it is likely
stresses build up, possibly because of reduced centrifugal force due to slowing rotation, until
the crust is forced to change shape and crack. Such cracking could produce deformations
or “mountains” on the surface. It is unclear how long it would take gravity to smooth out
these deformation after cracking, which could possibly leave long lived distortions [14].
Strong internal magnetic fields can also produce ellipticities in neutron stars. Such fields
if not aligned with the rotation axis, could produce non-axisymmetric deformations due tomagnetic tension, effectively pulling on the matter of the star. A strong dipolar field notaligned with the rotation axis in a star with a superconducting interior, which has expelled
all the magnetic field lines from the superconducting region, could lead to deformations oforder 10−6 [11]. This deformation would result in gravitational wave emission at both therotation frequency and twice the rotation frequency.
Torodial magnetic fields could also generate ellipticity. Differential rotation immediately
after the core collapse which formed the neutron star can produce large internal toroidal
88magnetic fields. The magnetic fields lines trapped with in the different layers are effectively
wrapped into a toroidal shape as they rotate at different speeds around the star. Dissipation
within the star would tend to drive the symmetry axis of this toroidal field toward the star’s
equator, the orientation which maximizes ellipticity. For toroidal magnetic fields on theorder of 1015 Gauss, an ellipticity of order ∼ 10−6 would occur [16].

5.3Energy considerations

Neutron stars when born will in general have some spin angular momentum. This angularmomentum acts as an energy reservoir which is used to power various phenomenon including
radio pulsation, ejection of particles from the surface, and gravitational waves. Each ofthese activities cause the star to spin down over time which will reduce the frequency of
any emitted gravitational waves. This evolution of the spin period turns out to be large
enough that it needs to accounted for during any gravitational wave searches of reasonable
length. It also gives insight into how the parameter space of ellipticity, frequency, andemitted gravitational wave strength is populated by isolated neutron stars. Neutron stars
with companions can undergo spin up via accretion and will tend to an equilibrium state
between the spin up and the spin down mechanisms, thus the following arguments do not
apply in such cases. In this section we will derive equations describing the spin down andthen apply them to the case of gravitational radiation.
Let us assume an isolated neutron star has a moment of inertia I and angular frequencyΩ so that we can write the kinetic energy as
1Ekin = IΩ2 .
2

(5.4)

If we take the time derivative of the energy we haveĖkin = IΩΩ̇ .

(5.5)

If we assume that all of the energy loss is due to a single emission mechanism, we canconsider a general relationship between the rate of change in the angular frequency, Ω̇, andthe angular frequency, Ω, of the form89Ω̇ = KΩn(5.6)

where K is the torque function which contains all the physics of the torque causing the spin
down and n is the braking index. The braking index has a value of 3 for spin down due todipolar magnetic radiation and a value of 5 for spin down due to gravitational radiation.
This can be seen from the fact that the power emitted in dipolar magnetic radiation goesas the 4th power of frequency while the power in gravitational quadrupole radiation goes as
the 6th power of frequency. Eq. (5.5) relates this power emission to Ω and Ω̇ to get n.
By taking the time derivative of Eq. (5.6) we can solve explicitly for the braking index
in terms of observables Ω, Ω̇, and Ω̈. The braking index is then just [43]

n=

Ω̈Ω
.
Ω̇2

(5.7)

If such values are known, by observation of radio pulsations for example, it gives us insight
into what mechanisms might be causing the spin down.
In the case of unknown frequency and spin down rate, we can find a relation between thefrequency and spin down if we know the approximate age of the neutron star and assume
the ellipticity is constant. By using Eq. (5.6) we can solve for Ω as a function of time.
Moving all the factors of Ω and Ω̇ to one side, we can integrate with respect to time withthe boundaries set at zero, the time of initial angular frequency Ω0 , and the current time
tage , which corresponds to angular frequency Ωt . This is written as- tage

Kdt =

0

- Ωt

dΩ
Ωn

2

Ωt

Ω0

(5.8)

and after performing the integration we get

Ktage =

−1(n−1)(n − 1)Ωt

1−

(n−1)
(n−1)

Ω0

3

.

(5.9)

If we assume that the current rotation frequency is small compared to the initial rotation
frequency, Ωt << Ω0 , we can write a simple relation between the current frequency, thecurrent rate of frequency change, and the approximate time since the neutron star wasborn (also known as the spin down age). This approximation is good to 1% in the case ofgravitational radiation, the n = 5 case, when Ωt /Ω0 < 0.3. Using this approximation and90

the fact that K = Ω̇/Ωn , from Eq. (5.6), we solve for tage to get
7 7
7 7
1 77 f 77
1 77 Ωt 77=.
tage =
n − 1 7 Ω̇t 7 n − 1 7 f˙ 7

(5.10)

We can now apply the above derivations to the specific case of gravitational radiation.
We know that the gravitational wave luminosity is equal to [40]Lgw =1 ... ...uv- I uv I . .
5(5.11)We already know I¨uv for a non-axisymmetric neutron star with some ellipticity ) from Eq.
(1.13) so we can substitute it in and simply get
Lgw =32 6 2 2Ω I ) .
5

(5.12)

If we make the assumption that spin down is due only to gravitational wave emission
we can simply set the energy loss, Ėkin equal to the gravitational luminosity, Lgw . This isa best case assumption as then all the energy leaving the star is in a form we can detect. Ifother energy loss processes are occurring this will increase the energy loss and spin down,
resulting in slower spinning stars and weaker signals. Having made this assumption, we cancalculate an upper limit on the gravitational wave strain at Earth if we know the distance,
frequency and spin down rate, such as for the Crab pulsar [43]. This is done in Section 5.6.
In the case of unknown frequency and spin down rate, we can use Eq. (5.13) to limit
the range of parameter space that needs to be consdiered if the age of the neutron star is
known. As can be seen by comparing Eqs. (5.5), (5.6), and (5.12), the braking index n isequal to 5 for gravitational radiation. This lets us rewrite Eq. (5.13) as
7 7
7 7
1 77 Ωt 77 1 77 f 77
tage = 7 7 = 7 7
4 Ω̇t
4 f˙

(5.13)

for the gravitational case. This relationship is useful in setting up searches for gravitational
waves from isolated neutron stars with partially known parameters. If we know an approximate age we can estimate the largest possible strain reaching a detector here at earth acrossall frequencies and what ellipticity of the neutron star makes it possible.

91

5.4

Differences between GW and EM pulses

For targeted searches with radio pulse data, such as the Crab pulsar, it is important toconsider physical situations in which the gravitational wave frequency might differ from theelectromagnetic frequency or multiples of it. If the frequencies are not exactly matched,then a wider search may be necessary to cover the parameter space sufficiently, so as not
to miss the true frequency and spin down of the gravitational radiation.

5.4.1

Free precession

The case of steady rotation, when the principle axis and the inertial axis I3 are aligned wasintroduced in the Chapter 1. This results in gravitational radiation at twice the rotation
frequency, ΩGW = 2Ω. In this case, a hot spot which is producing the radio pulses will
come around once per rotation of the pulsar, producing a pulse frequency of ΩEM = Ω. In
this simple case, we only need to look at exactly twice the electromagnetic pulse frequency.
If the rotation axis of the Crab pulsar is not aligned with the principle axis, it could
precess like a top [56] [30]. Let us consider the slightly more complex situation of biaxial
free precession. In this case, I1 = I2 < I3 . We define a constant angle θ between the body’s−→
third principle axis x3 and the total angular momentum vector J axis of rotation, which
has an inclination angle i relative to the direction of the observer as shown in Figure 5-1.
We then have two rotation periods described by
J
I1(5.14)

I3 − I1
φ̇
I1(5.15)φ̇ =andψ̇ = −

where φ̇ represents the inertial space x3 axis angular velocity and ψ̇ represents the body
frame precessional angular velocity.
In this case gravitational waves can be calculated to have the formh+ =+2I1 φ̇2 ) sin(θ) *(1 + cos2 (i)) sin(θ) cos(2φ̇t) + cos(i) sin(i) cos(θ) cos(φ̇t) ,r

92(5.16)

Figure 5-1: Diagram of a neutron star with a bump on one side. The angle θ is theangle between the I3 moment of inertia and the angular momentum vector. The red circle
represents the location where the EM pulse emission is being generated. The angle λis the angle between angular momentum vector and the point on the surface where theEM emission occurs. The angle i is the inclination angle, the angle between the angularmomentum vector and the vector towards the detector.

and
h× =

+
2I1 φ̇2 ) sin(θ) *cos(i) sin(θ) sin(2φ̇t) + sin(i) cos(θ) sin(φ̇t)
r(5.17)

with ) = (I3 − I1 )/I1 and r being the distance to the source.

However, the electromagnetic pulses have a more complicated frequency in this case.

Let us define the angle between the pulsar beam and the x3 axis as λ. If in general λ 0= 0

then there will be modulation of the electromagnetic pulse frequency.

When the angle θ is larger than the angle λ, the pulse would be seen when the x3 pulsar
axis passes close enough to the observer’s line of sight. During the body’s precession time
the observer would pass through the pulsar radiation beam from many different directions.

93For the precession to be invisible, the pulsar beam would have to both be nearly axisymmetric and could not have any observable linear polarization. Any net linear polarization
would rotate through 360◦ during a precession time, since the orientation of the electromagnetic emission source would rotate completely around the body axis. The electromagnetic
This effect has not be been observed [56]. In this case, the average electromagnetic pulse
frequency is just φ̇. This can be seen from Figure 5-2. The rotation about the star’s I3
axis primarily changes the orientation of the source while the rotation about the x3 inertial
frame axis brings the pulse around like a light house. For this case, we can write

θ > λ ⇒ -ΩEM . = φ̇ .

(5.18)

The angle λ being larger than the angle θ is more likely to be occurring without producing observable effects in the radio timing. In this case the mean electromagnetic pulse
frequency is the sum of precession angular velocity, ψ̇, and the inertia space x3 angularvelocity, φ̇. This can be seen from Figure 5-3, where the electromagnetic source point isbeing rotated about the star by both the inertial space angular velocity and the angularvelocity about the I3 axis. The observer always passes through the pulsar beam from approximately the same direction, so no significant changes in pulse profile or polarization
would be expected. A simple knife beam model of the pulsar radiation pattern gives theresult for small θ that during a precession time pulses arrive early and late by a phaseof up to θ/ tan(λ), with sinusoidally varying phase shift. We can write the time averaged
electromagnetic pulse frequency for this case as

θ < λ ⇒ -ΩEM . = φ̇ + ψ̇ .
By comparison, the gravitational wave emission would be observed at φ̇ and 2φ̇.

94

(5.19)

Figure 5-2: Exaggerated diagram of a biaxially symmetric neutron star with λ < θ. The
electromagnetic emission pulse frequency is determined by the angular velocity φ̇, as thestar rotates around the inertial space axis x3 . The rotation about the I3 moment of inertia
axis of the star will cause small variations in that pulse frequency, but which average out
over many precession cycles. The rotation about the I3 axis will also result in a changing
orientation, which if linear polarization is present, will result in that polarization rotating
through 360◦ over the course of a single precession.

95

Figure 5-3: Exaggerated diagram of a biaxially symmetric neutron star with θ < λ. The
inertial space x3 angular velocity, φ̇ and the body axis angular velocity, ψ̇, on average add
together to determine the total angular velocity of the electromagnetic wave source. Thisis because the rotation vectors are close and the electromagnetic source never passes in
between the x3 and I3 axes. The rotations are in the same direction in this case, where asin the case shown in Figure 5-2, the body axis angular velocity adds as often as it subtracts
from the overall angular velocity, which when averaged over time doesn’t modify the φ̇rotation rate. This also means the electromagnetic source is always in approximately thesame orientation when facing an observer. It can not flip over, relative to this diagram, bypassing in between the x3 and I3 axes.

96

In the more complicated case of triaxial precession, where I1 < I2 < I3 , one needs tomake some assumptions to be able to determine approximate waveforms. If we assume
θ << 1, which is just a small wobble angle, (I3 − I1 )/I1 << 1, which is small oblateness,and (I2 − I1 )/I1 << (I3 − I1 )/I1 , which means the non-axisymmetry is small compared tothe oblateness, then the wave form can be approximated by [55]h ∼ (I2 − I1 ) cos(2(φ̇ + ψ̇)t)+ ∼ (I3 − I1 )θ cos(φ̇t)+ ∼ (I3 − I1 )θ2 cos(2φ̇t) .

(5.20)

Each term in this equation is an approximation with different prefactors being neglected,
hence the somewhat unusual usage of ∼, where x+ ∼ y means x plus terms which look likey. From Eq. (5.20) we can see that we have gravitational wave emission at the frequenciesof 2(φ̇ + ψ̇), φ̇ and 2φ̇. For electromagnetic pulses we expect the same as in Eq. (5.19) andEq. (5.18).

5.4.2

Two component model

Some neutron stars are observed to have sudden increases in pulsation frequency, which areusually called glitches [37]. These spin ups generally decay away on various time scales.
One possible explanation for this is that pulsars consist of two components, such as acrust component and a core component, rotating at different rates. For the Crab and a
few other young pulsars approximately a fraction of 10−5 of the spin-down is reversed
during the glitch. The decay time varies on the order of days to months. We can use thisinformation to produce an estimate of the maximum frequency difference between the twocomponents and thus between the electromagnetic and gravitational frequencies, since theelectromagnetic pulse frequency will be tied to the crust while the gravitational frequency
could be tied to the core.
One way to approach this information is with a coupling-torque model. First for thecrust we define its rotation frequency f1 and its moment of inertia I1 . For the core we
similarly define its rotation frequency f2 and its moment of inertia I2 . If we allow the spin
down torque to act on just one component, say the crust, and couple the two components
with a friction-like torque we can write [29]97I1 f˙1 = −α(f1 − f2 ) − T(5.21)

I2 f˙2 = α(f1 − f2 )(5.22)

andwhere α parameterizes the strength of the coupling torque and T > 0 is the spin down
torque. The spin down torque T effectively is I Ω̇ from section 5.3, with the exact evolution ofΩ̇ dependent on whether the slow down is due to electromagnetic or gravitational radiationor perhaps some other mechanism. One can show that the steady state lag, %f , is then
T τc
I1

(5.23)(−1(5.24)%f = f1 − f2 = −
where

1
τc =
α

'

1
1
+
I1 I2

and is the coupling timescale on which transients are damped out. This could be expected
to come into play after a glitch occurs.
We now use the results of Section 5.3 to estimate a spin down age. This is the timescaleon which the pulsar slows down. We begn by using Eq. (5.13) and noting that the torqueT is T = 2π(f˙1 I1 + f˙2 I2 ). We define a new f such thatf=

I1 f1 + I2 f2.
I1 + I2(5.25)We then note that the torque T is T = 2π(f˙1 I1 + f˙2 I2 ) = 2π f˙(I1 + I2 ). We solve for f˙ andsubstitute into Eq. (5.13). This then lets us write the steady state spin down timescale asτss =1 2πf (I1 + I2 ).
n−1
T

(5.26)

This is just the timescale on which the pulsar takes to spin down to its current frequencyof f from some sufficiently fast initial frequency, as described in section 5.3.
We now use Eq. (5.26) to substitute for the torque in Eq. (5.23) and rewrite it as%f = −

1 2πf τc I1 + I2.
n − 1 τssI1

98

(5.27)

This relates the lag between the two components to two observed timescales, the glitch
recovery timescale and the spin down age. The steady state spin down age can be approximated by the age of the neutron star, which in the case of the Crab pulsar is 103 years.
Since we observe the neutron star returning to its original spin down rate on a timescale ofweeks or months, it is believed that this is also the timescale associated with coupling of
the two components.
We identify the torque T with the electromagnetic torque. There would be a gravitational wave torque, but we assume the electromagnetic dominates. Given that glitches in
the Crab pulsar always spin it up, this seems like a reasonable assumption. If we included
the effects of a gravitational wave torque in the above equations, it would simply make thelag between the two frequencies smaller.
Assuming I1 is of the order or larger compared to I2 makes (I1 + I2 )/I1 ) of order 1,
which is reasonable since we are associating component 1 with the crust [37]. In addition
as the difference between the two components is small compared to frequency of the crust’s
rotation, we can use f1 as a good approximation to the frequency f . Using all of the above
facts, plus the Crab pulsar’s braking index of n = 2.5, we find an upper estimate of %f of

3×10−3 for the Crab pulsar, which is in fact small compared to its 29.77 Hz pulse frequency.

5.5

Application to RX J1856.5-3754

In the specific case of RX J1856.5-3754, we did not have a known frequency or spin down
values. However we did have a reasonable estimate of the age, somewhere between 105 and106 years old, and its distance of about 120 pc. By equating the age of the star with its
spin down age, and assuming a typical moment of inertia, I = 1038 kg m2 , we can solvefor epsilon as a function of the neutron star’s possible current frequency, to determine if its
consistent with reasonable neutron star models. Similarly, we can calculate the best casestrain at any given current frequency to determine if its detectable within the LIGO band.
To begin with, we replace Ω̇ in Eq. (5.6) with (−1/4)(Ωt /tage ). We then solve for ) and
get

)=

'

5
1
128 tage IΩ4(12∝ Ω−2 .

(5.28)

Since ) goes as Ω−2 this will cancel with the normal frequency dependance to the 2nd power99

, so that the final maximum strain is not dependant on the frequency. Using Eqs. (5.28)
and (1.15) we can write the h+ and h× strains as a function of Ω to get|h+ | =
and

1
R

'

1
|h× | =
R

5 I
2 tage

(1

'

(1

5 I
2 tage2/20
1 + cos2 (i)(2 cos(i)) .

(5.29)(5.30)In the case of pure gravitational spin down, the strain is only a function of age, irregardless ofthe current spin frequency. In Figure 5-4 the ellipticity is plotted versus potential frequency
at three different possible ages for RX J1856.5-3754, along with the expected strain at eachage. In these plots, it is assumed that the neutron star has a typical 1038 kg m2 momentof inertia and the signal has been averaged over possible inclination angles. Also factors ofG, the gravitational constant, and c, the speed of light, were reintroduced to get the units
into more traditional form. Similarly in Figure 5-5 we plot the 1st spin down value, whichis the first derivative of frequency with respect to time, versus current frequency. In thenext chapter, we will look at the detectability of these strains.

100

−3

10

106 years old
5x105 years old
105 years old−4

Ellipticity

10

−5

10

−6

10

−7

10

50

12

100

150

200

250

300

Frequency [Hz]

350

400

500

−25

x 10

106 years old
5x105 years old
105 years old

10

h0

450

8
6
4
50

100

150

200

250

300

Frequency [Hz]

350

400

450

500

Figure 5-4: The top plot shows the expected ellipticity if all of the spin down of RX
J1856.5-3754 was due to gravitational radiation for three different possible ages. The lower
plot shows the best case strain of the gravitational waves reaching Earth. Other spin downmechanisms at work in addition to gravitational radiation would move the ellipticity lines
down, requiring less ellipticity but at the same time, reducing the strain reaching the Earth.

101

8

−11

x 10

106 years old
5x105 years old
105 years old

7

Spin down [Hz/s]

6
5
4
3
2
1
0
0

200

400

600

Frequency [Hz]

800

1000

Figure 5-5: The expected 1st spin down value if all the spin down was due to gravitationalradiation as a function of current frequency. The younger ages have higher spindown since
the neutron star will have have had less time to spin down from its initial frequency. To
reach the same current frequency as an older age estimate, but in less time, it must be
spinning down faster.

5.6

Application to the Crab pulsar

The Crab has been observed enough by radio telescopes that we know its frequency f , its
1st spin down parameter, which is df /dt, and also its 2nd spin down parameter, which is
d2 f /dt2 . The Crab has a frequency of 29.77 Hz, a 1st spin down of −3.73 × 10−10 Hz s−1 ,and a 2nd spin down of 1.24 × 10−20 Hz s−2 , taken from the Jodrell Pulsar Timing Bank

near the beginning of LIGO’s fifth science run [28]. So far we have been working in units
where c = G = 1, but it is easy to convert to more conventional units for the following
calculations. If we assume the Crab pulsar has a typical moment of inertia of I = 1038
kg m2 we can calculate the necessary ellipticity ) needed to cause the spin down seen.
Ω̇ = −3.7294045 × 10−15 Hz/s = −102

32 G 2 2 5
I ) Ω
5 c5

(5.31)

Solving for ) gives)=2−5 c5 Ω̇ 1
32 G Ω5 I

31
2(5.32)

and a value of 7.5 × 10−4 , again assuming a standard I = 1038 kg m2 [43]. It then becomesa simple matter to calculate the strain using Eq. (1.15) here at Earth if we know the Crab’s
distance, which is about 2 kpc. This provides an upper limit on the strain of 1.4 × 10−24 .
It is interesting to note that while we have been assuming a moment of inertia I = 1038

kg m2 there have been suggestions that the Crab’s moment of inertia could be up to threetimes larger than this typical value [7] [34]. This would in decrease the necessary ellipticity
to explain the spin down by a factor of I 1/2 , down to 4.3 × 10−4 , and increase the upper
limit on the strain by the same factor, up to 2.4 × 10−24 .

We also have information for the 2nd spin down value and can use it to calculate thebraking index, n. When we calculate it value from Eq. (5.7) we come up with somethingclose to 2.5, rather than the 5 of gravitational emission, or even the 3 of dipolar magnetic
radiation. This indicates that some additional energy loss mechanisms must be occurring.
In 2000 Palomba published a paper using the braking index of 2.5, along with the ageof the Crab, to derive upper limits on the ellipticity and gravitational wave emission. He
began by assuming the total spin down,Ω̇ was due to two componentsΩ̇ =Ω GW + Ωother(5.33)

where ΩGW is due to gravitational radiation and has a braking index of nGW = 5 while theΩother component is due to some unknown emission and which has a braking index notherwhich is allowed to vary below 3.
He then defined a ratio

Y (Ω) =Ω̇GW
KGW (5−nother )=Ω.
KΩ̇otherother(5.34)

By solving for Ω̇other in terms of the ratio Y (Ω) and substituting into Eq. (5.33) he wroteΩ̇ = KGW Ω5'

1 + Y (Ω)
Y (Ω)((5.35)

which can be solved numerically to produce a set of solutions for Ω. This set of solutions

103

depend on the ellipticity, which is contained in KGW , on nother and on the initial angularvelocity Ω0 . By using the known age of the Crab and choosing the solution with the smallest
nother consistent with an overall n = 2.5, he was able to arrive at an upper limit on ellipticityof 3 × 10−4 . This corresponds to an upper limit on h0 of 5.5 × 10−25 [43].

104Chapter 6

Search methods
This chapter describes several continuous gravitational wave search methods and discusses
their merits in directed searches.

6.1

Search codes and constraints

When looking for a continuous wave gravitational source there are several limiting factors
which determine the final sensitivity one can achieve. The fixed constraints include thequality and quantity of data available, and also the amount of computation power available.
These are combined with the size of the parameter space, which includes position in the sky
and frequency parameters of the source, to determine the final sensitivity. Larger searches
in parameter space necessarily turn up more outliers due to noise, which in turn degrades
any upper limits that might be made from them. Thus, any continuous wave search must
weigh these different constraints to determine the optimal method of search to be used.
One can use coherent or incoherent methods to sift the data for a weak, but consistently
present source. A coherent search will create a signal template with the known parameterseof the source and convovle it with the data. This will produce the most sensitive search since
one is looking for the signal with an exact match and utilizes all the data available [19] [27].
An incoherent search will break the data up into shorter time segments and Fourier transform them with the fast Fourier transform algorithm (FFT) and then add up the power at
individual frequencies from each data segment. These shorter transformed data chunks areusually referred to as short Fourier transforms (SFTs) [39]. Since phase information is lost
between segments, the sensitivity is necessarily worse, but gains in computational speed.
105

6.2

Detector response function

In order to estimate our final sensitivity, it is necessary to understand how an incoming
gravitational wave interacts with the our detector. We will follow the derivation by Jaranowski, Krolak and Schutz in [27]. This will then factor in to our calculation of overallsensitivity of the LIGO detectors to RX J1856.5-3754 and the Crab pulsar.
The dimensionless detector’s response function h is defined as the difference between the
wave induced relative length changes of the two interferometer arms and can be computedfrom the following formula [27]
*
+ 1
*
+
1
h(t) = n̂1 · h̃ab (t)n̂1 − n̂2 · h̃ab (t)n̂2
2
2

(6.1)

where n̂1 and n̂2 are the unit vectors parallel to the interferometer arms. The order ofarms is defined such that n̂1 × n̂2 points outwards from the surface of the earth. h̃ab is the

3-dimensional matrix of the spatial metric perturbation produced by the wave in the proper
reference frame of the detector, or in other words, just the spatial components of hµν as
defined in Eq. (1.5) but transformed to the detectors frame. We can compute h̃ab byh̃ab = M (t)hab (t)M (t)T

(6.2)

where M is the 3-dimensional orthogonal matrix of transformation from the wave Cartesian
coordinates, written (xw , yw , zw ), to the Cartesian coordinates in the detector’s proper
reference frame, written (xd , yd , zd ), and T indicates transposition of the matrix. If wechoose the direction of the gravitational wave propagation as the +zw direction we canwrite in the wave’s frame that
h+ (t)h× (t)

0







hab (t) = h× (t) −h+ (t) 0 ,


0
0
0

(6.3)

where we have absorbed all the time dependence and distance factors into h+ (t) and h× (t),which represent the two independent wave polarizations. If we consider Eqs. (6.1) through(6.3) it becomes apparent that the detector’s response function h is a linear combination of
the functions h+ (t) and h× (t). This lets us write106h(t) = F+ (t)h+ (t) + F× (t)h× (t)(6.4)

where F+ and F× are called the beam-pattern functions.
The Earth’s rotation rotates the detectors along with it causing the beam-patterns F+
and Fx to be periodic functions of time with a period equal to one sidereal day. Ideally
we want to determine the time dependence of F+ and F× and also their dependence on
the source’s position and orientation, in terms of right ascension α, declination δ, and thewave’s polarization ψ.

Figure 6-1: Diagram of a Interferometer located on the Earth’s surface. Ωr is the Earth’s
rotation rate, λ is the latitude, and φr is a reference phase which defines the position of the
Earth in its diurnal motion at t = 0. The inset shows a pair of interferometer arms as if
looked down on from above. The angle ζ is the angle between the arms (90◦ in the case of
LIGO) while γ is the angle between East and the arm bisector.

We can write the matrix M as
M = M3 M2 M1T

107

(6.5)

where M1 is the transformation matrix from the wave frame coordinates to the celestial
frame coordinates, M2 is the transformation matrix from celestial coordinates to the cardinal coordinates and M3 is the transformation matrix from cardinal coordinates to thedetector proper reference frame coordinates. The definition of these coordinate systems are
as follows: In the celestial sphere coordinates the z axis coincides with the Earth’s rotation
axis and points towards the North pole, and the x and y axes lie in the Earth’s equatorial
plane, with the x axis pointing towards the vernal point. In cardinal coordinates the (x, y)plane is tangent to the surface of the Earth at the detector’s location with the x axis alignedin the North-South direction and the y axis aligned in the West-East direction, and the z
cardinal axis along the Earth’s radius pointing outwards from the surface of the Earth. In
detector coordinates the z axis coincides with the z axis of cardinal coordinate and the xaxis is along the first interferometer arm. With these definitions we can write M1 , M2 , andM3 as follows
sin α cos ψ − cos α sin δ sin ψ
M1 = − sin α sin ψ − cos α sin δ cos ψ− cos α cos δ− cos α cos ψ − sin α sin δ sin ψ
cos α sin ψ − sin α sin δ cos ψ− sin α cos δ

sin λ cos(φr + Ωr t)

sin λ sin(φr + Ωr t) − cos λ
M2 =  − sin(φr + Ωr t)cos(φr + Ωr t)

cos λ cos(φr + Ωr t) cos λ sin(φr + Ωr t)


0

sin λ


0




M3 = − cos(γ + ζ/2) − sin(γ + ζ/2) 0 .


0
0
1
− sin(γ + ζ/2)







cos δ sin ψcos δ cos ψ − sin δ
(6.6)

(6.7)

cos(γ + ζ/2)

(6.8)

In the above equations λ is the latitude of the detector’s site, Ωr is the rotational angularvelocity of the Earth, φr is a deterministic phase which defines the position of the Earth init diurnal motion at t = 0, such that φr + Ωr t coincides with the local sidereal time of thedetector’s site, γ is the angle measured counter-clockwise from East to the bisector of theinterferometer arms, and ζ is the angle between the interferometer arms. These angles areshown in Figure 6-1 for reference. The vectors n̂1 and n̂2 in the detector’s reference frame108are n̂1 = (1, 0, 0) and n̂2 = (cos(ζ), sin(ζ), 0).
By using Eqs. (6.1)-(6.8) and the relevant definitions we can algebraically determine
the following expressions for F+ (t) and F× (t):F+ (t) = sin(ζ)[a(t) cos(2ψ) + b(t) sin(2ψ)](6.9)

F× (t) = sin(ζ)[b(t) cos(2ψ) − a(t) sin(2ψ)](6.10)

and

where
1
sin(2γ)(3 − cos(2λ))(3 − cos(2[α − φr − Ωr t])
16
1− cos(2γ) sin(λ)(3 − cos(2δ)) sin(2[α − φr − Ωr t])
4
1+ sin(2γ) sin(2λ) sin(2δ) cos(α − φr − Ωr t)
4
1− cos(2γ) cos(λ) sin(2δ) sin(α − φr − Ωr t)
2
3+ sin(2γ) cos2 (λ) cos2 (δ)4a(t) =(6.11)

andb(t) = cos(2γ) sin(λ) sin(δ) cos(2[α − φr − Ωr t])+

1
sin(2γ)(3 − cos(2λ)) sin(δ) sin(2[α − phir − Ωr t])
4(6.12)+ cos(2γ) cos(λ) cos(δ) cos(α − φr − Ωr t)+

1
sin(2γ) sin(2λ) cos(δ) sin(α − φr − Ωr t) .
2

With the explicit equations for the F+ (t) and F× (t) in hand, it is possible to look at the
response of the detectors to an average source. The averages of F+2 (t) and F×2 (t) are both

equal to 15 , when averaged over right ascension, declination, polarization, and integrating
for sufficiently long periods (longer than a few days).

109

6.3

Signal form in terms of Doppler and amplitude parameters

We can write a general, elliptically polarized gravitational wave in the reference frame ofthe source as [47]h+ (τ ) = A+ cos(Φ(τ ))(6.13)

h× (τ ) = A× sin(Φ(τ ))where A+ and A× are amplitudes containing the detector’s response function that depends
on the wave’s polarization and source inclination angle and τ is the time at which the wave
left the source. If we assume a quasi-monochromatic signal with slowly-varying intrinsic
frequency f (τ ), the signal phase Φ(τ ) can be Taylor expanded as

Φ(τ ) = φ0 + φ(τ ), φ(τ ) = 2πs5f (k) (τref )k=0(k + 1)!

∆τ k+1(6.14)

where ∆τ ≡ τ − τref and τref is the reference time at which the initial phase φ0 and thes + 1 spin parameters f (k) ≡ dk f (τ ) are defined, i.e. the 1st spin down parameter, the 2nd
spin down parameter, etc.

From an Earth based detector’s frame the signal amplitude of an isolated neutron staris modulated by the time dependent antenna patterns F+ (t) and F× (t) and also Dopplermodulated by the relative motion of the detector with respect to the source. We can writea relation between the detector arrival time t of a wave-front that left the source at timeτ (t). If we consider a neutron star at sky position n̂, defined in equatorial coordinates asn̂ = (cos δ cos α, cos δ sin α, sin δ) in terms of right ascension α and declination δ, and at a
distance d(t) then we can writeτ (t; n̂) = t + ∆$ (t; n̂) −d(t)c(6.15)

where∆$ (t; n̂) =r(t) · n̂+ ∆E$ (t; n̂) + ∆S· (t; n̂)c110(6.16)

and where r(t) is the vector from the solar system barycenter to the detector on Earth.
Here ∆S$ is the general-relativistic Shapiro delay caused by the gravitational field of thesun and ∆E$ is delay caused by the gravitational redshift and time dilation at Earth [49].
Both of these effects are much smaller than the r(t)·n̂
Roemer-delay term, which is simplycthe difference in the propagation time of the wave to reach the solar system barycenter andthe detector. [47]. In the case of searching for a signal, the time dependent distance d(t) is
typically treated as constant and neglected, which involves no loss of generality since any
effects can be absorbed into the frequency evolution of the source. We can now write themeasured strain signal h(t) ash(t; A, λ) = F+ (t; n̂, ψ)A+ cos(φ0 + φ(t; λ))+ F× (t; n̂, ψ)A× sin(φ0 + φ(t; λ))(6.17)

where we define the set of four “amplitude parameters” as A ≡ {A+ , A× , ψ, φ0 } and we

define a set of “Doppler parameters” λ ≡ {f (k) , n̂}.

Using 6.9 and 6.10 we can separate h(t)’s dependence on the amplitude and Doppler

parameters and writeh(t; A, λ) =45Hµ hµ (t; λ)(6.18)µ=1

in terms of four basis waveforms

h1 (t; λ) = a(t; n̂) cos[φ(t; λ)]h2 (t; λ) = b(t; n̂) cos[φ(t; λ)]h3 (t; λ) = a(t; n̂) sin[φ(t; λ)]h4 (t; λ) = b(t; n̂) sin[φ(t; λ)]and the amplitude vector H µ is defined as111(6.19)

H 1 = A+ cos(φ0 ) cos(2ψ) − A× sin(φ0 ) sin(2ψ)
H 2 = A+ cos(φ0 ) sin(2ψ) + A× sin(φ0 ) cos(2ψ)
H 3 = −A+ sin(φ0 ) cos(2ψ) − A× cos(φ0 ) sin(2ψ)
H 4 = −A+ sin(φ0 ) sin(2ψ) + A× cos(φ0 ) sin(2ψ) .

(6.20)In the case of a non-precessing triaxial neutron star, as described in Section 1.2, theamplitudes A+ and A× can be written as
1
A+ = h0 (1 + cos2 (i)), A× = h0 cos(i),2(6.21)

where h0 is the overall amplitude. From Section 1.2, it is 64π 2 f 2 )I33 /R. In more traditional
units and using fGW = 2f (i.e. looking at a frequency twice the rotation rate) we can rewrite
this ash0 =2
4π 2 G )I33 fGW.
c4
R(6.22)It should be noted that for other types of continuous wave generation, different relations
hold between the rotation frequency and the gravitational wave frequency. For example,
free precession emits near the rotation frequency and twice the rotation frequency. However,
the above representations of A+ and A× can be used without loss of generality since themapping between {h0 , cos(i)} and {A+ , A× } is one to one.

6.4

The F statistic: a coherent search method

The use of the F statistic as a search method was developed and first presented by Jaranowski, Krolak, and Schutz in [27]. The search code was developed by the LIGO Continuous
Wave pulsar group [46]. I will present a short summary here of the method.
One method to extract that signal is the method of maximum likelihood detection. A
key part of this method is the likelihood function Λ, a kind of probability distributionthat expresses the likelihood that our data is described by a particular model or set ofparameters. By finding the signal parameters which maximize Λ, by taking the derivative112

of Λ with respect to those parameters and setting it equal to zero to find the extrema, we
effectively find the best match in the data to our possible signal. The signal parameterswhich maximize Λ are called the maximum likelihood estimators. The maximum of Λ
determines the probability of detection of the signal. If the maximum of Λ exceeds acertain threshold determined by the false alarm probability we can afford, we say the signal
is detected.
We begin by making the assumption that the noise in the detector is an additive, stationary, Gaussian and zero-mean process. Then the data x we get from the detector whena signal h is present is

x(t) = n(t) + h(t)(6.23)

where n(t) is our noise. The log likelihood function has the form1log Λ = (x|h) − (h|h)
2(6.24)

where the scalar product (•|•) is defined by(x|y) ≡ 4*

- ∞0x̃(f )ỹ ∗ (f )df
Sh (f )

(6.25)

where ∼ denotes a Fourier transform, ∗ denotes complex conjugation, and Sh is the one

sided spectral density of the detector noise.

The gravitational wave signal consists of a narrowband component around the frequency
f , twice the rotational frequency. We can assume that over the bandwidth of the signal
Sh (f ) is nearly constant, stationary, and equal to Sh (f0 ) where f0 is the frequency of thesignal h at t = 0. With this assumption the detector noise term can be factored out of theinner product integral. By using the defintion of the Fourier transformation, we can change
the frequency domain intergral of Eq. (6.25) into a time domain integral, a more convienient
form for periodic source analysis. Thus the above scalar product can be approximated by(x|h1 ) ∼=2
Sh (f0 )- T0 /2x(t)h(t)dt(6.26)−T0 /2where T0 is the observation time starting at t = −T0 /2. We introduce a new scalar product,113

(x||y) ≡2
T0- T0 /2

x(t)y(t)dt ,(6.27)−T0 /2

without the detector noise term and normalized by the observation time. We instead include
these terms explicitly in the log likelihood function. The log likelihood function for thissignal with this new scalar product is approximately given bylog Λ ∼=

89
T0
1(x||h) − (h||h) .
Sh (f0 )2(6.28)The maximum likelihood estimators can be found by maximizing the following normalized
log likelihood function
1
log Λ&1 = (x||h) − (h||h) .
2(6.29)The normalized log likelihood function has had its explicit dependence on the spectral
density of the noise in the detector removed.
The signal h depends linearly on the four amplitudes H µ . These in turn depend on
the parameters h0 ,θ,ψ,i, and φ0 and are independent. In a search where not all of theseparameters are known, we must find the maximum of log Λ as a function of the unknown
parameters.
If we insert Eq. (6.18) into Eq. (6.29) we can write
log Λ& (x; A, λ) = Aµ xµ −1 µ ν
A A Mµν2(6.30)

wherexµ ≡ (x||h/mu )(6.31)Mµν (λ) ≡ (hµ ||hν ) .

(6.32)

andWith these we can now maximize logΛ over Aµ to obtain the maximum likelihood estimators
AµM L from the data x(t) by∂ log Λ&= 0 =⇒ AµM L = M µν xν
∂Aµ

114

(6.33)

where M µα Mαν = δνµ . Placing these maximum likelihood estimators back into Eq. (6.30)
we obtain a new partially maximized detection statistic

2F (x; λ) = xµ Mµν xν(6.34)

which is called the F-statistic. Since we have maximized over the amplitude parameters,
the F-statistic only depends on the Doppler parameters λ. In the presence of a signal, theexpectation value of the F-statistic with perfectly matched Doppler parameters, λ = λsig ,is given by 4 + ρ2 , where ρ is the optimal signal to noise ratio. It is shown in [27] that
ρ=and that this scales as √hS0

h

√

.

(h|h)

(6.35)

Tobs for a single detector. One can show, again in [27] that 2F

is a random variable with a χ2 distribution with 4 degrees of freedom and a non-centrality
parameter ρ2 . The probability density function of this is12
p1 (F ) = e−(F +ρ )/2
2

4

.
F
I
(
(F ρ2 ))
1
ρ2

(6.36)

where I1 is the modified Bessel function of the first kind of order one. In the absence of
signal, when ρ = 0, this reduces to a central χ2 distribution

p0 (2F ) =

F −F
e
.
2

(6.37)

With these probability distributions we can compute the false alarm and false dismissal
probabilities for a threshold 2Fc and a signal to noise ρ. The false alarm probability issimply the integration 6.37 from the threshold 2Fc to infinity. In a Frequentist framework
it represents the limiting fraction of 2F values which exceed the threshold, given an infinite number of identical trials on Gaussian, stationary, white noise. We can perform thisintegration to get

PF A (2Fc ) =- ∞p0 (2F )d(2F ) =2Fc(1 + Fc ) −Fce.
2(6.38)For example, if we choose a 1% false alarm rate for a single template search, the necessary
cutoff is 2Fc ≈ 13.3 [46]. The false dismissal rate is the integration of 6.36 from negative115

infinity to the threshold 2Fc for the given signal to noise ρ. In general this needs to be
integrated numerically. As an example, if we use the cutoff derived from a 1% false alarmfor a single template search and require a false dismissal rate of 10%, we find a ρ ≈ 4.5.

6.5

Coherent vs incoherent templates and unknown parameters

When determining what type of search to perform, it is important to consider how thecomputation costs scale and the relative gains in sensitivity for that cost. The computation
cost is most easily expressed in the number of templates needed to cover a given parameter
space for a given type of search. Searching each template, which defines the sky postion
and frequency evolution of the source to be looked at, takes the same amount of time, andthus the computation scales directly with the number of templates.
We also need to consider the sensitivity scaling of the search, which is a function of thecoherence time, Tcoh , which is effectively the length of time over which we integrate, keeping
track of all the parameters of the signal including both phase and power. In the case ofa fully coherent search, this Tcoh is equal to the total amount of data analyzed, Ttotal . In
the case of an incoherent search, where short segments of data are converted via FFT andthen combined ignoring phase, Tcoh is the length of the individual segments. In that case,Ttotal = N Tcoh , where N is the number of segments.
As we integrate, the mean power in the noise goes as 1/Tcoh . The signal to noise interms of power therefore goes as Tcoh . Note that the strain we want to detect, h0 , is an√amplitude rather than a power. The detectable strain is therefore proportional to 1/ Tcoh ,remembering that a smaller signal is easier to detect with higher signal to noise. So for a√
fully coherent search which simply uses Tcoh = Ttotal the detectable h0 goes as 1/ Ttotal .
In the case of an incoherent search of N segments of fixed length Tcoh , the individual
segments have a mean noise power of 1/Tcoh , but when added incoherently the power of the√noise scales as 1/( N Tcoh ). The signal to noise of the incoherent search therefore goes as√√
N Tcoh = Ttotal Tcoh . The detectable strain then scales as 1/(Tcoh Ttotal )1/4 . The coherent
search is therefore more sensitive than the incoherent search. However, depending on thesize of the parameter space, the coherent search may require orders of magnitude morecomputation time.
116

In order to calculate the number of templates, we begin with defining the resolution of
the search in terms of frequency, which sets the scale of how far templates can be in theparameter space before losing information. Then we must consider that for a sufficiently
large f˙, or other higher order derivative, the frequency of the source will move by morethan the resolution over the length of a search. That makes it necessary to cover the 1st
spin down space with templates. Similarly, how well the source location is known needs tobe compared to the resolution in sky position.
The resolution in frequency for either the coherent or the incoherent search, %f , which

1
determines the maximum spacing in frequency of the templates is Tcoh
. With this spacing, asignal exactly matched to one template would show up with half the power in an adjacently
placed template. Thus the number of frequency points needed to reasonably cover a search−fminfrom a minimum frequency fmin and a maximum frequency fmax is just fmax)f. Mostblind continuous wave LIGO searches use a minimum frequency of 50 Hz, motivated by theperformance of the detectors at low frequency, and a maximum frequency of 1 kHz, which
is motivated by the fastest pulsars seen.
Next, we need to consider how a signal might evolve over the course of the measurement.
We start from Eq. (6.14) and derive the Taylor expansion of the frequency as a function oftime at the solar system barycenter (SSB) by writingf˙f¨dΦ= f = f0 + (T − T0 ) + (T − T0 )2 + ...
dT
1!
2!

(6.39)

where Φ is the phase of the signal, T is the time at the SSB, and T0 is the start time at the
SSB. We can translate this to the detector frame of reference here on Earth by using Eqs.
(6.15) and (6.16) to write

dΦ dT
dΦ
=
= f⊕ =dt
dT dt'

3v · n̂
1+
c

3
(2
'
(
'
(2
f˙
3r · n̂f¨
3r · n̂f0 +t+− T0 +t+− T0 + ...
1!
c
2!
c
(6.40)

where 3v is the earth’s velocity relative to the SSB, n̂ is the unit vector pointing from the
Earth to the source, 3r is the vector from the Earth to the SSB, and t is the time at thedetector on Earth. The above treatment neglects the daily rotation that the detector will
under go on the surface of the earth, but the exact same methodology can be applied to117

include those effects. This also neglects the small general relativity corrections.
So with this expansion we now consider grid points with differing f˙ which are necessary
to map the spin down parameter space. It is important to note that we consider eachparameter of the parameter space separately. We hold all the other parameters fixed while
calculating the spacing of that single parameter. We begin by considering the mth grid
spacing in f˙ space while holding the frequency, f0 , the higher order derivatives such as f¨,and the sky position all constant. This can be written as

f (m) ='

3v · n̂
1+
c

3
(2
'
(
'
(2
f˙ + m % f˙
3r · n̂f¨
3r · n̂f0 +t+− T0 +t+− T0 + ...
1!
c
2!
c
(6.41)

where f˙ is the first templates frequency time derivative and %f˙ is the spacing between

templates in f˙ space. The difference between the mth template and mth + 1 cannot belarger than the resolution %f . If we choose a spacing such that they are equal then we have%f = f1 (m + 1) − f1 (m) ='3v · n̂
1+
c

(

'
(
3r · n̂r30 · n̂˙
%f t+
− t0 −
cc(6.42)The maximum value for )rc·n̂ is ∼ 480 seconds and thus the difference could be at most

twice this at ∼ 960 seconds. This compared to the days or months of most LIGO observation

runs, defined as Trun = t−t0 , will be quite small and can be neglected. Similarly, the Earth’s
orbital velocity relative to the SSB is small when compared to the speed of light, so 1 + )vc·n̂
1
can be approximated as 1. Using %f = Tcoh
we get the maximum spacing in f˙ space.

Simply

%f
1
%f˙ ∼
=
=
Trun
Tcoh Trun(6.43)

However the boundaries to search within in f˙ space are not quite as clear as in the fspace case. The maximum f˙ we might be expected to see from a star spinning down only
to gravitational radiation is discussed in 5. However, for the general case of f˙ proportional
7),7
7 Ω(t) 7
1
to nth power of frequency is n−1
7 Ω̇(t) 7. Thus a reasonable upper estimate is to drop the1prefix n−1
and set f˙max equal to ftmax
where tage is spin down age below which we will notage

look for sources. We then a number of 1st spin down points equal to ftmax
Tcoh Trun .
age118

For sufficiently long runs or sufficiently young targets a 2nd spin down parameter will
be necessary (and in principle even higher orders but for the searches considered here the
2nd spin down is the highest order necessary). Following a similar procedure as for the 1st
spin down case, it can be shown that the spacing is %f¨ = Tcoh2T 2

runand an upper limit on

2

Tcoh Trunthe necessary number of points is ftmax
.
2
2
age

Lastly we need to consider the resolution in terms to of points on the sky. We once
again go back to Eq. (6.40), noting we can represent 3v · n̂ with |v| cos θ where θ is simply

the angle between the Earth’s velocity and the vector pointing from Earth to the source.
Let us now consider the mth template, but with the spacing in the θ rather than f˙, while

holding the frequency parameters constant. We can write

f (m) ='

|v| cos (θ + m % θ)
1+
c

3
(2
'
(
'
(2
f˙
3r · n̂f¨
3r · n̂f0 +t+− T0 +t+− T0 + ...
1!
c
2!
c
(6.44)

where %θ is our spacing in angle. We again set equal the different between the mth andmth + 1 points equal to %f , neglect )rc·n̂ , but also assume sin(%θ) ∼= %θ to get%f =,|v| sin (θ) % θ )f0 + f˙Trun + ... .
c(6.45)

1.
vmax fmax Tcoh

(6.46)

Solving for %θ and using fmax for the largest contributing term of f0 + f˙Trun + ... wefind%θmin =

c

We can now look at the total number of templates necessary for coherent and incoherent
searches. In the case of incoherent searches, since the Tcoh is fixed, the templates only scale
3 , one factor of Tstnd spinas Trunrun from the 1 spin down and two factors of Trun from the 2
8 since T∼
down. However in the case of coherent searches, it scales as Truncoh = Trun . A single

template using either an incoherent or coherent search code takes approximately 1µs per
1800 second SFT when run on a single node of a cluster, such as those at the Hanford LIGO
site or the Caltech cluster. Given that the time to run a single template is proportional
to the amount of data in both coherent and incoherent searches, the total computational
4
9 . In directed
cost scales as Trunfor incoherent searches while coherent searches scale as Trun

119searches, if the position of the target is known to better than %θ then the coherent search
7 .
scales as Trun

If the parameter space can be kept sufficiently small so that the coherent search iscomputationally possible, then because its sensitivity scales as the square root of the amountof data available, compared to the fourth root of the amount of data available for incoherent
searches, it is the better choice. For the searches presented here, it was possible to keep
the computation time to days or weeks on a cluster, which is what prompted the use of thecoherent F statistic code over an incoherent method..

6.6

Exact grid spacing

We have just discussed the scaling of the number of templates with the search parameter
space, but we began with the statement that δF is simply 1/Tcoh . We need to consider
how to quantify how good that spacing is, and whether we need to increase or decrease it
from this nominal value. The overall scaling will remain the same, but it affects the exact
number of templates we choose.
For the case of the F-statistic coherent search method, we begin to consider this bydefining the mismatch µ, the difference between the F value exactly at the signal, which wewill call Fmatch and its value at the different position where our search grid told us to placea template, which we will call Fsearch . The small offset in template space we will label dxµsuch that x0 = f, x1 = α, x2 = δ, x3 = f˙, x4 = f¨. We can then writem≡Fmatch − Fsearch
Fmatch

(6.47)In the local limit as the grid spacings dxµ goes to 0 and neglecting effects due to amplitude parameters, the mismatch in Eq. (6.47) can approximately be written as

m = gµν dxµ dxν

(6.48)The metric components were worked out by Ian Jones, Ben Owen, and David Whitbeck,and later tested by Reinhard Prix [45]. To do so they used an epicyclic (or Ptolemaic) motionof the detectors to account the effects of the Earth’s motion on the incoming signal. Thiswas done so that an analytic expression could be used to derive the metric components

120explicitly and reduce the amount of numerical computation during searches. We can write
the metric components of interest (f, f˙, f¨) using the shorthand fk = dk f /dtk as(πT )23(2π)2 T k+2 f
gf fk =
(k + 1)(k + 3)
(2πf T )2 T j+k
gfj fk =.
(j + 1)(k + 1)(j + k + 3)
gf f =(6.49)(6.50)
We can create a new metric with the frequency projected out, using the analytic formula

γµν =

'

gf µ gf ν
gµν −
gf f

(

.

(6.51)

Explicitly substituting in 6.49 into 6.51 we can solve for the diagonal metric components,
which we then can use to define our spacings in the individual subspaces of the overall
parameter space. We get
γµν =(2π)2 T j+k+2(j + 2)(k + 2)(3 + j + k)

(6.52)and the spacings are just

df =
df˙ =
df¨ =

1
4

4

m
gf f
m
γf˙f˙

(6.53)

m.
γf¨f¨(6.54)

There is a slight discrepancy in the definitions between the code and this spacing. For...
example the code uses f (t) = f + f˙t + f¨t2 /2 + f t3 /6... when calculating the final frequency
to look at, while the metric does not take into account the additional factor of 2 or 6 for thehigher order spin downs. Effectively a prefix of n! should be added to the spacings where n121

is order of the derivative associated with the spacing.

6.7

Interferometer sensitivities

The overall sensitivity we can detect is highly dependent on the sensitivity of the interferometers used since their noise hides the actual gravitational wave signals. One methodof estimating Sh (f ), the single-sided power spectral density, is to take the SFTs, which
are simply 1800 second long Fourier transforms, averaging the absolute magnitude at eachfrequency and normalizing by the 1800 seconds. ThusN

Sh (f ) =1 5 |SF TN (f )|2N
TSF T(6.55)

i=1

where |SF TN (f )| is the absolute value of the N th SFT at frequency f and TSF T is the√length of time used to make each SFT. Typically, the noise is reported as the Sh .

6.8

Estimating limits of detection

We now have developed sufficient tools in the chapter to be able to make estimates for thesmallest signal a single template or a multiple template search could find.
We begin with the optimal signal to noise ratio ρ. Using Eq. (6.35) and the same
reasoning leading up to Eq. (6.26) we can write

ρ=

.

:

2
(h|h) ∼=
Sh (f0 )- T0 /2[h(t)]2 dt−T0 /2;1/2(6.56)

where f0 is our gravitational wave frequency, typically twice the rotation frequency. We
then use Eqs. (6.17) and (6.21) to re-write this while dropping the cos and sin terms which
oscillate about the frequency of the gravitational wave so that
:

1
ρ2 ∼
(1 + cos2 (i))2=4- T0 /2−T0 /2F+2 dt + cos2 (i)- T0 /2−T0 /2

F×2

;

h20
.
Sh (f0 )(6.57)For an all sky survey, to find the average optimal signal to noise ratio, we could average thisequation over all right ascensions, declination, polarizations and inclination angles which
would give122

< ρ2 >α,δ,ψ,i ∼=4 h20 T0.
25 Sh (f0 )(6.58)

Lastly, we can then solve this equation for h0 in terms of the observation time, the noisein the detector, and the required signal to noise ratio for 1% false alarm and 10% false
dismissal rates to get

h0 ∼
=

4

25ρ2 Sh (f0 ) ∼= 11.44
T0

4

Sh (f0 )T0(6.59)

where we have used the ρ ∼= 4.5 for a single template search from section 6.4.
The more interesting case is that for the targeted search, where we is average over all
right ascensions, polarizations and inclinations while fixing the declination of our source, foreither the Crab pulsar or RX J1856.5-3754. In this case, results depend on the detector’slocation, whether Hanford, Washington or Livingston, Louisiana.
For the Hanford detectors and the Crab pulsar we obtain2

h T0< ρ2 >α,ψ,i ∼= 0.1418 0
Sh (f0 )(6.60)

while for the Livingston detector we get< ρ2 >α,ψ,i ∼= 0.1670

h20 T0.
Sh (f0 )(6.61)In the case of RX J1856.5-3754 and the Hanford detectors we have2

h T0
< ρ2 >α,ψ,i ∼= 0.2127 0
Sh (f0 )(6.62)

and for the Livingston detector we have< ρ2 >α,ψ,i ∼= 0.1951

h20 T0.
Sh (f0 )(6.63)

Thus, for a single template search, the final detectable signal strength for the Crab would be
66
Sh (f0 )Sh (f0 )∼roughly h0 ∼
11.6
while
for
RX
J1856.5-3754
it
would
be
about
h
10.1
=
0 =
T0
T0 .

123

THIS PAGE INTENTIONALLY LEFT BLANK

124

Chapter 7

Searches and results
This chapter describes the use of directed search of RX J1856.5-3754 as a test to compare theversions 1 and 2 of the ComputeFStatistic code. Version 2 of the ComputeFstatistic codeis then used for a search of the Crab pulsar over a physically motivated parameter space.
These results are presently under review, and should, therefore, be considered preliminary.

7.1

RX J1856.5-3754 directed search

The ComputeFStatistic code, whose basic principles were described in Chapter 7.2.1, hasbeen used in several previous searches [2]. At the time a search for RX J1856.5-3754
was under consideration, a newer multi-interferometer search code was becoming available,
which we call the version 2 code. The fundamental difference between these codes was theversion 2 code took the data from all three interferometers simultaneously, to calculate asingle 2F value, while the original code could only deal with a single interferometer at atime. As validation test of this newer code, we ran both the original search code and theversion 2 code and compared their results.

7.1.1

Parameter space chosen

The search parameter space was chosen to extend from 50 Hz to 250 Hz, using 13 days ofdata (624 SFTs). This was based on the estimated computation time of ∼ 2 weeks on the

LIGO Hanford computer cluster and a total of 7×1010 templates to be searched. It included
the frequency band of highest sensitivity of the interferometer, around 130 Hz to 150 Hz. In
that band it was estimated the search would be able to place better direct upper limits on125

the gravitational radiation than the theoretical upper limit calculations from Section 5.5,
assuming RX J1856.5-3754 was 105 years old and spinning down purely to gravitationalradiation. In addition, the 1st spin down (dF/dt) parameter space was selected so that itcovered ages of 105 years and older, by using Eq. (5.13). This meant the 1st spin down
parameter space covered increased linearly with the frequency searched.

7.1.2

Search results

For a search using 7.1 × 1010 templates, and assuming Gaussian white noise, we would

expect a 1% false alarm 2F cutoff value of 66. The overall search would have an expectedlargest 2F value of 58. The calculation methods are described in Section 7.2.1.
The largest 2F values for every 0.25 Hz band for H1 and L1 for the version 1 searchcode are shown in Figures 7-1 and 7-2, and for the combined search with the version 2 code
in Figure 7-3. In each case, the floor of 2F of approximately 40 is expected based on thenumber of templates in each 0.25 Hz search band. That estimate is based on the probability
density function for the F statistic when only Gaussian noise is present. The slight trend
upwards is due to the fact that the number of templates increases at higher frequencies, so
as to cover a sufficient 1st spin down space.
By itself H1 has three large templates above what might be expected from Gaussian
noise at the 1% false alarm level. These features are associated with known lines present in
the instrument itself during the S4 run. They correspond to 60 Hz line noise, a 16 Hz comb
(which is a collection of narrow lines at all multiples of 16 Hz caused by noisy electronics)
with a particularly large spike at 112 Hz, and the leakage from one of the hardware injected
test pulsars near 194 Hz. It is not clear why the 7th harmonic of the 16 Hz comb was so large
relative to the background noise, when compared to the other harmonics. L1 individually
has a very complex “forest” of events, due to known instrument lines. In this case, theeasiest way to reduce the number of lines was the coincident step, which is described in thenext section.

126

Figure 7-1: Plot of the largest 2F values every 0.25 Hz for the H1 search with the version 1
code. The expected value for any particular 0.25 Hz band is around 40, with slightly higher
values expected at higher frequencies because there are more 1st spin down values covered
and thus more templates searched per 0.25 Hz band. The line at 60 Hz is due to 60 Hz
power line noise, the line at 112 Hz is due to a 16 Hz comb, and the line at 194 Hz is due
to a hardware injected pulsar.

Figure 7-2: Plot of the largest 2F values every 0.25 Hz for the L1 search with the version
1 code. The forest of lines are due to many narrow and strong instrument lines, such as a
37 Hz comb, 60 Hz power lines, and other combs. As in the H1 case, the expected value isroughly 40, increasing at higher frequencies as the number of templates increases per 0.25
Hz band.

127

80

Largest Events from V2 code every 0.25 Hz

70

2F value

60
50
40
30
20
50

100

150
Frequency

200

250

Figure 7-3: Plot of the largest 2F values every 0.25 Hz for the combined H1 and L1 searchwith the version 2 code. The forest of lines we see is mostly due to the instrumental lines
present in L1. The expected value is roughly 40, increasing at higher frequencies due toadditional templates searched 0.25 Hz band.

7.1.3

Version 1 coincidence step

To improve the upper limit results of two individual interferometer searches, one can set
a coincidence window in frequency space and find the largest coincident events. This step
resulted in a 5 − 10% increase in sensitivity with a coincidence window of 10−5 Hz, approx-

imately 10 bins to either side in frequency space. The mean 2F value for the largest H1
templates every 1 Hz before requiring coincidence was 43.6, but dropped to 39.4 for eventsin coincidence with L1. Similarly, the mean 2F value for the largest L1 templates every
1 Hz was 62.6 before the coincidence step and 56.5 after. Figure 7-4 show the largest 2F
templates remaining after the coincidence step. The coincident largest templates still havea few remaining large events. Near 60 Hz is the usual 60 Hz line noise. The lines at 74
and 111 Hz correspond to a 37 Hz comb present in the L1 data. The 194 Hz line was theinjected test pulsar leakage.

128

Figure 7-4: Plot of the largest 2F values surviving the coincidence requirement, every 0.25
Hz. There are several lines surviving the coincidence step such as the 60 Hz power line
which is present in both interferometers, lines corresponding to a 16 comb, and also withthe hardware pulsar injection at 194 Hz.

7.1.4

Upper limit injections

Both versions of the code used Monte Carlo injections to produce upper limits in 1 Hz wide
search bands. These injections are done by choosing an h0 injection strength, randomizing
the signal parameters, and injecting a fake pulsar signal into the data. This is then searched
for with the same search method as the full search, but restricted to a small volume ofparameter space surrounding the injection. If the resulting search produces a value above thecutoff, it is considered found. After enough injections (approximately 4,000) are performed,
the confidence of the upper limit can be calculated from the total number of injections founddivided by the total number of injections. The injection sets are repeated with different h0
values until the h0 value which produced a confidence of 0.95 was found in each 1 Hz band.
In the version 1 case, the cutoff was determined by the sum of the log of the largest
coincident 2F values in the 1 Hz band. This was compared to the sum of the log of the 2F
values from an identical injection into both H1 and L1’s data. In the version 2 case, thecutoff was simply the largest 2F value found within the 1 Hz band.

129

7.1.5

Final comparison

The version 1 and 2 upper limits for each 1 Hz sub-band are shown in Figure 7-5. For
reference, the spectrum of the H1 and L1 interferometers at a time of good sensitivity
are also shown in Figure 7-5, scaled by a factor of 100. For a single interferometer the.
expected upper limits would be 37 Sh /Tobs . This was calculated by taking into accountthe declination of RX J1856.5-3754, the detector response functions, the 1% false alarm

cutoff and a 95% confidence for the upper limits. The method of this calculation is detailedin Section 7.2.3.
The ratio of the version 1 upper limits to the version 2 upper limits are shown in Figure√
7-6. An improvement on the order of 2 is expected from the version 1 to the version 2
code since it uses both sets of data to compute the final 2F value, effectively integrating thenoise floor for twice as long. The exceptions where the version 1 code does better occur inthe presence of extremely strong instrument lines present in only one of the interferometers.
For example, near 147 Hz and 221 Hz the version 2 code produces unusually high upper
limits. This is due to a comb of narrow lines at multiples of 37 Hz present in L1. Nearthese two points, the L1 data dominated the 2F calculation, and thus resulted in unusuallyhigh cutoff values. The version 1 coincidence search did not see these as H1 did not have
any large coincident templates. These narrow lines in L1 were eventually removed when an
oscillator was replaced on March 10, 2005.
What this tells us is that when using the version 2 code, we need to check for lines inindividual interferometers to ensure we are not seeing very loud events which are in reality
restricted to only one interferometer. With that caveat however, the version 2 code performs
better than the coincidence method with the version 1 code. These lessons were ported overto the next targeted search performed, that of the Crab pulsar.

130

Figure 7-5: Plot of the upper limits for the version 1 and version 2 searches. Plotted forreference are the strain of the H1 and L1 interferometers scaled down by a factor of 100.
We find that a majority of the very large upper limit values correspond to line features
seen in the interferometers. The unusually high upper limits at 148 Hz and 221 are due tothe extreme 2F values found in the search for those bands, which in turn translates into ahigher upper limit. These large values were due to a 37 Hz comb of lines in L1.

131

Ratio of V1 upper limit to V2 upper limit

3

2.5

2

1.5

1

0.5

0
50

100

150
Frequency [Hz]

200

250

Figure 7-6: Plot of the ratio of the version 1 upper limits to the version 2 upper limits. The
few exceptional points where the version 2 code does worse than the version 1 code werein locations where a very strong and narrow line feature determined the largest 2F value
found. This is turn resulted in larger upper limits, since the largest 2F value determines
how large the injections need to be to be counted for upper limits. The fact that theseparticular lines were not present in H1 meant that the coincidence method was better at
those particular points. In cases where large features are present in both interferometers,such as near multiples of 60 Hz, the version 2 code does significantly better.

132

7.2

Crab wide parameter search

A time domain search has already been carried out using LIGO data targeted on the exact
radio ephemeris available for the Crab pulsar [1]. However, as noted in chapter 5, there
are several physical mechanisms by which the electromagnetic pulse frequency could differ
from the gravitational wave frequency. At a frequency difference of 3× 10−3 Hz between theelectromagnetic pulse and gravitational wave frequencies, this search would be roughly 105
times more sensitive than the exact single template search, simply because of the resolutionof the search for 9 months of data is about 4 × 10−8 Hz. Thus, the wide parameter searchpresented here on the Crab pulsar is complimentary to that search effort.

7.2.1

Chosen search parameter space and results

Chapters 5 and 7.2.1 describe the competing constraints in determining the final parameter
space to search, namely the computational cost of the search, the expected sensitivity, andphysical motivations.
If we consider the free precession models described in Eqs. (5.16) through (5.20) we finda maximum difference between 2fEM and fGW of order 2)fEM . For ) we can consider the
discussion from Section 5.2 and take the largest reasonable values of order 10−5 for hybrid
stars. These combined would give a band to search of order 6 × 10−4 given the Crab’s radio

pulse frequency of 29.777 Hz. The two component model using Eq. (5.27) and the Crab’s
parameters also gave an estimate on the order of 6 × 10−3 . These both have an overall formof

fGW = 2fEM (1 + δ)

(7.1)

where δ is corresponding to the small variation these models induce in the gravitational
wave frequency, and is of the order 10−4 for these models. The associated search space is%fGW = δ2fEM . One way to estimate the variation in 1st and 2nd spin downs is to realizethat

f˙GW = 2(ḞEM (1 + δ) + fEM δ̇ .

(7.2)

If we assume that fEM /f˙EM is of the order of δ/δ̇ or smaller, then we are left with a largest

133

1st spin down search band of%f˙GW ∼ 3f˙EM δ.

(7.3)

The above assumption corresponds to assuming the gravitational wave frequency spins
down at a rate close to the electromagnetic wave frequency. In particular, for the case of
free precession we are assuming that the geometry of the precession, essentially the wobble
angle, evolves only on the spin-down timescale. For the two component model this would
correspond to the steady state lag that occurs after all the glitch transients have died out.
If we take another derivative with respect to time we get
f¨GW = 2f¨EM (1 + δ) + 4f˙EM δ̇ + 2fEM δ̈ .

(7.4)

If we are willing to assume that f˙EM /f¨EM and δ̇/δ̈ are of the same order as fEM /f˙EM , wecan determine a 2nd spin down band of%f¨GW = 8f¨EM δ .

(7.5)

This is equivalent to assuming the rate of change of the gravitational wave frequency spin
down is close to that of the rate of change of the electromagnetic wave frequency. Plugging
in the Crab frequency parameters from Table 7.1 into f˙EM /f¨EM and fEM /f˙EM , we do findthat they are explicitly of the same order, roughly 3 × 1010 to 8 × 1010 .

Given δ ≈ 10−4 and the Crab pulsar’s frequency parameters, this suggests choosing aparameter space 6 × 10−3 Hz wide in frequency space, 1.2 × 10−13 Hz wide in 1st spin down

space, and 1 × 10−23 wide in 2nd spin down space, all centered on twice the radio pulse

frequency and frequency evolution of the Crab pulsar. The relevant Crab parameters forthe start of the S5 run are detailed in Table 7.1 [28]. A single sky point was all that wasnecessary due to the well known position of the Crab pulsar.
At the time of this writing, we have access to a majority of the data from LIGO’s
fifth science run, which has been running since November 4, 2005. However, a large glitch
occurred in the Crab pulsar on August 23, 2006 and its uncertain what effect this would
have on the gravitational wave frequency evolution. That glitch is thus a natural stopping
point in data consideration. This leaves approximately nine months of data available to be134

searched. From that nine month data stretch we created 8798 SFTs from H1, 6649 SFTs
from L1, and 9877 from H2.
Using the discussion from Chapter 7.2.1, we find this parameter space requires roughly
8.1 × 106 templates when using all the available SFTs. This is with a mismatch parameterof 0.15, which injections show causes a loss of roughly 5% in sensitivity from an exactly
matched template. The full search on all three LIGO interferometers with the necessaryMonte Carlo upper limit injections takes approximately one day on 100 nodes on the LIGO
Caltech computing cluster.
As for final sensitivity, this is affected by both the number of templates and the regionof frequency space needed from the LIGO interferometers. For Gaussian noise, the final
expected sensitivity scales slowly with templates, as shown in Figure 7-7. However as we
increase the parameter space we also add in more noise from the surrounding frequency
bins, especially that caused by the 60 Hz lines. This is apparent in the case of L1 which is
discussed in the following sections.

Figure 7-7: Plot of the relative sensitivity of a multi-template search relative to a single
template search as a function of the number of templates

135

Table 7.1: Crab parameters at the start of S5
Frequency
1st Spin down
2nd Spin down
Right Ascension
Declination

29.7772 Hz−3.7292 × 10−10 Hz/s
1.2426×−20 Hz/s2
1.4597 radians
0.3842 radians

Expected and actual largest templates
Given our parameter space with 8.1 × 106 templates, we can estimate the expected largest

2F value that will be found, and also a 1% false alarm cutoff value, assuming Gaussian noise.
Starting from the noise probability density function, p0 in Eq. (6.37), we can calculate theprobability Pbelow that a single template is at or below a cutoff 2F of 2Fc . That is
Pbelow (Fc ) =- Fc

p0 (F )dF =0

1 (1 + Fc ) −Fc
−
e
.
2
2

(7.6)

To find the probability of N templates being below 2Fc we simply raise Pbelow (Fc ) to the
N th power. To find the probability of one or more templates being above the cutoff we only
need to calculateN
PF A = 1 − Pbelow(Fc ) .

(7.7)

We can then numerically solve for the 2Fc such that PF A is equal to 0.01. We find that
2Fc = 46 for N = 8.1 × 106 . This is effectively a 3σ level of confidence requirement on a
positive detection result.

We can also calculate the expected largest 2F value found. In this case we want N − 1

templates being below our expected 2Fe , our expected 2F value, and one template being
between 2Fe and 2Fe + dF . We can write the probability distribution for 2Fe in N trials as
pN (2Fe ) =N
Fe e−Fe (1 − Pbelow (Fe ))N −1 .
2

(7.8)

We can then compute the expected maximum 2Fe numerically in N trials as- ∞2Fe pN (2Fe )d(2Fe )0which for N = 8.1 × 106 is 39.
136

(7.9)

When we look at the actual result from the full search we find it to be very consistent
with Gaussian noise with no signal present. We find an maximum 2F value of 38.27 which
is below our threshold of detection and very close to our expected value of 39.

7.2.2

Upper limits

Monte Carlo injections
With our largest 2F values being below a 1% false alarm cutoff, we now turn to setting
upper limits on the maximum gravitational wave strain coming from the Crab pulsar. Thisis done by Monte Carlo injections. The following methodology is applied.
Choose an initial injection strength, h0 , that we think is near the minimum detectablelevel. We fix right ascension and declination to that of the Crab pulsar. Then randomly
choose polarization, inclination and initial phase from all possible values. Randomly choose
frequency, 1st spin down, and 2nd spin down within the parameter space of the search.
With the signal now defined, inject it into the actual data via software. Search for thisinjected signal. To save time we search only a small space surrounding the signal, including±2 template spacings in frequency, 1st spin down, and 2nd spin down. If the largest 2F

value found in this injection search is greater than the largest 2F value found in the actual
search, we declare the signal found. We then repeat until we have reached our desired
number of injections, at which point we calculate our confidence of that injection run. The
confidence is simply the number of injections declared to be “found” divided by the total
number of injections made. If the confidence of the injection run is over 0.95 we decrease
our injection strength and repeat the run. If the confidence is less than 0.95, we increase
the injection strength. We also decrease our step size if we cross the 0.95 confidence level
between injection runs. This eventually settles on the injection level which is found 95% of
the time. In this way we find the 95% confidence upper limit value.
It is important to note how many injections are sufficient for a run. The number of
injections are sufficient if over multiple injection runs at the same strength, the confidence
level does not vary too much, say of the order 1%. Having performed a test run with 36000
injections and breaking it into 9 sets of 4000, we find that the confidence value of each small
set stayed within 95.3% and 96.2% when the confidence value from all the injections was
95.6%. This is shown in Figure 7-8. This suggests that 4, 000 injections are sufficient to137

determine the confidence level to within 1%.

1

Confidence

0.98
0.96
0.94
0.92
0.9

2

4
6
Injection run number

8

Figure 7-8: Plot of the confidence for 9 sets of 4000 injections compared to the confidence
for all 36000 injections.

Upper limit result
Having performed the upper limit injections on the search, the upper limit on the strain
coming from the Crab pulsar was found with h0 = 1.6 × 10−24 at a confidence of 95.5%

with 20000 injections. This upper limit holds over the parameter space of the search.

7.2.3

Checks on the search result and upper limits

Several checks were performed to determine if the search and upper limits were performed
correctly. First, data from each individual interferometer (i.e. H1, H2 and L1) was run
through the entire analysis pipeline by itself. These were then compared to a set of software
generated Gaussian white noise also run through the analysis pipeline.
SFT noise
Our cuts on the data set used to generate our SFT segments were minimal. The only data
removed was from periods out of lock, had bad or non-existent calibration lines, was during138periods of high winds over 30 mph, or within 30 seconds of a lock loss. This means thatwhen we look at individual SFTs it is no surprise that within our frequency band of interest,
there are periods were the noise was significantly louder than others. Figure 7-9 shows themean noise in a frequency band from 59.52 to 59.58 Hz in terms of the one sided power
1/2

spectral density. This is the band size the search code considered when calculating Sh .

139

H1

−21

h[f], [1/Sqrt( Hz )]

x 10
0.8
0.6
0.4
0.2
0

h[f], [1/Sqrt( Hz )]

1

2000

3000 4000 5000 6000
SFT number (time)

7000

8000

H2

−21

x 10

0.8
0.6
0.4
0.2
0

1
h[f], [1/Sqrt( Hz )]1000

1000 2000 3000 4000 5000 6000 7000 8000 9000
SFT number (time)
L1

−21

x 10

0.8
0.6
0.4
0.2
0

1000

2000

3000
4000
SFT number (time)

5000

6000

Figure 7-9: Plot of spectral density of strain in the 59.52 to 59.58 Hz band versus SFT (inchronological order) for H1 (top), H2 (middle), and L1 (bottom). The many lines coming
out of the noise floor for each interferometer shows the non-stationarity of the noise overthe course of the S5 run. In order to get a good estimate of the noise floor a noise weighting
scheme is necessary in order to deal with this non-stationarity.

140

These large variations are dealt with in the search code by applying a weight factor
based on a running median with a window size of 50 bins. Effectively, for a given data
segment, the noiser the frequencies surrounding the one we are interested in the less weight
we give to that portion of the data. Since each SFT corresponds to 1800 seconds of data,
each bin is 1/1800 = 5.55 × 10−4 Hz wide. The median value of these bins is effectively set
as the noise floor for the frequency under consideration. In the analysis the calculation isweight by 1/w(f ) where wα (f ) =< Sα,k (f ) >k where <>k denotes the median over k bins
and Sα,k denotes the noise of k bins of the αth SFT. The weighting minimizes the impact
excessively noisy segments have on the search.
In order to estimate the effective noise under this weighting scheme, we need to performa similar weighting on the SFTs. This is done by calculating the mean value of all the bins
within a window size of 50 bins and associating that value with the central frequency. We
denote this as < SF Ti (f ) >α indicating the ith SFT and an average of α bins. We thencalculate < 1/ < SF Ti (f ) >α >−1where in this case <>i denotes averaging over all SFTs.
i
This is effectively estimates the double sided power spectral density. Finally we convert tothe one sided power spectral density and take the square root to get it into the usual form
1/2

of Sh . This method produces the plot in Figure 7-10.
In addition to running on data from the three LIGO interferometers, the above was also
carried out on a Gaussian white noise data set, which was treated as if being generated at
1/2

Hanford and using the same number of SFTs as H1. This data set was created with a Sh

value of 4 × 10−22 with the code lalapps M akef akedata. When looked at in the same way

as above, it produces Figures 7-11 and 7-12. From Figure 7-11 we see that the SFTs are of
the correct strain and in Figure 7-12 shows that our noise is effectively white, varying byless than 0.1% over the frequency band of interest.

141

Figure 7-10: Estimate of spectral density of strain for the all three LIGO interferometersin the frequency band of interest for the Crab pulsar. H1 has the best noise performance,
followed by L1 and then H2. However, L1 has a large trend upwards with increasing
frequency indicating the presence of an instrumental line feature.

142

Figure 7-11: Plot of spectral density of strain versus a Gaussian white noise data set SFT(in chronological order). This particular data set has been generated with an expected
4 × 10−22 spectral density of strain.

Figure 7-12: Estimate of spectral density of strain versus frequency for the entire Gaussian
noise fake data set. This shows the estimation method is working properly, since theestimated strain shown here is within 0.5% of the expected value of 4 × 10−22 .
143Individual interferometer search results
The individual interferometers were run through the search pipeline and none of themproduced a template with a 2F value beyond the 1% false alarm cutoff of 46 for 8.1 × 106

templates. The largest 2F value for H1 was 43.67, for H2 was 37.02, and for L1 was 30.43.
The largest 2F value for the Gaussian Noise data set was 35.83. These can be comparedwith the overall search’s largest 2F value of 38.27 and also the expected largest 2F value
assuming 8.1 × 106 templates of 39.

Using Eq. (6.37) we can estimate the expected number of templates with a given 2F

value assuming Gaussian noise and normalizing by the total number of templates in the
search. This can be plotted directly against the actual number of templates falling within aband of 2F values. Because the interferometers spanned slightly different time periods dueto their duty cycle, the number of templates used by each individual search was slightly
different. The H1 search used 7.9 × 106 templates, the H2 search used 8.1 × 106 templates,

and the L1 search used 7.3 × 106 templates. The Gaussian Noise test was based on SFTs
with the same time stamps as the H1 SFTs and thus also used 7.9 × 106 templates. The

Figure 7-13 shows the expected plotted against the actual in each of these cases. Note thatthe full search does not record templates with a 2F value below 20 for data storage reasons.

144

H1

4

ActualExpected

3

10

2

10

1

10

0

10

20

25

30 35
2F

45

3

2

1

10

0

20

25

2F

1

10

4

10

10

2

10

10

ActualExpected1030

Number of results

Number of results

40

30

20

25

30
2F

35

40

Full Search: H1 H2 L1
ActualExpected

3

10

2

10

1

10

0

10
20

35

ActualExpected

10

10

L1

4

10

H2

4

10

Number of results

Number of results

10

25

30
2F

35

40

Figure 7-13: For H1 (upper left), H2 (upper right), L1 (lower left) and the full multiinterferometer search (lower right) are shown plots of the number of templates returned by
the search with 2F values within a band versus the expected number. In the case of L1
there are far fewer large templates than one would expect primarily due to non-random
features in the L1 spectrum. The full search being below expected is mainly due to this
disturbance in L1.

145

Figure 7-14: Plot of the number of templates returned by a search on a Gaussian noise fakedata set with 2F values within a certain band versus the expected number. The fact that
the Gaussian noise matches the expected value implies that deviation from the expected in
Figure 7-13 for L1 is due to the non-Gaussian nature of the noise rather than a bias in the
search pipeline.

146

The first thing to notice in these Figures is the actual interferometer results tend to be
below the expected value, while the Gaussian noise data set is at the expected level, moststrikingly apparent in the case of L1. Since the Gaussian noise data does not show any bias,
this must be an effect of the non-Gaussian and non-stationary nature of the interferometer
noise. By comparing the extreme case of L1 to the H1 case we can gain some insight.
We first go and look at the surrounding noise, as shown in Figure 7-15, and compare
it with the somewhat cleaner case of H1 in Figure 7-16. There is the obvious large 60 Hz
peak in both H1 and L1 caused by 60 Hz line noise. In addition, there appears to be morestructure in the L1 case. If we go back to the specific band of interest in Figure 7-10 wenote that the L1 noise increases by about 15% over the band relative to its mean value,
where as in H1 and H2 the variation is less than 3% of its mean value. Suggesting that
the L1 search is more on top of 60 Hz “wings” and associated noise than the respective
searches in H1 and H2, but all are obviously non-white to some degree. In this case, ourassumptions of Gaussian stationary white noise is poor for L1 hence leading to a break
down of our predictions based on Eqs. (6.36) and (6.37). The final upper limits, discussed
in Section 7.2.3, are not affected by more than a few percent because we use the largest 2F
value found in the search as the cutoff, rather than a theoretically calculated cutoff. The
direction of the bias comes from the fact that a non-random instrument line brings up theeffective noise floor without affecting the Gaussian distribution of noise on top of it. Since
the F statistic is proportional to the square of the optimal signal to noise, this leads to acorresponding drop in the F statistic values found.

147

Figure 7-15: Plot of the weighted noise in L1 from 59 to 60 Hz. Distinct steps are visible
in the noise, indicating some non-random instrumental effect is raising the noise floor.

Figure 7-16: Plot of the weighted noise in H1 from 59 to 60 Hz. The really only visible lineis the 60 Hz line due to power line noise, while around 59.5 Hz the noise seems effectively
flat.

148

2F distribution in parameter space
We can also take a look at the distribution of the 2F values within the parameter space
searched, to check if it looks like a random distribution as opposed to clumping near an
obvious signal or line. These are shown in Figures 7-17 and 7-18. The templates returned
with large 2F do look uniformly distributed except possibly in the case of L1, which asalready noted in Section 7.2.3 seems to have a broad feature in the noise.
If a strong narrow line were present, it would manifest itself as a diagonal line of large(redder in the plot) values from the upper left trailing to the lower right. This is becauseof the negative spin down values are being applied to the templates starting frequency overthe course of the run to determine what frequency to look at. Larger spin down values (inan absolute sense) results in lower frequencies being searched for that particular template,meaning a narrow line feature will be seen by templates with high frequencies and large
spin downs or low frequencies with small spin downs. This suggests that there is no narrow
line feature in the searched frequency band causing unusually large results.

149

Figure 7-17: For H1 (upper left), H2 (upper right), L1 (lower left) and the full multiinterferometer search (lower right) are shown plots of the distribution of 2F values greaterthan 25 in the parameter space searched. The color signifies the 2F value, red is larger
and blue is smaller. For H1, H2 and the multi-interferometer search the distributions lookuniform, while L1 looks different due to a combination of fewer results to work with andthe broad noise feature present in the L1 spectrum. No strong narrow line features appear
to be present in any of the interferometers.

150

Figure 7-18: Plot of the distribution of 2F values greater than 25 for a Gaussian noise fakedata set. The color signifies the 2F value, red is larger and blue is smaller. This is usefulas a comparison to the Figures in 7-17 since this fake data set was created without any lineor broad spectrum features.

151Expected individual interferometer upper limits
Given the maximum 2F values found for each individual interferometer and the Gaussian
noise test case, one can use Eq. (6.36) to estimate the injection strength necessary to have
95% of the injections exceed the respective maximum 2F values. To do so also requires
taking into account the detector response functions since the injections have varying polarizations and inclination angles which non-trivially affects the effective detectability of theinjection. Lastly, Eq. (6.36) applies to templates perfectly matched to the signal, which is
not the case in our search or Monte Carlo injections.
In order to determine the effect of perfectly matching templates, an additional Monte
Carlo injection set was performed on the H1 search but with the follow up searches eachplaced exactly at the injection. The 95% confidence level for strain was found to be 2.12 ×

10−24 Hz−1/2 , compared to the actual H1 95% confidence level of 2.25 × 10−24 Hz−1/2 . Thissuggests that the grid spacing worsens the upper limit by making it larger by roughly 5%,
which should be taken into account when using Eq. (6.36) to make predictions.
We can use Eq. (6.57) along with the detector response functions with the known
positions of the Hanford and Livingston detectors, shown in Table 7.2 [4], and the fixedright ascension and declination of the Crab pulsar, to write the following
h2 T< ρ2 >= A(ψ, i) 0
Sh

(7.10)

where A(ψ, i) encodes the detector response function dependance on just the polarizationangel ψ and inclination angle i.
The function A(ψ, i) can be calculated from the detector response functions [27] by fixing
the declination of the source and averaging over the right ascension. The function A(ψ, i)for the Hanford and Livingston locations are plotted in Figures 7-19 and 7-20. It effectively
represents the changing response of the interferometers for different possible polarizationsof the wave and for the different possible inclination angles of the source relative to theinterferometers. By considering this function over the entire injection space and using the
calculated ρ in Eq. (6.36) we can calculate the overall expected 95% confidence level, or at
any other desired confidence level. It should be noted because of the 95% confidence level
requirement, the lowest 5% of values from Figures 7-19 and 7-20 determine the effective
detection threshold, as opposed to an average over the whole parameter space. In any
152case, these expected values can then be directly compared to upper limit injections at andaround the 95% confidence level derived from actually performing the injections. These areshown in Figures 7-21. As seen in the plots, the difference between the expected and actual
confidence values are less than 5%.
The good agreement between the actual and expected values shows that the analysis
pipeline is working and that the results are consistent with being generated by mostly
Gaussian noise. The upper limits seem to be reasonably robust in the face of the noisefeatures in L1 since we used the largest 2F value found of 30 which effectively factors in thereal distribution of the noise into the upper limits. So while the noise in L1 is not perfectly
Gaussian, actual signals above the noise floor are still detected without problems.
Table 7.2: Positions of Hanford and Livingston detectors
Detector
Hanford
Livingston

Latitude (degree)
46.45 N30.56 N

153

Longitude (degree)
119.41 W
90.77 W

Figure 7-19: Plot of the function A(ψ, i) for Hanford. Effectively, the physical response of
the interferometer as a function of the incoming wave’s polarization and source inclination,with the sky position fixed to that of the Crab pulsar. Redder colors correspond to higher
values of A(ψ, i).

Figure 7-20: Plot of the function A(ψ, i) for Livingston. Effectively the physical response of
the interferometer as a function of the incoming wave’s polarization and source inclination,with the sky position fixed to that of the Crab pulsar. Redder colors correspond to higher
values of A(ψ, i).

154

H1
0.95

0.96
Confidence

Confidence

0.9
0.85
0.8
0.75
0.7

1

1.5
2
Injection Strength
L1

1

1

Search UL data
Expected
3.4

3.6
3.8
4
Injection Strength x 10−24

Gaussian Noise Fake Data

0.95
Confidence

0.8
Confidence

0.92

0.88
3.2

2.5

−24

x 10

0.94

0.9

Search UL data
Expected

0.65

0.6
0.4
0.2
1

H2

0.98

0.85
0.8

Search UL data
Expected
1.5

0.9

0.75
2

2
2.5
3
3.5
Injection Strength x 10−24

Search UL data
Expected
2.5
3Injection Strength

3.5

−24

x 10

Figure 7-21: For H1 (upper left), H2 (upper right), L1 (lower left) and a search on a setof Gaussian noise fake data (lower right) are shown plots of the expected confidence versus
injection strength of the Monte-Carlo injections. The position of the curves are determinedby the Crab pulsar’s sky position, the function A(ψ, i), the noise in the individual interferometers, along with the largest 2F value found in the individual searches. The shape of thecurves are determined by Eq. (6.36). The difference between the expected and actual values
is in general less than the uncertainty in the upper limit result for runs of 4000 injections,which gives us confidence in the upper limit values produced by the search pipeline.

1557.2.4

Constraining energy loss in the Crab pulsar

With a strain upper limit of 1.6 × 10−24 that we have confidence in, we can now turn to thequestion of what we can say about the Crab pulsar.

Equation 6.22 relates our h0 upper limit to the moment of inertia, ellipticity, distanceand spin frequency of the Crab pulsar. We know the Crab pulsar is located 2kpc distant,
and we know the radio frequency is 29.777 Hz. If we make the assumption that the Crabpulsar has the a typical neutron star’s moment of inertia of 1045 g cm2 , we can calculate themaximum ellipticity the Crab pulsar could have. In this case, we calculate )max = 8.5×10−4 .
This corresponds to a gravitational luminosity of 4.6 × 1038 erg/s. These values are larger
than the indirectly calculated values of )max = 3 × 10−4 and 1.6 × 1038 erg/s estimated by

Palomba [43].

However, there is a good deal of uncertainty in the moment of inertia of the Crab pulsar,
and as noted earlier it might be up to a factor 3 larger than the canonical value [7] [34].
For a value of I of 1.59 × 1038 kg m2 or higher we begin beating the canonical upper limiton the Crab pulsar’s gravitational wave emission. For a value of I of 3 × 1038 kg m2 we

can constrain the energy loss due to gravitational waves within the search parameter space
to less than 53% of the total spin down energy loss of the Crab pulsar. The relationshipbetween ellipticity and the moment of inertia is plotted in Figure 7-22. The ratio of theenergy loss due to gravitational radiation versus the total spin down energy loss is plottedin Figure 7-23.

156

Figure 7-22: Plot of the upper limits generated by this search on ellipticity as a function
moment of inertia compared to that of the canonical upper limit calculation. The area
above the green line is effectively excluded by theoretical calculations, while the area above
the blue line has been excluded by direct measurement in this search.

157

Figure 7-23: Plot of the ratio of gravitational energy loss upper limit to total spin downenergy loss as a function of moment of inertia. Ratio values greater than 1 indicate weare not constraining the Crab pulsar’s energy budget of its spin down at that moment ofinertia.

158

Figure 7-22 effectively shows the excluded regions of ellipticity and moment of inertia
parameter space. The area above the green line is excluded by theoretical calculations,while the area above the blue line is excluded by direct measurement in this search. The
wedge to the right of the crossing point is effectively the additional information the search
is providing.
In Figure 7-23 values above one are effectively unphysical and thus the search is effectively providing no information in that region. However, for the higher values of the
moment of inertia, we are providing a directly measured constraint on a portion of the
Crab pulsar’s energy budget of the spin down. This direct measurement is interesting, because even though there are estimates of the Crab pulsar’s spin down energy budget, theuncertainties involved in those estimates leave a significant portion of the budget available
for gravitational wave emission.
We will briefly consider the total spin down energy budget of the Crab pulsar. If we takethe Crab pulsar’s frequency and 1st spin down values, and let the moment of inertia range
from 0.5 × 1038 to 8 × 1038 kg m2 [15], the total energy available is Ė = 4π 2 If f˙ = 2 × 1038

to 1.6 × 1039 erg/s. This energy loss can be made up of several components in addition to

the gravitational wave emssion, including the observed electromagnetic emission from thepulsar and the surrounding nebula, and also the nebula’s accererating expansion.
The Crab pulsar’s black-body thermal emission is a very minor component, less than
∼ 3 × 1034 erg/s, assuming a 10 km radius star [26]. The total bolometric luminosity of the

Crab nebula is roughly 1 − 2 × 1038 erg/s [17]. Lastly, the energy necessary for the nebula
expansion could be between 3 × 1038 and 9 × 1038 erg/s [8] [9]. The uncertainties in the

nebula expansion estimates are large due to the uncertainty in the mass of the Crab Nebula
itself, which is between 2.8 and 6.4 M$ [21]. Using these values, we can estimate the total
loss to non-gravitatoinal wave mechansims as being between 4 × 1038 and 10 × 1038 erg/s.

In comparing the possible range of rotational energy loss, 2 × 1038 to 1.6 × 1039 erg/s

based on possible moments of inerita, to the non-gravitational wave losses, 4 × 1038 to

10 × 1038 erg/s, we see there is a large variation in the difference, which we might attributeto gravitational radiation. From this comparison, we might estimate that anywhere from anegligble to 75% of the energy loss might be in the form of gravitational radiation. This thenmakes limits from the search presented here or from the narrow time domain search [44]useful in understanding the overall energy budget.
159

Part IV

Conclusion

160

Conclusion
In the first half this thesis, we describe the LIGO interferometers and the phase camera
subsystem. We discuss how the phase camera was used to understand an output modecleaner placed at the AS port of the interferometer and how it was shown to be working as
expected. We also discuss the thermal compensation system, and how it was a necessarycomponent for the interferometers to reach high power (and higher sensitivity) operation.
We examine the impact of the thermal compensation system on the AS port light throughthe use of the phase camera. We compare different states of thermal compensation in orderto understand how the light changes to reduce signal in the Iphase of the AS port photodiodes and reduce the need for a correction servo, whose limit had previously determined
the maximum laser power LIGO could use.
In the second half of this thesis, we performed a single target search for continuous
gravitational wave emission from the nearby star RX J1856.5-3754 using data from LIGO’s
4th science run. While a periodicity in RX J1856.5-3754’s X-ray emission was found below the LIGO search band, the search itself was still a useful test of a newly developed
multi-interferometer F-statistic code and an application of targeted search methods. This
experience was directly applicable to a second search looking for continuous gravitational
wave emission from the Crab pulsar, searching over a wide parameter space to complimentvery narrow targeted searches already performed. This parameter space was chosen basedon physical motivations, computational constraints, and expected final sensitivity. Using
data from the first nine months of LIGO’s 5th science run to perform the search, no gravitational wave signal was found. Monte Carlo injections were used to place a Frequentist upper
limit on the gravitational wave strain from the Crab pulsar of h0 = 1.6 × 10−24 at a 95%

confidence level. This result is currently under review by the LIGO Scientific Collaboration,
and as such should be treated as preliminary.

161

As LIGO’s 5th science run finishes, we can look forward to examining the entire dataset, which extends the possible depth of targeted continuous wave searches. In addition aperiod of upgrades to the LIGO interferometers aimed at improving their sensitivity by afactor of 2 [22] is scheduled to start soon after, extending the reach of gravitational wave
searches even further.

162

Appendix A

Table of acronyms
Definitions of acronyms
ACRONYM DEFINITION
RM
Recycling Mirror
BS
Beam Splitter
ITMX
Input Test Mass, X-Arm
Input Test Mass, Y-Arm
ITMY
ETMX
End Test Mass, X-Arm
ETMY
End Test Mass, Y-Arm
TCS
Thermal Compensation System
PSL
Pre-Stabilized Laser
PMC
Pre-Mode Cleaner
PC
Pockels Cell
MC
Mode Cleaner
OMC
Output Mode Cleaner
AOM
Acoustic-Optical Modulator
PZT
Piezoeletric Transducer
NSPOB
Normalized Sideband Power Pick-off Beamsplitter
AS I
Anti-symmetric Port Photodiode In-phase Signal
AS I Corr
Anti-symmetric Port Photodiode In-phase Correction
USB
Upper Side Band
LSB
Lower Side Band
In-phase
IphaseQphaseQuadrature-phase
Table A.1:

163

THIS PAGE INTENTIONALLY LEFT BLANK

164

Appendix B

Phasecamera demodulation andcontrol board schematic

165

D

C

B

P0P402P6

GND

P0P302P5

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P0P602P8

GND

P0P502

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C106
1.0u

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1

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2P0U3602
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MC7915CT
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1.0u

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C41
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IN

2

3
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P0U507

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P0U2907

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P0R5901

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P0U1901

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Inductor
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P0U1205

6

P0U1206

4

5

P0U1104

P0U1105

6

P0U1106

4

5

6

P0U1004

P0U1005

P0U1006

4

5

P0U904

P0U905

6

P0U906

4

5

P0U804

P0U805

6

P0U806

4

P0U704

5

P0U705

6

P0U706

4

5

6

P0U604

P0U605

P0U606

GND

GND

GND

GND

GND

GND

GND

GND

TP1

TP2

TP3

TP4

TP5

TP14

TP15

TP16

R19

C53

Res1
200

P0R2301

R23

Cap
0.1uF

R56
Res1
200

P0R5601

GND

5

R43
P0R4302

P0R5102

1

3

2

P0U2601

P0U2603

P0U2602

Cap
820pF

P0C8801 P0C8802

Res1
25K
C88

P0R5101

R51

1

P0U2401

3

P0U2403

2

P0U2402

Cap
820pF

P0C7601 P0C7602

Res1
25K
C76

P0R4301

1

3

2

P0U2201

P0U2203

P0U2202

Cap
820pF

P0C5501 P0C5502

Res1
25K
C55

P0R3102

1

3

P0U2001

P0U2003

2

P0U2002

Cap
820pF

R31
P0R3101

GND

GND

P0R5602

P0R4802

Cap
0.1uF

GND

GND

P0R3702

GND

P0R1602

Res1
25K
C26

R16

P0C2601 P0C2602

P0R1601

R12
Res1
12K

GND

GND

P0R2302

P0R5402 P0C8601 P0C8602

C86

Res1
200

P0R4801

R48

Cap
0.1uF

P0R4502 P0C7401 P0C7402

C74

Res1
200

P0R3701

R37

Cap
0.1uF

P0R3302 P0C5301 P0C5302

Res1
50

P0R5401

C24
P0R1902 P0C2401 P0C2402

R54

Res1
50

P0R4501

R45

Res1
50

P0R3301

R33

Res1
50

P0R1901

P0R702GND

P0U302

2

3

P0U303

4

Res1
NC
8P0U308

GND

5

R7
Res1
12K
7P0U307
1P0U301
P0U304

5

6

8

5

6

8

5

8

5

8

6

-15

GND

a15

-15

GND

GND

a15

-15

GND

GND

a15

-15

GND

C91
Cap
0.1uF

P0R670W

-15

GND

R67
8-20k Multi-turn

C83
Cap
0.1uF

C79
Cap
0.1uF

P0R650W

R65
8-20k Multi-turn

C71
Cap
0.1uF

C67
Cap
0.1uF

U26
OP27GS

C90
Cap
1uF

P0U2608

P0U2606

P0U2605

C82
Cap
1uF

C78
Cap
1uF

P0U2408

GND

a15

P0P201GND

R63
8-20k Multi-turn

C48
Cap
0.1uF

C36
Cap
0.1uF

P0R630W

U24
OP27GS

P0U2406

6

P0U2405

C70
Cap
1uF

C66
Cap
1uF

P0U2208

P0U2206

C18
Cap
0.1uF

C14
Cap
0.1uF

P2

R61
8-20k Multi-turnP0R610W

U22
OP27GS

P0U2205

C47
Cap
1uF

C35
Cap
1uF

P0U2008

P0U2006

U20
OP27GS

P0U2005

C13
Cap
1uF

Res1
200

U3
AD620
R10 P0R1002
P0R1001

C17
Cap
1uF

6P0U306

P0C502
P0C501
P0C1301 P0C1302

GND

P0P202

P0C602
P0C601
P0C1401 P0C1402

P0R302

C54

Res1
200

P0R2401

R24

Cap
0.1uF

Res1
200

P0R5701

R57

Cap
0.1uF

P0R1702

6P0U406

P0R402

P0R3202

P0R4402

P0R5202

1

3

2

P0U2701

P0U2703

P0U2702

Cap
820pF

P0C8901 P0C8902

Res1
25K
C89

P0R5201

R52

1

P0U2501

3

P0U2503

2

P0U2502

Cap
820pF

P0C7701 P0C7702

Res1
25K
C77

R44
P0R4401

1

3

2

P0U2301

P0U2303

P0U2302

Cap
820pF

P0C5601 P0C5602

Res1
25K
C56

R32

1

3

P0U2101

P0U2103

2

P0U2102

Cap
820pF

P0C2701 P0C2702

Res1
25K
C27

R17

GND

P0R3201

GND

6

Res1
1.2K

R4

P0R401

P0R1701

GND

GND

P0R5702

P0R4902

GND

GND

GND

GND

GND

P0R3802

P0R5502 P0C8701 P0C8702

C87

Res1
200

P0R4901

R49

Cap
0.1uF

P0R4602 P0C7501 P0C7502

C75

Res1
200

P0R3801

R38

Cap
0.1uF

P0R3402 P0C5401 P0C5402

Res1
50

P0R5501

C25

GND

P0R2402

R14
Res1
12K

2

3

P0U402

P0U403

R8
Res1
12K

P0R2002 P0C2501 P0C2502

R55

Res1
50

P0R4601

R46

Res1
50

P0R3401

R34

Res1
50

P0R2001

R20

Header 8

1
P0JP1102
2
P0JP11033
P0JP1104
4
P0JP1105
5
P0JP1106
6
P0JP1107
7
P0JP11088

JP11
P0JP1101

GND

4

R3

5

P0R301

7P0U407
1P0U401
P0U404

P0C702

2
1
P0JP202

C15
Cap
1uF

8

5

8

6

5

8

5

8

C92
Cap
1uF

GND

a15

-15

GND

GND

a15

-15

GND

C93
Cap
0.1uF

P0R680W

-15

GND

R68
8-20k Multi-turn

C85
Cap
0.1uF

C81
Cap
0.1uF

U27
OP27GS

P0U2708

P0U2706

6

P0U2705

C84
Cap
1uF

GND

a15

-15

GND

R66
8-20k Multi-turn

C73
Cap
0.1uF

C69
Cap
0.1uF

P0R660W

U25
OP27GS

C80
Cap
1uF

P0U2508

6

P0U2506

P0U2505

C72
Cap
1uF

GND

a15

R64
8-20k Multi-turn

C50
Cap
0.1uF

C38
Cap
0.1uF

P0R640W

U23
OP27GS

C68
Cap
1uF

P0U2308

P0U2306

P0U2305

C49
Cap
1uF

C20
Cap
0.1uF

-15

GND

GND

a15

R62
8-20k Multi-turnP0R620W

U21
OP27GS

C37
Cap
1uF

P0U2108

P0U2106

6

C16
Cap
0.1uF

C19
Cap
1uF

P0U2105

5

C8
Cap
0.1uF

P0JP201

Header 2

JP2

U4
AD620

P0C701

P0C802
P0C801
P0C1601 P0C1602

GND

8P0U408
P0U405C7
Cap
1uF

P0C1901 P0C1902

6

GND
R5
Res1
12K

R11
Res1
12K

GND

2

3

P0U102

P0U103

4

a15

Res1
NC

R1

P0R1012
1P7

P0JP702

P9

2
1

P10

GND

P0P1001

P0JP901

P0JP902

Header 2

JP9

GND

P0P901

P0JP701

Header 2

2
1

GND

JP7

P0P701

P0JP601

P0JP602

Header 2

2
1

P0P1201

P0JP301

P12

GND

JP6

P0P1101

P0JP302

Header 2

JP3

GND

6P0U106

P0R102

P11

GND

5

C6
Cap
0.1uF

7P0U107
1P0U101
P0U104C5Cap
1uF

8P0U108
P0U105

P0R701
P0R1202
P0R1201

7
P0C102

7

Res1
1K

P0R5301

R53

GND

Res1
1K

R47
P0R4701

P0R5302

Res1
1K

P0R5001

P0R5002

P0R4202

P0R3502

P0R2802

P0R2502

P0R1802

C10
Cap
0.1uF

R50

P0R4702

Res1
1K

R42

P0R4102

P0R4201

GND

Res1
1K

R41
P0R4101

Res1
1K

R35
P0R3501

GND

Res1
1K

P0R2801

R28

Res1
1K

R25

Res1
1K

P0R2501

GND

C2
Cap
0.1uF

P0R902R18
P0R1801

C9
Cap
1uF

Res1
200

P0R901

U1
AD620
R9

C1
Cap
1uF

P1

-15

GND

P0P101GND

a15

P0P102

P0C202
P0C201
P0C1001 P0C1002

51
2
P0JP1003
3
P0JP1004
4
P0JP1005
5
P0JP1006
6
P0JP1007
7
P0JP1008
8

JP10

Header 8

P0JP1002

P0JP1001

GND

R13
Res1
12K

2

3

P0U202

P0U203

R6
Res1
12K

Date:
File:

D

Size

Title

GND

4

P0R502
P0R501
P0R1102
P0R1101

7P0U207
1P0U201
P0U204

P0R802
P0R801
P0R1402
P0R1401A

P0C6102 P0C6101

7
42

P0U2807

P0U2804

1

P0R690CW
P0R690CCW

P0C9401 P0C9402
P0C9801 P0C9802

P0C9501 P0C9502
P0C9901 P0C9902

P0TP101
P0TP201
P0TP301
P0TP401

7
4
7
4
7
4

P0TP501
P0TP1401
P0TP1501
P0TP1601

P0U2007

P0U2004

P0U2207

P0U2204

P0U2407

P0U2404

P0P1102

Res1
1.2K

R2

P0R201

6P0U206

P0R202

2
1

P0C302

J1
P0J1035
35
36
P0J1036
P0J1037
37
38
P0J1038
P0J1039
39
P0J1040
40
P0J1041
41
P0J104242
43
P0J1043
P0J1044
44
45
P0J1045
P0J1046
46
47
P0J1047
P0J1048
48
P0J1049
49
50
P0J1050
P0J1051
51
52
P0J1052
P0J1053
53
54
P0J1054
P0J1055
55
P0J1056
56
57
P0J1057
P0J1058
58
59
P0J1059
P0J1060
60
61
P0J1061
P0J1062
62
63
P0J1063
64
P0J1064
P0J1065
65
66
P0J1066
P0J1067
67
68
P0J1068

GND

C11
Cap
1uF

U2
AD620

8

C4
Cap
0.1uF

34
33
32
31
30
29
28
27
P0J1026
26
P0J1025
25
P0J1024
24
P0J1023
23
P0J102222
P0J1021
21
P0J1020
20
P0J1019
19
P0J1018
18
P0J1017
17
P0J1016
16
P0J1015
15
P0J1014
14
P0J1013
13
P0J101212
P0J1011
11
P0J101010
P0J1099
P0J1088
P0J107
7
P0J106
6
P0J105
5
P0J104
4
P0J103
3
P0J102
2
P0J101
1

GND

GND

Connector 68

GND

GND

GND

GND
GND

GND

GND

GND

GND
P0J1028
P0J1027

GND

-15

GND

GND

a15

P0J1029

P0J1030

P0J1031

P0J1032

P0J1033

P0J1034

C12
Cap
0.1uF

P0JP101

P0JP102

Header 2

JP1

C3
Cap
1uF

P0C4028

7/20/2007
Sheet of
C:\Documents and Settings\..\Phasecam_V6.SchDoc
Drawn By:

Number

GND

GND

GND

GND

GND

GND

GND

GND

GND
GND

GND

5

P0C101
P0C902
P0C901

8P0U208
P0U205

P0C301
P0C1101 P0C1102

P0C401
P0C1201 P0C1202

P0R602
P0R601
P0R1302
P0R1301

P0U305

7
4

P0C8201 P0C8202
P0C9001 P0C9002

P0U2607

P0U2604

P0C1701 P0C1702
P0C3501 P0C3502
P0C4701 P0C4702
P0C6601 P0C6602

P0R610CW
P0R610CCW
P0R630CW
P0R630CCW
P0R650CW
P0R650CCW

P0C7001 P0C7002
P0C7801 P0C7802

P0C1801 P0C1802
P0C3601 P0C3602
P0C4801 P0C4802
P0C6701 P0C6702
P0C7101 P0C7102
P0C7901 P0C7902
P0R670CW
P0R670CCW

P0R620CW
P0R620CCW

7
4
7
4
7
4

P0C2001 P0C2002
P0C3801 P0C3802

P0U2107

P0U2104

P0U2307
P0U2304

P0U2507

P0U2504

P0C3701 P0C3702
P0C4901 P0C4902
P0C6801 P0C6802

P0C5001 P0C5002
P0C6901 P0C6902
P0C7301 P0C7302

P0R640CW
P0R640CCW
P0R660CW
P0R660CCW

P0C7201 P0C7202
P0C8001 P0C8002

P0C1501 P0C1502

7
4

P0C8301 P0C8302
P0C9101 P0C9102

P0U2707

P0U2704

P0C8101 P0C8102
P0R680CW
P0R680CCW

P0C8401 P0C8402
P0C9201 P0C9202

P0C8501 P0C8502
P0C9301 P0C9302

69
P0J1069
70
P0J1070

P0P1202
P0P702
P0P902
P0P1002

166

Revision

D

C

B

A

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