Hartmann Wavefront Sensors for
Advanced Gravitational Wave
Interferometers

byAidan F. Brooks

Thesis submitted for the degree ofDoctor of Philosophy
in
The University of Adelaide
Department of Physics
School of Chemistry and Physics
July, 2007

2For my parents

2

Contents
Abstractvii

Statement of Originalityix

Acknowledgements

xi

List of Symbols

xiii

List of Figures

xv

List of Tables

xix

1 Gravitational wave astronomy
1.1 A better understanding of the Universe . . . . . . . . . . . . .

1
1

1.1.1

Gravitational waves . . . . . . . . . . . . . . . . . . . .

1

1.1.2

Sources of GW and potential new science . . . . . . . .

2

1.2 The challenge of GW interferometry . . . . . . . . . . . . . . .

3

1.2.1

First generation GW interferometry . . . . . . . . . . .

3

1.2.2

Advanced GWI enabling gravitational astronomy . . .

6

1.2.3

Higher stored power leads to wavefront distortion . . .

9

1.2.4

Adv. GWI: thermally-induced performance reduction . 11

1.3 Active compensation of WD . . . . . . . . . . . . . . . . . . . 12
1.3.1

Thermal compensation techniques . . . . . . . . . . . . 12

1.3.2

Wavefront sensor requirements for thermal compensation 14

1.4 Wavefront sensors for TCS . . . . . . . . . . . . . . . . . . . . 14
1.5 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17i

ii

CONTENTS

2 Interferometric test of H-V theory
19
2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Design and choice of test optic . . . . . . . . . . . . . . . . . . 20
2.3.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Choice of material . . . . . . . . . . . . . . . . . . . . 21
2.3.3 Induced WD vs. H-V predicition . . . . . . . . . . . . 22
2.4 Measurement and analysis of WD . . . . . . . . . . . . . . . . 25
2.4.1 Multiple heating beam sizes . . . . . . . . . . . . . . . 26
2.4.2 Experiment design . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Measurement procedure . . . . . . . . . . . . . . . . . 29
2.4.4 Analysis of MZ interference patterns . . . . . . . . . . 30
2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.1 Background noise . . . . . . . . . . . . . . . . . . . . . 36
2.5.2 Temporal development of measured wavefront distortion 36
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Hartmann sensor development
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Hartmann sensor basics . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Applications of the Hartmann sensor . . . . . . . . . .
3.2.2 Overview of Hartmann sensor . . . . . . . . . . . . . .
3.2.2.1 Absolute or differential operation . . . . . . .
3.2.3 Hartmann plate - CCD configurations . . . . . . . . . .
3.2.4 Variants of the traditional Hartmann sensor . . . . . .
3.2.4.1 Shack-Hartmann . . . . . . . . . . . . . . . .
3.2.4.2 Other variations . . . . . . . . . . . . . . . .
3.2.5 Temperature dependence of Hartmann sensors . . . . .
3.3 Hartmann sensor for advanced GWI . . . . . . . . . . . . . . .
3.3.1 Light source - coherent versus incoherent . . . . . . .
3.3.2 Optimization of Hartmann plate design . . . . . . . . .
3.3.2.1 Modelling . . . . . . . . . . . . . . . . . . . .
3.3.2.2 Simulations of cross-talk . . . . . . . . . . . .
3.3.2.3 HWS construction . . . . . . . . . . . . . . .

43
43
44
44
46
50
52
54
54
57
59
60
60
63
64
70
74

CONTENTS

iii
3.3.2.4

3.3.3

3.3.4

3.3.5

3.3.6

Estimated temperature dependent defocus error . . . . . . . . . . . . . . . . . . . . . . . . 74

CCD camera . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.3.1

Expected noise in CCD . . . . . . . . . . . . 77

3.3.3.2

CCD: Measurement of noise . . . . . . . . . . 78

3.3.3.3

Pixel size measurement. . . . . . . . . . . . 79

Centroiding algorithms . . . . . . . . . . . . . . . . . . 80
3.3.4.1

Weighted center of gravity (WCoG) . . . . . . 82

3.3.4.2

Fractional pixel centroiding (FPC) . . . . . . 83

3.3.4.3

Comparison of algorithms . . . . . . . . . . . 85

Wavefront reconstruction algorithm . . . . . . . . . . . 87
3.3.5.1

Modal wavefront reconstruction . . . . . . . . 87

3.3.5.2

Zonal wavefront reconstruction . . . . . . . . 88

3.3.5.3

Error propagation - From gradient to wavefront 91

Summary of noise sources . . . . . . . . . . . . . . . . 92

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4 Testing the sensor

95

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Lever arm calibration . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.1

Discounting other solutions due to pattern degeneracy

98

4.3 Incoherent vs. coherent sources . . . . . . . . . . . . . . . . . 100
4.4 Temperature dependence of HWS . . . . . . . . . . . . . . . . 106
4.5 Noise floor of the Hartmann sensor . . . . . . . . . . . . . . . 108
4.6 Accuracy test using known wavefront change . . . . . . . . . . 111
4.6.1

Analytic form of known wavefront change. . . . . . . 111

4.6.2

Experiment design . . . . . . . . . . . . . . . . . . . . 113

4.6.3

Calibration of the origin for z0 . . . . . . . . . . . . . . 115

4.6.4

Mitigating cyclic temperature fluctuations . . . . . . . 118

4.6.5

Experiment procedure . . . . . . . . . . . . . . . . . . 118

4.6.6

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

iv

CONTENTS

5 HOPTF WD measurement

127

5.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.2

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3

Description of measurement system . . . . . . . . . . . . . . . 128

5.4

5.5

5.6

5.3.1

The high-optical-power cavity at HOPTF . . . . . . . . 128

5.3.2

Installation of the HWS at HOPTF . . . . . . . . . . . 130

5.3.3

Reduction of environmental noise coupling into HWS . 132

Measurement procedure and analysis . . . . . . . . . . . . . . 135
5.4.1

Measurement procedure . . . . . . . . . . . . . . . . . 135

5.4.2

Analysis of wavefront distortion . . . . . . . . . . . . . 138
5.4.2.1

Off-axis WD to on-axis WD . . . . . . . . . . 138

5.4.2.2

On-axis WD to cavity-mode WD . . . . . . . 141

5.4.2.3

Defocus of cavity-mode WD . . . . . . . . . . 143

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5.1

Measured off-axis wavefront distortion . . . . . . . . . 145

5.5.2

Temporal development of cavity defocus . . . . . . . . 145

5.5.3

Measured distortion and stored power correlation . . . 148

5.5.4

Measured distortion and cavity mode size correlation . 150

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6 Off axis measurement of distortion

155

6.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.2

Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.3

Analysis of off-axis WD . . . . . . . . . . . . . . . . . . . . . . 156
6.3.1

6.4

6.5

6.6

A zonal representation of T (r, z) . . . . . . . . . . . . . 158

Numerical simulation of voxel analysis . . . . . . . . . . . . . 161
6.4.1

Simulation procedure . . . . . . . . . . . . . . . . . . . 161

6.4.2

Simulation results . . . . . . . . . . . . . . . . . . . . . 163

Proof-of-principle of off-axis analysis . . . . . . . . . . . . . . 165
6.5.1

System layout . . . . . . . . . . . . . . . . . . . . . . . 165

6.5.2

Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.5.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

CONTENTSv

7 Conclusion
171
7.1 Review of aims . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.2 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . 172
7.3 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . 173
A Additional derivations
175
A.1 Analytic form of Hello-Vinet solution . . . . . . . . . . . . . . 175
A.2 Variance of a digitized value . . . . . . . . . . . . . . . . . . . 177
A.3 ABCD cavity calculation . . . . . . . . . . . . . . . . . . . . . 178
B Computer code181
B.1 Hartmann plate optimization code . . . . . . . . . . . . . . . . 181
B.2 Cross-talk analysis: diffraction propagation . . . . . . . . . . . 189
B.3 FEM of ITM and CP thermal lens at the HOPTF . . . . . . . 192
C Relevant papers
203
C.1 Gen. Relativ. Gravit. 37, 1575-1580 (2005) . . . . . . . . . . . 203
C.2 Opt. Express 15 (16), 10370-10375 (2007) . . . . . . . . . . . 210

vi

CONTENTSAbstract
The upgrade of gravitational wave interferometers from first- to secondgeneration will enable gravitational wave astronomy to become a reality,
providing a new view of our Universe. One of the numerous upgrades inmoving to the second-generation is the increased intra-cavity laser power in
the interferometer optical cavities, which should reduce the shot-noise in theinterferometer. Unfortunately, small, but finite, absorption of optical powerin the intra-cavity optics will result in thermal gradients within the interferometer optics, which causes wavefront distortion of the cavity eigenmode.
Numerical modelling, based on an analytic solution for the thermal gradient,has shown that the effect of the accumulated wavefront distortion on thecavity mode is to degrade the sensitivity of the interferometer, and it may
cause instrument failure. The wavefront distortion must, therefore, be compensated and actuators for this purpose have been proposed but a suitablewavefront sensor is required.
In this thesis, I describe an interferometric validation of the analytic solution for the thermal gradient which, as discussed above, is central to numerical modelling of advanced GWI. The bulk of the thesis, however, concentrates
on the development and testing of a wavefront sensor for wavefront compensation. I shall describe a Hartmann wavefront sensor that can measure
wavefront changes, such as those due to substrate and coating absorption in
GWI, with unprecedented precision and accuracy. It is simple to optimize
and reliable, and appears very suitable for measuring wavefront distortion inadvanced gravitational wave interferometers.
vii

viii

Abstract
I shall also describe the deployment of the Hartmann wavefront sensor

at the High Optical Power Test Facility in Western Australia to measure
wavefront distortion in the optics of a high-optical-power cavity. Despite significantly increased environmental noise, the sensor measured the wavefront
distortion in the optics with a precision necessary for deployment in an advanced gravitational wave interferometry. The Hartmann wavefront sensor
measurements were verified using measurements of the intra-cavity mode size
and intra-cavity power.
The potential use of the wavefront sensor for off-axis measurement ofwavefront distortion in a gravitation wave interferometer prompted an investigation into reconstruction of the on-axis wavefront distortion. The proof-ofprinciple of the resulting single-view off-axis analysis is demonstrated using
simulated and actual measurements.

Statement of Originality
This work contains no material which has been accepted for the award of
any other degree or diploma in any university or other tertiary institution
and, to the best of my knowledge and belief, contains no material previously
published or written by another person, except where due reference has been
made in the text.

I give consent to this copy of my thesis being made available in the University
Library.

The author acknowledges that copyright of published works contained within
this thesis (as listed below) resides with the copyright holder/s of those works.

SIGNED: ..................................................

DATE: .....................

Supervisors: Peter J. Veitch and Jesper Munch.

ix

xStatement of Originality

Acknowledgements
First and foremost I would like to thank my supervisors, Peter Veitch andJesper Munch. I have really enjoyed working with you all these years. Your
support and guidance has been invaluable and I am truly grateful for all yourassistance.
To Thu-Lan Kelly: thank you so much for all your help on the Hartmann
sensor over the years. You were always there for me when I needed thosequestions answered about your code.
To David Blair, David McClelland and everyone in ACIGA: your help andassistance at AIGO is most appreciated. I’d especially like to thank Chunnong Zhao and Yan Zewu: I couldn’t have got my results without your help
and all those late nights in the lab were so much more bearable with you guys
around. Thanks also to Phil Willems for your assistance at AIGO. I’ll missdiscussing the Simpsons and Family Guy whilst simultaneously performing
large scale Gaussian beam size measurements.
To Murray Hamilton; thanks for all your words of wisdom. I’ll missdemonstrating the Photonics II experiments with you.
I’d like to thank Trevor Waterhouse and Bob Hurn for all your help over
the years. If I ever needed a chat, you guys were always available.
To my fellow students in the Physics Department, thanks for all yoursupport and cheer over the years. In particular, Brony, Skye and Catherine:
thanks for always being up for a drink at the Unibar.
I’d like to acknowledge my colleagues in the Optics Group: Tom Rutten, Nick Chang, Miftar Ganija, Shu-Yen Lee, Keiron Boyd, Sean Manning,
xi

xii

Acknowledgements

Damien Mudge, Francois Jeanneret, David Ottaway and Alex Hemming. You
guys made it fun to be at work. I’ll miss those long afternoon discussions in
the lunch room when we should have been working. In particular, I’d like tothank Blair Middlemiss for all the help in the workshop and for always beingaround when I needed a reality check.
To David Hosken and Matthew Heintze: quite simply, I wouldn’t havemade it without your help. I wish you all the best in the future and hope
our paths will cross again soon.
To Peter and the crew at The Coffee Club: you kept me awake. Thanks
for all the cheer and smiles over the last few years.
To my friends: Matthew, Sarah, Poh, Zoe, Andrew, Lily and Paul: I love
you guys. You were always there for me, especially in those darkest hours.
Thank you.
To my family: John, Megs, Lizzy, Peter and Jack: I can’t believe howmuch we’ve all matured over the years. We’ve moved to different houses andcities, but have grown closer as a family. You will always be part of myworld.
Lastly to Mum and Dad: your love and support over this time of trial
has been beyond measure. Words cannot express how much it has meant tome and so I dedicate this thesis to you.

List of Symbols

Throughout this thesis, several symbols will be used repeatedly to represent
specific quantities or parameters, the following is a list of these symbols andshort descriptions for the readers convenience. This list is not exhaustive
but every effort has been made to maintain conformity of symbols used here.
Wherever possible standard symbols and notation have been used which appear in most laser texts.

Iobj (x)...

Intensity profile of object beam

Iref (x)...

Intensity profile of reference beam

Iint (x)...

Interference pattern between two beams

ψ(x, ti )...

Wavefront distortion at time ti

L...

Distance between Hartmann plate and CCD: lever arm

W...

Wavefront∆W...

Wavefront change (aberration)

S...

Primary aberration defocusk...

Order of Parabolic function

hp

...

Distance between holes on Hartmann plate

Sthermal...

Apparent defocus due to thermal effects

Np

...

Number of photons in a given time interval

ps

...

Pixel size

xc

...

x centroid
xiii

xiv

List of Symbols
C∆W

...

Southwell noise coefficient (variance)

σ∆W

...

RMS wavefront difference between adjacent holes

Nref

...

Number of samples in reference centroid set average

Navg

...

Number of images averaged over

z0

...

nominal distance from fibre-couple output of SLD
to Hartmann plate

∆z

...

displacement of fibre-couple output of SLD(off )...

2D off-axis wavefront distortion profile(off)
WDSAPH...

vertical cross-section through WD(off ) correspondingWD

to sapphire optic
(off)
WDFS

...

vertical cross-section through WD(off ) corresponding
to fused silica optic

KE1 (t)...

coefficient describing the elongation of the sapphire
distortion

KE2 (t)...

coefficient describing the elongation of the fusedsilica distortion

KC1 (t)...

contribution of the fused-silica distortion to thesapphire distortion cross-section

KC2 (t)...

contribution of the sapphire distortion to the
fused-silica distortion cross-section(on)
WDSAPH
(on)
WDFS

...

on-axis wavefront distortion due to sapphire...

on-axis wavefront distortion due to fused-silica

WDcavity...

wavefront distortion experienced by cavity mode

wITM (S)...

cavity mode size at ITM as function of S

wETM (S) ...

cavity mode size at ETM as function of S

Vi j (r, z)...

i j-th annular volume element

Vi j (r, z)...

i j-th annular volume element

wi j (x, y)...

path integral through Vi j (r, z) projected onto
plane Pexit

List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7

Schematic of a first generation GWI . . . . . . . . . . . . . . .
Sensivity curve for Initial LIGO . . . . . . . . . . . . . . . . .
Schematic of a second generation GWI . . . . . . . . . . . . .
Absorption in ITM causes temperature distribution . . . . . .
Radiative heating from a shielded heating ring . . . . . . . . .
Staring heating of ITM . . . . . . . . . . . . . . . . . . . . . .
Advanced LIGO TCS schematic . . . . . . . . . . . . . . . . .

5
7
8
10
13
13
15

2.1 Schematic showing volumetric absorption optical power . . . . 20
2.2 Photograph of Schott BG20 filter glass cylinder . . . . . . . . 23
2.3 FEM prediction of thermal lensing with different levels of absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Schematic of experiment to measure thermal lensing in a cylindrical test optic . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Linear carrier analysis demonstration: Part 1 . . . . . . . . . . 33
2.6 Linear carrier analysis demonstration: Part 2 . . . . . . . . . . 34
2.7 Linear carrier analysis filtering examples . . . . . . . . . . . . 34
2.8 Linear carrier analysis demonstration: Part 3 . . . . . . . . . . 35
2.9 Transient WD profiles measured by MZ interferometer at t1 = 2.5 s
and t1 = 8.5 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.10 Transient WD profiles measured by MZ interferometer at t1 = 29.2 s
and t1 = 100.0 s. . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.11 Temporal development of a) optical path difference and b)
defocus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 The original Hartmann test . . . . . . . . . . . . . . . . . . . 45
xv

xvi

LIST OF FIGURES
3.2
3.3
3.4
3.5
3.6
3.7
3.8

An aberrated wavefront incident on a Hartmann sensor . . . .
Example Hartmann spot patterns . . . . . . . . . . . . . . . .
Rayces wavefront distortion figure . . . . . . . . . . . . . . . .
Hartmann plate - CCD configurations . . . . . . . . . . . . . .
Schematic of a Shack-Hartmann sensor . . . . . . . . . . . . .
Schematic of thermal expansion of a Hartmann plate . . . . .
Coherence length measurement of Agilient HFBR-1414 fibre
coupled LED . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Theoretical Hartmann plate diffraction pattern - poor result .
3.10 Theoretical Hartmann plate diffraction pattern - good result .
3.11 Microscope views of the optimum Hartmann plate . . . . . . .
3.12 Single spot - Simulated intensity patterns with crosstalk . . .
3.13 Multiple spots - Simulated intensity patterns with crosstalk .
3.14 Hartmann sensor construction . . . . . . . . . . . . . . . . . .
3.15 Relative error in digital number in CCD vs digital number . .
3.16 Lateral translation of a diffraction pattern on the CCD . . . .
3.17 Pixel size measurement - diffraction pattern cross-sections . .
3.18 Schematic of the fractional centroiding algorithm . . . . . . .
3.19 Schematic of nearest neighbours . . . . . . . . . . . . . . . . .
3.20 Noise coefficient in wavefront reconstruction . . . . . . . . . .
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

47
49
51
53
55
58
62
67
68
69
72
73
75
80
81
81
86
89
92

Schematic of the experiment used to calibrate the lever arm. . 97
Centroid pattern displacement in the experiment to measure
the lever arm . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Measured and simulated Hartmann spot cross-sections at different lever arms . . . . . . . . . . . . . . . . . . . . . . . . . 101
Schematic of experiment to test the precision of the HWS . . . 102
Distribution of centroids when illuminated with incoherent
and coherent sources . . . . . . . . . . . . . . . . . . . . . . . 103
RMS error in the centroids vs time when illuminated withincoherent and coherent sources . . . . . . . . . . . . . . . . . 105
Temperature of HWS and defocus vs time . . . . . . . . . . . 107
Defocus vs. Temperature . . . . . . . . . . . . . . . . . . . . . 108
Vector field and wavefront map of background noise in HWS . 110

LIST OF FIGURES

xvii

4.10 Reduction in the noise in the HWS as measurement is averaged
over N images . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.11 Schematic showing the change in direction of a Hartmann ray
in the presence of defocus . . . . . . . . . . . . . . . . . . . . 112
4.12 Schematic of experiment used to measure a known wavefrontchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.13 Cross-section of interference fringes used to measure ∆z indefocus measurement . . . . . . . . . . . . . . . . . . . . . . . 115
4.14 Calibrating the translation stage . . . . . . . . . . . . . . . . . 117
4.15 Cyclic temperature fluctuations appearing in defocus measurement, phase measurement and temperature sensors when measuring a known wavefront change . . . . . . . . . . . . . . . . 121
4.16 Matching defocus measurements with peak and troughs in asinusoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.17 Measured gradient vs position for a known wavefront change . 123
5.1 Schematic of the Fabry-Perot cavity with high stored power,
located at the HOPTF. . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Schematic showing HWS installed to measure thermal lensing
in the ITM and CP. . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3 Photograph showing interior of the HOPTF vacuum system . 132
5.4 ITM and suspension system at the HOTPF . . . . . . . . . . 133
5.5 Aperture through ITM and suspension system . . . . . . . . . 134
5.6 RMS noise in HWS measurement versus time upon installation
at HOPTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.7 Photographs of beam tubes surrounding probe beam . . . . . 136
5.8 Residual background wavefront difference . . . . . . . . . . . . 137
5.9 Simulated off-axis optical path distortion . . . . . . . . . . . . 139
5.10 Conversion parameters for off-axis distortion . . . . . . . . . . 141
5.11 Distortion experienced by cavity mode and probe beam . . . . 143
5.12 Vector fields showing the gradient of the wavefront distortionin sapphire and fused silica . . . . . . . . . . . . . . . . . . . . 146
5.13 Contour plots of wavefront distortion in sapphire and fusedsilica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

xviii

LIST OF FIGURES

5.14 Vertical cross-sections of distortion in ITM and CP . . . . . . 148
5.15 Temporal development of the measured thermal lenses . . . . 149
5.16 Finite element prediction of total defocus using measured intracavity power and measured defocus . . . . . . . . . . . . . 151
5.17 Cavity mode size versus time - measured and predicted from
HWS measurement . . . . . . . . . . . . . . . . . . . . . . . . 152
6.1
6.2
6.3
6.4
6.5
6.6
6.7

Schematic showing off-axis transmission through an axiallysymmetric refractive index distribution . . . . . . . . . . . . . 157
Cut-away of a cylinder containing annular voxels . . . . . . . . 159
An example annular volume element and associated wavefront
distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Simulation: voxel reconstruction of temperature distribution
and OPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Schematic of the experiment used to test the voxel analysis . . 166
Contour plots of on-axis and off-axis wavefront distortion . . . 168
On-axis measured and off-axis reconstructed wavefront distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

List of Tables
2.1 Properties of the BG20 glass cylinder used for measuring absorption induced wavefront distortion . . . . . . . . . . . . . . 22
3.1 A comparison of the relative merits of Shack-Hartmann andHartmann sensors. . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Properties for Agilent fibre-coupled LED . . . . . . . . . . . .
3.3 Best five Hartmann plate configurations as rated by the figureof merit (FOM). . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Final design parameters for the Hartmann sensor . . . . . . .
3.5 Cross talk estimation for different lever arms . . . . . . . . . .
3.6 Parameters of Dalsa 1M60 CCD camera . . . . . . . . . . . .
3.7 Noise sources in CCD . . . . . . . . . . . . . . . . . . . . . . .
3.8 Digitization noise in CCD . . . . . . . . . . . . . . . . . . . .
3.9 Precision of centroiding algorithms in the presence of realistic
noise sources . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57
61
69
69
74
76
78
79
86

4.1 Summary of the measurements of known wavefront change . . 124
5.1 Physical parameters of the HOPTF Fabry-Perot cavity optics 129
5.2 Sizes of thermoelastic deformation and the elastooptic effectrelative to the thermooptic effect. . . . . . . . . . . . . . . . . 142

xix

xx

LIST OF TABLES

Chapter 1
The rewards and challenges ofgravitational wave astronomy
1.1

The promise of new science and a better
understanding of the Universe

1.1.1

Gravitational waves

Gravitational waves (GW) are ripples in spacetime, predicted by Einstein in1918 [1] as a solution to the General Theory of Relativity. Although they
have never been directly observed, there is excellent indirect evidence for
their existence. The pulsar PSR B1913+16, discovered by Hulse and Taylor
in 1974 [2], shows a decay in its orbital period that is consistent with loss ofenergy through the emission of gravitational waves as predicted by General
Relativity [3]. This observation earned Hulse and Taylor a Nobel Prize in
1993.
The generation of GW is analogous to the generation of EM waves. EM
waves are generated by the acceleration of charges and GW are generated
by the acceleration of masses. Whilst EM waves are dipolar in nature, thelowest order GWs possible are quadrupolar in nature (monopolar GW anddipolar GW would violate conservation of mass-energy and conservation of1

2

CHAPTER 1. GRAVITATIONAL WAVE ASTRONOMY

momentum, respectively [4]). Thus a passing quadrupolar GW exerts a tidal
strain on space-time, stretching it along one axis and squeezing it along theperpendicular axis. This effect is characterized by the tidal strain amplitude,h(t), such that a passing gravitational wave of amplitude h(t) causes the
distance, Ltest , between two fiducial test masses to fluctuate by a time-varying
differential amount, ∆Ltest (t) = h(t) Ltest , [5]. The observable quantity h(t)decays as 1/r, where r is the distance between the source and the pointof observation, a fact that becomes very important in the discussion of themeasurement of GW.
The interaction between GW and matter is extremely weak and, therefore, the Universe is effectively transparent to GW and so the information
they carry from their sources is uncontaminated [4]. Given that they may
originate from areas, discussed in the following section, from which there islittle to no direct EM radiation, GW illuminate the Universe in an entirely
new and rich way.

1.1.2

Astrophysical sources of GW and the new science they may reveal

There are a variety of predicted astrophysical sources of GW. These sources
each promise to reveal different types of new science in their GW radiation.
The following is a short list of the main types of sources and the associatednew science.
• Inspiral sources. Compact binary systems, such as PSR B1913+16,
containing either, two neutron-stars (NS), a NS and a black-hole (BH),or two BHs radiate orbital energy as GWs with a characteristic ’chirp’signal [5]. When the first direct detection of GW occurs it will mostprobably be from the final decay and merger of the two objects in acompact binary system as these sources are very powerful, are predicted to occur frequently enough to be seen and as the time dependence of the GW signal emitted during the inspiral is believed to be1.2. THE CHALLENGE OF GW INTERFEROMETRY

3

well-known. Detection of such a source would reveal new astrophysical information about the late inspiral and merger epochs of such an
inspiral. Conversely, non-detection would say that the current models
of GW generation are incorrect.
• Continuous sources, generally rotating asymmetric neutron stars (which
may additionally be observable as pulsars), would reveal observationsof extremely dense states of matter [4] and maybe reveal an information
about free-quark matter [6].
• Burst sources. Detection of GW from the catastrophic core collapseof massive stars would help our understanding of these violent events,

especially when combined with, for example, neutrino detections of thesame event [4].
• Stochastic sources. Random source, for example, the gravitational-

wave background from cosmological inflation [4], enabling testing of
new unification physics.

The observation of GW holds great promise in terms of increasing ourunderstanding of high energy astrophysical processes and non-linear General
Relativity. Additionally, it holds the promise of potentially revealing evidence of new types of physics. As such grativational waves are a prominent
topic in the scientific community and several large scale projects are underway to enable their direct detection and the development of observational
gravitational wave astronomy.

1.2

The challenge of GW interferometry

1.2.1

First generation GW interferometry

The tidal strain effect of GWs on fiducial test masses can be observed bymeasuring the differential length change between a set of two test masses
separated by a distance L1 along one line and a second set of two test masses

4

CHAPTER 1. GRAVITATIONAL WAVE ASTRONOMY

separated by a similar distance L2 along a perpendicular line. This measurement can be made using interferometry, hence the name gravitational wave
interferometry (GWI).
A schematic of a GWI is shown in Figure 1.1. It is, essentially, an Lshaped Michelson interferometer formed from a laser, a beamsplitter, twoarms of length L1 and L2 at right angles and a photo-diode measuring theoutput of the interferometer [7] [8]. The signal-to-noise is maximized byaveraging for a dark fringe at the PD, and thus most of the power propagatesback towards the power recycling mirror (PRM) which reflects it back into
the interferometer. A GW, of correct polarization, passing through the device
will compress one arm and distend the other. The change in the path length
difference between the two arms, ∆Larm (t) = L2 (t) − L1 (t), is equal to thetidal strain amplitude, h(t), multiplied by the average arm length, Larm .

Hence, there is a time-varying phase change between the two arms and atime-varying change in the intensity on PD, both of which are proportional
to the tidal strain amplitude.
The interferometer arms are kilometer-scale length (4 km in Initial LIGO)
in order to increase the length of time that the light in the arms is exposed toa GW [7]. This has the effect of increasing the phase-delay between the twoarms. The phase-delay is further increased by the addition of Fabry-Perotcavities.
Ignoring other noise sources (seismic and suspension thermal noise), theminimum strain, hmin , that can be measured is determined by photon shotnoise at the beam splitter, which is inversely proportional to the squareroot of the number of photons, Np , incident on the beam splitter in a given
exposure time [7]. Therefore, hmin , is given byhmin ≈λ1q
B Larm Np(1.1)

where B is the mean number of times the light bounces back and forthin the Fabry-Perot cavities (proportional to the cavities’ finesse) [7]. At

1.2. THE CHALLENGE OF GW INTERFEROMETRY

5

End Test
Mirror (ETM)

4 km Fabry−Perot cavities
Mode Cleaner (MC)Input Test
Mirror (ITM)

ETM

Beamsplitter (BS)
Laser

6W

Power Recycling
Mirror (PRM)

100 W

13 kW

Photodiode (PD)GW readout

Figure 1.1: Schematic of a first generation GWI. A power-recycled FabryPerot long-baseline Michelson interferometer.

6

CHAPTER 1. GRAVITATIONAL WAVE ASTRONOMY

low frequencies, (< 200 Hz), the minimum detectable strain is degraded byseismic and suspension thermal noise. The minimum detectable strain athigh frequencies, (> 200 Hz), can be decreased by increasing the number ofphotons incident on BS, accomplished by placing the PRM between the laser
and BS to form a resonant cavity between PRM, ETM1 and ETM2 .
The target strain sensitivity for Initial LIGO is approximately 3 × 10−22@ 150 Hz [5]. The current sensitivity of Initial LIGO, expressed in units of√m/ Hz, is shown in Figure 1.2. Converting to strain, by dividing by the 4
km length of the interferometer arms and multiplying by the square root of
the frequency, shows that Initial LIGO has achieved its design stain sensitivity of approximately 3 × 10−22 @ 150 Hz. The strain sensitivity of Initial
LIGO is such that there is a reasonable, though not certain, chance of detect-ing a GW in a year’s worth of observations [8]. However, the aim of Initial
LIGO is to demonstrate the up-scaling of laboratory interferometers to longbaselines with the projected improvement in sensitivity. In order to increase
the probable rate of detection it necessary to move to improve the sensitivity of the interferometers, which requires the next-generation ’advanced’
interferometers.

1.2.2

Advanced GWI enabling gravitational astronomy

The purpose of second-generation or ’advanced’ interferometers is to increase
the rate of detection of GWs to a point where GW astronomy becomes feasible. The justification for trying to accomplish this is related to the 1/r decayof h(t). If a second-generation interferometer is N times more sensitive thanfirst-generation one, it can detect a ’standard candle’ N times further away.
The beauty of this is that it operates in all three dimensions and the volumeof space that is searched, and hence the event rate of GW, increases as N 3 .
Therefore, if an advanced interferometer can achieve a 10× increase in sensitivity, the number of detectable sources increases by a factor of 1000, andthe estimated rate of detection of GWs will go from less than one detection
every year to an average rate of one detection a day! [5]1.2. THE CHALLENGE OF GW INTERFEROMETRY

7

Figure 1.2: The blue curve, L1:LSC-DARR
ERR, shows the current sensitiv√
ity of Initial LIGO in units of m/ Hz, recorded 11th July 2007 at 04:56:07
UTC [9]. The sensitivity is limited by photon shot noise for frequencies >about 200 Hz.

8

CHAPTER 1. GRAVITATIONAL WAVE ASTRONOMY
ETM

4 km Fabry−Perot cavities
Input Mode
Cleaner (IMC)
ITM

PRM
BS
Laser

180 W

4 kW

ETM

2 kW
2 kW

800 kW

Signal Recycling
Mirror (SRM)

PD

Output Mode
Cleaner (OMC)
GW readout

Figure 1.3: Schematic of a second generation GWI. A power-recycled signalrecycled Fabry-Perot long-baseline Michelson interferometer, including theapproximate power levels expected in the input laser and interferometer cavities.
A schematic for an advanced GWI is shown in Figure 1.3. It largely
resembles the first-generation GWI and the majority of the improvements tothe device have been highlighted. The improvements targeted for Advanced
LIGO are discussed below and divided into two groups, the first group arethose associated with reducing noise sources not discussed in detail in thisthesis, and the second group are those associated with increasing the numberof photons at the BS and hence decreasing photon shot noise.
• The improvements associated with other noise sources are:
– Better seismic isolation and suspension systems to reduce seismic
motion coupling into the test masses

1.2. THE CHALLENGE OF GW INTERFEROMETRY

9– Heavier test masses made of higher quality material to decrease
Brownian thermo-elastic noise and reduce radiation pressure noise.
– An output mode cleaneradded to improve the spatial profile ofany light incident on the PD.
• The improvements associated with decreasing shot noise:– Increasing the laser power from from 10 W to 180 W [5]– Increasing the power recycling factor from 30 to 80 [5] to increase
the total stored power in the PRC from 100 W to ≈ 4 kW.
– Adding a signal recycling mirror to enable resonant sideband extraction. In this system, the interferometer is turned into a tunable resonant cavity. This is useful for monitoring continuous
sources at known frequencies [10].
As a result of the improvements Advanced LIGO is expected to be sensitive enough to measure strains as small as ≈ 3 × 10−23 [11] in the range

60-300 Hz [5], resulting in an approximately 1000× increase in event rate for
expected sources [11].

1.2.3

Higher stored power leads to wavefront distortion

The increase in optical power, while reducing the shot noise at the BS, also
has some side-effects which adversely affect the interferometer. The high reflectivity (HR) coatings, anti-reflection (AR) coatings and optical substrates
absorb a small but finite amount of the optical power in the PRC and thearm cavities, as illustrated in Figure 1.4. This is particularly significant forthe substrates and the HR and AR coatings of the ITMs, the substrate andcoating of the BS and the HR coating of the ETMs [12].
The absorbed power causes local heating of the optic and results in aninternal temperature gradient. An analytic solution for the temperature

10

CHAPTER 1. GRAVITATIONAL WAVE ASTRONOMY

HR coating
Arm Cavity
Mode (800 kW)

Power Recycling
Cavity Mode (2 kW)

T(r, z)AR coating

ITM

Figure 1.4: Absorption in ITM causes temperature distribution
distribution, T (r, z), which has not been directly validated, was reported by
Hello and Vinet [13]. The full form of this solution is outlined in Appendix
A.1. The integral of T (r, z) along the z-axis through the optic is also given
there. As discussed by Lawrence [12], the distribution changes the opticalproperties in three ways:

1. Thermo-optic effect: the refractive index changes with temperature
and, therefore, there is a change in the optical length along a path S
through the material given byψ (r)TO =dn Z
T (r, z) ds
dT S

where dn/dT is the thermo-optic coefficient. This effect is manifested
within the substrate of the optic.

2. Elasto-optic effect: thermal expansion results in mechanical strain which
results in local refractive index changes. The change in optical length1.2. THE CHALLENGE OF GW INTERFEROMETRY

11

along a path S through the material is given byψ (r)EO ≈ α p1 1

ZS

T (r, z) ds

where α is the coefficient of thermal expansion and p1 1 is the componentof the elasto-optic tensor along the probe beam polarization axis. Thiseffect is also manifested within the substrate of the optic.
3. Thermo-elastic effect: The surfaces of the optic expand along the optical axis and the resulting change in optical path length is given byψ (r)TE ≈ noptic α

ZS

T (r, z) ds

where noptic is the refractive index of the optic. It also changes thecurvature of the surfaces, which affects reflection. Obviously, this effectis also manifested at the surfaces of the optic.
The resulting changes to the mirror properties adversely affect the performance of the interferometer, as discussed in the next section.

1.2.4

Advanced GWI: thermally-induced performance
reduction

Essentially, the thermal distortion in the cavity optics scatters power fromthe fundamental spatial cavity mode into higher order spatial modes that are
not resonant, as discussed by Strain et al. [14]. This has several effects on
the behaviour of the interferometer.
1. Power scattered out of the carrier mode in the arm and power recycling
cavities reduces the number of photons at the beam splitter.
2. Differential thermal lensing between the two arms of the interferometerincreases the phase noise at the dark port [12] and results in a mismatch
between the overlapping wavefronts at the beamsplitter [14].

12

CHAPTER 1. GRAVITATIONAL WAVE ASTRONOMY
3. Power is scattered out of the RF sidebands whose uses include, but are
not limited to, measurement of the length of the PRC and also, in an
AC readout scheme, readout of the GW signal itself [15].
These effects have been predicted in numerical models of high opticalpower cavities and advanced GWI (using the MELODY code [16]) and thethermal lensing has been shown to ultimately drive the advanced GW interferometer to failure [12]. It should be noted than these models are predicatedon the assumption that the Hello-Vinet theory is correct.
The problem in a nutshell: the ability to investigate new physics withadvanced GWI is limited, in part, by thermal lensing in the optics of the
GWI.

1.3

Active compensation of absorption-induced
wavefront distortion

1.3.1

Thermal compensation techniquesCompensation of wavefront distortion has been suggested as a solution to
the problem of performance reduction of the interferometer and various techniques have been demonstrated. A brief summary of the suggested techniques
is given below.
• Radiative heating from a shielded heating ring of the areas of a distorted

optic not heated by the cavity mode, as illustrated in Figure 1.5. Thiswas shown to be effective in reducing the overall distortion in a heated
optic by Lawrence et al. [17].

• Conductive heating of the barrel of a compensation plate in a highpower optical cavity was shown to produce a negative lens and com-pensate for wavefront distortion in the intracavity optics by Zhao et al.
[18].

1.3. ACTIVE COMPENSATION OF WD

13

Heating ringTransmissive optic

Cavity mode

Aluminium shielding

Figure 1.5: Cross-section showing radiative heating from a shielded heating
ring
Compensation Plate

CO2 laser + mask

Figure 1.6: Staring heating of ITM• A CO2 laser beam with a specific intensity profile that ’stares’ at a distorted optic, as illustrated in Figure 1.6, has been shown to be effective
in compensating wavefront distortion in Initial LIGO [19].
• A scanning CO2 raster pattern was proposed by Ryan Lawrence [12].

The initial design of Advanced LIGO does not include such an actuator
within the Thermal Compensation System (TCS), but the design is left
open to this installing this option [20].

All of the actuators listed above, however, require an accurate and sensitive wavefront sensor to measure the distortion and compensation, in order

14

CHAPTER 1. GRAVITATIONAL WAVE ASTRONOMY

to function as part of a closed-loop active thermal compensation system.

1.3.2

Wavefront sensor requirements for thermal compensation

The requirements for an accurate and sensitive wavefront sensor for Advanced
LIGO have been investigated by Willems [21]. In Advanced LIGO, “theextraction efficiency of the GW sidebands through the signal recycling cavity(SRC) sets the most strigent requirement on TCS”. The maximum acceptablephase error for the sidebands per round trip is 0.08 radians @ 1064 nm =13.5 nm. This will ensure that no more than 0.1% of power is scattered outof the sidebands per round trip, or no more than 5% in total. The following
requirements were therefore set for a wavefront sensor.
1. Sensor should have precision at least 10× better than this = 1.35 nm(or λ/467 @ 632.8 nm) [21]2. The spatial resolution must be such as to not miss a feature from apoint absorber [21]. This requires a 1 cm × 1 cm grid to cover acircular region 224 mm in diameter, approximately 23 × 23 sample

points.

3. The sensor may need to be installed in an off-axis configuration [12].
The current TCS design for Advanced LIGO [20] requires an active thermal compensation system with dedicated wavefront sensors and actuators foreach optic, as illustrated in Figure 1.7.

1.4

Wavefront sensors for TCS

Two types of potential wavefront sensors for TCS are interferometric andHartmann-type sensors, the relative merits of which are presented in thefollowing discussion.

1.4. WAVEFRONT SENSORS FOR TCS

15

ETM
CO2 laser heater
Wavefront sensor beam (WFSB)WFSB 1

Ring heater

WFSB 2

Shielded ring heater
ITM

WFSB 5WFSB 4WFSB 3WFSB 6

ETM
PRM

BS
ITM

SRM

WFSB 7

Figure 1.7: Schematic of initial design for wavefront sensors and TCS for
Advanced LIGO [20]

16

CHAPTER 1. GRAVITATIONAL WAVE ASTRONOMY
An interferometer is capable of functioning as a wavefront sensor when its

output is read by a CCD array. Phase shifting interferometry has achievedvery high accuracy (1 nm) and precision (0.1 nm) [22]. This was accomplished by a attaching piezo-electric transducer (PZT) to a mirror in thereference arm of the interferometer, appyling a sinusoidally varying voltage
to the PZT, and measuring the time-varying interference pattern on the
CCD. The main drawback of an inteferometer is that it is highly intolerantof mechanical vibrations [23] and alignment issues. Additionally, the interferometric measurement of phase distortion induced by an object requires
two separate path matched laser beam paths, increasing system complexity.
Hartmann-type sensors, discussed in detail in Chapter 3, can be summarized here as being based upon a geometrical technique that samples thegradient of the wavefront profile at a series of discrete points. Given thatthey only require a single path to measure wavefront distortion induced byan object, they are relatively insensitive to alignment (in comparison to aninterferometer) and they offer a simpler alternative to interferometry. A
Hartmann sensor has been reported with a precision of 2.1 nm (λ/500 @
1064 nm) [24]. Additionally, commercially available variants called ShackHartmann sensors, also discussed further in Chapter 3, are quoted as having
an accuracy of λ/150 @ 633 nm in an array of 25 × 19 lenslets [25].
The difference in the operation of the two sensors is summarised as follows: an interferometer measures the difference between two different paths
recorded at the same time and a Hartmann-type sensor measures the difference between the same path recorded at two different times. Koch et al.
[26] compared a SH sensor and a phase shifting interferometer (PSI) and determined that they had comparable performance. They also suggested PSIs
could be replaced with SH sensors in order to reduce system complexity.
Given the comparable performance of SH and Hartmann sensors, the conclusions of Koch et al. can be assumed to also apply to traditional Hartmamnn
sensors.
Given the existing complexity of GWI design (see Figures 1.3 and 1.7),

1.5. THIS THESIS

17

the relative simplicity and promising performance of traditional Hartmann
sensors make them an attractive choice for inclusion in an active thermalcompensation system. As such, they were chosen for further study and development, described in this thesis, to determine their suitability for such asystem.

1.5

This thesis

This thesis addresses several of the issues presented in the previous discussion. A verification of the theory of Hello and Vinet that underlies the numerical modelling of advanced GWI, described in Section 1.2.4, is presented
in Chapter 2. This verification was performed in laboratory conditions using a Mach-Zehnder interferometer as the accuracy and sensitivity of theHartmann sensor had not been established at the time of the test. The
development of an ultra-sensitive and accurate Hartmann wavefront sensor,
suitable for measurement of absorption-induced wavefront distortion in advanced GWI, is described in Chapters 3 and 4. The application of this sensor
to the measurement of wavefront distortion in the input-coupling mirror ofa Fabry-Perot cavity suspended in a vacuum system at the HOPTF is thenpresented in Chapter 5. This test was designed to produce a wavefront distortion similar to that expected in Advanced LIGO and is the first direct
measurement of absorption-induced wavefront distortion in a GWI-like environment. Finally, the use of the HWS for single-view tomographic analysisof axially symmetric temperature distributions is described in Chapter 6.
The overall effectiveness of the Hartmann wavefront sensor for measurementof thermal lensing in advanced GWI and potential future developments are
summarised in Chapter 7.

18

CHAPTER 1. GRAVITATIONAL WAVE ASTRONOMYChapter 2
Interferometric test of
Hello-Vinet model
2.1

Background

The analytical models for coating and substrate absorption-induced wavefront distortion (WD), published by Hello and Vinet [13], are used in the
MELODY code which is used to model the behaviour of advanced gravitational wave interferometers. These time dependent models have not beenvalidated however. Nevertheless, some aspects of WD theories have been investigated experimentally. Mansell et al. [27] measured absorption-induced
wavefront distortion in a test sample, but compared the result to an simpler
model proposed by Strain et al. [14] that does not include boundary conditions or any axial or temporal information. Lawrence et al. [17] verified thesteady-state coating absorption analytic model by heating a cylindrical test
optic with a CO2 laser beam with a Gaussian profile. The temporal modelof Hello and Vinet was indirectly validated by Zhao et al. [18] when they
induced a thermal lens in one the mirrors of a Fabry-Perot optical cavity
and observed its effect on the cavity mode. However, no direct validation
with a wavefront measurement of either the coating or substrate temporal
Hello-Vinet models, or the substrate steady-state Hello-Vinet model have
19

20

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY
Glass test optic

diameterGaussian heating beamheightFigure 2.1: A cylindrical test optic exposed to a Gaussian beam expected toabsorb optical power in the substrate and produce a wavefront distortion as
described by Hello and Vinet
been performed to date.

2.2

Objective

The aim of this chapter, therefore, is to describe a bench-top test of thetemporal and steady-state Hello-Vinet model for substrate absorption using
a glass test optic, as depicted in Figure 2.1, which produces a wavefront
distortion that is similar in magnitude to that expected in Advanced LIGO
and satisfies the assumptions of Hello and Vinet (H-V). The design and choiceof the test optic is discussed in Section 2.3. The system used to produce
the wavefront distortion and its measurement using a Mach-Zehnder (MZ)
interferometer are described in Section 2.4. The results of the measurement
are reported in Section 2.5.

2.3

Design and choice of test optic

Why is this objective difficult to meet? The primary difficulty in this experiment is creating an absorption-induced distortion in the test optic that is2.3. DESIGN AND CHOICE OF TEST OPTIC

21

measurable on a benchtop with, ideally, an amplitude similar to Advanced
LIGO. A secondary difficulty is controlling the experiment and environment
such that the signal is capable of being measured by an interferometer.

2.3.1

DesignThe size of the volumetric-absorption-induced wavefront distortion expected
in Advanced LIGO across the width of the cavity mode is, to first order,
approximately λ, where λ = 1064 nm. This is determined using the formulaeof Hello and Vinet (H-V) and knowing the power incident on the 20 cm
thick fused silica optics in Advanced LIGO is of the order of 2 kW, theexpected volumetric absorption in these optics is of the order of 50 ppm per
cm [28] [29] (giving a total absorbed power of approximately 2 Watts) andthe thermo-optic coefficient, dn/dT , is approximately 6 × 10−6 .

The maximum power available in the Gaussian beam for this benchtop

test was 500 mW @ 1064 nm. In order to produce a distortion of similar
size to Advanced LIGO, it was necessary to find a material with a higher
coefficient of volumetric absorption, α, than the fused silica used in Advanced
LIGO. Additionally, this coefficient could not be so large that the Hello-Vinet
assumption of constant optical flux along the optical axis was no longer valid;a concern that is addressed in Section 2.3.3.

2.3.2

Choice of material

Filter glasses typically have a well-defined absorption spectrum and are available with many different values of volumetric absorption @ 1064 nm.
Schott BG20 filter glass [30] was chosen because the volumetric absorption@ 1064 nm (0.33% per mm) was large enough to produce a thermal lensapproximately 10% - 20% of the size expected in Advanced LIGO, but not
so large as to completely negate the assumption in Hello-Vinet of uniform
optical flux through the material, as discussed in Section 2.3.3. The thermal
conductivity was not specified, but was expected to be in the range 0.9 −1.05

22

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY
Property
Diameter
Average Height
Heat capacity
Density
Thermal conductivity
Emissivity
Volumetric absorption

Value
65.0 mm29.1 mm
727 J kg−1 K−1 [31]2848 ± 0.5 kg m−3 (measured)
0.9 − 1.05 W m−1 K−1 @ 18◦C [31]0.9 [35]
0.33% mm−1 (measured)

Table 2.1: Properties of the BG20 glass cylinder used for measuring absorption induced wavefront distortion
W m−1 K−1 @ 18◦ C [31] because of its chemical composition (50-60% SiO2 , 1120% Na2 O, 11-20% Pr:Nd, 1-10% B2 O3 , 1-10% BaO [32]). The thermo-optic
coefficient, dn/dT , is typically in the range 1-6 ×10−6 [33] [34]. Accurate

knowledge of this coefficient is not required, however, as the Hello-Vinet

theory was tested using the temporal development of the shape of the thermallens, as discussed in more detail in Section 2.4.1.
Two cylinders of BG20 glass were cut and polished. The first had adiameter-to-length ratio (65 mm : 20 mm) similar to that for an Advanced
LIGO fused silica mass. The second was 9.1 mm thicker in order to absorb
more power and produce a larger thermal lens. This latter cylinder is depicted in Figure 2.2 and its relevant properties are listed in Table 2.1. Thiscylinder was used as the glass test optic (GTO) in the wavefront distortion
experiment.

2.3.3

Comparison of induced WD with H-V predicition

The H-V model assumes that the optical flux and beam-size are uniform
along the optical axis. In reality, the intensity, I, at any point along theoptical axis is given by2.3. DESIGN AND CHOICE OF TEST OPTIC

23

Figure 2.2: Schott BG20 filter glass cylinder. The absorption spectrum issuch that the glass appears pink under natural or incandescent light andappears green under florescent light; hence the pink edges where the camera
flash is illuminating the material and the green transmission in the upper left
hand side. The twisted wire around barrel of the glass cylinder is attached toa temperature probe that was not used in the experiment described in thischapter.

24

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY

I (z) ==

Iinc exp (− α z)
Iinc

≈ Iinc(α z)2
1 − αz +− ...
2for α z << 1!

where Iinc is the incident intensity. For Advanced LIGO, the assumption
α z << 1 is valid. For larger α, such as in this experiment, higher orderterms are required to maintain accuracy in a Taylor-series expansion of theabsorbed intensity, for example:
Iabs (z) ≈ α Iinc (1 − α z) .
The effect of this z dependence on the absorption-induced WD was modelledusing a finite element simulation, a variant of the one described in Appendix
B.3, and compared to the predictions of H-V. Two H-V scenarios were modelled using the parameters listed in Table 2.1. In the first simulation, thevolumetric absorption was 0.33% mm−1 , the same as that in the BG20 glass.
In the second simulation, the absorption was 1000× smaller than this. The
incident power was adjusted in the second simulation such that the total
absorbed power was the same as that in the first simulation. Note that thesimulation code described in Appendix B.3 assumed constant flux throughout the glass, whereas the code for these simulations assumed an exponential
decrease in the flux as it was absorbed by the glass. The code used for thesesimulations is identical in all other respects to the code in Appendix B.3.
The results from the finite-element model: the temperature distribution
along the z-axis and the total WD are shown in Figure 2.3 a) and b), respectively. Also shown are the magnitude of the differences between thehigh-absorption and low-absorption scenarios. As expected, there is a slightvariation in the temperature distribution with the high-absorption coefficientscenario being slightly warmer toward the input face (h = −14.5 mm). The2.4. MEASUREMENT AND ANALYSIS OF WD(a) Axial temperature

25(b) Radial WD

Figure 2.3: Predictions of the finite element model of absorption-induced
WD: a) Temperature along cylindrical axis for low-absorption (blue curve)
and high-absorption (red curve) coefficient scenarios. The green curve shows
30 × the difference between the two results. b) Radial WD (dn/dT × integrated temperature distribution) as a function of radius. The actual absorption curve (red) lies on top of the low absorption curve (blue). The green
curve shows 1000 × the difference between the two.

difference in the overall WD is negligible. Hence the BG20 glass cylinder isappropriate for testing the H-V wavefront distortion equation.

2.4

Measurement and analysis of wavefront
distortion in test optic

In this section, I start by describing the system used to produce the WD in theglass test optic (GTO), then I describe the measurement of the WD usinga Mach-Zehnder interferometer and finally the analysis of the interference
pattern.

26

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY

2.4.1

Multiple heating beam sizes

The heating beam size is one of the easiest1 parameters in the H-V solution
to vary so as to change the size and shape of the expected wavefront distortion. As a result, the experiment was designed to be run with three different
heating beam sizes and all other parameters kept constant to provide a more
rigorous test of the Hello-Vinet solution. Additionally, this allowed one of
the measurements to be used to calibrate the thermo-optic coefficient, since
the absolute size of the wavefront distortion predicted by Hello-Vinet is directly proportional to the thermo-optic coefficient and this was not precisely
known.

2.4.2

Experiment design

Figure 2.4 illustrates the experiment used to make an interferometric measurement of absorption induced WD in a cylindrical transmissive test mass.
The 65.0 mm × 29.1 mm BG20 glass test optic was mounted on four small

point contact sapphire balls to reduce conductive cooling and placed in atent to reduce convective cooling so radiative cooling was the dominant heatloss mechanism.
A Mach-Zehnder (MZ) interferometer was created with a HeNe laser beam
(λ = 632.8 nm) with the object and reference beams expanded to 75mm and
50mm by telescopes T2 and T3, respectively. The object and reference beams
had intensity profiles Iobj and Iref , respectively. The object beam was directedalong the axis of the BG20 glass, demagnified by 4× by telescope T4 and thenrecombined with the reference beam by beamsplitter BS2. The interference
pattern of the two beams at the exit plane of the glass test optic was imaged
onto the CCD by lens L1 and a PC recorded the digital output from the
CCD.
A mirror in the reference arm was tilted slightly, by an angle ∆θx , tointroduce a carrier signal in the interference pattern, as required by the1

The easiest parameter to vary is the incident power, but the only effect of this is toscale the wavefront distortion without varying the shape.

2.4. MEASUREMENT AND ANALYSIS OF WD

27

Fourier analysis described in Section 2.4.4. The analytic expression for thecentral horizontal cross-section interference pattern between the two beams,
Iint (x), is thus given by
Iint (x) = Iobj + Iref + 22π( ψ (x) + ∆θx x)
Iobj Iref cos
λ

q





(2.1)

where ψ (x) is the wavefront distortion introduced by the BG20 glass.
The laser beam from the Innolight 500 mW 1064 nm Mephisto NPRO [36]was passed through the PC-controlled shutter, S1, expanded by a telescope(T1) and directed along the axis of the BG20 glass cylinder. Immediately
before the BG20 glass, the 1064 nm and 632.8 nm beams were combined bya dichoric beam splitter (DCBS) which reflected 1064 nm and transmitted
632.8 nm. After the BG20 glass, the 1064 nm beam was absorbed by an
infra-red filter (IRF) that allowed visible light to pass through.
The telescope T1 was made using 150 mm and 50 mm lenses. The experiment was repeated three times, varying the configuration of T1 each time tochange the heating beam size to test more rigourously the Hello-Vinet theory,
as discussed in Section 2.4.1. The telescope configurations were varied:
1. 150 mm lens then 50 mm lens: heating beam 1/e2 -radius ≈ 1.1 mm at
the exit plane of the GTO.

2. No telescope: heating beam 1/e2 -radius ≈ 5.8 mm at the exit plane of
the GTO.

3. 50 mm lens then 150 mm lens: heating beam 1/e2 -radius ≈ 10.6 mm
at the exit plane of the GTO.

In the second configuration (no telescope), the heating beam propagateda distance of approximately 2.5 m from the NPRO to the GTO without
passing through any intervening lenses. The waist size of the heating beam at
the NPRO was approximately 150 µm. Hence, the beam was not completely
collimated at the GTO because of the natural divergence of a Gaussian beam.
However, the width varied by only 1-2% over the length of the glass test

28

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY

S1

T1

GTO

PBS

T4

BS2

IRF

CCD

object beam

λ/2

BD

L1
DCBS
T2

500mW NPRO

reference beam

10mW HeNe
BS1

T3

Figure 2.4: Schematic of experiment to measure thermal lensing in a cylindrical test optic. The output of a 500 mW 1064 nm laser is passed through ahalf-wave plate, λ/2, and a polarizing beam splitter (PBS) and is incident ona shutter S1. When the shutter is open, the beam passes through a telescopeand is transmitted through the BG20 glass test optic (GTO). The remainderof the beam is absorbed by an IR filter, IRF, in front of the CCD. The output
of a 10 mW HeNe laser is the input to a Mach-Zehnder interferometer, theobject arm of which contains the BG20 optic. The object and reference arms
are expanded by telescopes, T2 and T3, respectively. The object arm wasdemagnified by telescope T4 and recombined with the reference arm at thesecond beamsplitter, BS2. The interference pattern at the exit plane of theglass test optic was imaged onto the CCD with lens L1. A dichroic beam
splitter, DCBS, immediately before the glass test optic was used to combine
the two different wavelengths.

2.4. MEASUREMENT AND ANALYSIS OF WD

29

optic. This Gaussian divergence is also the reason that the beam size in
configuration 2 was not 3× the beam size in configuration 1 and the beam
size in configuration 3 was not 3× the beam size in configuration 2.
The flux of the 632.8 nm laser beam was at least 100× smaller than the1064 nm laser beam in configuration 3 and over 10,000× smaller in configuration 1. Additionally, the absorption of the 632.8 nm light in the BG20
glass is of the same order of magnitude as the absorption of the 1064 nm
light [30]. As such, the thermal lensing due to the absorption of the HeNe
was not considered.

2.4.3

Measurement procedure

1. Adjust T1 and DCBS to direct an approximately collimated 1064 nm
beam along the axis of the glass test optic. [1064nm ON, 632.8nm
OFF].
2. Adjust λ/2 plate so that most of the 1064 nm is reflected from thepolarising beam splitter (PBS) into the beam dump (BD). Remove
IR filter (IRF) and open shutter. Record an image of the beam size.
[1064nm Low Power, 632.8nm OFF].
3. Close shutter and adjust λ/2 plate so that all the power is transmitted
through the PBS onto S1. [1064nm ON & Blocked, 632.8nm OFF].
4. Place an object of known size at the exit plane of the glass test optic.
Measure the size of the image of the object on the CCD to determine
the magnification. [1064nm Blocked, 632.8nm ON]. Remove the objectof known size.
5. Start to record the interference patterns between the object and reference beams, Iint (x, ti ), at times ti , at a rate of 10Hz. At this stage i isdesignated as negative (i < 0) and these patterns are used to determine
the rms background noise. [1064nm Blocked, 632.8nm ON].

30

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY
6. Use the PC to open the shutter and allow the 1064 nm onto the glass
test optic. This designates the time t0 = 0 s of the zeroth interference
pattern. [1064nm ON, 632.8nm ON].
7. Continue to record interference patterns (i > 0) on the CCD at a rateof 10Hz. The ith pattern, recorded at time ti , is designated Iint (x, ti ).
[1064nm ON, 632.8nm ON].
8. The last and M th pattern, designated tM , is the first one recorded aftert = 1000.0 s. [1064nm ON, 632.8nm ON].

2.4.4

Analysis of MZ interference patterns

The following is a summary of phase profile extraction from the MZ interference patterns using Fourier transforms, described in detail by Malacara[37]. This discussion begins with the interference pattern at time ti due toa wavefront distortion ψ (x, ti ) with no tilt angle, ∆θx , between the object
and reference beams:

Iint (x, ti ) = Iobj:ti + Iref:ti + 2

2π
Iobj:ti Iref:ti cosψ (x, ti )
λ

q





(2.2)

where Iobj:ti is the intensity profile of the object beam, Iobj:ti is the intensity
profile of the reference beam and ψ (x, ti ) is the absorption-induced wavefront
distortion to be extracted. Equation 2.2, can be rearranged as a sum ofcomplex exponential terms
Iint (x, ti ) = g (x, ti ) + hψ (x, ti ) + h∗ψ (x, ti )(2.3)

g (x, ti ) = Iobj:ti + Iref:ti

(2.4)

where

is the background intensity in the interference pattern and2.4. MEASUREMENT AND ANALYSIS OF WDhψ (x, ti ) =2πψ (x, ti )Iobj:ti Iref:ti exp i
λ


q

31



(2.5)

is the complex form of the fringes in the interference pattern induced by the
wavefront distortion in the object arm of the interferometer.
The Fourier transform, S (fx , ti ), of the interference pattern is thus

S (fx , ti ) =Z +∞Iint (x, ti ) ei fx x dx−∞

= G (fx , ti ) + Hψ (fx , ti ) + Hψ∗ (fx , ti )(2.6)

where G (fx , ti ), Hψ (fx , ti ) and Hψ∗ (fx , ti ) are the Fourier transforms ofg (x, ti ), hψ (x, ti ) and h∗ψ (x, ti ), respectively. For the discrete intensity datafrom the CCD, this is done using a discrete FFT. In IDL, the code is:
Fw = FFT(Iint)
The spectral components, Hψ (fx , ti ), of the intensity-encoded phase information in hψ (x, ti ) and the spectral components of the background intensity
distribution, g (x, ti ), are both centered around fx = 0 and are therefore
indistinguishable. Separation of these signals into two distinct peaks in frequency space is achieved by frequency shifting one signal with respect to theother.
To achieve such a frequency shift, a linear carrier signal, 2π (f∆θx x),
where f∆θx = ∆θx /λ, is applied to the interference pattern by tilting a mirror
in the reference arm so that the angle between object and reference beams
becomes ∆θx . The complex representation of the fringes becomes2π( ψ (x, ti ) + ∆θx x )λ= hψ (x, ti ) exp [i 2π f∆θx x]

hc (x, ti ) =q

Iobj:ti Iref:ti exp iand the interference pattern becomes32

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY

Iint (x, ti ) = g (x, ti ) + hc (x, ti ) + h∗c (x, ti )= g (x, ti ) + hψ (x, ti ) exp [i 2π f∆θx x] + h∗ψ (x, ti ) exp [−i 2π f∆θx x] .
An example of such an interference pattern measured by the interferometeris shown in Figure 2.5 a).
The Fourier transform, Hc (fx , ti ), of hc (x, ti ) can be determined using
the Fourier transform of the Dirac delta function
g (x) = exp [−i 2π f0 x]⇒ G (f ) = δ (f − f0 )and the Convolution Theorem
F {g (x) h (x)} = G (f ) ∗ H (f )
=

Z +∞
−∞

G (α) × H (f − α) dα,

where F {g (x) h (x)} denotes the Fourier transform of g (x) × h (x), giving
Hc (fx , ti ) = Hψ (fx − f∆θx , ti )(2.7)

and the Fourier transform of the interference pattern becomesSc (fx , ti ) = G (fx , ti ) + Hψ (fx − f∆θx , ti ) + Hψ (fx + f∆θx , ti ) .
The spectral components, Hψ (fx , ti ), of the intensity-encoded phase information in hψ (x, ti ) are thus centered around f∆θx , as illustrated by the amplitude spectrum in Figure 2.5 b). The spectral components of the phase
information, the two lobes at ±f∆θx , are now clearly separated from thespectral components of the background intensity, G (fx , ti ), centered aroundzero.
A band-pass filter, illustrated in Figure 2.6 a), is applied to one of the offset lobes to select the interference fringes and remove high spatial frequency

2.4. MEASUREMENT AND ANALYSIS OF WD(a) Interference pattern cross-section

33(b) FFT of interference patternFigure 2.5: Demonstration of phase extraction from interference pattern a)
Original interference pattern, b) FFT of interference pattern.
noise. That lobe is then demodulated to zero frequency by f∆θx as illustrated
in Figure 2.6 b). The minimum width of the band-pass filter was determinedby applying this analysis to the H-V model with the BG20 parameters and aheating beam width of 1.1 mm. The result of filtering the H-V signal with too
narrow a bandwidth is a smoothing of the WD and the loss of high-frequency
information and is illustrated in Figure 2.7 a). Shown in Figure 2.7 b) is theresult of filtering the H-V with the bandwidth used in this experiment: there
is virtually no smoothing of the signal.
The filtering and demodulation of the discrete data, using a full bandwidth of 70 spatial frequency units2 as in Figure 2.7 b), was done in IDL
using the following code:
bandwidth = 35.0
filter = exp(-2 * ((freq - fxc)/bandwidth)^8 )Fw_filt = filter * Fw
Fw_demod = SHIFT(Fw_filt, fxc)
where freq is an array of indices corresponding to the spatial frequencies and2

If the interference pattern is sampled at N points with an interval of ∆x, then thespatial frequency unit of the complex values returned by the FFT is 1/(N ∆x). In this case,N = 1024 and ∆x = 12 µm is the width of one pixel and hence the sampling frequency
unit is 0.0814 mm−1 .

34

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY(a) Super-Gaussian bandpass filter(b) Demodulation of carrier signal

Figure 2.6: a) Super-Gaussian bandpass filter, b) Demodulation of filteredcarrier signal in frequency space. The dashed curve is the original signal andthe solid curve is the demodulated signal.

(a) Fullwidth of filter = 10 frequency units(b) Fullwidth of filter = 70 frequency units

Figure 2.7: a) Image filtering of sharpestH-V signal with bandwidth of 10
spatial frequency unit. b) Image filtering with bandwidth of 70 spatial frequency units. Original signal (red), filtered signal (blue) and a) magnitudeof the difference (green) and b) 10× the magnitude of the difference (green).
1 spatial frequency unit = 1/(1024 × 1 pixel) = 0.0814 mm−1 .

2.4. MEASUREMENT AND ANALYSIS OF WD

35

Figure 2.8: The reconstructed and filtered phase profile is given by the arctangent of the inverse FFT yields of the demodulated and filtered carrier
signal.
fxc is the index of one of the offset lobes. The function SHIFT(x, n) moves
all the values along in the array x by n units, which is the digital equivalenthiof demodulation. Note that the filter is a super-Gaussian, exp − (fx /∆fx )8 .

After the filtering and demodulation, the inverse Fourier transform isapplied using
hWD_filt = FFT(Fw_demod, /INVERSE)
to yield hWD, filt (x). The phase profile is extracted by taking the arctangent(

Im (hWD, filt (x))
2π
ψ (x) = arctanλRe (hWD, filt (x)))

an example of which is illustrated in figure 2.8 a). The range of the arctangent
function is {−π, +π} and hence the return value 2π
ψ (x) is ”wrapped”
λbetween these values. Unwrapping and extracting ψ (x) is a straightforward
process that is not described here.
The wavefront distortion in the ith pattern, at time ti , relative to thezeroth pattern, at time t0 = 0 s, WDti (x), is given by

36

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY

WDti (x) = ψti (x) − ψt0 (x)(2.8)

Once the wavefront distortion was found the residual tilt/prism in the x
and y directions was removed.

2.5

Results

2.5.1

Background noise

The rms background noise was determined by measuring the temporal fluctuations in the interference patterns before the shutter was opened, Iint (x, ti )with i < 0 in Step 5, using the Fourier analysis described above and wasapproximately λ/150 for λ = 632.8 nm.

2.5.2

Temporal development of measured wavefront
distortion

The measured wavefront distortion profiles at times t = 2.5 s, 8.5 s, 29.2 s and100.0 s and for beam sizes ω = 1.1, 5.8 and 10.6 mm are plotted in Figures 2.9
and 2.10. The quasi-steady state at t = 100.0 s, in Figure 2.10 b), shows good
agreement with the H-V predicted profile for which a thermal conductivityof 0.98 W m−1 K−1 was used. The measured profiles at the remaining times
appear to have the same shape as the predicted profiles within the region of
the heating beam albeit with marginally reduced magnitudes.
To illustrate the diffusion of heat through the material, the optical path
difference between the center and x = ±GTO diameter /4 = 16.25 mm for themeasurements and the Hello-Vinet prediction is shown in Figure 2.11 a). The
temporal evolution of the OPD for ω = 1.1 mm was used to determine thebest-fit value of thermal conductivity, giving 0.98 W m−1 K−1 . As it has
been used to fit a parameter, the red plot in Figure 2.11 a) cannot be used
to validate the model. The data for the w = 5.8 mm and 10.6 mm agree

2.5. RESULTS

37(a) t1 = 2.5 s(b) t2 = 8.5 s

Figure 2.9: The wavefront distortion measured by the interferometer when
the test mass was heated with a laser beam of three different beam sizes:
w = 1.1 mm (red), w = 5.8 mm (blue), w = 10.6 mm (green). The results
were recorded at time t1 from when the shutter was opened, where a) t1 = 2.5s, b) t1 = 8.5 s. Note the different vertical scales on the figures. The blacklines show the prediction of H-V for each beam size.

38

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY(a) t1 = 29.2 s(b) t2 = 100.0 s

Figure 2.10: The wavefront distortion measured by the interferometer when
the test mass was heated with a laser beam of three different beam sizes:
w = 1.1 mm (red), w = 5.8 mm (blue), w = 10.6 mm (green). The results
were recorded at time t1 from when the shutter was opened, where a) t1 =29.2 s, b) t1 = 100.0 s. Note the varying scales on the figures. The blacklines show the prediction of H-V for each beam size.

2.6. CONCLUSION

39

reasonably well with the H-V prediction.
To illustrate the development of a thermal lens, the average defocus acrossa beam width (calculated from the measured WD) and the H-V predictionof the development of the defocus is plotted in Figure 2.11 b). There is good
agreement between the temporal development of the measured and predicted
thermal lenses for beam sizes ω = 1.1 mm and 5.8 mm and reasonable agreement for the very weak thermal lens when ω = 10.6 mm.
These measurements show good agreement with the predictions of Helloand Vinet for longer times, however, there are small systematic differences
between the predictions and the measurements. These discrepancies are mostobvious in the central sections of the wavefront distortion profiles in Figures
2.9 a) and b) and Figures 2.10 a) and also in the blue curve in Figure 2.11a) and, to a lesser extent, the green curve in Figure 2.11 b). It appears
from Figures 2.9 and 2.10 that there is less thermal lensing than expected
at earlier times. It is unlikely that this was due to less laser power/heating
at earlier times as the intensity of the NPRO was observed to be very stablewith time. A possible explanation is that, given the test optic is in a tent tominimize air currents and not a vacuum to eliminate them, there is a small
amount of convective cooling of the glass test optic - a cooling mechanism
not included in the Hello-Vinet model.

2.6Conclusion

This chapter has described an interferometric measurement, with a precisionof λ/150 for λ = 632.8 nm, to validate the formula of Hello and Vinet. A
completely independent evaluation proved to be difficult because the valueof thermal conductivity of the test material was not known well enoughand had to be determined from one of the measurements. The remaining
measurements show good agreement with the predictions of Hello and Vinet.
It is possible that the small deviations from the predicted results are due toconvective cooling of the glass test optic.

40

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORY(a) OPD at wi(b) Defocus at wi

Figure 2.11: a) The optical path difference between x = 0 and x =diameter/4 measured by the interferometer and the predicted result from
Hello-Vinet and b) the average defocus across the heating beam width andthe predicted result from Hello-Vinet. In both plots the three different beam
sizes are w = 1.1 mm (red), w = 5.8 mm (blue), w = 10.6 mm (green) andthe H-V predictions are the corresponding black curves.

2.6. CONCLUSION

41

The validation of the H-V model presented in this chapter increases theconfidence in the validity of numerical models of GWI, such as MELODY [16],that employ the Hello-Vinet model, within the limitation of the systematic
error discussed. A more strenuous validation of the Hello-Vinet theory would
be possible with a stronger laser source, which would allow one to use a more
precisely characterized test optic, and construction of the experiment in a
vacuum chamber to eliminate convective cooling. Additionally, one could use
a Hartmann wavefront sensor, the construction and development of which is
described in the following chapter.

42

CHAPTER 2. INTERFEROMETRIC TEST OF H-V THEORYChapter 3
Hartmann wavefront sensor development
3.1

Introduction

The Hartmann Wavefront Sensor (HWS) was first used in 1900 [38] as away to measure aberrations in spherical mirrors and has since then been
applied to many more general wavefront sensing applications. Its popularity
has dramatically increased in recent years due to the incorporation into its
design of cheap and fast solid-state arrays [39]. This chapter discusses theoperation of the HWS and describes its development for measuring thermal
lensing in advanced Gravitational Wave Interferometers.
Section 3.2 begins by describing several applications of Hartmann-type
sensors to familiarise the reader with the scenarios in which they can beused. The basic physics and operation of a HWS are described in Section
3.2.2 and potential configurations of a HWS in an experiment are discussed
in Section 3.2.3. With the basic physics and operation covered, variantsof the standard Hartmann theme are described in Section 3.2.4, including
the popular Shack-Hartmann sensor. The suitability of these variants formeasuring thermal lensing in GWIs is contrasted with the traditional HWS.
Lastly, a short description of potential temperature induced limitations is43

44

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

described in Section 3.2.5.
The development of the HWS for an advanced GWI is described in Section
3.3. The design of the hardware components (the light source, the Hartmann
plate and the CCD) and the noise sources associated with these components
are described in Sections 3.3.1 to 3.3.3. The software components (the centroiding and wavefront reconstruction algorithms) and the errors associatedwith these components are described in Sections 3.3.4 and 3.3.5. Finally, thetotal accumulated noise and error in the sensor are discussed in Section 3.3.6.

3.2

Hartmann sensor basics

3.2.1

Applications of the Hartmann sensor

The original purpose of the Hartmann sensor was to characterise aberrations
in curved mirrors as illustrated in Figure 3.1 ( reproduced from Ghozeil[40]). In this figure, a light source illuminates a plate containing a series ofapertures, the Hartmann plate (HP), that is directly in front of a test optic(in this case, a mirror). The rays from the apertures are reflected from themirror and are incident on a photographic plate forming a pattern of spots.
The displacements of the spots, from the positions expected for an ideal optic,
reveal the aberrations in the mirror. This process of characterizing reflective
or transmissive optics with a HWS has been used since 1904 [41] [42] [43][44] [45] [26] [46] [47]. Note that the light from the source that reflects off
the mirror is often referred to as a “probe beam”, a label that is used often
in the latter parts of this chapter.
Of the myriad of other applications of Hartmann-type sensors, a particularly salient one for this thesis is the measurement of thermal lensing in
optics. It is interesting to note that one might consider this application to
be quite old; Plaskett employed a Hartmann sensor in 1920 to determine
the change in the optics of a 72-inch mirror due to temperature [48]. Morerecently, Mansell et al. [27] measured the effect on laser beam quality witha Shack-Hartmann wavefront sensor with a reproducibility of λ/100 @ 633

3.2. HARTMANN SENSOR BASICS

45

Hartmann plate

Photographic plate

Test optic

Light source
Figure 3.1: The original Hartmann test. A light source illuminates a Hartmann plate that is directly in front of a test optic. The resulting rays arereflected onto the photographic plate, forming a pattern of spots. Aberrations in the spot pattern reveal the aberrations in the test optic. Reproduced
from Ghozeil [40]nm. Also, Lawrence et al. [17] described a measurement of a radiative thermal corrector designed to compensate thermal lensing induced in an optic byabsorption of a Gaussian beam. Lawrence et al. used a Wavefront Sciences
CLAS-2D Shack-Hartmann sensor with a sensitivity of approximately λ/100@ 633 nm [25]. While these examples illustrate the application of Hartmann
sensor technology to thermal lensing problems, the sensors that were used
do not meet the sensitivity requirement for use in Advanced LIGO (better
than λ/470 @ 633nm) established in Section 1.3.2.
Hartmann type sensors have also been used in:• Adaptive optic systems in large ground-based optical telescopes [49][50] [51] [52]. Light from stars and other stellar bodies accumulates

phase aberrations upon passage through atmospheric turbulence. The
resolution of telescopes viewing this light has been improved by measuring the aberrations using Shack-Hartmann sensors and actuating
upon the telescopes’ optics to compensate for the effects.
• Ophthamology. Modified Hartmann tests have reflected rays off anaberrated cornea have been used to measure its topography [53] [54].

46

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT• The characterization of laser-beam parameters, such as M 2 , divergenceand beam widths [55] [56].

• The automatic alignment of a synchrotron beamline using a Hartmann
sensor (operating at 0.414 nm) has been demonstrated at the SOLEIL
synchrotron in France [57].
• The triangulation of a point-like source [58].

3.2.2

Overview of Hartmann sensor

Stated simply, a Hartmann sensor measures the change in the gradient of a
wavefront relative to a reference wavefront. The gradient change is numerically integrated to give the wavefront change. The following description of
the operation of a Hartmann sensor is a summary of the original Hartmann
papers [38] [59] and the work by Ghozeil [40].
A Hartmann sensor consisting of an opaque plate containing an array of
apertures and an intensity recording medium is shown in Figure 3.2. Originally, the recording medium was a photographic plate, but it is now almost
exclusively a solid-state photo-sensitive array. The basic operation of a Hartmann sensor can be described succinctly by:
1. The wavefront, W′ , or wavefront change, ∆W = W′ − W, to be mea-

sured is incident on the Hartmann plate (HP) which divides it into a
set of rays, known as Hartmann rays.
In a traditional Hartmann sensor the apertures are simply
holes in a screen from which the rays diffract. There are avariety of commonly used arrangements of apertures: radial
patterns [60], square arrays, or hexagonally closed-packed arrays. The process of optimizing the apertures in the Hartmann plate for measuring thermally-induced wavefront distortion in GWIs is discussed in Section 3.3.2.

3.2. HARTMANN SENSOR BASICS

W

47

L

W′

xi
∆xi
x′i

x

z
y
(out of page)

HP

CCD

Figure 3.2: An aberrated wavefront W′ is incident on a Hartmann plate(HP). The resulting rays propagate a distance L, normal to the wavefront,
and are incident on a CCD. The spot position, x′i , is determined by thecentroid of that spot’s intensity profile. The reference spot positions, xi ,(either measured using a non-aberrated wavefront, W or calculated using
the hole positions in the HP) are indicated by the intersection of the dottedlines and the CCD. The gradient of the wavefront at the ith aperture is givenby the displacement, ∆xi , from the reference position divided by L.

48

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT
2. The rays propagate a known distance L, normal to the wavefront, andare incident on the CCD.
The process of optimization of L is also discussed in Section
3.3.2. The lever arm needs to be accurately known in orderto prevent systematic errors in the gradient of the wavefrontchange. Fortunately, it can be calibrated by measuring an
accurately known gradient as demonstrated in Section 4.2.
3. The pattern of spots on the CCD is recorded as a digital image.
An example image of a recorded pattern of spots is shown in
Figure 3.3. The operation of the CCD, the digitization andthe noise are discussed in Section 3.3.3.
4. The position of the ith spot, x′i , is determined by a centroiding algorithm.
The optimization of centroiding algorithms is discussed in
Section 3.3.4.
5. The displacement of each spot, ∆xi , from a previously measured reference position, xi , for wavefront W is calculated.
6. The gradient of the wavefront change is calculated using
∂∆W
∆xi
=
∂x
L
The angle between the ith Hartmann ray and its reference ray
is equal to the displacement, ∆xi divided by the lever arm,
L. Rayces [61] showed that this angle is approximately proportional to the gradient of the wavefront aberration. Figure
3.4 shows an aberrated wave-front PC and a spherical reference wavefront SC of radius R centered at Q. The wave-

3.2. HARTMANN SENSOR BASICS

49(a) Hartmann spots on a CCD(b) Hartmann spots - closer view

Figure 3.3: a) An example of a Hartamnn spot pattern and b) a closer viewof some of the spots.

50

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT
front change (aberration), ∆W, is defined as the distance PS.
Rayces determined the following relation:

∂∆W
|QT |
= −
∂x
|QS| − ∆W(3.1)

which can be simplified to

∆x
∂∆W
≈ −
∂x
L

(3.2)

if one assumes L ≪ R and ∆W ≪ L. For absorption-induced

wavefront distortion in GWIs, ∆W ≈ 10−7 m, L ≈ 10−2 m

and the reference wavefront is approximately flat (R ≈ ∞).

That is, these assumptions and thus Equation 3.2 are valid.

7. The wavefront change, ∆W, is calculated by integrating the discrete
gradient field.
The techniques used to integrate the field and the propagtionof errors in the integration are based largely on the work ofSouthwell [62] and are discussed in Section 3.3.5.

3.2.2.1

Absolute or differential operation

In a conventional HWS, an absolute measurement of that wavefront canbe made by comparing the measured spot positions with the reference spot
positions (Step 5) which are determined using the positions of the holes in the
Hartmann plate, typically known with an accuracy of ±1µm. Alternatively,

the positions of those holes could be calibrated by illuminating the HWS
with a flat wavefront and measuring the spot centroids on the CCD - exceptthat the flatness of this wavefront must be verified with another method. The3.2. HARTMANN SENSOR BASICS∆W

P(x,y,x)S

51

T(X,H+Y,0)
∆x
x

y

Q
H
O

z

C

L

Figure 3.4: Reference sphere SC and wave-front PC (propagating to theright), QS = QC = R, PS = ∆W. Reproduced and revised version of the
figure from Rayces [61].

52

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

uncertainty of the absolute positions of the reference spots therefore limits theaccuracy of the HWS when used in the conventional absolute measurement.
For absorption-induced wavefront distortion in GWI, however, one is interested only in the differential wavefront change, ∆W, from an undistorted
wavefront. The reference spots produced by the undistorted wavefront at
an earlier time are measured accurately by the sensor. This removes a potential systematic error and allows much greater accuracy. All of the HWS
measurements within this thesis are differential.

3.2.3

Hartmann plate - CCD configurations

When measuring the wavefront change in a large diameter beam, some demagnification of the beam is necessary to fit the region of interest onto the
CCD of the HWS. For example, the wavefront sensor in Advanced LIGO
must probe a 224 mm diameter region (see Section 1.3.2). In order to fit thisonto a 12 mm × 12 mm CCD it must be demagnified by a factor of 18.7.
Similarly, the beam probing the distortion in the 65 mm diameter BG20 glass

test optic described in Section 2.3.2 must be demagnified by a factor of 5.4.
To facilitate this, the Hartmann plate, CCD and demagnifying optics can be
configured in several ways, as shown in Figure 3.5.
In Configuration A, a Hartmann plate is placed at the plane HPO immediately after the test optic and before the telescope. The rays from theHartmann sensor propagate through the telescope, which demagnifies themby M, and then are incident on the CCD. The effective lever arm of theHartmann sensor, L1 , is the distance between the Hartmann plate and the
conjugate plane of the CCD, CCDI .
In Configuration B, a different Hartmann plate is placed at the plane HPI ,the conjugate plane of HPO . The wavefront from the test optic propagates
through the telescope and is incident on the Hartmann plate.
Configurations A and B are equivalent if the Hartmann plate in B is ademagnified version of the plate in A and L2 = L1 /M 2 . Thus, if the test
optic introduces the primary aberration defocus, S, for example, given by3.2. HARTMANN SENSOR BASICS

53

L1
W

W’O
xI

xO

CCD O

Test Optic

T1
HPO
CCD I

Configuration A
W

W’O
L2

W’I

HPI
CCD O

Test Optic

T1
HPO

Configuration B
Figure 3.5: Two configurations of the Hartmann sensor. Configuration A:
the Hartmann plate is placed at HPO immediately after a test optic thatchanges the wavefront from W to W′ . The beam propagates through atelescope, which demagnifies it by M onto a CCD. The effective lever arm of
the Hartmann sensor, L1 , is the distance between the Hartmann plate and the
conjugate plane of the CCD, CCDI . Configuration B: the beam propagates
through the test optic and telescope and is incident on the Hartmann plateat HPI , the conjugate plane of HPO , where the lever arm is L2 = L1 /M 2 .

54

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

W=

S 2
x
2

then the size of the aberration, S, at HPI is M 2 times that at HPO . In
other words, if the test optic introduces a wavefront change with a radiusof curvature RO then the Hartmann sensor will measure a wavefront changewith a radius of curvature RI = RO /M 2 . This is important in the analysis
of a thermal lens in a high optical power cavity and its effect on the cavity
mode, discussed in Chapter 5.

3.2.4

Variants of the traditional Hartmann sensor

3.2.4.1

Shack-Hartmann

In the S-H sensor, the holes in the HP were replaced by a lenslet arrayby Shack [63]. Such a lenslet array is illustrated in Figure 3.6. A ShackHartmann sensor operates in largely the same way as a standard Hartmann
sensor, except that the wavefront propagating through the lenslets are focussed onto the CCD rather than diffracting from the aperture. This resultsin spots that are smaller than those from apertures of the same diameter in a
Hartmann sensor. The advantages and disadvantages of the Shack-Hartmann
sensor are described below and summarised in Table 3.2.4.1.
• Flexibility: the lever arm distance between the array and the CCD may
be varied easily in the Hartmann sensor, whereas in a Shack-Hartmann

sensor it is approximately equal to the focal length of the lenslets [64].
• Light collection: The Shack-Hartmann array has a higher transmission

per unit area and also concentrates the light onto the CCD, which isan advantage in astronomical adaptive optics applications when there
are only a limited number of photons available [65].

• Spatial resolution: The spots in the Hartmann and Shack-Hartmann

sensors need to be separated from one another to prevent cross-talk3.2. HARTMANN SENSOR BASICS

55

Wavefront

Lenslet array

CCD surface

Figure 3.6: A distorted wavefront incident on a Shack-Hartmann lenslet
array. The light through the each lenslet is focussed to a position on the
CCD determined by the average gradient of the wavefront over the lenslet.
The dotted lines show the reference positions of the spots when the incident
wavefront is flat.

56

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT
and the subsequent reduction in sensitivity. The area on the CCD occupied by a spot in a Hartmann sensor is larger than the area occupied
by a spot in a Shack-Hartmann sensor (assuming the Hartmann aperture and Shack-Hartmann lenslet diameters are equal). Therefore thelenslets in a Shack-Hartmann sensor can be packed closer togther thanin a Hartamnn sensor and achieve a higher spatial resolution.
• Both are susceptible to the misalignments between the array and thedetector identified by Pfund et al. [66]. Of the misalignments identified(translation of the array in the x or y directions, translation of the array
along the z axis, rotation of the array about the z axis and rotation of
the array about the x and y axes) only rotation about the x and y axes
poses any problems for a differential measurement [67] and this can be
mitigated by careful assembly of the Hartmann sensor, as discussed in
Section 3.3.2.3. For the orientation of the axes, see Figure 3.2.
• The spots in a Hartmann sensor will typically cover more pixels than a
Shack-Hartmann sensor. This may average out the effects of any nonuniform response in the pixels of the CCD, but it does not improve theaccuracy of the centroid. This can be shown using the theoretical result
higlighted by Thomas et al. [68]: the minimum possible centroid noise
in a Gaussian spot with rms size σspot and pure photon noise, is equaltoσspot
σNph = qNphwhere Nph is the average number of photons per spot. If the photon flux
on the CCD is large enough, then the number of photons per spot will2
be proportional to the area of the spot, which is proportional to σspot,

and hence σNph will be a constant. Note that this only applies if a spotcovers an area greater than one pixel. If a spot is smaller than a single
pixel there is only a single data point and the location of the center of3.2. HARTMANN SENSOR BASICS
Property
Best sensitivity at time of design
Lever arm is variable
Higher transmission per unit area
More pixels per spot
Subject to lenslet aberrations
Ease of manufacture

57

Shack-Hartmann
Hartmann
λ/150 @ 633 nm [25] λ/500 @ 1064 nm [24]
No
Yes
Yes
No
No
Yes
Yes
No
MediumEasy

Table 3.1: A comparison of the relative merits of Shack-Hartmann and Hartmann sensors.
the spot cannot be determined with sub-pixel precision. In this case,a Hartmann sensor with larger spots will offer an improvement on a
Shack-Hartmann sensor with single pixel-sized spots.
• A Hartmann sensor is a simpler device and is easier and cheaper tomanufacture than a Shack Hartmann lenslet array.

• A Shack-Hartmann array may suffer from aberrations in the lenslets[67]. Clearly, this will not be a problem for a Hartmann sensor as it
contains no lenslets.

Ultimately, the large number of available photons, flexibility, low cost andthe relative ease of manufacturing a Hartmann plate to exact specifications
led to its choice over a commercially available Shack-Hartmann sensor.
3.2.4.2

Other variations

Other minor variations on the Hartmann sensor theme include:• The scanning Hartmann sensor in which an incoming wave front isscanned by a moving aperture that is created using a programmableliquid-crystal display [69] [70] or a single beam is scanned across a CCD
using a galvanometer-based rotating mirror [71]. In this configuration,
cross-talk is not an issue. However, because of the sequential scanning,real-time operation is not possible making it unsuitable for use in a
GWI.

58

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

Figure 3.7: Apertures on the Hartmann plate move further apart as thetemperature of the plate increases. The dashed circles represent the original
positions and the solid circles represent the new positions after an increasein temperature.

• The interferometeric Hartmann sensor [52] [72]. The spots in the Hartmann sensor are deliberately brought closer together to create inter-

ference fringes between neighbouring spots. Changes in the fringes areused to extract the wavefront change. Using this method West [52]reliably measured wavefront distortions as small as 10 nm, which does
not satisfy the Advanced LIGO sensitivity requirement of 1.35 nm =λ/467 @ 633 nm.

3.2. HARTMANN SENSOR BASICS

3.2.5

59

Temperature dependence of Hartmann sensors

As discussed by Chernyshov et al. [73], if the temperature of the Hartmann
sensor changes then there will be thermal expansion in the components of thesensor. Thermal expansion of the HP causes an increased distance between
the holes, as shown in Figure 3.7, which results in an increase in the distance
between the spots on the CCD which increases the change in the wavefront
gradient measured by the sensor. Thermal expansion of the CCD results ina decrease in the apparent distance between the spots.
For the HWS discussed in this chapter, the thermal effect is dominated bythermal expansion of the brass HP and leads to a systematic error equivalent
to defocus, S, as shown below.
Assume that there is a uniform (and small) increase in temperature, ∆T ,of the Hartmann plate. The increased distance between two Hartmann holes,∆xthermal , is given by∆xthermal = α hp ∆T(3.3)

where hp is the nominal distance between holes on the Hartmann plate. The
distance between the two corresponding spots on the CCD increases by thesame amount and the apparent change in gradient, ∂ (∆W) /∂x, is∂ (∆W)∆xthermal=∂xL
α hp ∆T
=
L 

α ∆T
=
hp
L

(3.4)

or more generallyα ∆T∂ (∆W)
=
∂x
L




x

(3.5)

for the distance, x, between any two points. The apparent change in gradient

60

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

is linear in x and, therefore, the apparent change in wavefront is quadratic
in x. That is, thermal expansion of the HP results in a measurement of the
primary aberration defocus [37], with a coefficient Sthermal , given by

Sthermal =

3.3

α ∆T
L

(3.6)

Hartmann sensor for advanced GWI

As discussed in Section 1.3.2, the requirements for the wavefront sensor to
be used in the thermal compensation system in Advanced LIGO [20] [74] are
1. A sensitivity of at least λ/785 @ 1064 nm
2. Spatial resolution sufficient to cover a 224 mm diameter circle withsampling points that are separated by less than 10 mm.
This section describes the design of a Hartmann sensor that exceeds theserequirements. The description of the sensor hardware includes the probe
beam light source in Section 3.3.1, the optimization of the Hartmann plateand lever arm distance in Section 3.3.2 and the choice of the CCD camera
in Section 3.3.3. The analysis software is discussed in the following two
subsections, and includes the description of a novel centroiding algorithmand discussion of the wavefront reconstruction algorithm. Finally, Section
3.3.6 collates the errors identified in earlier sections to determine the overallsensitivity.

3.3.1

Light source - coherent versus incoherent

The probe beam is an integral part of the use of a HWS to measure thermal
lensing in an optic. The effect of intensity noise in the probe beam should
thus be included in the overall noise of the sensor.

3.3. HARTMANN SENSOR FOR ADVANCED GWIProperty
Product name
Emmiting Material
Fibre Output Power
Peak Wavelength
Coherence Length (FWHM)
Fibre size

61

Value
Agilent HFBR-1414 Transmitter
AlGaAs60 µW
820 nm
5.5 µm50/125 µm

Table 3.2: Properties for Agilent fibre-coupled LED
The intensity of the light source will fluctuate due to photon shot noise
and also due to time-varying coherent effects, such as interference fringes,that are present in the intensity distribution.
Photon shot noise, σphoton , is simply the Poissonian fluctuations in thenumber of photons, Np , arriving in a given time interval, given by
σphoton =

q

Np

(3.7)

This will be quantified in the discussion about the CCD in Section 3.3.3.
Coherent fluctuations, σcoherent , in the intensity pattern depend entirelyon the particular system under test. They are more likely to appear in asystem with many transmissive optics, in which multiple reflections of thebeam may occur, as opposed to one with purely reflective optics. As such,no attempt will be made to quantify them at this point.
It is possible to circumvent these effects entirely by using a source witha short coherence length. The feasibility of using an incoherent light source
was discussed by Roddier in 1990 [75]. Additionally, an incoherent source
will reduce the inteferometric cross-talk between neighbouring spots. This ismodelled and discussed in Section 3.3.2.
Thus, a fibre-coupled AlGaAs LED, with the properties listed in Table
3.2, was chosen as the primary light source for the HWS. The coherence
length of this source was measured using a path-matched Michelson interferometer and observing the Michelson visibility [76], VM , given by

62

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

Figure 3.8: A plot of the Michelson visibility in the output of the interferometer as a function of the optical path difference. The horizontal dashed line
is due background fluctuations in the object and reference intensity profiles.
The best fit Gaussian, indicated by the dashed line, has a FWHM coherence
length of 5.5 µm.

3.3. HARTMANN SENSOR FOR ADVANCED GWI

VM =

63

I1 − I2
I1 + I2of interference fringes as a function of optical path difference, where I1 and I2
are the maximum and minimum intensities of the fringes, respectively. The
results are plotted in Figure 3.8.

3.3.2

Optimization of Hartmann plate design

The optimum Hartmann plate design for measuring thermally induced wavefront changes in the optics of a GWI is found by determining the best combination of hole size, hs , hole spacing, hp , and lever arm, L, that give a spatial
resolution and sensitivity consistent with the Advanced LIGO requirements
stated in Section 1.3.2.
For example, longer lever arms will yield greater displacements of the
spots on the CCD in response to a given wavefront change, resulting in better
sensitivity for a constant uncertainty in the centroid positions. However,diffraction from the holes increases the size of the spots on the CCD and,if the lever arm was long enough then there would be substantial cross-talkbetween neighbouring spots, reducing the sensitivity. Of course, cross-talk
could be reduced by increasing the hole spacing of the holes in the array but
this would decrease the spatial resolution. Alternatively, cross-talk could be
reduced by varying the hole size: either increasing the hole size if L is large
and there is substantial diffraction, or decreasing the hole size if L is small
and there is minimal diffraction. The optimum balance between these, hs ,hp and L must be found.
Unfortunately, the optimum balance cannot be found analytically because a typical Hartmann sensor is arranged such that the CCD is in thenear field of the holes in the plate. Therefore, numerical simulation of thediffraction patterns for various Hartmann plate configurations was used tofind the optimum solution.

64

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT
The modelling simulated a Hartmann sensor operating in Configuration

A, shown in Figure 3.5. It calculated the wavefront change induced by absorption of a 8 mm (1/e2 -radius) Gaussian beam in a 30 mm high, 50 mm
diameter cylinder of Schott FK51 glass, which was a potential candidate glass
for the thermal lensing experiment described in the previous chapter at thetime of design.

3.3.2.1

Modelling

The simulation, described below, determined the RMS noise, σsim , in the centroid measurements for a Hello-Vinet thermal lens for a variety of Hartmann
sensor configurations. The smallest noise occurs when the separation of theholes, hp , in the Hartmann plate is large and the lever arm is long, but thisis undesirable because of the low spatial resolution. Ideally, the RMS noise
should be small, i.e. 1/σsim should be large, and the density of holes, ∝ 1/h2p ,should also be large. Therefore, a figure of merit (FOM) was defined by

FOM = σsim h2p

−1.

The configuration with the largest figure of merit was then manufactured.
Note that only close-packed hexagonal arrays were modelled because thedensity of spots is approximately 15% larger than in square arrays of thesame hole spacing, improving the spatial resolution and reducing the error
in the reconstruction of the wavefront as discussed in Section 3.3.5.
The simulation used the physical optics propagation package in the ZEMAX
Optical Design program [77] [78]. The code used for this simulation is listed
in Appendix B.1.
The procedure used for each simulation was:

1. Choose the parameters for the Hartmann plate and lever arm fromthe following sets :3.3. HARTMANN SENSOR FOR ADVANCED GWI

65hs= 0.2, 0.4, 0.6, 0.8 or 1.0 mm radius2
hp = 2.25, 2.50, 2.75, 3.00, 3.25, 3.50, 3.75 or 4.00 mm
L = 100, 300, 500, 700 or 900 mm.
2. Create a 1024 × 1024 numerical mask representing a 60 mm × 60 mm
square Hartmann plate, slightly larger than the 50 mm diameter of
the FK51 glass cylinder, with 4 rings of holes in a close-packed
hexagonal arrangement (61 holes). Weight each pixel in the array by
the fraction of its area that is covered by an aperture.
3. Create a 1024 × 1024 array representing the phase front incident on
the Hartmann plate in Mathematica.

4. Calculate the phase across the array, given by 2π/λ times the optical
path distortion due to an 8 mm (1/e2 -radius) laser beam heating acylindrical piece of FK51 Schott Glass using Hello-Vinet.
5. Combine the mask and phase front to calculate the electric field
immediately after transmission through the Hartmann plate.
6. Use the diffraction propagation code in ZEMAX to propagate thefield array through the chosen lever arm distance to an output plane.
7. Convert the field at the output plane to intensity and digitize into an
8-bit TIFF image.
8. Calculate spot centroids, using Equation 3.10 (see Section 3.3.4), andcentroid displacements from their nominal position.
9. Determine the magnitude of the measured gradient as a function ofradial position.

66

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

10. Compare to the known theoretical gradient of the Hello-Vinet
distortion, calculated by the gradient mutiplied by the lever arm anddivided by the pixel size.
11. Calculate the standard deviation of the difference between themeasured and known gradients.

The intensity patterns, resulting from Step 7, for hs = 2.0 mm, hp = 2.75
mm, L = 300 mm is shown in Figure 3.9 a) and for hs = 0.8 mm, hp = 2.25mm, L = 900 mm is shown in Figure 3.10 a). Plots for the simulated (points)
and known (line) spot displacements, resulting from Step 10, for hs = 2.0mm, hp = 2.75 mm, L = 300 mm, and for hs = 0.8 mm, hp = 2.25 mm, L =900 mm are shown in Figure 3.9 b) and Figure 3.10 b).
The five best Hartmann sensor parameter sets and their figures of merit
are listed in Table 3.3. The parameter set: (hole diameter of 0.8 mm diameterand hole spacing of 2.25 mm) was chosen for fabrication as a Hartmann plate.
Subsequently, it was decided that the sensor would be more robust if theHartmann plate was physically attached to CCD, as shown in Configuration
B in Figure 3.5. Thus the hole spacing and hole size were demagnified by afactor, M ≈ 5.3, the demagnification of the telescope in Figure 3.5, and thelever arm was demagnified by M 2 . The parameter sets for Configurations A
and B are listed in Table 3.4.
A hexagonally-close-packed array of 150 µm diameter holes spaced 430 µm
apart was laser drilled in a 50µm thick piece of brass plate to cover a 12.5 ×

12.5 mm area. The laser drilling process could position holes to within ±1 µm[79].

The plate, as viewed under a microscope, is shown in Figure 3.11. The
small diameter holes have a tendency to get clogged with dust when the plateis stored in a dirty environment. The two images in this figure show thatplate before and after cleaning in an ultrasonic bath. Also indicated is thehexagonal unit cell.

3.3. HARTMANN SENSOR FOR ADVANCED GWI

67

(a)

(b)

Figure 3.9: a) Numerically determined diffraction pattern from a Hartmann
plate with hs = 2.0 mm, hp = 2.75 mm, L = 300 mm and b) the simulated(points) and known (solid line) centroid displacements.

68

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

(a)

(b)

Figure 3.10: a) Numerically determined diffraction pattern from a Hartmann
plate with hs = 0.8 mm, hp = 2.25 mm, L = 900 mm and b) the simulated(points) and known (solid line) centroid displacements.

3.3. HARTMANN SENSOR FOR ADVANCED GWI

69Hole diameter, hs (mm)
0.8
0.8
0.8
0.8
0.8

FOM
92.1
67.5
66.1
57.8
55.9

Hole Spacing, hp (mm)
2.25
2.25
2.75
2.50
2.50

Lever Arm, L (mm)
900
700
900
500
700

Table 3.3: Best five Hartmann plate configurations as rated by the figure of
merit (FOM).

Hole diameter
Hole Spacing
Lever Arm

Configuration A
0.8 mm2.25 mm≤ 900 mm

Configuration B
150 µm
430 µm≤ 31 mmTable 3.4: Final design parameters for the Hartmann sensor

Figure 3.11: Hartmann plate as viewed under microscope. a) as delivered, b)
after cleaning in an ultrasonic bath and with hexagonal unit cells highlighted.
Note that the second image is brighter because the plate was more highly
illuminated.

70

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

3.3.2.2

Simulations of cross-talk

If the array of holes in the HP is is illuminated by a spatially varying intensity
and/or wavefront, then centroid pulling due to cross-talk will occur and resultin systematic errors in the centroid displacements. The magnitude of theseerrors will also be affected by the length of the lever arm and the coherence
length of the light source, particularly if spurious interference fringes areproduced by transmissive optics between the light source and HWS.
Numerical simulations were used to quantify the effect of cross-talk for
Configuration B. The simulations used two different lever arms, 31 mm and
10 mm, and coherent light, with a wavelength of 820 nm, and incoherent
light with a central wavelength of 820 nm and a FWHM spectral width of
120 nm (the width of a source with a coherence length of 5.5µm).
The systematic error due to cross-talk was calculated by comparing thedisplacement of a spot centroid due to wavefront defocus for an isolated spot
and a spot at the centre of 6 hexagonally-coordinated neighbouring spots.
The procedure used for this simulation was:
1. Single hole HP, shown in Figure 3.12 a)(a) Illuminate the HP with a flat wavefront and simulate numericallythe diffraction pattern at the lever arm distance using Fourier
diffraction propagation [80] (see Appendix B.2 for the source codeof this simulation). Determine the centroid.
(b) Repeat step 1 (a), but illuminate with a wavefront aberrated bya small, randomly-selected, amount of the primary aberration defocus that is centered at a random position within the vicinity of
the hole in the HP.
2. Seven hole HP, shown in Figure 3.13 a)(a) Repeat step 1 (a).
(b) Repeat step 2 (a), but illuminate with the aberrated wavefront
used in step 1 (b).

3.3. HARTMANN SENSOR FOR ADVANCED GWI

71

The resulting diffraction patterns for the single hole HP are shown in
Figure 3.12 b) and c) coherent and incoherent, respectively, for the 31 mm
lever arm and d) and e) coherent and incoherent, respectively, for the 10
mm lever arm. The diffraction patterns for the seven hole HP are shown in
Figure 3.13 b) - e), arranged in the same order as the previous figure.
The cross-talk is defined to be that part of the spot displacement that isdue to the presence of neighbouring spots. The two quantities required to
calculate this are:
• The displacement of the spot for the single hole HP, ∆x1h , given by

the difference between the centroid calculated in step 1 (b) and thecentroid calculated in step 1 (a).

• The displacement of the central spot for the seven hole HP, ∆x7h , givenby the difference between the centroid calculated in step 2 (b) and thecentroid calculated in step 2 (a).

Recall that ∆x1h and ∆x7h are simulations of the same wavefront curvature. Thus, the cross-talk, ǫCT , is given byǫCT =∆x7h − ∆x1h
∆x1h

(3.8)

The results of the cross-talk simulations are summarised in Table 3.5. The
cross-talk shown in this table is the result of averaging over many defocus
aberrations. They show that the contribution of cross-talk is much moresignificant with a lever arm of 30 mm than 10 mm. They also show that better
results are obtained with the incoherent source, which is not unexpected
because of the reduced interference between neighbouring spots.
As a result of these simulations, the nominal lever arm in the design was
reduced to 10 mm. It should be noted that this is a departure from theoptimized design reached in Section 3.3.2.1, however, the reduction of thelever arm by a factor of only 3 in order to reduce the systematic cross-talkfor the incoherent source by a factor of 18.75 was deemed to be worthwhile.

72

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT(a) Hartmann mask(b) Coherent: L=31mm(c) Incoherent: L=31mm(d) Coherent: L=10mm(e) Incoherent: L=10mm

Figure 3.12: a) Simulated Hartmann plate with a single hole. b) calculateddiffraction pattern from the Hartmann plate at a distance of 31 mm whenilluminated with a coherent 820 nm light source and c) with an incoherent 820
nm light source with a coherence length of 5.5 µm. d) diffraction pattern at
a distance of 10 mm when illuminated with a coherent 820 nm light sourceand e) with an incoherent 820 nm light source with a coherence length of
5.5 µm.

3.3. HARTMANN SENSOR FOR ADVANCED GWI73(a) Hartmann mask(b) Coherent: L=31mm(c) Incoherent: L=31mm(d) Coherent: L=10mm(e) Incoherent: L=10mm

Figure 3.13: a) Simulated Hartmann plate with multiple holes. b) calculateddiffraction pattern from the Hartmann plate at a distance of 31 mm whenilluminated with a coherent 820 nm light source and c) with an incoherent 820
nm light source with a coherence length of 5.5 µm. d) pattern at a distanceof 10 mm when illuminated with a coherent 820 nm light source and e) withan incoherent 820 nm light source with a coherence length of 5.5 µm.

74

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT
Lever Arm Length
10 mm
31 mm

Crosstalk - Coherent
0.18%
2.7%

Crosstalk - Incoherent
0.08%
1.5%

Table 3.5: Level of cross-talk error in the centroid displacement of a single
Hartmann spot for coherent and incoherent sources at two different lever arm
lengths
3.3.2.3

HWS construction

An exploded view of the final Hartmann sensor is illustrated in Figure 3.14.
A 4 mm thick spacer containing a 36 mm diameter hole is bolted to the frontof the CCD camera (the CCD is discussed in detail in Section 3.3.3). A front
plate, also containing a 36 mm diameter hole, is bolted to the spacer. The
Hartmann plate is sandwiched between the spacer and the front plate in a
30 µm recessed region. The CCD is nominally 6 mm behind the front faceof the camera body. In combination with the spacer, this yields a lever arm,
L, of approximately 10 mm.
This construction addresses problems of CCD-HP alignment. Rotationsof the HP about the x or y axes will cause errors in the HWS measurement.
Either of these rotations cause the CCD and HP not to be parallel. Therefore,
the lever arm distance between the CCD and HP will not be constant acrossthe face of the HP. This is not a problem here because, as shown in Figure
3.14, the CCD is nominally parallel to the front face of the camera housing
and the sandwich construction should minimize misalignments between the
HP and the CCD.

3.3.2.4

Estimated temperature dependent defocus error

For a brass Hartmann plate (with a linear thermal expansion coefficient,αbrass , of 20.3 µm m−1 K−1 [81]) and a lever arm length of 10.43 mm (themeasurement of which is described in the next chapter), the temperaturedependent defocus error can be determined using Equation 3.6

3.3. HARTMANN SENSOR FOR ADVANCED GWIPower connector
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CCD

30µm recess

Figure 3.14: A 4mm wide spacer is bolted to the front of the Dalsa 1M60
camera. The Hartmann plate is positioned in a 30 µm recess in the front
plate which is then bolted onto the camera. 36mm diameter holes in bothplates allow the the rays to propagate onto the CCD. The inset shows theicon used to represent the HWS in schematic diagrams.

αbrassSthermal== 0.00195 m−1 K−1
∆T
L

3.3.3

(3.9)

CCD camera

The CCD records a digital image of the intensity distribution of the Hartmann spots by accumulating photoelectrons, produced by the incident lightin an Nx × Ny array of picture elements (pixels). Once exposure is complete,

the CCD transfers the accumulated charge in each pixel out of the array andconverts the output into a measurable voltage [82]. This voltage is passed
through an n-bit A/D converter to yield the digital numbers (DN) that makeup the digital image. The recording, transfer and digitization processes canintroduce noise which degrades the image fidelity and reduces the ability ofHartmann sensor to locate accurately the centroids of the each of the spots
in the image.
One additional problem during exposure is the leakage of charge from a

76

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT
Parameter
Array size
Pixel size, psDigitization
Quantum efficiency @ 820 nm
Full well capacity (linear operation)
Dark current @ 45◦ C
Random Readout noise (rms)
Fixed pattern noise
Random non-uniformity
Antiblooming

Value1024 × 1024 pixels
12 µm × 12 µm
12 bits≈ 6%
350,000 electrons
1.3 × 10−16 A per pixel
1.2 DN
3.8 % rms
0.3 % rms
> 100× saturation level

Table 3.6: Specifications of Dalsa 1M60 CCD camera using an FTT 1010-M
image sensor [85] [86]. Note that the operating temperature of the CCD
array is approximately 45◦ C when the room temperature is approximately
20◦ C.

full pixel well into the wells of adjacent pixels. This effect is known as blooming and it will reduce the quality of the digital image. Fortunately, modern
scientific grade CCDs have antiblooming features that use an overflow drain
below each pixel to prevent blooming for reasonable exposure levels. Charge
flows into the drain before it is able to flow into adjacent pixels and is constantly removed [82] [83] [84]. The magnitude of the anti-blooming in a CCD
refers to the multiple of the saturation level that can be applied to a pixel
before any charge leaks into a neighbouring pixel. Ultimately, this featureallows the CCD to be used close to saturation level, improving the sensitivity,
without any danger of adjacent pixels contaminating one another.
A Dalsa 1M60 CCD camera [85], the relevant parameters of which arelisted in Table 3.6, was chosen as the image sensor in the Hartmann wavefront sensor. The substantial technical and diagnostic information available
on this CCD indicates that it should be shot-noise limited (see below). A
measurement that tested and verified this conclusion is described in this section after a brief discussion of the various noise sources in the CCD. The
magnitudes of these noise sources are collated in Table 3.7 for convenience.

3.3. HARTMANN SENSOR FOR ADVANCED GWI
3.3.3.1

77

Expected noise in CCD

The dominant noise sources, in order of their appearance in the measurement
process, are:• Photon shot noise. This arises because of the non-uniform arrival of
the photons at the pixels. It can be calculated by dividing the numberof electrons per pixel, Ne , by the quantum efficiency (QE ≈ 6%) todetermine the approximate number of photons, Np , required to produce
Ne electrons. For the Dalsa 1M60, Np ≈ 5.8 × 106 when Ne is equal

to the maximum value of 350,000. The rms noise in the number ofphotons isq

Np . This can be expressed in electrons by multiplying by

the QE. The maximum rms fluctuation due to photon shot noise is 145
electrons.
• Photoelectron shot noise arises because of the random nature of the√photoelectric conversion process and is given by Ne . For the maximum number of electrons in a pixel (Ne = 350, 000) the rms fluctuation
due to electron shot noise is 592 electrons.
• Readout noise is due to random noise sources within the sensor, includ-ing dark current and amplifier noise. Dark current refers to electrons
which are thermally generated, rather than photo-generated. The HWS
typically uses a short integration time (≤ 50 ms) and thus dark current
can be ignored. Amplifier noise is simply the noise introduced by thesensor electronics. The specified value [85] in Table 3.6 is 1.2 digital
numbers (DN). This can be converted into electrons assuming a gain of
4096 DN per 350,000 electrons. The readout noise is thus 103 electrons.

• Digitization noise is the uncertainty in the measurement of the numberof electrons in a pixel because of quantization into an n-bit scale. Thisis calculated using Equation A.4:
1 Nemax
σdigital = √
3 2n

78

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT
Noise TypeRMS Value (el)Photon shot noise
145
Photoelectron shot noise
592
Random readout noise
103
Digitization noise @ 12-bits
49.3
Fixed Pattern noise
3.9
Random non-uniformity
0.3% of total variance5.40 %
91.2 %
2.76 %
0.63 %
0.004%
2 × 10−5 %Table 3.7: Noise from each source expressed in electrons.
which gives a value of 49.3 electrons when Nemax = 350, 000 electrons
and n = 12.
The remaining noise sources specified by the camera manufacturer, fixedpattern noise and photoresponse non-uniformity, were small enough, see Table 3.7, to be safely ignored.
It is clear from Table 3.7 that digitization noise is negligible and thatthe total noise is dominated by photoelectron shot noise for the Dalsa 1M60
CCD. The contribution of the digitization noise for a reduced number of bits
is shown in Table 3.8, indicating that 10-bits, as opposed to 12-bits, would
not increase the total noise significantly. The total relative error, σtot /Ne , inthis Table is calculated using
σtot
=
Ne

q

2
2
σothers+ σdigitalNe2
where σothersis the total variance from all the other noise sources as presented

in Table 3.8. In summary, using a CCD with more than 10-bits of digitization
is worthwhile only if the photoelectron shot noise is also reduced. This would
be accomplished by increasing the maximum number of electrons per pixel.
3.3.3.2

CCD: Measurement of noise

To confirm that the CCD was indeed shot noise limited, it was illuminatedat three different intensity levels by the LED described in Section 3.3.1. The
RMS fluctuations in the DN of the pixels were measured and are plotted as3.3. HARTMANN SENSOR FOR ADVANCED GWI
Number of bits Total var. (el2 ) % of total var.
Other sources
3.82 × 105
58
10.0 × 10
62%
95.38 × 105
29%
5
10
4.21 × 10
9.3%
5
11
3.92 × 10
2.5%
5
12
3.85 × 10
0.63%

79

Total relative error
0.29%
0.21%
0.19%
0.18%
0.18%

Table 3.8: The contribution of digitization noise to the total variance in the
pixel measurement using 8 to 12-bit digitization. All other noise sources are
assumed to be fixed at the levels shown in Table 3.7. Also shown is theoverall relative error in the measurements in each pixel.
a function of digital number in Figure 3.15 on a log-log scale. A line withslope -1/2, the dependence expected for Poissonian statistics, is also plottedin Figure 3.15. The good agreement indicates that the noise in the CCD is,indeed, Poissonian and therefore that the CCD is random. From this plot,
the relative error at DN = 4095 is estimated to be 2.5 × 10−3 (≈ 10−2.6).

This is equivalent to the noise in a Poissonian distribution with a mean of≈ 160, 000 counts, assumed to be the equivalent number of electrons in the

pixel well when full. This value is slightly less than the nominal full wellcapacity of 350,000 electrons.
3.3.3.3

Pixel size measurement

Since all centroid positions are calculated in units of pixels, the wavefront
reconstruction is only as accurate and precise as the pixel size, ps . Therefore,
the pixel size was measured as outlined below.
The CCD was illuminated with the diffraction pattern from the edgeof a razor blade and the image recorded. The CCD was then translated
horizontally and perpendicular to the illuminating beam by 6.001 ± 0.003

mm using a micrometer-controlled translation stage. A second image of thediffraction pattern was recorded. The two images are shown in Figure 3.16.
Horizontal cross-sections of each image are plotted in Figure 3.17. The second
fringe in the diffraction pattern was chosen as a well defined feature to locate

80

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

Figure 3.15: RMS fluctuation in intensity vs intensity. The dashed line hasa slope of −1/2.
on the CCD when it was in the two positions and its peak was found for both
images. The data points marked by the square boxes in Figure 3.16 indicate
the data that were used to calculate the peaks. The difference between thetwo peaks was 500.9 pixels. This yielded an average apparent pixel size of
11.975 ± 0.005 µm.

3.3.4

Centroiding algorithms

As discussed in Section 3.2.2, a centroid algorithm is normally used to locate
the centres of the spots in a Hartmann image. The conventional form of thisalgorithm is Pi = N max Pj = M maxmmini = Nmmax
i = Nmmini = Nm(xc , yc )m = ps  Pmmin
j = Mmmax
j = Mmmin
j = Mm

P

i Iij
Iij

max Pj = M max
Pi = Nmmmin
i = Nmmax
i = Nmmini = Nm, P

min
j = Mmmax
j = Mmmin
j = Mm

P

j Iij
Iij




(3.10)

where ps is the pixel size, i and j are the CCD pixel indices in the horizontal

3.3. HARTMANN SENSOR FOR ADVANCED GWI

81Figure 3.16: Fixed diffraction pattern on the CCD before (top) and after(bottom) a lateral translation of the CCD by 6.001 mm as measured on a
micrometer-controlled translation stage.

(a) Position 1(b) Position 2

Figure 3.17: Average horizontal cross section of 1.0 mm vertical slice of both
images shown in Figure 3.16 in the vicinity of the edge of the shadow. The
peak of the second fringe in the diffraction pattern was determined for both
images over the range indicated by the square data points.

82

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

and vertical directions, respectively, and Ii j is the intensity of the ij th pixel.
This algorithm can give rise to significant systematic errors dependingon the limits of summations, especially if the centroid of the spot is notat the center of the detection area [87]. Many variations on this algorithm
have therefore been developed to minimize systematic and random errors in
the centroid. Much of this work was concerned with the performance of thealgorithm (and any variations) for a low photon flux [88] [89], such as might beused in adaptive optics in astronomy. However, much brighter light sources
can be used for metrology and thermal lens measurement. Therefore, rather
than considering all the various centroiding algorithms, this discussion will belimited to those that perform best in high photon flux: the weighted centerof gravity algorithm [87] [88] [68] and a novel ’fractional pixel’ algorithm of
the author’s design.
3.3.4.1

Weighted center of gravity (WCoG)

The WCoG algorithm applies a mask to the intensity distribution such that aweighting, (Fw )i , is given to different pixels depending on their digital value.
The basic 1-D form is
Pimax

i Ii (Fw )i
i=imin Ii (Fw )imin
xWCoG = ps Pi=i
imax

(3.11)

The weighting function is quite often chosen to be a Gaussian that iscentered around an a priori estimate of the spots center [68]. Alternatively,
it can chosen to be the intensity in the pixel raised to a power P − 1 [87]:(Fw )i = IPi −1
and thus
Pimax

i IPi
.
P
i=imin Ii

min
xWCoG = ps Pi=i
imax

(3.12)

Note that the summation range imin ≤ i ≤ imax about a spot is chosen

3.3. HARTMANN SENSOR FOR ADVANCED GWI

83

such that the central pixel is that pixel with the maximum intensity. Additionally, the summation range extends by the same amount, n1 , in thepositive and negative directions from the central pixel, resulting in 2 n1 + 1
pixels, an odd number, in the range.
3.3.4.2

Fractional pixel centroiding (FPC)

An iterative algorithm to locate the centroid of a symmetric intensity distribution was developed. A vaguely similar method was suggested by Topa [90]for use in a Shack-Hartmann sensor.
The premise for this algorithm is derived from the problem identified byTopa [90] of a symmetric spot being located asymmetrically about a pixelon the CCD array. The center of the range of summation for the centroid
calculation is not coincident with the center of the spot and there is a systematic error in the centroid calculation. In general, it is very unlikely thatthe true center of a spot will lie in the center of a pixel. If, however, therange of summation can be altered such that the true center lies at the center of the range of summation, then the systematic error should be removed.
Therefore, we must treat the data from the CCD as a continuum and consider fractional pixels. Thus, the true centroid is defined to be that centroid
which is coincident with the center of the rectangular region (boxcar) used to
calculate it.
Since the coordinates on the CCD are considered to be a continuum,
but the individual pixels still need to be identified with discrete indices,
the following scheme applies. The coordinates in the continuum, (ī, j̄), arelabelled with a bar to signify their continuous nature and the indices of the
pixels remain the same, i and j. In the ij th pixel, therefore, the continuum
ranges from (ī, j̄) in the bottom left corner to (ī + 1, j̄ + 1) in the top rightcorner.
The algorithm used to locate the true centroid, (illustrated schematicallyin figure 3.18) is:
1. Define a boxcar of ranging from i¯0 to i¯1 in the horizontal direction and

84

CHAPTER 3. HARTMANN SENSOR DEVELOPMENTj¯0 to j¯1 in the vertical direction, that is approximately centered over a
spot in the digital image.
2. Apply equation 3.13 to determine the centroid. The numerator of theequation is the sum of a series of values. Each value in that sum isthe weighted intensity, Iij , of the ij th pixel multiplied by the fractionof the area, Aij , of that pixel that is covered by the boxcar and thecenter, (īmid , j̄mid ), of the intersection between the pixel and the boxcar.
The denominator is the sum of the weights, Aij , multiplied by theintensities, Iij . If the boxcar range is restricted to the discrete indices,
this centroid algorithm becomes the same as the conventional centroid
algorithm in Equation 3.10.

(xc , yc ) = ps F C (Iij : i¯0 , i¯1 , j¯0 , j¯1 ) Pceiling(i¯ ,j¯ )1= ps 

1

i,j=floor(i¯0 ,j¯0 ) īmid Aij Iij,
Pceiling(i¯1 ,j¯1 )
A
I
ij
ij
¯
¯
ij=floor(i0 ,j0 )(3.13)
Pceiling(i¯1 ,j¯1 )where īmid and j̄mid are defined ¯
i0 + (i + 1)



2


for

 ¯
j0 + (j + 1)



2


j = floor (j¯0 )for ceiling (j¯0 ) ≤ j < floor (j¯1 )for
j = floor (j¯1 )īmid = j̄mid =
i + 12i + i¯12

j + 21j + j¯1
2

i = floor (i¯0 )for ceiling (i¯0 ) ≤ i < floor (i¯1 )for
i = floor (i¯1 )forand Aij is defined

Aij =



(i + 1) − i¯0




1






i¯1 − ii = floor (i¯0 )for ceiling (i¯0 ) ≤ i < floor (i¯1 ) ×
fori = floor (i¯1 )for

i,j=floor(i¯0 ,j¯0 ) j̄mid Aij Iij 
Pceiling(i¯1 ,j¯1 )ij=floor(i¯0 ,j¯0 ) Aij Iij

3.3. HARTMANN SENSOR FOR ADVANCED GWI
(j + 1) − j¯0




1






j¯1 − jj = floor (j¯0 )for ceiling (j¯0 ) ≤ j < floor (j¯1 )for
j = floor (j¯1 )

85

for

.

3. Center the new boxcar at this centroid.
4. Repeat steps 2 and 3 until the difference between the calculated centroid of a boxcar and the center of that boxcar differ by less than aspecified error value.

3.3.4.3

Comparison of algorithms

A numerical simulation was used to compare the accuracy of the WCoG
and FPC algorithms in determining the centroid of a 2D Gaussian intensity
distribution with a 1/e2 full-width of 8.4 pixels, in the presence of realistic
noise sources. The width of 8.4 pixels was chosen because this is the width
for the best fit Gaussian to a spot 10 mm from the Hartmann plate designed
in Section 3.3.2.
A Gaussian spot with a known center, xG1 , was produced. Shot noise (SN)
was added at the realistic levels determined in Section 3.3.3.2 and readout
noise (RN) at the levels in Table 3.7 were added and the spot was quantizedusing 12-bit digitization. The centroid of the spot was found with both algorithms for a given summation range in the WCoG algorithm and a boxcarof the same width in the FPC algorithm. In this instance the optimum summation range/boxcar width was determined to be 19 pixels. The Gaussian
spot was moved to a new location, xG2 , and the procedure was repeated.
The difference between the centroids from the first and second spots was determined for both algorithms and was compared to the known displacement.
The results are summarised in Table 3.9.
The FPC is, ultimately, limited in precision because all pixels are weighted
by the same amount. Those pixels with higher levels of shot noise, due to lesselectrons in the pixel well, will increase the overall noise in the centroid calcu-

86

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

Figure 3.18: A cartoon of the pixels on a CCD and the integration regions.
The centroid is determined for a boxcar of fixed width that is positioned
to include fractional parts of pixels. The true center is illustrated by thecross. The center of each boxcar is the centroid of the previous boxcar. The
successive boxcars and their numbered central spots are seen approaching
the true center.

Algorithm
Fractional Pixel Centroiding
Weighted Center of Gravity

SN & RN
0.34 % pixel
0.19 % pixel

Table 3.9: RMS error in the FPC and WCoG centroiding algorithms measurements of displacements in the presence of realistic noise sources, shotnoise (SN) and readout noise (RN).

3.3. HARTMANN SENSOR FOR ADVANCED GWI

87

lation. Conversely, the WCoG algorithm preferentially weights those pixels
with higher intensities, and, therefore, lower levels of shot noise reducing theoverall noise in the centroid calculation.
The FPC algorithm was used for earlier HWS measurements describedin Section 4.3 and in Chapter 6. Later measurements, the remaining HWS
measurements in Chapter 4 and all measurements in Chapter 5, used the
WCoG algorithm.

3.3.5

Wavefront reconstruction algorithm

For most applications the discrete gradient field output from the HWS must
be converted, via numerical or analytic integration, into a wavefront map
for additional post-processing1 . Most of the material published on wavefront reconstruction is based on the work of Southwell [62] who applied
reconstruction techniques that were developed for shearing interferometers
to Hartmann-type sensors and described the two basic types of wavefront
reconstruction algorithms: modal and zonal.
3.3.5.1

Modal wavefront reconstruction

In modal reconstruction the wavefront is represented by the sum of continuous functions
W (x, y) =nXai fi (x, y)nX~ i (x, y)
ai ∇f

i=1

and its gradient is represented by~ (x, y) =∇W

i=1

The coefficients ai are found by performing a least-squares fit of the gradient~ i (x, y), to the discrete gradient field measured by the HWS.
functions, ∇f
1

Sometimes full wavefront reconstruction is unnecessary and the analysis can be donewith the gradient data directly, for example in the measurement of defocus [91].

88

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT
Zernike polynomials are a particularly popular choice for SHWS used in

adaptive optics [92] [93] and ophthalmology [94] because they are orthogonalover circular apertures, or approximately orthogonal over a discretely sampled circular aperture. However, unless many polynomials are used, whichbecomes computationally expensive, the result is generally a smooth wavefront map.
The smooth nature of the reconstructed wavefront tends to mask localized
thermal blooming. Ryan Lawrence [12] found that Zernikes were inappropriate for describing point absorbers and scanning CO2 laser compensation.
Recall from the requirements for the wavefront sensor, Section 1.3.2, that after wavefront reconstruction one must be able to resolve the thermal lensing
from a point absorber. For this reason modal reconstruction is considered to
be inappropriate for use in a Hartmann sensor measuring thermal lensing in
the optics of a GWI.
3.3.5.2

Zonal wavefront reconstruction

Zonal wavefront reconstruction determines the best fit wavefront at the centerof each of the holes in the Hartmann plate. The advantage of this technique isthat it typically provides richer information than the modal technique [95].
Numerical integration can produce rounding errors when there are sharp
peaks in the wavefront. The following is a description of a reconstruction
technique reported by Southwell, which uses the Gauss-Seidel iteration.
An iterative algorithm can be used to reconstruct the wavefront, Wm ,at each of the M points on the Hartmann sensor from the discrete gradient
field, Gm , given by(

Gm = (x, y) ,

∂W ∂W
,
∂x ∂y

!)

.
m

First, the nearest neighbours of each point must be found. The nearest
neighbours of a point are defined as those points the distance to which lies
within the range hp ± ε/2, as illustrated in Figure 3.19, where hp is the hole

3.3. HARTMANN SENSOR FOR ADVANCED GWI
21

22

23

89
24

ε
16

17

18

19

20

hp
10

11

5

12

13

6

7

1

2

14

15

8

3

9

4

Figure 3.19: Hartmann spots arranged in a hexagonal grid and numberedfrom 1 to 24. The nearest neighbours of the 13th spot are those lying within
the dashed annular region of inner radius hp − ε and outer radius hp + ε.
spacing of the array of spots. This can be expressed as an M × M matrix,

Λm n :

Λm n =

 1
 0

if | ~xm − ~xn | − hp ≤ εif| ~xn − ~xn | − hp > εwhere ~xm designates the mth point, ~xn designates the nth point and where
the tolerance value, ε, is included to allow for any small deviations from ahexagonal array.
Next, a matrix of wavefront differences between nearest neighbours, ∆wneighbours
is defined:neighbours
∆wm
=
n


 ∆w

mn



0

if | ~xm − ~xn | − hp ≤ εif| ~xm − ~xn | − hp > ε,where the wavefront difference, ∆wm n , between a point, ~xm , and its neigh-90

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

bour, ~xn , is found using the Improved Euler-Cauchy method (Heun’s method)
[96]:

∆wm n

1
=
2

"

∂W
∂x

!

∂W
+
∂x
m

! #

× (xm − xn ) +

1
2

"

∂W
∂y

!

∂W
+
∂y
m

! #

× (ym − yn )nn

(3.14)

The wavefront can be now be reconstructed using Gauss-Seidel iteration
as follows:
1. Begin with a flat wavefront map as a zeroth guess, such thatWm, 0 = 0
2. Determine the k th wavefront map, Wm, k , by updating each value inthe map, Wm, k−1 , using the matrix ∆wneighbours :Wm, k =P
neighbours+ N
j=1 ∆wm j
j=1 Λm j × Wj, k−1
PN
j=1 Λm j

PN(3.15)In the Jacobi method, the wavefront is updated as a whole. That is, all
the wavefront values, Wm, k are updated from a previous set of values{W1, k−1 , W2, k−1, . . . , Wm−1, k−1 , Wm, k−1, . . . , WN, k−1 }In Gauss-Seidel method, the wavefront value Wm, k is updated from a
set including the most recently updated values{W1, k , W2, k , . . . , Wm−1, k , Wm, k−1 , . . . , WN, k−1 }3. Repeat step 2 until the change between successive iterations is less thana previously defined error value.

3.3. HARTMANN SENSOR FOR ADVANCED GWI

91

4. Once the wavefront, Wm , has been reconstructed at the discrete points,~xm , use linear interpolation to convert the discrete hexagonal array of
wavefront values into a 2D Cartesian array suitable for display as acontour plot.
In certain cases, the wavefront is such that it appears likely that rounding
error will occur in the numerical integration. The wavefront may contain asharp peak, for example. In this case, the rounding error can be reduced
by artificially adding points into the discrete gradient field, such that a new
point lies between two of the original points in the gradient field and thegradient of the new point is determined by interpolating the gradients ofseveral of the surrounding points.
3.3.5.3

Error propagation - From gradient to wavefront2, deThe mean square error (variance) in the reconstructed wavefront, σW

pends on the variance in the wavefront difference between adjacent holes,
2σ∆W. It can be predicted using

2
2= C∆W σ∆W
σW

(3.16)

where C∆W is the Southwell noise coefficient, [62], and the variance in the2
wavefront difference between adjacent holes, σ∆W, is given by

2
σ∆W
=

"

hp
L



σxc + σyc2

#2

(3.17)

where σxc and σyc are the rms errors in the x and y centroids, respectively,
hp is the distance between the holes in the Hartmann plate and L is the leverarm of the Hartmann sensor.
As pointed out by Southwell [62], C∆W is dependent on the number andarrangement of holes. Following Southwell’s methodology, the noise coefficient was numerically determined as a function of the number of holes in ahexagonal close-packed arrangement and the results are shown in Figure 3.20.

92

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

Figure 3.20: The Southwell noise coefficient of the Gauss-Seidel iterative
matrix wavefront reconstruction versus the number of holes in the HP. A
hexagonal close-packed arrangement of spots was used. Note that this result
assumes that the noise in the gradient field is uncorrelated.
This coefficient allows one to predict the noise in the overall reconstructed
wavefront if the noise in the gradient field is uncorrelated. The actual noise
is determined by analyzing the reconstructed wavefront. Note that C∆W < 1
for the approximately 700 holes required for a HWS for advanced GWI andthus the zonal reconstruction reduces the effect of uncorrelated noise in thegradient field on the reconstructed wavefront.

3.3.6

Summary of noise sources

The noise sources discussed in the previous sections are summarized below
for convenience.
• Coherent Noise: An incoherent source was chosen to mitigate the effects

3.3. HARTMANN SENSOR FOR ADVANCED GWI

93

of noise due to parasitic interference fringes caused by transmissive
beam-control optics.
• Cross-talk: It was established in Section 3.3.2 that the level of cross-talk between neighbouring spots is about 0.08% of the signal size, or
one part in 1250. Hence, for a signal with a maximum size of 200 nm,
the cross-talk is not expected to produce an error greater than 0.16 nm.

• Random noise: (shot noise, readout noise and digitization). The error

in the centroid due to these random noise sources is expected to beabout 0.19% pixel using the WCoG algorithm and 0.34% using the FPC
algorithm, (see Table 3.9). Using Equations 3.16 and 3.17 and a noise
coefficient for the variance of approximately 0.4 (see Figure 3.20 for 700
gradient measurements) the rms wavefront error is estimated to be 0.6
nm for the WCoG algorithm and 1.1 nm for the FPC algorithm. Since
these noise sources are random, this can be improved by averaging.

• Sthermal (T ): One can specify the maximum permissible temperaturefluctuation, ∆Tmax , such that the error from this source is not greaterthan Wεmax = λ/467 @ 633 nm = 1.35 nm.

∆Tmax = Wεmax ≈ 0.04 K

2
d2 Sthermal∆TThe largest source or error in the sensor is expected to be random noise in
the CCD (shot noise, readout noise and digitization uncertainty). This willcontribute an error of approximately 0.6 nm, or about λ/1050 @ 633nm, to a
wavefront measurement of thermal lensing in a GWI, which is substantiallybetter than the sensitivity required for the wavefront sensor in Advanced
LIGO. If, in addtion, there is a signal of approximately 200 nm to be measured, cross-talk will also contribute and the total error will be approximately

94

CHAPTER 3. HARTMANN SENSOR DEVELOPMENT

0.76 nm or about λ/830 @ 633nm.

3.4

Conclusion

The design and development of a HWS for advanced GWI has been presented
in this chapter. Random noise in the CCD has been identified as the major
error source in the HWS limiting the sensitivity to about λ/1050 @ 633nm a value that can be improved with averaging due to the random nature of thenoise. In the next chapter, diagnostic testing of and measurements using thissensor are described and the improvement of the sensitivity with averaging
is demonstrated.

Chapter 4
Testing the sensor
4.1

Introduction

This chapter focusses on the calibration and testing of the Hartmann wavefront sensor (HWS) described in Chapter 3. The accuracy of the sensor isdetermined by that of the pixel spacing, ps , and the lever arm, L, used toconvert the transverse displacement of the Hartmann spots to the gradient
field corresponding to the wavefront change. The measurement of ps was
described in Section 3.3.3.3, yielding 11.975 ± 0.005µm. The measurementof L will be discussed in Section 4.2. The sensitivity (precision) of the HWSis expected to be affected by the light source used for the HWS, see Section
3.3.1, and fluctuations in the temperature of the HWS, see section 3.3.2.4.
In Sections 4.3 and 4.4, therefore, a comparison of the noise floor of thecentroids in the HWS for different light sources and a measurement of its
temperature sensitivity are described.
For a suitable light source and temperature stability, the sensitivity of
the HWS is expected to be limited by photoelectron shot noise. Thus, inSection 4.5, a measurement is described of the noise floor of the HWS and theimprovement that can be obtained by averaging the centroid positions overmultiple Hartmann images. The measurement of a small known wavefrontchange to determine the accuracy of the HWS is described in Section 4.6.
95

96

CHAPTER 4. TESTING THE SENSOR

4.2

Lever arm calibration

The nominal value of the lever arm, L, the distance between the Hartmann
plate and the CCD, in the fully assembled Hartmann sensor is approximately
10 mm. Given that every gradient measurement of the sensor relies on thisvalue, any error in it reduces the accuracy of the sensor. As this error iscommon to all measurements it thus represents a systematic error. It wastherefore necessary to refine the value of the lever arm.
An experiment that precisely and accurately determines the lever arm of
the Hartmann wavefront sensor is illustrated in Figure 4.1. Vertical needles
that could be bolted coaxially to tapped holes were designed. These werebolted, at various times, into the regular grid of holes in a Newport RS Series
Research Grade Optical Table at positions A, B, C and D. Position E, shownin Figure 4.1, contained no vertical needle, but was the intersection of theline through BD and the line parallel to AB that passed through C. The
HWS was placed at the point of intersection of the line through AB and theline through CD and was approximately normally incident to AB, the precise
angle of incidence not being critical.
A HeNe laser beam was aligned to needles at A and B by overlapping
their shadows. The needles were removed and the resulting spot pattern wasrecorded and centroided 25 times. The mirrors M1 and M2 were then used
to align the beam to needles at positions C and D, again by overlapping theirshadows. The needles were removed and the new spot pattern was recorded
and centroided 25 times. This procedure was repeated 4 times.
Figure 4.2 shows the horizontal translation of the measured centroids.
The average measured horizontal displacement, ∆x, was 286.1 ± 0.3 pixels,

approximately 4.6× the horizontal-hole-spacing. Using this value and similar
triangles, it is straightforward to determine L

L = ∆x ps|AB||BD| − |AC|

4.2. LEVER ARM CALIBRATION

97

175.0 mm
HWS
A

M3
B
E

C
57.5 mm

M2

HeNe Laser

DM1(a) Lever arm measurement experiment

CCD

Bolt
Face plate
Hartmann plate
Spacer∆x

ACL(b) Lever arm measurement experiment - zoom

Figure 4.1: Schematic of the experiment used to calibrate the lever arm.

98

CHAPTER 4. TESTING THE SENSOR= ∆x ps

|AB|
|DE|

(4.1)

where ps is the previously determined value of the pixel size (see Section
3.3.3.3). Since |AB| = 1750.0 ± 0.5 mm and |DE| = 575.0 ± 0.5 mm, L =10.43 ± 0.02 mm. The errors in |AB| and |DE| are given by the error in
the placement of the tapped holes in the optical table and is set, rather
conservatively, to 0.5 mm.
The assumption of similar triangles is limited by the degree to which the
HWS is not normally incident to the beam |AB|. If the HWS is rotatedaway from normal incidence by a small angle δθ, then there will be a scaling

error in the similar triangles assumption of the order of δθ2 /2 when δθ isgiven in radians. For instance, if δθ = 2.5 degrees then the scaling error isapproximately 0.1%. Since the HWS can easily be aligned to better than 2.5
degrees the similar triangles assumption is assumed to be completely valid.
The accuracy of L is determined by the accuracy of ∆x, ps , |AB| and|DE|. However, in a wavefront measurement, transverse spot displacements,∆xi , are converted to wavefront gradients, ∂Wi /∂x, using∂Wi∆xi ps∆xi |DE|==∂x
L∆x |AB|

It is clear that the accuracy of a gradient measurement is limited by ∆xi ,|AB| and |DE|.

4.2.1

Discounting other solutions due to pattern degeneracy

The other possible, though unlikely, solutions, corresponding to spot pattern displacements of 3.6× and 5.6× the horizontal-hole-spacing, would giveL = 8.17 mm and L = 12.71 mm. These solutions were discounted by comparing the spot profile with numerical predictions as described below. The
cross-sections of measured spots at two different wavelengths, λ = 632.8 nm

4.2. LEVER ARM CALIBRATION

99

Figure 4.2: The measured centroids on the CCD for the HeNe aligned withneedles A and B (red) and with needles C and B (blue). Note that thecircular mask was digitally added after the measurement as a visual aid.

100

CHAPTER 4. TESTING THE SENSOR(coherent) and λ = 820 nm (incoherent), were compared to the cross-sectionsof a simulation using ZEMAX for the three potential solutions for the leverarm. The results are shown in Figure 4.3. The best matches between themeasured and simulated diffraction patterns are in Figure 4.3 c) and d) corresponding to a lever arm of 10.43 mm, thus verifying the previous result.
Additionally, this result is consistent with a lever arm of approximately 11mm, determined using a schematic diagram of the CCD camera and a measurement of the spacer thickness.
The comparison of the measured and simulated diffraction patterns to verify the lever arm measurement provides some interesting information about
the diffraction of the rays from the Hartmann plate. However, it would havebeen unnecessary to verify the lever arm measurement using this comparisonin the absence of pattern degeneracy. A simpler measurement would havecovered all holes except for a single column and used the displacement of the
spots from that column to measure the lever arm.

4.3

Comparison of incoherent and coherent
light sources

The Hartmann sensor should have lower noise when illuminated with lightfrom a broadband source because this light is free of coherent effects that cancause fluctuations in the intensity pattern illuminating the sensor. To test
this, the distribution of centroid fluctuations in the Hartmann sensor wasinvestigated when it was illuminated with different light sources, coherent
and incoherent, and operated under different conditions as shown in Figure
4.4. The incoherent source was the fibre-coupled super luminescent diode(SLD) described in Section 3.3.1 and the coherent source was a HeNe laserbeam.
In Tests 1 & 2 the wavefront from a light source diverged from the source
onto a Hartmann sensor placed a short distance away. In Tests 3 & 4 thelight propagated through a series of transmissive optical elements before be-4.3. INCOHERENT VS. COHERENT SOURCES

101(a) L = 8.17 mm, λ = 632.8 nm(b) L = 8.17 mm, λpeak = 820 nm(c) L = 10.43 mm, λ = 632.8 nm(d) L = 10.43 mm, λpeak = 820 nm(e) L = 12.71 mm, λ = 632.8 nm(f) L = 12.71 mm, λpeak = 820 nm

Figure 4.3: Intensity cross sections from measured Hartmann spots (red)illuminated with a), c), e) 632.8 nm coherent light and b), d), f) 820 nm
incoherent light. Physical optics simulations by ZEMAX (blue) are alsoplotted. 5 × the magnitude of the difference is also shown (green). The best
match between measured and simulated is at 10.43mm.

102

CHAPTER 4. TESTING THE SENSOR

L

Test 1
Optical fibre
from SLD

Test 2

HeNe
Laser

W

Hartmann plate
Hartmann sensor

CCD(a) Tests 1 & 2 - no intermediate optics

Test 3

L

Optical fibre
from SLD

Test 4

Hartmann plate
Hartmann sensor

CCD

HeNe
Laser

(b) Tests 3 & 4 - multiple intermediate optics

Figure 4.4: Experimental setup to test the precision of the Hartmann sensor.
a) Test 1: An incoherent source (output from a fibre coupled SLD) directly
illuminates the Hartmann wavefront sensor. Test 2: A HeNe laser beamfocussed to form a waist that directly illuminates the Hartmann wavefront
sensor. b) Test 3: An incoherent source that propagates through multiple
transmissive optics illuminates the Hartmann wavefront sensor. Test 4: A
HeNe laser beam that propagates through multiple transmissive optics illuminates the Hartmann wavefront sensor.

4.3. INCOHERENT VS. COHERENT SOURCES

103(a) Test 1 - incoherent, no optics(b) Test 2 - coherent, no optics(c) Test 3 - incoherent, multiple optics(d) Test 4 - coherent, multiple optics

Figure 4.5: The distribution of the deviations of each of the M centroids fromits average value for each Test and best fit Gaussians. a) Test 1: incoherent
illumination and no optics between source and Hartmann sensor, b) Test 2:
coherent illumination and no optics, except a single imaging lens, between
source and Hartmann sensor, c) Test 3: incoherent illumination and multiple
optics between source and Hartmann sensor, d) Test 4: coherent source andmultiple optics between source and Hartmann sensor.

104

CHAPTER 4. TESTING THE SENSOR

ing incident upon the Hartmann sensor. A series of N images of the M
Hartmann spots was recorded over a short period of time for each set-up.
The patterns were analyzed using the FPC algorithm described in Section
3.3.4.2 to obtain a list of all centroids over all images. For simplicity, theanalysis only considered the x component of the centroids.
For each test, the average centroids for the j th spot from N images, xj ,was calculated using
xj =

PN

i=1

xi j

N

, where j = 1, . . . , M.

The deviations, ∆xi j , of each of the M centroids from its average value forall images were determined using∆xi j = xi j − xj , where i = 1, . . . , N.
The distributions of the deviations for the four tests are plotted in Figure
4.6 and are compared to a best fit Gaussian. The width of the distribution
in Tests 1-3 is consistent with the shot noise limit, 0.34% of a pixel for the
FPC algorithm, established in Section 3.3.4.3, and the good agreement with
the Gaussian distributions shows that the distribution of centroid deviations
are largely random. The distribution of centroid deviations in Test 4 is muchwider, indicating an additional noise source.
The RMS deviation of the centroids in the ith image, from those in theinitial image, is defined as

σcenti =

v
u PM
2
u
t j=1 (xi j − x0 j )

M −1

.

and the temporal variation of this RMS deviation is shown in Figure 4.6.
Since the graphs for Test 1,2 & 3 are independent of i, the fluctuations in
the centroids are random. Futhermore, the RMS deviation for these tests issimilar to the 0.34% shot-noise floor predicted in the previous chapter. For

4.3. INCOHERENT VS. COHERENT SOURCES

105(a) No/minimal optics(b) Optics

Figure 4.6: RMS centroid difference between each set of centroids and thefirst image set vs time. a) Tests 1 & 2: incoherent source (blue) and coherent
source (red), no optics between source and Hartmann sensor. b) Tests 3& 4: incoherent source (blue) and coherent source (red), multiple optics
between source and Hartmann sensor. Note that the incoherent source test
(blue) was run for approximately 50 s and the coherent source test (red) for
approximately 100 s.

106

CHAPTER 4. TESTING THE SENSOR

Test 4, however, it is clear that there is a large temporal variation in the RMS
deviation. This is further evidence for some non-random noise component.
The most plausible explanation for this component is the presence of interference fringes in the intensity pattern, acquired from multiple reflections in
the transmissive optics, resulting in fluctuations in the Hartmann spots thatare not due to the variations in phase of the wavefront.
These results clearly demonstrate that using an incoherent source as aprobe beam yields a more robust Hartmann wavefront sensor as the incoherent source remains shot-noise limited in scenarios where a coherent source
does not.

4.4

Temperature dependence of HWS

As discussed in Section 3.2.5, if Hartmann sensor measurements are made before and after a temperature change then there could be an apparent change
in the wavefront where none exists. This change would appear as the primaryaberration defocus.
A measurement was made of this temperature dependent error using asystem very similar to that shown in Figure 4.4 a). The HWS was turned on,
at t = 0, and illuminated with light from the SLD. Two temperature sensors,
one on the A/D converter and the other on the CCD sensor board, recorded
the temperature of the camera from t = 22.5 s. Centroiding the Hartmannpatterns and calculating the apparent defocus, relative to the first pattern,commenced at t = 34.0 s.
The results of these measurements are plotted in the semi-logarithmicplot in Figure 4.7. The red and blue data show the temperature at the A/D
converter and CCD sensor, respectively. Notice that the temperature of theA/D converter starts increasing immediately, while the intial increase in thetemperature of the CCD is much slower and ultimately settles on a lower
value. The initial increase in the resultant apparent defocus (green data) isdelayed further still. This indicates that the heat in the camera is generated

4.4. TEMPERATURE DEPENDENCE OF HWS

107

Figure 4.7: Temperature on the A/D board of the camera (red) and temperature on the CCD sensor (blue). Apparent defocus, relative to the initialstate, measured by HWS (green).

in the immediate vicinity of the A/D converter and diffuses first to the CCD
and, subsequently, to the Hartmann plate.
Figure 4.8 is a scatter plot of the defocus and the temperature on the A/D
converter for measurements that were recorded at the same time. Noticethat there is a one-to-one and approximately linear relationship between
defocus and temperature. Also shown is the line of best fit, with a slope of
0.0025 m−1 K−1 . Recall that an estimate, 0.00195 m−1 K−1 , of this coefficient
was determined in Section 3.3.2.4.
The approximate linear relationship between defocus and temperatureand the reasonable agreement between the expected and measured values of
the coefficient of the temperature dependence of defocus are strong evidence
for the existence of a temperature dependent error in the Hartmann sensor
that is consistent with the physical argument presented in Section 3.2.5.
Also illustrated by this experiment is the necessity of allowing the Hartmann
sensor approximately 3 hours to reach a steady state temperature before it
should be used.

108

CHAPTER 4. TESTING THE SENSOR

Figure 4.8: Scatter plot of the measured defocus vs temperature. The dashedline is the best linear fit and has a slope = 0.0025 m−1 K−1

4.5

Noise floor of the Hartmann sensor

The equivalent wavefront error due to noise in the sensor was measured using the system shown in Figure 4.4 a) using the 820nm fibre-coupled superluminescent diode (SLD). Spot centroids1 were calculated for consecutive
Hartmann images separated in time by 15 s and the average prism (tip/tilt)
in each image was removed. These zero-prism centroids were used to calculate the error in the discrete gradient field and subsequent wavefront change.
A typical gradient field and map of the wavefront change, which has an RMS
error of λ/1450, are shown in Figure 4.9 a) and b), respectively. The RMS
error for these maps varied between λ/1000 and λ/2000 which is consistent
with the shot noise limit, established in Section 4.3, and the Southwell noise
coefficient (see Section 3.3.5.3), C∆W = 0.2, calculated for Nholes = 263 usedin this example. In general, the RMS wavefront error for a zonal reconstruc1/2

tion across all holes, assuming uncorrelated noise, is given by C∆W × σ∆W ,where σ∆W is the RMS wavefront difference between adjacent holes, which

was defined in Equation 3.17 and is repeated here.
1

All the remaining experiments in this chapter used the WCoG algorithm to calculate
the spot centroids.

4.5. NOISE FLOOR OF THE HARTMANN SENSOR

σ∆W =

hp
σ∆y
L

109

(4.2)

where σ∆y is the average RMS error in the individual centroids.
If the statistical characteristics of the noise do not vary with time (i.e.
the noise is stationary) then the wavefront error should be reduced by averaging over multiple Hartmann images. To test this, a sequence of 2000
Hartmann images was recorded at 30 images/second and the global prism
was removed from each image. A set of reference centroids was then calculated by averaging over Nref = 1000 images, consisting of the first and last
500 images. The central 1000 images were used to calculate sets of centroids
averaged over Navg images, where Navg = 1, . . . , 990. This process ensured
that the reference and average centroids were statistically independent. The
procedure was repeated 5 times. The dependence of σ∆W on Navg is plottedin Figure 4.10, showing that σ∆W < λ/15, 500 at Navg = 990.
The prediction of a numerical simulation that assumes stationary random
noise given by22σ∆W(Navg ) = σ∆W,1

1
1+Navg Nref

!

(4.3)

is also plotted in Figure 4.10, where σ∆W,1 , the Navg = 1 error, is the only
1/2
free parameter. For small Navg , the error decreases as Navg
, as expected and1/2

it asymptotically approaches a value that is Nref below the Navg = 1 error,
the limit due to noise in the reference centroids. Note the good agreement
between the measurement and the numerical prediction, except for Navg >300 where the measured error is slightly larger than that predicted.
The RMS wavefront error was approximately λ/15, 500, which is larger
than the λ/21, 000 predicted using the Southwell noise coefficient appropriate
to this measurement. These discrepancies are probably due to the effect ofa low-frequency temperature fluctuation, which is described in Section 4.6.
These results demonstrate that the dominant noise in the Hartmann
wavefront sensor is random and well described by Poissonian statistics, which

110

CHAPTER 4. TESTING THE SENSOR(a) Vector plot of gradient field(b) Contour plot of reconstructed wavefront

Figure 4.9: Hartmann sensor measurement of the background shot-noise level
made by comparing two Hartmann patterns separated in time by approximately 15 s. Shown here are a) the gradient field of the apparent wavefrontchange and b) a reconstruction of the apparent wavefront change.

4.6. ACCURACY TEST USING KNOWN WAVEFRONT CHANGE 111

Figure 4.10: The improvement in HWS sensitivity due to averaging over
Navg Hartmann images. The solid curve shows the improvement predictedassuming only random, stationary noise in the spot centroids
is consistent with the HWS being shot-noise limited as predicted in Section
3.3.3. The sensitivity is limited when averaging, implying an additional small
noise source. It is plausible that thermal expansion of the HP may be a contributing factor to this noise.

4.6

Accuracy test using known wavefront change

A simultaneous measurement of the accuracy and precision of the HWS canbe tested by measuring a well-known, and preferably small, wavefront change.
Therefore, an experiment was developed to create a small quadratic wavefrontchange, the primary aberration defocus.

4.6.1

Analytic form of known wavefront change

Consider a wavefront diverging from a point source and incident on a Hartmann sensor a distance z0 away, as illustrated in Figure 4.11. The equation
for the wavefront at height h is

112

CHAPTER 4. TESTING THE SENSOR
∆y

HP

y0

h

z0
∆z
L
W W

′

CCD

Figure 4.11: Schematic showing the change in direction of a Hartmann ray
in the presence of defocus.

W (h) = z0 −q

z02 + h2 ,(4.4)

and the gradient of the wavefront in the paraxial limit is
h
∂W
= .
∂h
z0

(4.5)

If the source is translated toward the HWS by ∆z, the new wavefront W ′ ,equal to the original wavefront plus the wavefront change ∆W , has a gradientat height h given by
h
∂W ′
=
∂h
z0 − ∆z(4.6)

The gradient of the wavefront change is then given byhh∂ (∆W )−=
∂hz0 z0 − ∆z−∆z h=z0 (z0 − ∆z)
= Sh

(4.7)
(4.8)

4.6. ACCURACY TEST USING KNOWN WAVEFRONT CHANGE 113
where S is the primary aberration defocus. The gradient can also be expressed as∂ (∆W )
∆y
=
∂h
L
=

∆z
y0
(z0 − ∆z) (z0 + L)

(4.9)
(4.10)The measured wavefront change, which can be made arbitrarily small by
reducing ∆z, is characterized by plotting ∂ (∆W ) /∂h versus y0 , determining
the slope of the line-of-best-fit and the deviation of the data points about thisline. The predicted wavefront change is, therefore, known with the accuracy
with which z0 , L and ∆z are known.

4.6.2

Experiment design

The layout of the system used to measure ∆z and the resulting wavefrontchange is shown in Figure 4.12. The HWS was illuminated by light emitted
from the optical fiber coupled to the 820 nm Agilent HFBR-1414 superluminescent diode described in Section 3.3.1. The fiber was mounted on a
micrometer-controlled translation stage to allow the distance, z0 , between
the light source and the HWS to be adjusted precisely.
Translation of the fibre end, ∆z, was measured using a Michelson interferometer, in which the input beam was split at point A. The object beamof the interferometer was formed by the path A-Mobj -A, where Mobj was a
mirror mounted on the fibre holder on the translation stage. The reference
beam was formed by the path A-B-C-Mref -C-B-A, where Mref was a mirror
mounted on the front of the HWS. These beams were recombined on a CCD,
and the horizontal cross-section of the interference pattern was recorded as
rapidly as the CCD would allow, approximately 13 Hz. To assist in measuring ∆z, a small amount or horizontal tilt was introduced between the object
and reference beams resulting in vertical fringes on the CCD and a sinusoidal
intensity pattern in the horizontal direction, which moved across on the CCD

114

CHAPTER 4. TESTING THE SENSOR
Interferometerread−outL

z0

Optical fibre
from SLD

W

A

obj

M obj
ref

C
B

Hartmann plate

HeNe laser

M ref CCD

Hartmann sensor

Translation stage

Figure 4.12: A schematic of the Hartmann wavefront sensor and the systemused to test the sensor. The sensor was illuminated using a fiber-coupled
super luminescent diode (SLD), the free end of which was mounted on atranslation stage. Changes in the distance between the fiber end and theHartmann plate were measured using a Michelson interferometer.

in response to a translation ∆z. Fourier analysis was used to filter out all
the noise, leaving only the sinusoidal signal. An example of the interference
pattern measured by the CCD and the filtered signal are shown in Figure
4.13.
A change, ∆z, in the distance between HWS and the fibre holder results
in a horizontal translation of the sinusoid by a phase, ∆φ, given by∆φ =4 π ∆zλ(4.11)

If the phase change per CCD frame was less than π then the translationof the sinusoid per frame was unambiguous (i.e. avoided any modulo 2 π
ambiguity) and the change in phase and thus ∆z over time could be tracked.
Provided it was varied slowly, ∆z could be recorded with approximately 10
nm precision for small translations.

4.6. ACCURACY TEST USING KNOWN WAVEFRONT CHANGE 115

Figure 4.13: A cross-section of the interference pattern output from theMichelson interferometer measured on the CCD. The solid line shows themeasured fringes and the dotted line shows the filtered sinuosidal modulation
due to the tilt introduce into the reference beam. The vertical dashed line
indicates the position at which the phase is measured in every frame.

4.6.3

Calibration of the origin for z0

The accuracy of the defocus measurement can be determined by comparing
the measured defocus with that predicted using Equation 4.10, but this comparison requires an accurate determination of the origin of z0 . The nominal
distance between the HP and the fibre end, znom , can be measured with aruler with a precision of ±1 mm. A micrometer or calipers could not be used

to measure the nominal because the HP was not a rigid surface. The true
distance between the HP and the fibre end is z0 and will differ from thisnominal value by a systematic offset εz such thatz0 = znom + εz
Therefore, if εz can be determined and a measurement of znom exists, thenthe origin of z0 can be determined. This can be accomplished by exploiting
the non-linear dependence of the gradient of the wavefront change on z0 .
Equations 4.7 and 4.9 can be combined to give

116

CHAPTER 4. TESTING THE SENSOR

∆z
∆y
=
h.
L
z0 (z0 − ∆z)where the theoretical slope, mtheory , of ∆y/L versus h is thus

mtheory =mtheory (εz ) =∆z
z0 (z0 − ∆z)∆z.
[znom + εz ] ([znom + εz ] − ∆z)

If a translation, ∆z, from an initial nominal distance, znom , is made, with
HWS measurements of N spots taken before and after, then the slope ofthe line-of-best-fit, mfit , of (∆y/L)i versus hi , where 1 ≤ i ≤ N, can be

determined. The corresponding value of mtheory (εz ) can be determined at
this nominal distance.
Multiple, large translations were made, ∆z ≈ 1 mm, for a series of nomi-

nal distances, znom , between the HP and the fibre end. The resulting theoretical and fitted slopes, mtheory and mfit , respectively, were plotted versus thenominal distances znom plus an estimate of εz . This resulted in two non-linear
curves that were offset from one another. The value εz was varied to displace
the curves until they lay on top of one another, as illustrated in Figure 4.14,
thereby yielding a value for the systematic offset εz , and hence calibrating
the origin of z0 .
In this way, z0 for the measurement in Section 4.6.5 was determined to
be 91.7 ± 0.2 mm. Note that, although the best fit of z0 gives an errorof approximately ±0.03 mm, the error here is set to 0.2 mm because therelative error in the lever arm L, which is used in the calculation of z0 , isapproximately 0.2% (recall that L = 10.43 ± 0.02 mm).

4.6. ACCURACY TEST USING KNOWN WAVEFRONT CHANGE 117(a) εz0 = 0.0 mm low(b) εz0 = −0.23 mm low(c) εz0 = −0.63 mm highFigure 4.14: Multiple measurements of mtheory (red) and mfit (blue) at different approximate distances, znom +εz0 , from the HP. The only fitting parameteris the error in the distance from the HP, εz0 . a) Shows the comparison using
only the nominal value znom with εz0 = 0. By varying the fitting parameter,
εz0 , the measurements (blue) could be made to lie on top of the theory curve(red). In this way the distance from the fibre to the HP could be calibrated.
b) shows an example where this distance has been underestimated and c)shows an example where this distance has been overestimated.

118

CHAPTER 4. TESTING THE SENSOR

4.6.4

Mitigating cyclic temperature fluctuations

Since ∆W was intended to be very small to simultaneously test precision
as well as accuracy, the temperature dependence of the HWS, discussed in
Section 3.2.5, was expected to introduce a significant error in response to any
changes in temperature. Recall that the change in wavefront due to a change
in temperature appears to be quadratic - the same form as the signal to bemeasured in this defocus experiment!
Periodic fluctuations in the temperature of the CCD (±150 mK @ 0.5mHz) were observed due to the laboratory air-conditioning (if the air- conditioning was turned off the laboratory was subject to diurnal temperature
variations, approximately ±5, 500mK per day). The air-conditioning induced

temperature fluctuations were periodic and the translations could be timed
to coincide with a maxima or a minima such that the average temperature
before and after translation was the same and the net effect of these fluctuations would thus be minimal. This was done by performing multiple translations of the fibre end and then leaving the system for another 2-3 hours torecord the period and phase of the background fluctuations. Knowing thesevalues, the periodic signal could be traced back in time and the translationthat occurred closest to a turning point of the periodic signal could then be
selected.

4.6.5

Experiment procedure

The experimental procedure was:
1. texpt = -180:00 minutes. The HWS was turned on 3 hours in advanceof any measurements to allow it to reach a steady-state temperature.
2. texpt = 000:00 minutes. The Michelson interferometer started
recording interference patterns at approximately 13 Hz and ran
continuously for the 6 hour duration of the experiment.

4.6. ACCURACY TEST USING KNOWN WAVEFRONT CHANGE 119
3. texpt = 000:10 minutes. A temperature record of the HWS sensor wasstarted and ran continuously for the 6 hour duration of theexperiment. This was read from the the on-board sensors with afrequency of 33 mHz (every 30 s).
4. texpt = 000:20 minutes. The HWS sensor started and ran continuously
for the 6 hour duration of the experiment. Centroid patterns wererecorded at approximately 33Hz.
5. texpt ≈ 000:20 → 006:30 minutes. 10,000 centroids were recorded at
the intial position.

6. texpt ≈ 006:30 → 007:30 minutes. The fibre was translated

approximately 10µm closer to the HWS by adjusting the translation

stage. This was done extremely carefully and slowly to enablesuccessful tracking of the fringes in the interference pattern.
7. texpt ≈ 007:30 → 014:00 minutes. 10,000 centroids were recorded at
the new position.

8. texpt ≈ 014:00 → 143:00 minutes. Steps 6 and 7 were repeated afurther 18 times.

9. texpt ≈ 143:00 → 360:00 minutes. The equipment was left recording
data for a further 3.5 hours.

10. texpt = 360:00 minutes. All data recording ceased.

4.6.6

Results

The defocus measured in Step 9 is plotted in Figure 4.15 a), the OPD measured by the interferometer for this Step in Figure 4.15 c) and the temperature measured by the on-board temperature sensors on the HWS in Figure
4.15 e). A sinusoidal variation in defocus with a frequency of 0.5 mHz canbe seen in the defocus data, the OPD measurements and the on-board temperature. The spectral components of this signal are shown for the defocus,

120

CHAPTER 4. TESTING THE SENSOR

OPD and temperature in Figures 4.15 b), d) and f), respectively. Given thatit has been established, Section 3.2.5, that the HWS produces a temperaturedependent systematic error that appears to be defocus and that the temperature and defocus oscillations have the same frequency, it is concluded thatthe observed periodic fluctuations in defocus are caused by the cycling laboratory temperature. Periodic thermal expansion of the optical table would
also explain the fluctuations in OPD observed in Figure 4.15 c).
The defocus measurements were made by comparing measurements immediately before and after maxima or minima in the temperature cycle,
with the fibre translation itself coinciding with the maxima or minima. The
fluctuations in temperature were slowest at these points and the averagetemperature of the measurements before and after the displacement wereapproximately the same. The sinusoidal signal measured after the last measurement was traced backward in time to determine the displacement which
occurred closest to a maxima or minima and therefore had the smallest systematic error. This is illustrated Figure 4.16. The displacement at 68 minutes
coincided with a maximum of the sinusoid and hence the centroids immediately before and after this measurement were used in the following analysis.
The change in the optical path length, measured by the interferometer, was19.2 ± 0.1 µm, giving ∆z = 9.60 ± 0.05 µm.
The vertical gradient field of the wavefront distortion was determinedfrom the difference between the Hartmann centroids before and after the displacement at t = 68 minutes. This was done for centroids averaged over Navgmeasurements for Navg = 1, 100, 1000, 5000. Figure 4.17 shows plots of thevertical component of the gradient field for the four averages. The predictedvalue of vertical gradient is also plotted. These plots show a linear relationship between the local gradient and transverse position, y0 , as predicted by
Equation 4.10, and the improvement in sensitivity due to averaging.
The defocus due to the source translation can be calculated using theslope, m, of the line-of-best-fit to this data and S = m (z0 + L) /z0 and theresults are summarised in Table 4.1. Linear regression analysis was used to4.6. ACCURACY TEST USING KNOWN WAVEFRONT CHANGE 121(a) Hartmann defocus measurement(b) Hartmann defocus frequency spectrum(c) Interferometer measurement(d) Interferometer frequency spectrum(e) CCD temperature measurement(f) CCD temperature frequency spectrum

Figure 4.15: a) Hartmann sensor defocus measurement and b) its temporal
spectrum. c) Mach-Zehnder interferometer optical path distortion measurement and d) its temporal spectrum. e) Average temperature measured by
the two CCD temperature sensors and f) its temporal spectrum. These plotsshow a 0.5 mHz sinusoidal signal common to the defocus, interferometer andtemperature measurements.

122

CHAPTER 4. TESTING THE SENSOR

Figure 4.16: The defocus measured by the Hartmann sensor, relative to an
average background, versus time where the fibre end is displaced multiple
times by approximately 10µm every 7.5 minutes. The dotted lines are sinusoidal curves that have the same frequency and phase as the cyclic fluctuationof the laboratory temperature and are traced backward in time. Notice thatthe displacement at 68 minutes coincides with a peak.

4.6. ACCURACY TEST USING KNOWN WAVEFRONT CHANGE 123(a) Navg = 1(b) Navg = 100(c) Navg = 1000(d) Navg = 5000

Figure 4.17: Measured local gradient of the wavefront change versus spot
position at the CCD, y0 , due to translation of the fiber light source, averaged
over a) 1, b) 100, c) 1000 and d) 5000 Hartmann images.

124

CHAPTER 4. TESTING THE SENSOR

Navg
Defocus, S
1−1.3 ± 0.2 × 10−3 m−1
100−1.127 ± 0.036 × 10−3 m−1
1000−1.150 ± 0.008 × 10−3 m−1
5000−1.159 ± 0.007 × 10−3 m−1
Expected value −1.14 ± 0.01 × 10−3 m−1

Accuracy
λ/860
λ/3700λ/5200λ/3300-Precision
λ/330
λ/1800λ/8200λ/9300-

Table 4.1: Summary of the defocus measurements with the HWS averaged
over Navg measurements, where Navg = 1, 100, 1000 and 5000. Precision is
given by the 95% confidence interval.
determine the slope and precision of the line-of-best-fit. The uncertainty in
S when averaged over 5000 images is equivalent to an uncertainty in the
wavefront sag of 0.1 nm (λ/9300) over the CCD aperture (≈ 10 mm), which
is roughly twice the previously measured λ/15, 500 due probably to nonstationary noise during the extended acquistion time.
The accuracy of the defocus measurement over the aperture of the HWS(≈ 10 mm) is determined by comparing the measured defocus with the predicted value of −1.14 ± 0.01 × 10−3 m−1 , which differs from the measurement

by about 2 × 10−5 m−1 or 1.7%, for Navg = 5000. This error is equivalent to

λ/3300 and could be explained by a change in average temperature of the
HWS of order 10 mK.

These results show that the HWS is ideal for high precision and highaccuracy measurement of wavefront changes, and represent a large improvement in the state-of-the-art. The application to the measurement of a small
wavefront change due to defocus, S, with a precision of 7 × 10−6 m−1 (seeTable 4.1 for Navg = 5000 ) was demonstrated. It was also established thatthe sensor measures this aberration with an accuracy of about 2 × 10−5 m−1 .

4.7

Conclusion

The sensitivity and accuracy of the Hartmann sensor has been demonstrated
by the results reported in this chapter. In Section 4.3, the choice of an
incoherent light source was shown to be crucial in reducing the uncertainty

4.7. CONCLUSION

125

in centroid measurements to a shot-noise limit. A high precision and accuracy
calibration of the effective lever arm of the Hartmann sensor was describedin Section 4.2. In Section 4.5, the background noise in the Hartmann sensor
was shown to be dominated by Poissonian shot-noise and capable of being
reduced to at least λ/15,500 by averaging multiple measurements. Evidenceof another, potentially temperature related, noise source was observed at thislevel. An observation of the temperature dependent error in the Hartmann
sensor was described in Section 4.4 and a simple measurement of the sizeof this effect yielded a result relatively close to the expected value. Section
4.6 described the measurement of the accuracy using a novel experiment
to induce very small, yet well-defined, wavefront aberrations and the resultsfrom this experiment demonstrated the measurement of a very small defocus,
equivalent to an 850m lens, with an accuracy of 1.7% and a precision of 0.7%.

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CHAPTER 4. TESTING THE SENSOR

Chapter 5
Direct measurement ofwavefront distortion at theHigh Optical Power Test
Facility
5.1

Background

The High Optical Power Test Facility (HOPTF), operated by the Australian
Consortium for Interferometric Gravitational Astronomy (ACIGA), is a test
facility designed to investigate critical issues associated with high power optical cavities, such as the measurement and compensation of absorptioninduced wavefront distortion and the effect of parametric instabilities oncavity stability [97].
The high optical power cavity at the facility has previously been used
to investigate the effect of absorption-induced thermal lensing in the cavity
optics on the cavity mode size by Zhao et al. [18]. This was accomplishedby measuring the spatial profile of the light emitted from the cavity andcomparing it with that predicted by a FEM, assuming the dominant thermal
time constants expected for the intra-cavity optics and that the cavity power
127

128

CHAPTER 5. HOPTF WD MEASUREMENT

was constant. Whilst there appeared to be reasonable agreement between
measurement and model in that investigation, such a technique could not be
used to produce a signal suitable for a thermal compensation system (TCS)in Advanced LIGO, as described in Section 1.3 and illustrated in Figure 1.7,
as it a) estimates only the defocus (thermal lens) component of the wavefront
distortion (WD), and b) could not determine the WD of individual optics
within a GWI.

5.2

Objectives

In this chapter, I describe the first direct measurement of wavefront distortionin an optical cavity that has high stored power using a sensor that meets thesensitivity and spatial resolution criteria specified for Advanced LIGO (seeSection 1.3.2).
The installation of the sensor at the HOTPF and the properties of thecavity itself are discussed in Section 5.3, as well a measurement of the sensor
noise floor. The experimental procedure and analysis of the data from thesensor are discussed in Section 5.4. This discussion is particularly detailed
because the sensor measures the absorption-induced WD of multiple intracavity optics off-axis and this measurement must be converted to the on-axis
distortion that is experienced by the cavity mode. The results of the wavefront measurement and their validation using two independent measurements
are reported in Section 5.5 and concluding remarks are made in Section 5.6.

5.3

Description of measurement system

5.3.1

The HOPTF high-optical-power cavity in detail

The current optical configuration is a 77 m Fabry-Perot cavity in which thesubstrate of the input coupler or ’input test mass’ (ITM) is inside the cavity
to enhance the absorption-induced wavefront distortion in that mirror, as5.3. DESCRIPTION OF MEASUREMENT SYSTEM

129

Vacuum system
140 mminput
optics

5−6W
ITM

10 W

Nd:YAG
LASER

L3

7.0 − 8.4 kW

BP

ETM
CP
77 m

Figure 5.1: Schematic of the Fabry-Perot cavity with high stored power,
located at the HOPTF.
Material
DiameterThickness
Radius of curvature
HR transmission
AR reflectivity
Volumetric absorption

ITM
ETM
CP
Sapphire
Sapphire
Fused Silica
100 mm
150 mm
160 mm
46 mm
80 mm
17 mm
flat
720 ± 100 m
flat
1840 ± 100 ppm
20 ppmn/a
30 ± 20 ppm
12 ± 10 ppm
100 ppm≈ 51 ppm/cm
n/a≈ 2 ppm/cm

Table 5.1: Physical parameters of the HOPTF Fabry-Perot cavity optics
shown in Figure 5.1. The cavity also contains a fused-silica compensation
plate (CP) that can be heated via a heating coil wrapped around its barrel
surface, which is used by other members of ACIGA.
The optical and physical parameters of the ITM, CP and the end test
mass (ETM) are shown in Table 5.1. The cavity has a measured finesse of
approximately 1400 [98], which is consistent with the losses due to the AR
coatings and absorption of the CP and ITM substrates. A laser beam from a
10W Nd:YAG single frequency 1064 nm laser [99] is passed through a series
of input optics (a pre-mode cleaner, isolator, mode-matching and alignment
optics) and the remaining 5 - 6W is coupled into the 77 m Fabry-Perot cavity,
resulting in approximately 7.0 - 8.4kW of intra-cavity power. A small amountof the cavity mode is transmitted through the end test mirror (ETM) andis imaged onto a beam profiler (BP) with a demagnification = 2.92 ± 0.05,

thereby recording the cavity mode size and power level at the ETM.

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5.3.2

CHAPTER 5. HOPTF WD MEASUREMENT

Installation of the HWS at HOPTF

A HWS was installed at the HOPTF to measure the wavefront distortionin the ITM and the CP at an off-axis angle as illustrated in Figure 5.2.
The HWS was identical to the one described in Chapters 4 and 5, the only
modification being a more powerful light source (QPhotonics QSDM-790-2
2mW 800 nm fibre-coupled superluminescent diode [100]). The output of thefibre was collimated into an approximately 80 mm diameter beam, steered
into the vacuum system and through the ITM and the CP at an angle of10◦ . The interior of the vacuum system, the beam steering mirrors, M2, M3,
and M4, the CP and the ITM are shown in the photograph in Figure 5.3.
After transmission through the ITM and CP, the beam was steered out of thevacuum system, through an imaging lens, L2 (f = 500 mm), and was incident
on the HWS. The imaging lens was adjusted such that the exit plane of theITM, pobj , was imaged onto the Hartmann plate.
The ITM suspension assembly [101] can be seen surrounding the ITM in
Figure 5.3 and also in Figure 5.4 a). Part of the Hartmann probe beam was
obscured by several components of this suspension assembly, as illustrated in
the computer-generated rendering of the ITM and suspension assembly, fromthe point-of-view of the probe beam, shown in Figure 5.4 b). The top andbottom cross beams, on which the magnetic actuators used to align the ITM
are mounted, which cannot be removed, obscure a substantial fraction of theITM but a region about 3 × the cavity mode diameter around the centerof the cavity mode remains unobscured. This is sufficient for measuring the
wavefront distortion in the region of the cavity mode. Additionally, there
were two brackets to the left and right of the ITM center that obscured partof the HWS probe beam. The front LH and rear RH brackets were removed as
their removal had little to no effect on the suspension system (these brackets
housed two safety stops and their removal left 12 remaining stops). This final
configuration is illustrated in the computer-generated rendering in Figure 5.4
c).
The aperture is highlighted in the computer-generated rendering in Figure

77 m
L1

HWS
conjugate

M2
10 W

Beam
Profiler(BP)

5.3. DESCRIPTION OF MEASUREMENT SYSTEM

5−6W

M1

L3
ETM
CP
ITM

800 nm SLD

131

Figure 5.2: Schematic showing HWS installed to measure thermal lensing in
the ITM and CP.

011
1
00
11
0
1
00
111
0
00

window
Nd:YAG
LASER

7.0 − 8.4 kW
input
optics

M4
M3
HWS

Vacuum systemwindow
L2

132

CHAPTER 5. HOPTF WD MEASUREMENT

Figure 5.3: Photograph showing interior of the vacuum system, the beamsteering mirrors, M2, M3, and M4, the CP and the ITM.
5.5 a) with the positions of the centers of the ITM and CP indicated by red
and blue crosses, respectively. An image of the probe beam at pobj is shownin Figure 5.5 b).

5.3.3

Reduction of environmental noise coupling into
HWS

The wavefront distortion across the whole aperture was measured with the
Nd:YAG laser off so that there was no stored power in the cavity. The
RMS error when the HWS was first installed, averaged over 10 contiguous
Hartmann images, as a function of time is shown in Figure 5.6 in curvea), exhibiting much higher average RMS wavefront error than the theoreticalshot-noise limit for the sensor: λ/45 vs λ/3000. The additional noise was primarily due to air filters creating turbulence in the vicinity of the experiment.
Additionally, the volume of air through which the probe beam passed wassignificantly greater than for the shot-noise limited measurements describedin Chapter 4 and hence the contribution from air currents was greater.

5.3. DESCRIPTION OF MEASUREMENT SYSTEM

133

Figure 5.4: a) Photograph of the ITM installed in its suspension system. b)
Computer-generated rendering of the ITM and suspension system from thepoint-of-view of the probe beam. c) Computer-generated rendering of the
ITM and suspension system from the point-of-view of the HWS with thecorner brackets removed.

134

CHAPTER 5. HOPTF WD MEASUREMENT

(a)

(b)

Figure 5.5: a) computer-generated rendered view of ITM and suspension
system at 10 degrees with aperture highlighted and the positions of the ITM
and CP centers indicated by red and blue crosses, respectively. b) Image ofthe probe beam shadow at pobj .

Figure 5.6: RMS noise in HWS measurement versus time. a) With all air
filters on, b) with air filters off, c) with air filters off and beam tubes enclosing
the majority of the beam path and d) shot-noise limit (simulated).

5.4. MEASUREMENT PROCEDURE AND ANALYSIS

135

When the air filters were turned off to reduce air currents, the RMS
wavefront error decreased to an average value of λ/130. The time dependenceof the RMS wavefront error for that configuration is displayed in Figure 5.6
in curve b). The RMS wavefront error was further decreased by isolatinga substantial fraction of the probe beam with beam tubes, as illustrated in
the photographs in Figure 5.7. The final average RMS wavefront error was
λ/730 and this error as a function of time is shown in Figure 5.6 in curvec). Also shown in this figure, curve d), is the RMS wavefront error expected
for purely random noise, with an average of λ/3000. The large excursions
from the mean RMS level in curve c) indicate that there is probably stillsome correlated noise in the wavefront. Indeed, the wavefront difference forcurve c) at t ≈ 75 s indicates evidence of correlated noise as shown in Figure5.8. This is most likely due to residual air currents and the large volume of
air in the probe beam, although it could also be acoustic vibration of the
optical components. To alleviate this in future the probe beam could be
expanded inside the vacuum system reducing the volume of air to which it is
exposed. Nevertheless, the sensor as installed in this experiment still satisifes
the Advanced LIGO sensitivity specification.

5.4

Measurement procedure and analysis

5.4.1

Measurement procedure

The procedure for measuring the thermal lensing in the sapphire (ITM) andfused silica (CP) as a function of time, where texpt is the experiment time, isas follows:
1. texpt ≈ −15 s. Start recording Hartmann images to establish reference
centroids before the 1064 nm beam is coupled into the cavity.

2. texpt = 0 s. Couple 10W laser into cavity. Start recording the transmitted power and beam size with the beam profiler (BP).

136

CHAPTER 5. HOPTF WD MEASUREMENT

(a)

(b)

Figure 5.7: Photographs showing some of the (improvised) beam tubes that
attenuated background noise in the HWS.

5.4. MEASUREMENT PROCEDURE AND ANALYSIS

137

Figure 5.8: An example of correlated noise in the background wavefront
difference for curve c) in Figure 5.6 at t ≈ 75 s.
3. texpt = 0 s → texpt ≈ 300 s. Continue recording Hartmann images tomeasure the temporal development of thermal distortions in the ITM
and CP.

4. texpt ≈ 300 s. Turn off 1064 nm beam.
5. texpt ≈ 300 s → texpt ≈ 400 s. Continue recording Hartmann images toobserve the decay of thermal distortions in the ITM and CP.

6. texpt ≈ 400 s. Cease recording with the HWS and the BP.
This procedure was repeated on 13 separate occasions. The power levelin the cavity was noticeably more stable on some occasions than others due
to variations in the mode-matching of the 1064 nm laser into the Fabry-Perotcavity.

138

5.4.2

CHAPTER 5. HOPTF WD MEASUREMENT

Analysis of wavefront distortion

As indicated earlier, the Hartmann probe beam is incident on the CP and
the ITM with a 10◦ angle of incidence so that the wavefront distortions canbe measured separately. In Section 5.4.2.1, I show how this off-axis HWS
measurement of distortion is transformed into the distortion that would bemeasured by an on-axis probe beam. The on-axis WD is measured in transmission through the CP and ITM, however, while the cavity mode is reflectedfrom the high-reflecting surface of the ITM. Thus, in Section 5.4.2.2, I show
how the on-axis distortion can be used to determine the distortion experienced by the cavity mode. Lastly, in Section 5.4.2.3, I show how to determine,from the distortion experienced by the cavity mode, the thermally induced
defocus, which can be used to calculate the cavity mode size at the ETM.
In Section 5.5 the calculated mode size is compared with the measurementsfrom the BP to show that the wavefront distortion measurement is accurate.
Note that the off-axis to on-axis transformation described here is valid
only because the CP and ITM are separated by a distance that is large
compared to the thickness of the ITM and CP, and the angle of incidence issmall. More generally, a tomographic analysis should be used to achieve thistransformation as described in Chapter 6.
5.4.2.1

Converting off-axis distortion to on-axis distortion

A schematic illustrating the off-axis probe beam propagating through the CP
and ITM is shown in Figure 5.9 a). The results of a numerical simulation of
the expected off-axis wavefront distortion due to Hello-Vinet thermal gradient inside the optics is shown in Figure 5.9 b), where this off-axis distortion islabelled WD(off ) . Two lobes are clearly seen, indicating the distortion in thetwo optics. Due to the shallow viewing angle, a vertical cross-section througheach lobe is approximately equivalent to what would have been measured inthat component in an on-axis view. These cross-sections are indicated by
(off)

(off)

dashed lines in Figure 5.9 b) and are labelled WDSAPH and WDFS for the
sapphire and fused silica cross-sections, respectively.

5.4. MEASUREMENT PROCEDURE AND ANALYSIS

139

CP
ITM

10 degrees
BAprobe beam rays

140 mm(a)(b)

Figure 5.9: a) Off-axis probe beam refracting through sapphire ITM andfused silica CP. b) Simulated off-axis wavefront distortion, WD(off ) , showing
(off)
(off)
the two vertical cross-sections, WDFS and WDSAPH , used to determine thedefocus in the fused-silica and sapphire, blue and red respectively.

140

CHAPTER 5. HOPTF WD MEASUREMENTThe on-axis and off-axis distortions differ because of two effects. Firstly,
each lobe is smeared or elongated slightly in the horizontal direction due tothe axial extent of the thermal lenses. Secondly, each lobe is contaminatedslightly by the signal from the other lobe. This can be seen by considering
ray A in Figure 5.9 a): despite the fact that it is dominated by the strong
thermal lens at the center of the sapphire, there is still a contribution to thetotal distortion from the thermal lens in the compensation plate. Thus, thevertical cross-sections through each lobe represent a sum of on-axis wavefront
distortions which can be expressed as

(off)

(on)

(on)

WDSAPH = KE1 (t) WDSAPH + KC1 (t) WDFS
(off )

WDFS
(on)

(on)

(on)

= KE2 (t) WDFS + KC2 (t) WDSAPH

(5.1)
(5.2)

(on)

where WDSAPH and WDFS are the on-axis distortions of the sapphire andfused silica, respectively, KE1 (t) and KE2 (t) describe the elongation of thesapphire and fused silica distortion, respectively, and KC1 (t) and KC2 (t)describe the contribution of the fused-silica and sapphire distortion to the
opposite cross-section, respectively.
The parameters KE1 (t), KE2 (t), KC1 (t) and KC2 (t) can be determinedusing the geometry of the ITM-CP-probe-beam arrangement and assuming
the temporal development of the wavefront distortion as described by Helloand Vinet (see Appendix A.1) to first order.
The on-axis distortion is determined by solving the Equations 5.1 and 5.2
to yield

(off)

(on)
WDSAPH(off)

KE2 WDSAPH − KC1 WDFS=
KE1 KE2 − KC1 KC2
(off)
(off)
= b1 1 WDSAPH + b1 2 WDFS

(5.3)

5.4. MEASUREMENT PROCEDURE AND ANALYSIS(a) b1 1(b) b1 2(c) b2 2(d) b2 1

141

Figure 5.10: Conversion parameters for off-axis distortion, calculated using
the Hello-Vinet equation and known geometry of ITM, CP and probe beam
(off)

(on)

WDFS

(off)

KE1 WDFS − KC2 WDSAPH
KE1 KE2 − KC1 KC2
(off)
(off)
= b2 2 WDSAPH + b2 1 WDFS
=

(5.4)

where the conversion parameters, b1 1 , b1 2 , b2 1 and b2 2 , are plotted in Figure
5.10.

5.4.2.2

Converting on-axis WD to round-trip cavity-mode WD

As noted by Ryan Lawrence [12] and others, and discussed in Section 1.2.3,
absorption in the substrate of a mirror leads to wavefront distortion via thethermo-optic, elasto-optic and thermo-elastic effects. The first two effects

142

CHAPTER 5. HOPTF WD MEASUREMENT

Thermooptic Effect (SUB1, SUB2)
Elastooptic Effect (SUB1, SUB2)
Thermoelastic expansion (CS1, . . ., CS4)

Sapphire1
0.2
0.8

Fused Silica
1
-0.01
0.06

Table 5.2: Sizes of thermoelastic deformation and the elastooptic effect relative to the size of the thermooptic effect. SUB1, SUB2 and CS1, . . ., CS4,
refer to the locations at which these effects occur as indicated in Figure 5.11.

result in volumetric distortions that are acquired on transmission through thesubstrate, while thermo-elastic distortion leads to surface deformation andthus is acquired on reflection or refraction at the surface. The relative sizesof the thermo-optic, elasto-optic and thermo-elastic distortion for sapphire
and fused silica are listed in Table 5.2 (reproduced from Table 2.1 in Ryan
Lawrence’s Ph.D thesis [12]).
A schematic diagram showing the cavity mode and the Hartmann beamprobing these distortions is illustrated in Figure 5.11. With the exception of
the distortion at curved surface 4 (CS4), the cavity mode experiences every
distortion twice that the probe beam experiences once. Therefore, the majordifference between the distortion experienced by the cavity mode and probe
beam arises at CS4, due to the thermo-elastic effect, which the cavity mode
experiences on reflection and the probe beam on transmission. The relative
sizes of the distortions accumulated by the probe beam and the cavity modecan be determined using a simple ABCD matrix description of the reflectionof the cavity mode from and the transmission of the probe beam throughthe ITM and knowing the relative sizes of the thermo-optic, thermo-elastic
and elasto-optic effects (see Appendix A.3 for details). This modelling showsthat the wavefront distortion experienced by the cavity mode in the ITM isa factor of 2.53 larger than that measured by the HWS probe beam. Hence,the distortion experienced by the cavity mode, WDcavity , can be expressed
(on)

(on)

WDcavity ≈ 2.53 × WDSAPH + 2.0 × WDFS

(5.5)

5.4. MEASUREMENT PROCEDURE AND ANALYSIS

143

CP
ITM
probe beamcavity mode

SUB2
CS4

SUB1

CS3
CS2 CS1

Figure 5.11: The probe beam (blue) is transmitted through both substrates,
SUB1 and SUB2, and through all curved surfaces, CS1 to CS4. The cavity
mode (red) is transmitted through both substrates, SUB1 and SUB2, andthrough curved surfaces, CS1 to CS3 and is reflected from CS4.

5.4.2.3

Defocus of cavity-mode WD and ETM beam size

The characterization of the distortion by the defocus, S, is achieved using
the method of Arain et al.

[102] which determines the maximum overlap

integral, I (S), between the distortion experienced by the cavity mode and aspherically curved mirror, where

I (S) =

Z +∞ s 

2
π

1

×
wITM (S)"#!
22
exp −x
×
2
wITM
(S)

 
4π
4π
exp i
WDcavity (x) −S x2 dx
λ
λ
−∞

(5.6)

where wITM (S) is the calculated cavity mode size at the ITM and the defocus of the cavity, Scavity , is the value of S which maximizes the integral in
Equation 5.6 and is, therefore, a solution of

144

CHAPTER 5. HOPTF WD MEASUREMENT

d I (S)
= 0.
dS

(5.7)

The cavity mode can then be found from the cavity g-parameters (seeSeigman [103] for full details) given by
gETM = 1 −

Lcav
RETM

and
gITM (S) = 1 − S Lcav ,where RETM , RITM are the radii of curvature of the end test mass and input
test mass, respectively, and Lcav is the length of the cavity. The mode sizeat the ITM, wITM (S), is given by
wITM

v
s
u
u Lcav λ
t
(S) =

π

gETM
gITM (S) [1 − gITM (S) gETM ]Note that since the cavity mode size at the ITM, wITM (S), depends on thecalculated cavity defocus, S, this procedure must be iterated several times
to reach a convergent solution. It is first run using wITM (0) equal to the cold
cavity (nominal) mode size at the ITM to determine the new cavity modeand then iterated several times, using the updated cavity mode sizes at theITM. Once a convergent solution is found, the cavity mode size at the ETM,
wETM (S), is then calculated using
v
v
u
u
u
u Lcav λ utt
wETM (S) =π

5.5

gITM (S)gETM [1 − gITM (S) gETM ]Results

This section describes the results of the HWS measurement of wavefront
distortion. The comparison of the measured off-axis distortion with that ex-5.5. RESULTS

145

pected is discussed in Section 5.5.1, and the temporal development of verticalcross-sections are compared to H-V in Section 5.5.2. The measured distortion is also validated by comparing it to that predicted using a FEM and themeasured stored power and by comparing the predicted mode size with thatmeasured.

5.5.1

Measured off-axis wavefront distortion

The discrete gradient fields measured by the HWS at successively later times
are shown in Figure 5.12 a) - f), and the corresponding off-axis wavefront
profiles, WD(off ) , are shown in Figure 5.13 a) - f). The wavefront profiles
clearly show two wavefront distortions forming at different rates, where thelens in the sapphire (RHS lobe) is the larger and forms more quickly. The
distance between the centers of the two distortions (≈ 23 mm) is consistent
with the projection of the centers of the two optical elements separated by
140 mm and viewed at 10◦ , as previously illustrated in Figure 5.9 a).
(off)

(off)

The vertical cross sections of the distortions, WDSAPH and WDFS , attime texpt ≈ 300 s are plotted in Figure 5.14. The ratio of the wavefront

distortion on transmission due to the sapphire compared to that due to the
fused silica is 1.62. Assuming that this distortion is the steady-state value
and using the theory of Hello and Vinet (see Section A.1), the ratio of theabsorbed power in the sapphire and fused silica is 40, which is significantly
different to the ratio of 93 determined indirectly by Zhao et al. [18]. This
provides an additional demonstration that direct measurement of thermal
lensing with a wavefront sensor provides more accurate information about
the behaviour of the cavity than can be inferred without such a measurement.

5.5.2

Temporal development of cavity defocus

The cavity defocus produced by the measured off-axis wavefront distortionwas calculated using the procedure described in Section 5.4.2. The timedependence of the cavity defocus is shown in Figure 5.15 a). Also plotted

146

CHAPTER 5. HOPTF WD MEASUREMENT(a) texpt = 0.38s(b) texpt = 1.63s(c) texpt = 7.06s(d) texpt = 25.43s(e) texpt = 87.00s(f) texpt = 291.39s

Figure 5.12: Vector fields showing the gradient of the wavefront distortion insapphire and fused silica at various times from texpt = 0 s, at which the laser
was locked to the cavity. The magnitude of the gradient is proportional to
the length of a vector (from the tip of the arrow to the end of the line) atthat point. The numbers on the axes indicate the coordinates on the HWS
CCD in pixels (1 pixel = 12 µm) and the magnfication between the HWS
and its conjugate plane at the ITM is approximately a factor of 7, indicatingthat 1 pixel corresponds to ≈ 72 µm at the ITM.

5.5. RESULTS

147(a) texpt = 0.38s(b) texpt = 1.63s(c) texpt = 7.06s(d) texpt = 25.43s(e) texpt = 87.00s(f) texpt = 291.39s

Figure 5.13: Contour plots of the wavefront distortion in sapphire and fusedsilica, WD(off ) , at various times from texpt = 0 s, at which the laser was locked
to the cavity. In order to compare the wavefront distortion at different times,
all plots share the same colour scale. As such, parts of the distortion saturate
the scale at later times resulting in an apparent reduction of spatial extent
with time.

148

CHAPTER 5. HOPTF WD MEASUREMENT

Figure 5.14: Vertical cross-sections of wavefront distortion in the sapphire
(off)
(off)
ITM, WDSAPH , (red) and fused silica CP, WDFS , (blue). Some residual tilt
is present on the WD.
is the temporal development of the defocus predicted by the theory of Helloand Vinet. This theory, however, assumes that the power is constant, which
it is not in this case as shown in Figure 5.15 b). The difference between
the measured and theoretical curves illustrates the advantage of having a
wavefront sensor: the temporal development of the thermal lensing in thetwo optics may be measured directly and individually in each of the optics,
rather than inferred from the behaviour of the cavity and an idealized modelof the thermal lensing. This does not imply that the Hello-Vinet theory
is incorrect, merely that this scenario does not lie within the range of theassumptions of the theory. Therefore, it would be inappropriate to apply the
Hello-Vinet theory to this cavity to diagnose its properties.

5.5.3

Correlation of measured distortion and stored
power

A finite element simulation was run to model the temporal developmentof the thermal lenses in the sapphire and fused silica using the temporal5.5. RESULTS

149

(a)

(b)

Figure 5.15: a) Time dependence of the sapphire, fused silica and total (sapphire + fused silica) thermal lenses as measured by the Hartmann sensor andexperienced by the cavity mode. Also plotted is the prediction of the temporal development from the theory of Hello and Vinet. b) Time dependence ofthe power transmitted through the ETM, as recorded by the beam profiler.

150

CHAPTER 5. HOPTF WD MEASUREMENT

development of the intra-cavity power and beam size measured by the beamprofiler. The source code for the model can be found in Appendix B.3. The
normalized defocus of this lens is plotted as a function of time in Figure 5.16a) (red). Also plotted is the normalized defocus measured by the Hartmann
sensor (blue). The difference between the two curves is plotted in Figure 5.16b) and has an rms value of approximately 0.03. The curves were normalized
by setting the average value between texpt = 100 and texpt = 300 s to 1. This
was necessary because that the transmittance through the end mirror is notknown accurately and, hence, the absolute power in the cavity can not be
determined accurately. There is clearly good agreement between the finite
element model of the defocus and the defocus measured by the HWS sensor.

5.5.4

Correlation of measured distortion and cavity
mode size

The cavity mode size at the ETM, as measured by the beam profiler, and themode size predicted by the measured distortion were compared for a measurement that had particularly unstable mode-matching. The results are plottedin Figure 5.17 a). There is excellent agreement between the magnitudes ofthe predicted and measured cavity mode sizes and a clear correlation between fluctuations in the measured and predicted mode sizes. The difference
between the measured and predicted cavity mode size is plotted in Figure5.17 b) and has an rms value of approximately 0.05 mm.

5.6Conclusion

The results presented in this Chapter show that the HWS described in Chapters 3 and 4 can be used to measure absorption-induced wavefront distortionin optics that are suspended in a large vacuum system. The sensitivity ofthe sensor was λ/730 where λ = 800 nm (average of 10 frames), which wasworse than the shot-noise limited sensitivity, probably due to residual air

5.6. CONCLUSION

151

(a)

(b)

Figure 5.16: a) Finite element prediction of the total defocus using the measured of intracavity power versus time (red curve) and the measured defocus(blue curve). Both plots were normalized to remove any scaling issues. b)
the difference between the red and blue curves which has an rms value ofapproximately 3%.

152

CHAPTER 5. HOPTF WD MEASUREMENT

(a)

(b)

Figure 5.17: a) Cavity mode size versus time. The red curve is the measurement from the beam profiler. The blue curve is the prediction from themeasurement of the Hartman sensor. b) the difference between the red andblue curves which has an rms value of approximately 0.05 mm.

5.6. CONCLUSION

153

currents. The HWS was validated by showing that the measured distortionwas consistent with the predictions of a FEM of the ITM and CP and themeasured cavity power, and by showing that the cavity mode size predictedusing the measured distortion agreed to within 0.05 mm (rms), about 0.7%,
with the measured mode size.
These results thus indicate that the installed HWS is both sufficiently
sensitive for the measurements of absorption-induced wavefront distortion inadvanced GWI, concordant with the requirements stated in Section 1.3.2, andis accurate. Importantly, the HWS provides a direct, detailed measurementof the variation of the WD, rather than a measurement that relies on thevalidity of assumptions and models. Although the wavefront distortion wascharacterized by the defocus in this instance, the profile is rich in spatial
information that is necessary for any future directed compensation system,
such as the scanning CO2 heating laser described in Section 1.3.1.

154

CHAPTER 5. HOPTF WD MEASUREMENT

Chapter 6
Off axis measurement ofwavefront distortion
6.1

Background

The measurement of three-dimensional index-of-refraction distributions with
an optical wavefront sensor was described by Roggemann et al. [104]. They
framed the problem to cover arbitrary three dimensional index-of-refraction
distributions and, as such, required multiple views of the variations in accordance with traditional tomography. The solution presented in this chapter
takes advantage of the cylindrical symmetry inherent in the thermal lensing problem in a GWI to solve for the refractive index distribution in theglass test masses using only a single view approximately 20 − 40◦ off-axis.

Tomographic reconstruction of cylindrically symmetric refractive index dis-tributions using a single view has previously been discussed by Miranda et al.
[105]. However, their solution relied on a view that was perpendicular to theaxis of symmetry. This view is not possible with a temperature distribution
in a cylindrical glass optic and hence their solution is not applicable to thisproblem.
155

156

CHAPTER 6. OFF AXIS MEASUREMENT OF DISTORTION

6.2

Objective

In this chapter I will describe the measurement of absorption induced wavefront distortion (WD) using an off-axis HWS. I begin by discussing, in Section
6.3, the analysis of a measured off-axis wavefront distortion to determine theaxially-symmetric temperature distribution in a test mass, and thus calculate the on-axis wavefront distortion. The objective of the remainder of thechapter is the proof-of-principle of the analysis under ideal conditions using a
numerical simulation, described in Section 6.5, and under practical conditions
using a bench-top experiment, described in Section 6.5.3.

6.3

Analysis of wavefront distortion measured
off-axis

A probe beam transmitting through an axially-symmetric temperature distribution T (r, z) in a test optic is shown in Figure 6.1. The temperature
distribution results in a refractive index distribution, n(r, z) + n2 , which isa deviation from the nominal refractive index, n0 . The resulting wavefront
distortion, WD (x′, y′), is measured at the plane Pmeas . The projection ofthis wavefront distortion onto the exit plane of the optic, Pexit , is denoted
WD (x, y) and is determined by the path integral through T (r, z), where thepath, ~σ (z), is defined parametrically~σ (z) = {xp (z) , yp (z) , z}
xp (z) = x + z tan (θ)
yp (z) = y
where x and y are the coordinates of the path as it crosses the plane z = 0
and are related to the coordinates x′ and y′, by6.3. ANALYSIS OF OFF-AXIS WD

n1

157

θ n(r,z) + n2 n 1
r, x
z −ve

WD(x,y)z +ve

y(out of page)

ϕ

r, −x

WD(x’,y’)T(r,z)

x’ +vey’(out of page)Pexitx’ −vePmeas
Figure 6.1: Schematic showing off-axis transmission through an axially symmetric thermo-refractive index distribution, n(r, z) = dn/dT × T (r, z) in atest optic with nominal refractive index n2 . The accumulated wavefront distortion, WD(x′, y′), is measured at the plane Pmeas . The accumulated wavefront distortion projected onto the exit plane of the optic, Pexit , is denoted
WD(x, y).

158

CHAPTER 6. OFF AXIS MEASUREMENT OF DISTORTION

x′ = x cos (ϕ)
y′ = yThus, the path integral that returns the projected wavefront distortion,
WD (x, y), is givendn
WD (x, y) =
dT

Z z1

T (x (z) , y (z) , z) |~σ ′ (z)| dz

z0

where |~σ ′ (z)| is the infinitesimal length of a path element given by|σ ′ (z)| =
=

6.3.1

v
u
u
t

dxp (z)dz!2

dyp (z)
+
dz

!2

dz
+
dz

!2

q

tan2 (θ) + 1

A zonal representation of T (r, z)

In a zonal representation, the region of the temperature distribution is spannedby a set of discrete volume elements, or voxels, Vij (r, z), that are discontinuous in r and z and have a uniform internal density, defined as
Vij (r, z) =


 1

 0

for
otherwiseri ≤ r < ri+1 and zj ≤ z < zj+1

The boundaries of these functions in the cut-away of a cylinder are illustrated in Figure 6.2. For reference, the volume elements that were usedby Roggemann et al.

[104] were not cylindrically symmetric and had anon-uniform density defined by a three-dimensional Gaussian refractive index distribution within the region of the volume element. The path integral
through the function Vij (r, z) is given by:6.3. ANALYSIS OF OFF-AXIS WD

159

Figure 6.2: Cut-away of a cylinder containing 14 × 14 annular voxels

wij (x, y) ==Z z1

Vij (r, z) dz
z0
Z Z upper (x,y)
ijlower (x,y)
Zij

1 dz

(6.1)
(6.2)

where Zijlower (x, y) and Zijupper (x, y) are the limits of integration through theij th volume element for a ray passing through the coordinates (x, y, 0). Provided one chooses a fine enough mesh, one can ensure a good approximation
to all axially symmetric temperature distributions.
Physically, the functions wij (x, y) can be thought of as the wavefront distortion measured at the exit plane, Pexit , resulting from transmission througha uniform density annular volume element, Vij (r, z). An example of a single
annular volume element in a mesh is shown in Figure 6.3 a) and the corresponding wavefront distortion is shown as a greyscale image in Figure 6.3b).
To prevent large numbers of indices in the following discussion, the functions Vij (r, z) and wij (x, y) where 1 ≤ i ≤ Nr and 1 ≤ j ≤ Nz will be
re-labelled Vk (r, z) and wk (x, y) where 1 ≤ k ≤ Nr × Nz .

160

CHAPTER 6. OFF AXIS MEASUREMENT OF DISTORTION
Pexit
y
x

(a)

(b)

Figure 6.3: a) An example of a single annular volume element, Vij (r, z),
and b) a greyscale map of the corresponding wavefront distortion, wij (x, y),measured at the exit plane, Pexit .

Recovery of the temperature distribution from the off-axis measurement
becomes a straightforward process. The coefficients, Ak , of the best fit ofwk (x, y) to the measured wavefront distortion are determined. If the wavefront distortion is measured at m positions (xα , yα ), where 1 ≤ α ≤ m, thebest fit is determined using the fitting method described by Bevington [106].
The calculation of a least squares fit to a set of functions, wk (x, y), begins
by determining a vector, B, containing the overlap of the measured wavefront
distortion projected onto Pexit , the exit plane, WD (x, y), and each wk (x, y),where the Bk component of the vector is given by
Bk =X[WD (xα , yα ) wk (xα , yα )] .

(6.3)

α

Next one determines a square matrix M, representing the overlap of every
combination of wk (x, y), where the lk th component of the matrix, Ml k , isgiven byMl k =

Xα[wl (xα , yα) wk (xα , yα )] .

(6.4)

6.4. NUMERICAL SIMULATION OF VOXEL ANALYSIS

161

Thus, the coefficients, Ak , of the best fit of wk (x, y) to WD (x, y) are thecomponents of the vector A, given by
A = B M−1

(6.5)

where M−1 is the inverse of M. The fitted refractive index distribution,
nfit (r, z), is thennfit (r, z) =X

Ak Vk (r, z) ,

(6.6)

k

the temperature distribution, Tfit (r, z), is
Tfit (r, z) =

dn
dT

!−1

X

ak Vk (r, z) ,

(6.7)

k

and, finally, the fitted on-axis wavefront distortion, WDfit (r), is given by
WDfit (r) ==Z z1nfit (r, z) dz
z0
Z z1 Xz0

Ak Vk (r, z) dz.

(6.8)

k

6.4

Numerical simulation of voxel analysis

6.4.1

Simulation procedure

In this simulation, I compare the on-axis wavefront distortion, for a realistic
temperature distribution, by a) directly calculating the on-axis distortion,and b) calculating the off-axis distortion and then using the voxel reconstruction to determine the best-fit temperature distribution and its resulting
on-axis distortion.
1. Using the substrate absorption solution of Hello and Vinet, specify atemperature distribution, T (r, z), in a 200 cm diameter, 100 cm thickglass cylinder 1 heated by a Gaussian beam with a 1/e2 radius of 25 cm
1

The optic is large by realistic standards but does not affect the results of the voxel
analysis.

162

CHAPTER 6. OFF AXIS MEASUREMENT OF DISTORTION
propagating along the cylindrical axis. For this simulation, the glass
had the same thermal properties as the BG20 filter glass described in
Chapter 2.

2. Integrate T (r, z) along the z-axis to determine the on-axis wavefront
distortion, WDon (r).
3. Determine the off-axis path integral of T (r, z):(a) Determine the off-axis path integral projected onto, Pexit , the exit
plane of the optic, WDoff (xα , yα), through the distribution atangle θ = 32.6◦ by taking the sum of a series of axial slices of
the temperature distribution of thickness dz that are offset fromeach other in the x-direction by dz tan (θ) at the m coordinates(xα , yα ), where 1 ≤ α ≤ m. Note that the path integral (pro-jection) through a distribution at angle θ is known as the Radon
Transform of that distribution [107].
(b) Determine the aperture created by the front and back faces. Set
WDoff (x, y) to be zero outside this aperture.
4. Determine the functions, wk :(a) Specify the limits of each of the Nr ×Nz volume elements, Vk (x, y),where there are Nr elements in the radial direction and Nz ele-ments in the axial direction. Note that because of the aperturing,some areas of the cylinder are undersampled and some are notsampled at all. Hence the extent of the voxels in the radial direction is set to be less than the actual radius of the cylinder.
(b) Determine the path integral through each uniform density volume
element, wk where 1 ≤ k ≤ Nr × Nz , using Equation 6.1.
(c) Determine the overlap of WDoff (xα , yα ) with the functions wk ,using Equation 6.3 to yield B.

6.4. NUMERICAL SIMULATION OF VOXEL ANALYSIS

163

5. Determine the square matrix M, representing the overlap of every combination of wk (xα , yα ), using Equation 6.4.
6. Invert M to determine M−1 .
7. Determine the best fit coefficients A using Equation 6.5:
8. Reconstruct the temperature distribution Tfit (r, z) using Equation 6.7.
9. Determine the on-axis distortion, WDfit (r), by integrating the reconstructed temperature distribution along the axis using Equation 6.8.
10. Compare the reconstructed on-axis distortion, WDfit (r), with the known
on-axis distortion, WDon (r), calculated in Step 2.

6.4.2

Simulation results

The given and reconstructed temperature distributions from the simulation
are shown as false color images in the left and right cells, respectively, of Figure 6.4 a). Note that the full width of the temperature distribution can’t bereconstructed due to aperturing and thus the reconstructed temperature distribution does not span the full width and has grey edges. The reconstruction
qualitatively agrees with the given temperature distribution.
The directly calculated and reconstructed on-axis wavefront distortions
are shown in Figure 6.4 b). There is clearly very good agreement between
the two distortions within the constraint of the quantization inherent in theanalysis.
The results of the simulation demonstrates that the off-axis voxel analysis
is, in principle, capable of reconstructing the temperature distribution andon-axis distortion under ideal conditions.

164

CHAPTER 6. OFF AXIS MEASUREMENT OF DISTORTIONz (cm)

50

0

−50
0

50

100

r (cm)

050

100

r (cm)(a)(b)

Figure 6.4: Simulation results: a) (left) False colour image of temperature
distribution in simulated BG20 glass (arbitrary units) and (right) false colour
image of 14 × 14 voxel reconstruction of temperature distribution. Grey
area indicates the region not reconstructed. b) Wavefront distortion (WD):directly calculated on-axis distortion, WDon (r), (line) and reconstructed onaxis distortion, WDfit (r), (points).

6.5. PROOF-OF-PRINCIPLE OF OFF-AXIS ANALYSIS

6.5

165

Proof-of-principle demonstration of offaxis voxel reconstruction

6.5.1

System layout

The measurement system for the proof-of-principle of the off-axis voxel analysis is shown in Figure 6.5. The BG20 glass test optic (GTO) describedin Chapter 2 was placed in the object arm of a Mach-Zender interferometer that was constructed using a 10mW HeNe laser beam. A 3W Nd:YAG
cw laser beam was propagated along the axis of the GTO to produce wavefront distortion in the GTO. The on-axis WD was measured using the MZinterferometer in which the exit plane of the GTO was imaged onto CCDMZ .
The off-axis wavefront distortion, WD (x′, y′) at Pmeas , was measured using a HWS similar to that described in Chapters 3 and 4. The output fromthe 820 nm Agilent fibre-coupled SLD was expanded and collimated into abeam by the lenses in telescope T5. This probe beam was incident on the
GTO at an angle of 57.2 ± 0.2◦ and transmitted through the optic with aninternal angle of 32.8◦ ± 0.2◦ assuming the nominal refractive index is 1.55.
The probe beam was then demagnified with a Galilean telescope, T6, andwas incident on the HWS.

6.5.2

Procedure

The procedure for measuring the on-axis and off-axis WD in the GTO was:
1. Place a grid of holes at the exit plane of the GTO and record image of
632.8 nm transmission through the grid on CCDMZ to determine thescale of the GTO image on the CCD.
2. Repeat step 1 with the 820 nm transmission through the grid and record
an image on the HWS CCD but with the Hartmann plate removed, todetermine the scale of the GTO image at the HWS.

166

CHAPTER 6. OFF AXIS MEASUREMENT OF DISTORTION

S1

T1

Pexit

3W Nd:YAG laser
1064nm
T5

GTO

T4

BD

BS2

CCD MZ

object beam

DCBS
Fibre coupled LED
820nm

L1
HWS

T6

T2

reference beam

10mW HeNe
632.8nm

BS1

T3

Pmeas

Figure 6.5: Schematic of experiment to measure thermal lensing in a cylindrical test optic. The output of a 3W 1064 nm Nd:YAG laser is incident on
shutter S1. When the shutter is open, the beam passes through telescope,
T1, and is transmitted through the BG20 glass test optic. The 1064nm isincident on telescope T4 at an angle very close to normal incidence. A beam
dump with a small hole in it is placed at the focal plane of T4 and is adjusted
such that normally incident rays on T4 pass through the hole and those rays
at a small angle are blocked. The 1064 nm beam is blocked by the beam
dump. The output of a 10 mW HeNe laser is the input to a Mach-Zehnderinterferometer, the object arm of which contains the BG20 optic. The object and reference arms are expanded by telescopes, T2 and T3, respectively.
The object arm is normally incident on telescope T4, is demagnified andrecombined with the reference arm at the second beamsplitter, BS2. The interference pattern at the exit plane of the glass test optic is imaged onto the
CCD with lens L1. A dichroic beam splitter, DCBS, immediately before theglass test optic is used to combine the two different wavelengths. The output
from the 820 nm Agilent fibre-coupled SLD is expanded and collimated intoa beam by the lenses in telescope T5. This probe beam is incident on the
GTO at an angle of 57.2 ± 0.2◦. The transmitted probe beam is demagnified
with a Galilean telescope, T6, and is incident on the HWS. The exit andmeasurement planes described in Figure 6.1 are indicated on this figure by
Pexit and Pmeas , respectively.

6.5. PROOF-OF-PRINCIPLE OF OFF-AXIS ANALYSIS

167

3. Record a series of reference interference patterns, with no heating of
the GTO, with CCDMZ .
4. Record a series of reference measurements with the HWS (with theHartmann plate replaced).
5. texpt = 0 s. Turn on the 3W 1064 nm Nd:YAG laser to induce a thermallens in the GTO.
6. texpt ≈ 500 s. Record a series of distorted interference patterns with

CCDMZ . Simultaneously, record a series of distorted Hartmann images

with the HWS.
7. Turn off 1064nm laser and cease recording with CCDMZ and the HWS.

6.5.3

Results

The on-axis and off-axis wavefront distortions are shown in Figures 6.6 a)and b) respectively, where the scale was established by the images recordedin Step 1. The off-axis distortion is clearly elongated in the x direction due
to propagation through the temperature distribution at a large angle. The
shape of the aperture (tall and thin) seen in Figure 6.6 b) is consistent withviewing the GTO at a large external angle in the horizontal plane.
The off-axis wavefront distortion, WD (x′, y′), was projected onto thecoordinate system at exit plane of the optic, Pexit . The off-axis voxel analysis
was applied to the projected-off-axis-distortion, WD (x, y), to reconstruct the
on-axis distortion, WDfit (r), using a variety of different voxel configurations(Nr , Nz and rVmax ). Two reconstructions are plotted in Figures 6.7 a) and b),
respectively. Also plotted is the radial on-axis distortion, WDon (r), measured
by the interferometer. These plots show that the off-axis voxel analysis is, inprinciple, capable of reconstructing the on-axis wavefront distortion.
The accuracy of the reconstruction is dependent upon the configurationof the voxels. For example, the quality of the reconstruction improves when
the range in the radial direction is reduced. This is to be expected because

168

CHAPTER 6. OFF AXIS MEASUREMENT OF DISTORTION(a) Interferometer WD measurement(b) HWS WD measurement

Figure 6.6: Contour plots of a) on-axis and b) off-axis wavefront distortion
as seen at the plane Pmeas .

6.5. PROOF-OF-PRINCIPLE OF OFF-AXIS ANALYSIS

169(a) Nr = 10, Nz = 7, rVmax = 23.0 mm(b) Nr = 11, Nz = 4, rVmax = 13.5 mm

Figure 6.7: On-axis measured and off-axis reconstructed wavefront distortion.
a) Reconstruction with 10 annular × 7 axial volume elements to a radius of
23.0 mm with an RMS wavefront error of 11 nm. b) Reconstruction with
11 annular × 4 axial volume elements to a radius of 13.5 mm with an RMS
wavefront error of 4.5 nm.

170

CHAPTER 6. OFF AXIS MEASUREMENT OF DISTORTION

the aperture reduces the sampling of extreme radial values. Also, the qualityof the reconstruction decreases when the number of voxels is increased. Onceagain, this is an expected effect of trying to fit a higher spatial resolution
model to a finite spatial resolution measurement.

6.6

Conclusion

The off-axis voxel analysis described here has been demonstrated to work
under ideal conditions in the simulation presented in Section 6.4.1 and inpractice in the experiment results presented in Section 6.5.3. There is scopein future work to examine the efficacy of this technique under a variety ofconditions, such as the dependence on noise sources and the optimum model
configuration at a given angle. As such, this technique has been demonstrated
to be a promising area of research.

Chapter 7
Conclusion
7.1

Review of aims

The objective of the research described in this thesis is the development of a
wavefront sensor with sufficient sensitivity and spatial resolution for use in aclosed-loop active thermal compensation system in an advanced GWI. The
use of such a sensor contributes to solving one of the problems associatedwith upgrading from initial to advanced gravitational wave interferometry,
as required for the realisation of gravitational wave astronomy. The specificaims, therefore, were
1. to validate the Hello-Vinet solution for absorption induced wavefront
distortion. This solution has been used extensively in the modelling of
GW interferometers,
2. to develop and test a wavefront sensor with a sensitivity sufficient for
use in advanced GWI: at least λ/467 @ 632.8 nm with a spatial resolution of at least 23 × 23 sample points over the region of interest [21],and to investigate the limits of sensitivity possible with the wavefront
sensor,3. to deploy the wavefront sensor in a long-baseline high optical power
171

172

CHAPTER 7. CONCLUSION
cavity and measure the thermal lensing within the input-coupling mirror, or ITM, of that cavity,

4. and the development and proof-of-principle demonstration of a single
view tomographic algorithm for reconstruction of axially symmetricwavefront distortion.

7.2

Summary of results

A high-precision interferometric measurement of absorption-induced wavefront distortion was described in Chapter 2. The results showed good agreement with the predicitions of Hello and Vinet for longer times, but there wasa small systematic difference, possibly due to convective cooling of the glass
test optic - a cooling mechanism not included in the Hello-Vinet model. The
validation of this model, within this limitation, increases the confidence inthe validity of numerical models of GWI, such as MELODY [16], that employ
the Hello-Vinet model.
The design and development of a Hartmann wavefront sensor (HWS) foradvanced GWI was presented in this Chapter 3. Extensive modelling anddiffraction simulations were used to optimize the Hartmann sensor, including
an investigation of the systematic error introduced by cross-talk between
neighbouring spots in the HWS. In the final HWS design, random noise
associated with the photoelectrons in the pixels of the CCD was identified as
the major noise source, limiting the single-shot sensitivity to about λ/1050@ 633nm based on the nominal properties of the CCD.
The measurement of the sensitivity and accuracy of the HWS was discussed in Chapter 4. The choice of an incoherent light source was shown
to be effective in reducing the uncertainty in centroid measurements to the
Poissonian shot-noise limit, thereby enabling the wavefront error to be reduced to at least λ/15,500 by averaging multiple measurements. Evidence ofanother noise source, due probably to fluctuations in the temperature of the
HWS, was observed at this level. The measurement of a very small defocus,

7.3. FUTURE DIRECTIONS

173

equivalent to the change in wavefront curvature that would be caused byan 850m lens, with an accuracy of 1.7% and a precision of 0.7% was also
described.
Chapter 5 described the use of the HWS for the measurement of absorptioninduced wavefront distortion in optics that were remotely suspended in alarge vacuum system as part of a high optical power cavity. The sensitivity of the HWS was λ/730 at λ = 800 nm (average of 10 frames), limited
somewhat by the noisy environment and residual air currents. The HWS
measurements were validated by showing that the measured distortion wasconsistent with the predictions of a FEM of absorption-induced wavefront
distortion, using the measured cavity power. The cavity mode size predicted using the measured distortion agreed to within 0.05 mm (rms) withthe mode size measured by a beam profiler. The results in that chapter also
illustrated that the HWS provides a direct, detailed measurement of the variation of the absorption-induced wavefront distortion, rather than an indirect
measurement that relies on the validity of assumptions and models to infer
information about the intra-cavity optics.
In Chapter 6, a single-view off-axis tomographic technique was described.
A numerical simulation was used to demonstrate that a Hello-Vinet temperature and refractive index distribution could be recovered with good fidelity
using a single off-axis view. The technique was validated by showing that the
on-axis results of the tomographic analysis agreed well with the wavefront
distortion measured on-axis using a Mach-Zehnder interferometer.

7.3

Future directions

It was evident from the results presented in Chapter 4 that the temperaturedependence of the current Hartmann sensor design probably limits the longterm accuracy of the sensor. Work on future designs will most likely focus
on reducing this temperature dependent error. A passive reduction in thetemperature dependence can be made by replacing the brass Hartmann plate

174

CHAPTER 7. CONCLUSION

with one made from a material with a low coefficient of thermal expansion,for example, invar. Alternatively or concurrently, active stabilization of thetemperature, using a combination of high precision thermistors and thermoelectric coolers, could be employed to maintain a constant temperature in
the Hartmann sensor. Although this would require more development thanpassive solutions, it would most likely allow the Hartmann sensor to operatewith high precision accuracy in many more environments.
I would also suggest further investigation of the tomographic technique,presented in Chapter 6, to optimize its sensitivity and explore its application
under a variety of conditions. The generalization of this technique to nonaxially symmetric wavefront distortions may be important for identifying
point absorbers in test masses of advanced GWI or for the measurement ofwavefront distortion in the beam-splitter of a GWI.

“Our imagination has driven us to listen for the music of the cosmos. The opening movement entertains us with familiar themes,
but the remainder is an undiscovered symphony, infinitely richer
and more nuanced than we can possibly imagine.”
Aidan F. Brooks

Appendix A
Additional derivations
A.1

Analytic form of Hello-Vinet solution

Substrate absorption
Only the temporal Hello-Vinet substrate absorption case was consideredin Chapter 2, although that solution asymptotes to the steady-state case.
The solution is as follows.
For a Gaussian intensity distribution, I(r), of waist w and power P , givenby2P
r2
I(r) =
exp
−2
π w2
w2
"

#

transmitted through a cylindrical test optic of radius a, length h, volumetricabsorption α, density ρ, specific heat C, thermal conductivity K, and thermooptic coefficient dn/dT , and assuming only radiative boundary conditions,the total wavefront distortion, ψ (r, t), as a function of time is:
dn Z h/2[ T (t, r, z) − Text ] dz.
ψ (r, t) =dT −h/2
The temperature distribution, T (t, r, z), is given by
175

176

APPENDIX A. ADDITIONAL DERIVATIONS

T (t, r, z) = Text +

X

p, m

Cp′ m [1 − exp (−αp m t)] cos (up z/a) J0 (ζm r/a) ,where Text is the external temperature, where
4 α a pm
Cp′ m =
2K h ζm"sin (up h/2 a) τ cos (up h/2 a)−
2 + u2
up
ζm
p

#

1
,
cp

where αp m is given by

αp m =


K  2
2
ζ
+
u
p ,
ρ C a2 m

where up is the pth solution of"#uhu = τ cot,2awhere ζm is the mth solution ofx J1 (x) − τ J0 (x) = 0,
where pm is given by
2
r
1
1 Za
2 ζm
I(r)J0 ζm
pm = 22 2
2
ζm + τ J0 (ζm ) a 0a

where τ is given by3
τ = 4 σ ′ Text
a/K,

and, finally, where cp is given by
cp = 1 +aup hsin (up h/a) .



r dr,

A.2. VARIANCE OF A DIGITIZED VALUE

A.2

177

Variance of a digitized value

Determination of the variance in a digitized value. Suppose ones wishes todetermine the variance in a value, Ne , which has been digitized on an n-bit
scale. The gain, gD , of this digitization is the ratio of the highest possible
value of Ne , Nemax , and the total number of digital values, 2n .
gD =

Nemax
2n

(A.1)

All values between i gD and (i + 1) gD will be rounded down to i gD in the
digization process. Therefore the effective mean of these values is i gD . If a
series of M measurements, xei , are made, such that they are all rounded toi gD , then the variance is given by2
σdigital,M
=

M
X

j=1



xej − i gD
M −1

2

M 
2 g
1 X
D
xej − i gD=
gD j=1
M −1



(A.2)


In the limit that M → ∞, the set of values xej − i gD approaches acontinuum, x, that ranges between 0 and gD , the value gD /(M −1) approaches
dx, and the variance calculation becomes an integral

lim

M →∞

2
σdigital,M

1
=
gD
g2
= D
3

Z gD
0

x2 dx
(A.3)

Hence the standard deviation, σdigital is given by
1 Nemax
σdigital = √
3 2n

(A.4)

178

A.3

APPENDIX A. ADDITIONAL DERIVATIONSABCD cavity calculation

A.3. ABCD CAVITY CALCULATION

179

180

APPENDIX A. ADDITIONAL DERIVATIONSAppendix B
Computer code
B.1

Hartmann plate optimization code

The following Mathematica file produced a complex array representing thetransmission through a Hartmann plate of varying parameters of hole sizeand hole spacing, discussed in Steps 1 to 5 in Section 3.3.2.1.
The data array from Mathematica had some header information attached
to convert it into a .ZBF file [77] suitable for importing into ZEMAX. The
complex array was unchanged by this step.
The .ZBF file is loaded into ZEMAX via POP many.ZPL. The command
THIC(count) = i*200.0 + 100.0 increases the thickness of the count-th
surface in increments of 200.0 mm, from 100.0 mm to 900.0 mm. The
index count simply identifies the ’surface’ in ZEMAX which is being altered. In this case the surface is simply free space and, hence, its thickness is equal to the propagation distance or lever arm. The command POP
filename$ count performs the physical optics propagation which uses theFourier diffraction propagation discussed by Goodman [80]. This covers Step
6 in Section 3.3.2.1.
The output is saved to a .ZBF file from which the electric field is extracted
and converted into an 8-bit TIFF image as described in Step 7 in Section
3.3.2.1.

181

182

APPENDIX B. COMPUTER CODE

B.1. HARTMANN PLATE OPTIMIZATION CODE

183

184

APPENDIX B. COMPUTER CODE

B.1. HARTMANN PLATE OPTIMIZATION CODE

185

186

APPENDIX B. COMPUTER CODE

B.1. HARTMANN PLATE OPTIMIZATION CODE

187

POP_many.ZPL

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188

Printed by Aidan Brooks! ZEMAX test file 1
COPYFILE infile1$ "C:\Program Files\ZEMAX\POP\BeamFiles\input1.ZB! Clear All Surfaces
DelAll = NSUR()F"

IF (NSUR() > 2)
FOR i = DelAll − 1, 2, −1
DELETE (i)
NEXT
ENDIF

file3$OUTPUT SCREEN
PRINT infile1$
outfile$ = file1$ + holespace1$ + file2$ + holesize1$ +FORMAT 1.0! Create file to store locations
OUTPUT "centres.txt"

FOR i = 1, 5, 1
! Set propagation distance
THIC(count) = (i−1)*200.0 + 100.0! Set−up System Parameters
ATYP = 0
AVAL = 60.0
WAVL 1 = 0.6328
WWGT 1 = 1.0

!
!
!
!
!
!

ApertureSize = 60

IF (i < 10)filler$ = "00"
ELSE
filler$ = "0"
FORMAT 2.0
ENDIF! Loop to create fully apertured Hartmann Plate
count = 2

filler$ = "−PD−"
filenumber$ = $STR(THIC(count))
filename$ = outfile$ + filler$ + filenumber$ +INSERT (count)
APTP (count) = 0
THIC (count) = 1000.0

file4$! Run physical optics propagation and save to ap
propriate filename.
POP filename$ count! Remove result from C: drive and place on O:\ d

FOR j = 1, 8, 1
FOR k = 1, 5, 1
FOR q2 = 1, 2, 1

rive
outfile1$ = "C:\Program Files\ZEMAX\POP\BeamFiles\" + filename$! Loop through q2 to run DIST and UNDIST respectively
IF (q2 < 2)
file4$ = file4Y$
ELSE
file4$ = file4N$
ENDIF! Loop through j to run through all the hole spacings
holespace = 2.25 + (j−1)*0.25
FORMAT 3.2
holespace1$ = $STR(holespace)movedfile$ = "O:\PHYSICS\Optics\abrooks\HartFiles\" + filename$COPYFILE outfile1$ movedfile$
DELETEFILE outfile1$
NEXT
NEXT
NEXT
NEXT

! Loop through k to run through all the hole sizes
holesize = 0.2*k
FORMAT 2.1
holesize1$ = $STR(holesize)! Create the input filename.
infile$ = file1$ + holespace1$ + file2$ + holesize1$ + f
ile3$ + file4$
infile1$ = "C:\Program Files\ZEMAX\POP\BeamFiles\" + infile$

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APPENDIX B. COMPUTER CODEOUTPUT SCREEN
PRINT filename$file1$ = "Holes−Spacing−"
file2$ = "MM−HoleSize−"
file3$ = "MM−Radius"
file4Y$ = "−DIST.ZBF"
file4N$ = "−UNDIST.ZBF"

B.2. CROSS-TALK ANALYSIS: DIFFRACTION PROPAGATION

B.2

189

Diffraction propagation for cross-talk analysis

The source code listed in this section was used to model the cross-talk between neighbouring spots in Figures 3.12 and 3.13.
Chp4_Fresnel_diffraction_Broadband_centroid.pro
The initial electric field, EIn, is created from an intensity array, IntIn,of hexagonally arranged circular apertures and a phase array, phase, whichcontains a randomly set amount of defocus. This is propagated over thelever arm distance, deltaZ, using the Fourier transform technique discussed
by Goodman [80] to yield the output electric field, Eout. This is converted
to an intensity array.
For the coherent source, the final intensity array, outputArrPeak, is only
calculated for one wavelength. For the coherent source, the final intensity
array, outputArrFinal, is made from a sum of intensity arrays, where eacharray is weighted by the amplitude of that wavelength in the spectrum of theincoherent source.

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r = sqrt((x)^2 + y^2)
IntIn = IntIn + double(r le holesize/2.0)
if (HoleConfig eq 0) then begin
for i = 0, 5 do begin
angle = i*pi/3.0
pitch = 430E−3
r = sqrt((x − pitch*cos(angle))^2 + (y − pitch*sin(angle))^2; Chp4 − Fresnel approximation; Written by Aidan Brooks. 9th May 2007
seedL = 845739L
NTest = 10
time1 = systime(1)for test_number = 0, NTest−1 do begin; get the center and defocus of the wavefront distortion
xc = 1E−1*randomn(seedL, 2)
defocus = (−1.0)^(floor(2.0*randomu(seedL)))
defocus = defocus * (1.0D−5)*(randomu(seedL) + 1.0))
IntIn = IntIn + double(r le holesize/2.0)
endforendif;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
window, 0
window, 1

for displacement = 0, 1 do begin
for HoleConfig = 0, 1 do begin
time0 = systime(1); spectral widthdW = 100D−6
dW = 55.26D−6
dWTotal = 3.2*dW
ndW = 200dWStep = dWTotal/ndW

for k = −ndW, ndW do begin
wavelength = wavelengthPeak + k*dWStep;amplitudeE = exp(−1.0*((wavelength − wavelengthPeak)/(dW))^2.0)
amplitudeE = exp(−0.25*((wavelength − wavelengthPeak)/(dW))^2.0);−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; get propagator
PExponent = −1.0D*pi*wavelength*deltaZ*freqR2
TDeltaZ = COMPLEX(cos(PExponent), sin(PExponent), /DOUBLE);−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; get input field
phase = OPD*2.0*pi/wavelength
EIn = sqrt(IntIn) * COMPLEX(cos(phase), sin(phase), /DOUBLE);−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

outputArrFinal = dblarr(N1, N1)
outputArrPeak = outputArrFinal;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; Perform propagation to get output field and intensity
F1 = FFT(Ein, /DOUBLE)
F2 = TDeltaZ*F1
Eout = FFT(F2, /INVERSE, /DOUBLE)
IntOut = (abs(Eout))^2;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
if (k eq 0) then outputArrPeak = IntOut
outputArrFinal = outputArrFinal + IntOut
wset, 0plot, IntOut(N1/4:3*N1/4, N1/2)/max(IntOut(N1/4:3*N1/4, N1/2))
wset, 1
plot, outputArrFinal(*, N1/2)/max(outputArrFinal(N1/4:3*N1/4, N1;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; get spatial array coordinates
dx = (inputArrSize)/N1
x = (dindgen(N1)#replicate(1.0D, N1) − N1/2)*dx
y = (dindgen(N1)##replicate(1.0D, N1) − N1/2)*dx
OPD = (defocus/2.0)*((x − xc(0))^2.0 + (y − xc(1))^2.0)
if (displacement eq 0) then OPD = OPD*0.0D;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; get spectral array coordinates
f0 = findgen(N1) − (N1/2 − 1)
f1A = shift(f0, −(N1/2 − 1) )/(N1*dx);f1A = shift(f0, −511)/(N1*(dx*N1))
freqX = f1A#replicate(1.0D, N1)
freqY = f1A##replicate(1.0D, N1)
freqR2 = freqX^2 + freqY^2;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; get input mask
IntIn = 0.0*x/2))
timeNow = systime(1)print, [test_number, displacement, HoleConfig, k, wavelength, am
plitudeE^2, timeNow − time0, timeNow − time1]
wait, 0.001
endforwindow, 0
tvscl, IntIn(N1/4:3*N1/4, N1/4:3*N1/4)
window, 1

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APPENDIX B. COMPUTER CODE;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; specify parameters of model: length units − (mm)inputArrSize = (1.5D−3)*1024
wavelengthPeak = 820D−6;deltaZ = 31D0
;deltaZ = 10.43D0
deltaZ = 10.00D0
holesize = 150D−3
N1 = 1024
pi = acos(0.0D)*2.0D;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

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tvscl, IntOut(N1/4:3*N1/4, N1/4:3*N1/4)
wset, 0plot, outputArrPeak(3*N1/8:5*N1/8, N1/2)/max(outputArrPeak(3*N1/8:5*
N1/8, N1/2))
wset, 1
plot, outputArrFinal(3*N1/8:5*N1/8, N1/2)/max(outputArrFinal(3*N1/8:
5*N1/8, N1/2))
window, 2
x2 = dblarr(N1/8)
for i = 0, N1/8 − 1 do x2(i) = total(outputArrPeak(i*8:(i+1)*8 − 1,
N1/2−4:N1/2+3))x3 = dblarr(N1/8)
for i = 0, N1/8 − 1 do x3(i) = total(outputArrFinal(i*8:(i+1)*8 − 1,
N1/2−4:N1/2+3))N2 = N1/8
plot, x2(3*N2/8:5*N2/8)/max(x2(3*N2/8:5*N2/8))
window, 3
plot, x3(3*N2/8:5*N2/8)/max(x3(3*N2/8:5*N2/8));;

Final1 = outputArrFinal
Peak1 = outputArrPeak

dirData = ’D:\SciWord\Thesis\data\chapter_4\cross_talk_broadband_L3’
outfilename = dirData + ’\cross_talk_displ_test_number_’ + string(test_numb
er, format = ’(I3.3)’) $+ string(displacement, format = ’(I2.2)’) + $’HoleConfig_’ + string(HoleConfig, format = ’(I2.2)’) + ’.dat’
openw, outunit, outfilename, /GET_LUN
writeu, outunit, long(N1)writeu, outunit, Einwriteu, outunit, OPD
writeu, outunit, IntIn
writeu, outunit, outputArrFinal
writeu, outunit, outputArrPeak
free_lun, outunit;
writeu, outunit, long(N1);writeu, outunit, Final1;
writeu, outunit, Final2;
writeu, outunit, Peak1;
writeu, outunit, Peak2;
free_lun, outunit
endfor
endfor
endforend

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192

B.3

APPENDIX B. COMPUTER CODE

FEM of ITM and CP thermal lens at the
HOPTF

The source code listed in this section was used to produce the finite element
prediction of the wavefront distortion in the cavity at the HOPTF. The defocus was determined from this wavefront distortion and is plotted in Figure
5.16.
The five files listed here are as follows:
1. FEM_Using_Spiricon.pro
The main code for that runs the finite element model.
2. solvecrosssectionalareafem.pro
Solve the Cross Sectional Area on the end face.
3. Create_V_FEM.pro
A procedure to create a variable to hold the volume elements.
4. CreateSurfaceAreaArrayFEM.pro
Create an array storing the Surface Area elements seen by each cell inthe 3D mesh of the FEM.
5. CreateThermalConductM.pro
Procedure to determine the CrossSectional area between cells divided
by the length between nodes of cells

Jul 19, 07 0:24

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Jul 19, 07 0:24

FEM_Using_Spiricon.pro

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window, 4
N1k = 121
Fit1 = 0; if Ntype = 1 then set power to be constant
Ntype = 1
WType = 1
; if Wtype = 1 then set beam size to be constantmat1 = 1
if (mat1 eq 1 ) then material = ’sapphire’ else material = ’fused_silca’
if (STRCMP(material, ’sapphire’) eq 1) then begin;
ThermalConductivity = 33.0;
Text = 300.0;
density = 3980.0;
SpecificHeat = 757.0
ThermalConductivity = 46.0
Text = 300.0
density = 3970.0
SpecificHeat = 775.0
SBConstant = 0.75*5.67E−8
alpha = 3.6
radius = 0.05
height = 0.05
endif else begin
ThermalConductivity = 1.38
Text = 300.0
density = 2202.0
SpecificHeat = 745.0
SBConstant = 0.75*5.67E−8
alpha = 3.6
radius = 0.08
height = 0.017
endelse;radius = 0.05
;height = 0.05
waistX = 0.0083
waistY = 0.0083; determine cold cavity beam size
waistX = Get_Beam_Size(1D9, 720.0, 77.0, 1.064D−6)
waistY = waistX
if (WType eq 1) then begin
waistX = 0.0080
waistY = 0.0080
endif
NnodesR = N1k
NnodesZ = 27
dx = 2.0*radius/double(NnodesR−2.0)
dy = dx
dz = height/double(NnodesZ−2.0)
dNW = round(waistX/dx)

Thursday July 19, 2007

; Create the Surface Area Array
CreateSurfaceAreaArrayFEM, SurfaceArea, AreaEndOld, radius, CrossX, CrossZ
print, ’Starting thermal conductivity matrix’
wait, 0.001; Create the Thermal Conductivity Matrix
CreateThermalConductM, ThermalConductM, radius, CrossX, CrossZ, LocationArrXY, A
reaEndOld
print, ’ End thermal matrix’; Clear unnecessary variables;Delvar, LocationArrXY, AreaEndOld, AreaEnd;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

ThermalConductM = ThermalConductivity*ThermalConductM
SurfaceArea = SurfaceArea*SBConstant
TempArr = dblarr(NNodesR, NNodesR, NNodesZ) + Text
PArr = dblarr(NNodesR, NNodesR, NNodesZ);PArr(NNodesR/2−5:NNodesR/2+5, NNodesR/2−5:NNodesR/2+5, 1:NNodesZ−2) = 100000.0
OffsetXi = 0.000
OffsetXf = 0.000
Power = 1.0
time0 = systime(1)for i = 1, NNodesZ − 2 do begin
OffsetX = OffsetXi + (i−0.5)/(NNodesZ − 2.0)*(OffSetXf − OffSetXi)
OffsetY = 0.000
CrossX2 = CrossX#replicate(1.0D, NNodesR)
XMin = sqrt(2.0)*(CrossX2 − OffSetX − dx/2.0)/waistX
XMax = sqrt(2.0)*(CrossX2 − OffSetX + dx/2.0)/waistX
CrossY2 = CrossX##replicate(1.0D, NNodesR)
YMin = sqrt(2.0)*(CrossY2 − OffSetY − dx/2.0)/waistY
YMax = sqrt(2.0)*(CrossY2 − OffSetY + dx/2.0)/waistY

PArrEnd = 0.25*Power*(erf(Xmin) − erf(XMax))* $(erf(Ymin) − erf(Ymax));PArrEnd = PArrEnd * 0.5*(sin(2.0*CrossX2/0.005) + 1.0)
;PArrEnd = heat_Beam_FEM(NNodesR, Fit1)
PArrEnd = ParrEnd * double(AreaEndOld gt 0.0)
PArr(*, *, i) = height*alpha*PArrEnd(*, *)/(NNodesZ − 2.0)
endfor

FEM_Using_Spiricon.pro

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193

;dz = height/double(NnodesZ−2−1)
CrossX = − radius − dx + dx*(dindgen(NnodesR)+0.5)
CrossZ = −height/2.0 − dz + dz*(dindgen(NnodesZ)+0.5);CrossZ = −height/2.0 − dz + dz*(dindgen(NnodesZ));−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; Create the Volume and CrossSectional Area arrays
SolveCrossSectionalAreaFEM, AreaEnd, LocationArrXY, radius, CrossX
AreaEndOld = AreaEnd
AreaEnd = AreaEnd + double(AreaEnd eq 0.0)*10000.0
V = Create_V_FEM(AreaEnd, height, NNodesZ)

B.3. FEM OF ITM AND CP THERMAL LENS AT THE HOPTF

Printed by Aidan Brooks

Jul 19, 07 0:24

FEM_Using_Spiricon.pro

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time1 = systime(1)print, time1−time0;wait, 1000

Jul 19, 07 0:24

FEM_Using_Spiricon.pro

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194

Printed by Aidan Brooks

nSpirData = nLines;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
print, timeSpir;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
print, ’hi’

dt = 0.0005
Q = dblarr(NNodesR, NNodesR, NNodesZ)
diffT = dblarr(1)window, 0
window, 1
window, 2
TempOld = TempArr;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; load spiricon data
DirToAnalyzeHS = string(dblarr(13))
nDirToAnalyze = size(DirToAnalyzeHS, /N_ELEMENTS)
SpiriconDataFiles = DirToAnalyzeHS
ThursFiles = FILE_SEARCH("D:\Hartmann − Shared Documents\Spiricon\2007−03−08", ’2007*.txt’, CO
UNT = ncount1)
SpiriconDataFiles(0:ncount1−1) = ThursFiles(*)
FriFiles = FILE_SEARCH(’D:\Hartmann − Shared Documents\Fri Afternoon Folder’, ’2007−3*−9−*.txt’,
COUNT = ncount2)
SpiriconDataFiles(ncount1:ncount1+ncount2−1) = FriFiles(*)
FriFilesCP = FILE_SEARCH(’D:\Hartmann − Shared Documents\Fri Afternoon Folder’, ’2007−03−09.txt’, COUNT = ncount2)
SpiriconDataFiles(nDirToAnalyze−1) = FriFilesCP(*);−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; copy Spiricon file into a temporary directory and lose the time strings at the start
filenameSpir = SpiriconDataFiles(DirIndex)
filenameTemp = "c:\\temp\\Copied_Spiricon_file.txt"
strIn = ""nLines = FILE_LINES(filenameSpir)openr, outunit1, filenameSpir, /GET_LUN
openw, outunit2, filenameTemp, /GET_LUN
for i = 0, nLines − 1 do begin
readf, outunit1, strIn
strIn = STRMID(strIN, 11)
printf, outunit2, strIn
endforfree_lun, outunit2
free_lun, outunit1;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; open temporary file and load data
if (DirIndex eq 12) then nDims = 8 else nDims = 6
dataSpir = dblarr(nDims, nLines)
openr, outunit, fileNameTemp, /GET_LUN
readf, outunit, dataSpir
free_lun, outunit
timeSpir = dblarr(nLines)
timeSpir(*) = dataSpir(0, *)timeSpir = timeSpir − timeSpir(0)Thursday July 19, 2007

period = 200.0
w0 = 2.0*!Pi/period
TimeExpired = 0.0
TimeArr =[0.0]NLimit = 7L*10L^6TimeNew = 100.1
TimeNextRecord = 0.5
dtOld = 0.5
tCritical = density*SpecificHeat*radius^2.0/ThermalConductivity
RecordTempArrHot = 0
RecordTempArrCold = 0
RecordImage = 0
if (ntype eq 1) then begin
n1 = nLines/2
CurrentPower = dataSpir(1, n1)
endif else begin
CurrentPower = dataSpir(1, 0)
endelseCurrentTimeIndex = 0
OPDData = dblarr(4, NLimit);−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; Start iteration of FEM
for i = 1L, NLimit do begin;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; update energy−flow matrix
Q = 0.0*Q
Q(1 : NNodesR − 1, *, * ) = temporary(Q(1 : NNodesR − 1, *, * )) + (TempArr(
0 : NNodesR − 2, *, * ) − $TempArr(1 : NNodesR − 1, *, * ))*ThermalConductM(0, 1 : NNodesR − 1, *, * )Q(0 : NNodesR − 2, *, * ) = temporary(Q(0 : NNodesR − 2, *, * )) + (TempArr(
1 : NNodesR − 1, *, * ) − $TempArr(0 : NNodesR − 2, *, * ))*ThermalConductM(2, 0 : NNodesR − 2, *, * )FEM_Using_Spiricon.pro

2/4

APPENDIX B. COMPUTER CODE

DirIndex = 0
fileFEM = ’D:\Hartmann − Shared Documents\Spiricon\FEM_’ + string(DirIndex, format = ’(I3.3)’)
FILE_MKDIR, fileFEM

TestDiff1 = 0.0
TestDiff2 = 0.0
count = 1
EnergyArr = TempArr*density*V*SpecificHeat

Jul 19, 07 0:24

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Q(*, 1 : NNodesR − 1, * ) = temporary(Q(*, 1 : NNodesR − 1, * )) + (TempArr(*, 0 : NNodesR − 2, * ) − $TempArr(*, 1 : NNodesR − 1, * ))*ThermalConductM(3, *, 1 : NNodesR − 1, * )Q(*, 0 : NNodesR − 2, * ) = temporary(Q(*, 0 : NNodesR − 2, * )) + (TempArr(*, 1 : NNodesR − 1, * ) − $TempArr(*, 0 : NNodesR − 2, * ))*ThermalConductM(1, *, 0 : NNodesR − 2, * )Jul 19, 07 0:24

FEM_Using_Spiricon.pro

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CurrentBeamSize = dataSpir(3, CurrentTimeIndex)
if (CurrentBeamSize lt 3200.0) then begin; solve for beam size at ITM
mag = 0.00918612/(3145.0)
q1Inv = COMPLEX(−1/720.0, (1.064E−6)/(3.14159*(CurrentBeamSize*mag)^2.0))
q1 = 1.0/q1Inv
q2 = (q1 + 77.0)
q2Inv = 1.0/q2
ImgPart = IMAGINARY(q2Inv)
w2 = sqrt( abs((1.064E−6)/(3.14159*ImgPart)) )Q(*, *, 0 : NNodesZ − 2) = temporary(Q(*, *, 0 : NNodesZ − 2 )) + (TempArr(*, *, 1 : NNodesZ − 1 ) − $TempArr(*, *, 0 : NNodesZ − 2 ))*ThermalConductM(5, *, *, 0 : NNodesZ− 2)
Q(*, *, 1 : NNodesZ − 1) = temporary(Q(*, *, 1 : NNodesZ − 1 )) + (TempArr(*, *, 0 : NNodesZ − 2 ) − $TempArr(*, *, 1 : NNodesZ − 1 ))*ThermalConductM(4, *, *, 1 : NNodesZ− 1);−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

print, "Beam Size at ITM = ", w2*1000.0;waistX = w2
;waistY = w2
for k1 = 1, NNodesZ − 2 do begin
OffsetX = OffsetXi + (k1−0.5)/(NNodesZ − 2.0)*(OffSetXf − OffSetXi);−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; Add power to energy−flow matrix
Q = temporary(Q) + PArr*CurrentPower/100.0
Q = temporary(Q) − SurfaceArea*(TempArr^4.0 − Text^4.0)

OffsetY = 0.000
CrossX2 = CrossX#replicate(1.0D, NNodesR)
XMin = sqrt(2.0)*(CrossX2 − OffSetX − dx/2.0)/waistX
XMax = sqrt(2.0)*(CrossX2 − OffSetX + dx/2.0)/waistX
CrossY2 = CrossX##replicate(1.0D, NNodesR)
YMin = sqrt(2.0)*(CrossY2 − OffSetY − dx/2.0)/waistY
YMax = sqrt(2.0)*(CrossY2 − OffSetY + dx/2.0)/waistY;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; update energy and temperature arrays
EnergyArr = temporary(EnergyArr) + Q*dt
TempArr = EnergyArr/(density*SpecificHeat*V);−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

PArrEnd = 0.25*Power*(erf(Xmin) − erf(XMax))* $(erf(Ymin) − erf(Ymax));PArrEnd = PArrEnd * 0.5*(sin(2.0*CrossX2/0.005) + 1.0)
;PArrEnd = heat_Beam_FEM(NNodesR, Fit1)

wset, 0if (count le 5) then begin
TestDiff1 = TestDiff1 + max(abs(TempArr − TempOld)/TempOld)/10.0
endif else begin
TestDiff2 = TestDiff2 + max(abs(TempArr − TempOld)/TempOld)/10.0
endelsecount = count + 1
TimeExpired = TimeExpired + dt

PArrEnd = ParrEnd * double(AreaEndOld gt 0.0)
PArr(*, *, k1) = height*alpha*PArrEnd(*, *)/(NNodesZ − 2.0)
endfor;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; update FEM with real data
if (TimeExpired gt TimeSpir(CurrentTimeIndex + 1)) then begin
if (CurrentTimeIndex lt nSpirData−1) then begin
CurrentTimeIndex = CurrentTimeIndex + 1
if (nType eq 1) then CurrentPower = CurrentPower else CurrentPower = d
ataSpir(1, CurrentTimeIndex)
fileOUT = fileFEM + ’\\FEM_constW_’ + material + string(ntype, format =’(I2.2)’) + ’_PRL_’ + string(CurrentTimeIndex, format = ’(I4.4)’) + ’.raw’Thursday July 19, 2007;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; record data at appropriate times
if ((TimeExpired ge TimeNextRecord) OR (RecordImage eq 1)) then begin
RecordImage = 0
TimeNextRecord = TimeNextRecord * (tCritical/dtOld)^(1.0D/50.0)
TempTotal = (total(TempArr, 3)*dz*(2.6*10.0^(−6.0)))/(6.33*10.0^(−7.0))
endif;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

if (i mod 10) eq 0 then begin

FEM_Using_Spiricon.pro

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195

openw, outunit, fileOUT, /GET_LUN
writeu, outunit, byte(N1k)writeu, outunit, TimeExpired
writeu, outunit, double(CrossX)
OPD = total(TempArr(*, NNodesR/2, *), 3)
OPD = OPD − OPD(1)writeu, outunit, OPD
free_lun, outunit
print, systime(0)

endif
endif
endif
;−−−−−−−−−−−−−−−−−−−−−−−−−−−−

B.3. FEM OF ITM AND CP THERMAL LENS AT THE HOPTF

Printed by Aidan Brooks

Jul 19, 07 0:24

FEM_Using_Spiricon.pro

Page 7/8

Jul 19, 07 0:24;−−−−−−−−−−−−−−−−−−−−−−−−−−−−; check that time interval of FEM is small enoughif (TestDiff2 lt TestDiff1) then dt = dt*1.1 else dt = dt/1.8
if (dt le 0.0005) then dt = 0.0005
if (TestDiff2 lt 1.0E−14) then i = NLimit + 20L;−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Page 8/8;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; plot any remaining data
if (i mod 10) eq 0 then begin
wset, 1
plot, OPDData(0, 0:i−1), OPDData(1, 0:i−1)
wset, 4
plot, OPDData(0, 0:i−1), OPDData(2, 0:i−1)
oplot, OPDData(0, 0:i−1), OPDData(3, 0:i−1), THICK = 2
endif;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
endforend

endif
TEmpOld =TempArr
result = total(FINITE(TempArr, /NAN)+FINITE(TempArr, /INFINITY));print, result
wait, 0.001
OPD = total(TempArr(*, NNodesR/2, *), 3)
OPD = OPD − OPD(1)result = poly_fit(CrossX(NNodesR/2 − dNW:NNodesR/2 + dNW), OPD(NNodesR/2 − d
NW:NNodesR/2 + dNW), 2, STATUS = statRes)
OPDData(0, i−1) = TimeExpired
OPDData(1, i−1) = CurrentPower
OPDData(2, i−1) = result(2)*2.0
offset = 0.12*tan(10.0*acos(0.0D)/90.0)
offsetX = round(offset/dx)
OPD = total(TempArr(*, NNodesR/2 + offsetX, *), 3)
OPD = OPD − OPD(NNodesR/2)
result = poly_fit(CrossX(NNodesR/2 − dNW:NNodesR/2 + dNW), OPD(NNodesR/2 − d
NW:NNodesR/2 + dNW), 2, STATUS = statRes)
OPDData(3, i−1) = result(2)*2.0

Thursday July 19, 2007

FEM_Using_Spiricon.pro

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APPENDIX B. COMPUTER CODE;−−−−−−−−−−−−−−−−−−−−−−−−−−−−; print and plot updated data
print, i, dt, TimeExpired, total(SurfaceArea*(TempArr^4.0 − Text^4.0))/total(PArr) , $max(TempArr) − TempArr(1, NNodesR/2, 1), max(TempArr) − Text
TestDiff2 = 0.0
TestDiff1 = 0.0
count = 1
wset, 0plot, TempArr(1:NNodesR−2, NNodesR/2, NNodesZ/2) − TempArr(1, NNodesR/2,
NNodesZ/2)
oplot, TempArr(1:NNodesR−2, NNodesR/2, NNodesZ/2) − TempArr(1, NNodesR/2,
NNodesZ/2), PSYM = 6
oplot, TempArr(NNodesR/2, 1:NNodesR−2, NNodesZ/2) − TempArr(NNodesR/2, 1,
NNodesZ/2); print, k1;print, TempArr(NNodesR/2, NNodesR/2, 3) − Text
wait, 0.001
diffT = [diffT, TempArr(NNodesR/2, NNodesR/2, NNodesZ/2) − Text]
TimeArr = [TimeArr, TimeExpired];diffT = [diffT, alog10((max(TempArr) − TempArr(1, NNodesR/2, 1))/total(SurfaceArea*(TempArr^4.0 − Text^4.0))) ];wset, 1;
plot, TimeArr, diffT;dt = dt*1.05; print, dt
wset, 2
plot, TempArr(NNodesR/2, NNodesR/2, *) − Text;−−−−−−−−−−−−−−−−−−−−−−−−−−−−

FEM_Using_Spiricon.pro

196

Printed by Aidan Brooks

Mar 26, 07 21:38

solvecrosssectionalareafem.pro

Page 1/1; Solve the Cross Sectional Area on the end face
pro SolveCrossSectionalAreaFEM, AreaEnd, LocationArrXY, radius, CrossXs = size(CrossX)
Nnodes = s(1)dx = CrossX(1) − CrossX(0)x = CrossX#replicate(1.0, Nnodes)
y = CrossX##replicate(1.0, Nnodes)r = sqrt(x^2 + y^2)dxL = 2.0*radius/(NNodes − 3.0)
xLoc = −radius − dxL + dxL*findgen(NNodes)
LocationArrXY = dblarr(2, Nnodes, Nnodes)
LocationArrXY(0, *, *) = x
LocationArrXY(1, *, *) = y;LocationArrXY(0, *, *) = xLoc#replicate(1.0D, NNodes);LocationArrXY(1, *, *) = xLoc##replicate(1.0D, NNodes)
rInside = double(r le radius)
area = rInside
rInSm = smooth(rInside, 3)
SumA = total(rInSm eq 1.0)*(dx*dx)
rEdge = (rInSm gt 0.0) AND (rInSm lt 1.0)
EdgeList = where(rEdge)s = size(EdgeList)
s1 = s(1)N3 = 800.0
for i = 0, s1 − 1 do begin
xc = x(EdgeList(i))
yc = y(EdgeList(i))
XYPosn = ARRAY_INDICES(x, EdgeList(i))
x1 = (dx*(dindgen(N3) + 0.5)/double(N3) − dx/2.0 + xc)#replicate(1.0D, N3)
y1 = (dx*(dindgen(N3) + 0.5)/double(N3) − dx/2.0 + yc)##replicate(1.0D, N3)
r1 = sqrt(x1^2 + y1^2)
rTemp = double(r1 le radius)
TempSum = total(rTemp)*(dx*dx)/double(N3^2)
SumA = SumA + TempSumarea(EdgeList(i)) = TempSum/(dx*dx)
if (total(rTemp) ne 0.0) then begin
LocationArrXY(0, XYPosn(0), XYPosn(1)) = total(rTemp*x1)/total(rTemp)
LocationArrXY(1, XYPosn(0), XYPosn(1)) = total(rTemp*y1)/total(rTemp)
endif
print, i, " of ", string(s1, format = ’(i4)’), " − Calculating surface area"
wait, 0.0001
endfor

B.3. FEM OF ITM AND CP THERMAL LENS AT THE HOPTF

Printed by Aidan Brooks

AreaEnd = double(area*dx*dx)

solvecrosssectionalareafem.pro

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197

end

Monday July 23, 2007

Nov 01, 05 16:04

Create_V_FEM.pro

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198

Printed by Aidan Brooks; function to create variable V to hold the volume elementsfunction Create_V_FEM, AreaEnd, height, NnodesZs = size(AreaEnd)
s1 = s(1)V = dblarr(s1, s1, NnodesZ)for i = 0, NnodesZ − 1 do begin
V(*, *, i) = AreaEnd(*, *)endforreturn, V*height/(NNodesZ − 2.0)
end

Create_V_FEM.pro

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APPENDIX B. COMPUTER CODE

Monday July 23, 2007

Oct 24, 05 19:32

CreateSurfaceAreaArrayFEM.pro

Page 1/4

Oct 24, 05 19:32

CreateSurfaceAreaArrayFEM.pro

Page 2/4

endelse; Create an array storing the Surface Area elements seen by each cell.
pro CreateSurfaceAreaArrayFEM, SurfaceArea, AreaEndOld, radius, CrossX, CrossZ
dx = CrossX(1) − CrossX(0)dz = CrossZ(1) − CrossZ(0)if (radius^2.0 ge xmax^2.0) then begin
xmaxYVals = [sqrt(radius^2.0 − xmax^2.0), −sqrt(radius^2.0 − xmax^2.0)]
endif else begin
xmaxYVals = [10.0^6.0, 10.0^6.0]
endelseif (radius^2.0 ge ymax^2.0) then begin
ymaxXVals = [sqrt(radius^2.0 − ymax^2.0), −sqrt(radius^2.0 − ymax^2.0)]
endif else begin
ymaxXVals = [10.0^6.0, 10.0^6.0]
endelsesR = size(CrossX)
sZ = size(CrossZ)
Nr = sR(1)Nz = sZ(1); Create an array of the EDGES of the node boxes;x0 = CrossX(0) − 0.5*dx;z0 = CrossZ(0) − 0.5*dz

if (radius^2.0 ge ymin^2.0) then begin
yminXVals = [sqrt(radius^2.0 − ymin^2.0), −sqrt(radius^2.0 − ymin^2.0)]
endif else begin
yminXVals = [10.0^6.0, 10.0^6.0]
endelsex = CrossX#replicate(1.0, Nr)
y = CrossX##replicate(1.0, Nr)crossPts = dblarr(2, 4)
crossN = 0; surface area is nominally zero, except for those nodes whose grid cells intersect the radius; determine the area in each cell by working through the list of cells given above; Option A: Get list of fractional area elements from AreaEndOld. Search for the
cross−sections from ListOut2; Option B: Determine the cells which border the intersection points in ListOut2; Option A
AreaEndNew = AreaEndOld/(dx*dx)
EdgeList = where( (AreaEndNew lt 0.999999) AND (AreaEndNew gt 0.0) );ListIndices = (ListOut2 − x0)/dx
s = size(EdgeList)
s1 = s(1);SurfaceArea = AreaEndOld*0.0
for i = 0, s1 − 1 do begin; get centre and edges of cell
xc = x(EdgeList(i))
yc = y(EdgeList(i))
xmin = xc − dx/2.0
xmax = xc + dx/2.0
ymin = yc − dx/2.0
ymax = yc + dx/2.0

Monday July 23, 2007

if ((xmaxYvals(0) ge ymin) AND (xmaxYVals(0) le ymax)) then begin
crossPts(*, crossN) = [xmax, xmaxYVals(0)]crossN = crossN + 1
endif
if ((xmaxYvals(1) ge ymin) AND (xmaxYVals(1) le ymax)) then begin
crossPts(*, crossN) = [xmax, xmaxYVals(1)]crossN = crossN + 1
endif
;−−−−
;−−−−
if ((yminXVals(0) ge xmin) AND (yminXVals(0) le xmax)) then begin
crossPts(*, crossN) = [yminXVals(0), ymin]crossN = crossN + 1
endif
if ((yminXVals(1) ge xmin) AND (yminXVals(1) le xmax)) then begin
crossPts(*, crossN) = [yminXVals(1), ymin]crossN = crossN + 1
endif
if ((ymaxXVals(0) ge xmin) AND (ymaxXVals(0) le xmax)) then begin
crossPts(*, crossN) = [ymaxXVals(0), ymax]
crossN = crossN + 1
endif
if ((ymaxXVals(1) ge xmin) AND (ymaxXVals(1) le xmax)) then begin
crossPts(*, crossN) = [ymaxXVals(1), ymax]
crossN = crossN + 1
endif;−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−CreateSurfaceAreaArrayFEM.pro

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199

; Determine the intercepts of the gridlines and the circle
if (radius^2.0 ge xmin^2.0) then begin
xminYVals = [sqrt(radius^2.0 − xmin^2.0), −sqrt(radius^2.0 − xmin^2.0)]
endif else begin
xminYVals = [10.0^6.0, 10.0^6.0];−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−; Check positions of intercepts − do they line on cell edges?
if ((xminYvals(0) ge ymin) AND (xminYVals(0) le ymax)) then begin
crossPts(*, crossN) = [xmin, xminYVals(0)]crossN = crossN + 1
endif
if ((xminYvals(1) ge ymin) AND (xminYVals(1) le ymax)) then begin
crossPts(*, crossN) = [xmin, xminYVals(1)]crossN = crossN + 1
endif

B.3. FEM OF ITM AND CP THERMAL LENS AT THE HOPTF

Printed by Aidan Brooks

Oct 24, 05 19:32

CreateSurfaceAreaArrayFEM.pro

Page 3/4; rearrange values so that the first two form and arc and the second two form
an arcif (crossN gt 2) then begin
crossPtsX = dblarr(4)crossPtsY = dblarr(4)crossPtsX(*) = crossPts(0, *)crossPtsY(*) = crossPts(1, *)Oct 24, 05 19:32

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200

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SurfaceAreaOut(*, *, i) = AreaEndOld(*, *)endif
SurfaceAreaOut(*, *, i) = SurfaceAreaOut(*, *, i) + SurfaceArea(*, *)endfor
SurfaceArea = SurfaceAreaOut
end

XTrue = 0
YTrue = 0
Val = 0for j = 0, 2 do begin
for k = j+1, 3 do begin
if (crossPtsX(j) eq crossPtsX(k)) then begin
XTrue = 1
Val = [j, k]
endif
if (crossPtsY(j) eq crossPtsY(k)) then begin
YTrue = 1
Val = [j, k]
endif
endfor
endfor
rX = sqrt((crossPtsX − crossPtsX(Val(0)))^2.0 + (crossPtsY − crossPtsY(Val(0)))^2.0)
rX(Val(1)) = 100.0*radius
NearList = sort(rX)crossPts = crossOut
endif; get the arc length − taking into account the fact that the arc might cross
2Pi
ArcLength = abs( atan(crossPts(1, 0), crossPts(0, 0)) − atan(crossPts(1, 1),crossPts(0, 1)) )if (ArcLength ge !Pi) then ArcLength = 2.0*!Pi − ArcLength; if there are two points where the circle crosses the cell wall, then add the two arcs
if (crossN gt 2) then begin
ArcLength2 = abs( atan(crossPts(1, 2), crossPts(0, 2)) − atan(crossPts(1,3), crossPts(0, 3)) )if (ArcLength2 ge !Pi) then ArcLength2 = 2.0*!Pi − ArcLength2
ArcLength = ArcLength + ArcLength2
endif
SurfaceArea(EdgeList(i)) = ArcLength*radius*dz
endfor
SurfaceAreaOut = dblarr(Nr, Nr, Nz)
for i = 1, Nz−2 do begin
if ((i eq 1) OR (i eq Nz−2)) then begin

Monday July 23, 2007

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APPENDIX B. COMPUTER CODE

crossOut = 0.0*crossPts
crossOut(*, 0) = crossPts(*, Val(0))
crossOut(*, 1) = crossPts(*, NearList(1))
crossOut(*, 2) = crossPts(*, NearList(2))
crossOut(*, 3) = crossPts(*, Val(1))

Nov 05, 05 16:54

CreateThermalConductM.pro

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Nov 05, 05 16:54

CreateThermalConductM.pro

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endelse; procedure to determine the CrossSectional area between cells divided by the length between nodes of cells
pro CreateThermalConductM, ThermalConductM, radius, CrossX, CrossZ, LocationArrX
Y, AreaEndOld
dx = CrossX(1) − CrossX(0)dz = CrossZ(1) − CrossZ(0)sR = size(CrossX)
sZ = size(CrossZ)
Nr = sR(1)Nz = sZ(1); Northif (InsideR(1) AND InsideR(2)) then begin
CrossSectionBetweenCells(1, i, j) = dx
endif else begin
if ( (ymax^2.0 le radius^2.0) AND ( abs(abs(ymax) − abs(radius))/abs(radius) ge 1.0E−5 )) then begin
MidPt = sqrt(radius^2.0 − ymax^2)
MidPt = [MidPt, −MidPt]
Select = where((MidPt lt xmax) AND (MidPt ge xmin))if (InsideR(1) OR InsideR(2) OR abs(MidPt(0) − MidPt(1)) le dx ) then b
egin
if (InsideR(1)) then begin
CrossSectionBetweenCells(1, i, j) = abs(xmin − MidPt(Select(0)) )endif
if (InsideR(2)) then begin
CrossSectionBetweenCells(1, i, j) = abs(xmax − MidPt(Select(0)) )endif
if (abs(MidPt(0) − MidPt(1)) le dx ) then begin
CrossSectionBetweenCells(1, i, j) = abs(MidPt(0) − MidPt(1) )
endif
endif
endif

ThermalConductM = dblarr(6, Nr, Nr, Nz)
CrossSectionBetweenCells = dblarr(4, Nr, Nr)
for i = 0, Nr − 1 do begin
for j = 0, Nr − 1 do begin

xmin = CrossX(i) − dx/2.0
xmax = CrossX(i) + dx/2.0

endelseymin = CrossX(j) − dx/2.0
ymax = CrossX(j) + dx/2.0; [South−West, North−West, North−East, South−East
InsideR = [sqrt(xmin^2.0 + ymin^2.0) le radius, sqrt(xmin^2.0 + ymax^2.0)
le radius, $sqrt(xmax^2.0 + ymax^2.0) le radius, sqrt(xmax^2.0 + ymin^2.0)
le radius]; Westif (InsideR(0) AND InsideR(1)) then begin
CrossSectionBetweenCells(0, i, j) = dx
endif else begin
if ((xmin^2 le radius^2.0) AND ( abs(abs(xmin) − abs(radius))/abs(radius) ge 1.0E−5 )) then begin
MidPt = sqrt(radius^2.0 − xmin^2)
MidPt = [MidPt, −MidPt]
Select = where((MidPt lt ymax) AND (MidPt ge YMin))
if (InsideR(0) OR InsideR(1) OR abs(MidPt(0) − MidPt(1)) le dx ) then b
egin
if (InsideR(0)) then begin
CrossSectionBetweenCells(0, i, j) = abs(ymin − MidPt(Select(0)) )endif
if (InsideR(1)) then begin
CrossSectionBetweenCells(0, i, j) = abs(ymax − MidPt(Select(0)) )endif
if (abs(MidPt(0) − MidPt(1)) le dx ) then begin
CrossSectionBetweenCells(0, i, j) = abs(MidPt(0) − MidPt(1) )
endif
endif
endif

if (InsideR(2) OR InsideR(3) OR abs(MidPt(0) − MidPt(1)) le dx ) then b
egin
if (InsideR(2)) then begin
CrossSectionBetweenCells(2, i, j) = abs(ymax − MidPt(Select(0)) )endif
if (InsideR(3)) then begin
CrossSectionBetweenCells(2, i, j) = abs(ymin − MidPt(Select(0)) )endif
if (abs(MidPt(0) − MidPt(1)) le dx ) then begin
CrossSectionBetweenCells(2, i, j) = abs(MidPt(0) − MidPt(1) )
endif
endif
endif
endelse; Southif (InsideR(3) AND InsideR(0)) then begin
CrossSectionBetweenCells(3, i, j) = dx
endif else begin
if ((ymin^2.0 le radius^2.0) AND ( abs(abs(ymin) − abs(radius))/abs(radius) ge 1.0E−5 )) then begin
MidPt = sqrt(radius^2.0 − ymin^2)
MidPt = [MidPt, −MidPt]

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Monday July 23, 2007

; Eastif (InsideR(2) AND InsideR(3)) then begin
CrossSectionBetweenCells(2, i, j) = dx
endif else begin
if ((xmax^2.0 le radius^2.0) AND ( abs(abs(xmax) − abs(radius))/abs(radius) ge 1.0E−5 )) then begin
MidPt = sqrt(radius^2.0 − xmax^2)
MidPt = [MidPt, −MidPt]
Select = where((MidPt lt ymax) AND (MidPt ge ymin))B.3. FEM OF ITM AND CP THERMAL LENS AT THE HOPTF

Printed by Aidan Brooks

Nov 05, 05 16:54

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Page 3/4

Select = where((MidPt lt xmax) AND (MidPt ge xmin))if (InsideR(3) OR InsideR(0) OR abs(MidPt(0) − MidPt(1)) le dx ) then b
egin
if (InsideR(3)) then begin
CrossSectionBetweenCells(3, i, j) = abs(xmax − MidPt(Select(0)) )endif
if (InsideR(0)) then begin
CrossSectionBetweenCells(3, i, j) = abs(xmin − MidPt(Select(0)) )endif
if (abs(MidPt(0) − MidPt(1)) le dx ) then begin
CrossSectionBetweenCells(3, i, j) = abs(MidPt(0) − MidPt(1) )
endif
endif
endif
endelseNov 05, 05 16:54

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202

Printed by Aidan Brooks; Fill in the back surface (except between cells 0 and 1 in z direction −no thermal conduction)if ( i eq 4) then begin
if (m ne 1) then begin
ThermalConductM(i, *, *, m) = AreaEndOld(*, *)/dz
endif
endif
; Fill in the front surface (except between cells Nz−2 and Nz−1 in z dirn
− no thermal conduction)if ( i eq 5) then begin
if (m ne Nz − 2) then begin
ThermalConductM(i, *, *, m) = AreaEndOld(*, *)/dz
endif
endif
endfor
endfor;print, ’hi’
;wait, 10

endforwait, 0.001
endforend;ThermalConductM = [West, North, East, South, Back, Front]

for m = 1, Nz − 2 do begin
for i = 0, 5 do begin; Fill in the sides
if (i le 3) then begin
if (i eq 0) then Dist = Westif (i eq 1) then Dist = Northif (i eq 2) then Dist = East
if (i eq 3) then Dist = SouthImin1 = Dist(0, 0)
Imax1 = Dist(1, 0)
Jmin1 = Dist(0, 1)
Jmax1 = Dist(1, 1)
Imin2 = Dist(0, 2)
Imax2 = Dist(1, 2)
Jmin2 = Dist(0, 3)
Jmax2 = Dist(1, 3)
LArr = (LocationArrXY(*, Imin1:Imax1, Jmin1:Jmax1) − LocationArrXY(*, I
min2:Imax2, Jmin2:Jmax2))^2.0
Larr2 = dblarr(Imax1−Imin1+1, Jmax1−Jmin1+1)
LArr2(*, *) = sqrt(Larr(0, *, *) + Larr(1, *, *))ThermalConductM(i, IMin1:IMax1, Jmin1:Jmax1, m) = CrossSectionBetweenCells(i, Imin1:Imax1, Jmin1:Jmax1)*dz/Larr2
endif

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APPENDIX B. COMPUTER CODE

West = [[1, Nr−1], [0, Nr−1], [0, Nr−2], [0, Nr−1]]
South = [[0, Nr−1], [1, Nr−1], [0, Nr−1], [0, Nr−2]]
East = [[0, Nr−2], [0, Nr−1], [1, Nr−1], [0, Nr−1]]
North = [[0, Nr−1], [0, Nr−2], [0, Nr−1], [1, Nr−1]]

Appendix C
Relevant papers
C.1

An off-axis Hartmann sensor for the measurement of absorption-induced wavefront
distortion in advanced gravitational wave
interferometers

A. Brooks, P.J. Veitch, J. Munch and T.L. Kelly, Gen. Relativ. Gravit.,
37,1575-1580 (2005)
This paper was superseded by the work discussed in Chapter 6.

203

204

APPENDIX C. RELEVANT PAPERS

C.1. GEN. RELATIV. GRAVIT. 37, 1575-1580 (2005)

205

206

APPENDIX C. RELEVANT PAPERS

C.1. GEN. RELATIV. GRAVIT. 37, 1575-1580 (2005)

207

208

APPENDIX C. RELEVANT PAPERS

C.1. GEN. RELATIV. GRAVIT. 37, 1575-1580 (2005)

209

210

C.2

APPENDIX C. RELEVANT PAPERS

Ultra-sensitive wavefront measurement using a Hartmann sensor

Aidan F. Brooks, Thu-Lan Kelly, Peter J. Veitch, and Jesper Munch, Opt.
Express 15 (16), 10370-10375 (2007).

C.2. OPT. EXPRESS 15 (16), 10370-10375 (2007)

211

Ultra-sensitive wavefront measurement
using a Hartmann sensor
Aidan F. Brooks, Thu-Lan Kelly, Peter J. Veitch, and Jesper Munch
The University of Adelaide, Adelaide, SA, 5005
aidan.brooks@adelaide.edu.au

Abstract: We describe a Hartmann sensor with a sensitivity of λ /15, 500
at λ = 820 nm. We also demonstrate its application to the measurement of
an ultra small change in wavefront and show that the result agrees with that
expected to within λ /3, 300.
© 2007 Optical Society of America
OCIS codes: (120.4640) Optical instruments; (010.7350) Wave-front sensing; (350.4800) Optical standards and testing

References and links
1. J. Hartmann, “Bemerkungen uber den Bau und die Justirung von Spektrographen,” Zt. Instrumentenkd. 20, 47(1900).
2. T. L. Kelly, P. J. Veitch, A. F. Brooks and J. Munch, “A differential Hartmann wavefront sensor for accurate andprecise optical testing,” Appl. Opt. 46, 861–866 (2007).
3. C. Castellini, F. Francini and B. Tiribilli, “Hartmann test modification for measuring ophthalmic progressive
lenses,” Appl. Opt. 33, 4120 (1994).
4. F. Roddier, ed., Adaptive Optics in Astronomy (Cambridge U. Press, Cambridge, England, 1999).
5. J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida and D. H. Reitze,“Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack-Hartmann wavefront sensor,” Appl. Opt. 40, 366–374 (2001).
6. R. Lawrence, D. Ottaway, M. Zucker and P. Fritchel, “Active correction of thermal lensing through externalradiative thermal actuation,” Opt. Lett. 22, 2635-2637 (2004).
7. M. Smith and P. Willems, “Auxiliary Optics Support System Conceptual Design Document, Vol. 1: Thermal
Compensation System,” (2006), http://www.ligo.caltech.edu/docs/T/T060083-00/T060083-00.pdf .
8. A. Chernyshov, U. Sterr, F. Riehle, J. Helmcke and J. Pfund, “Calibration of a Shack-Hartmann sensor for absolute measurements of wavefronts,” Appl. Opt. 44, 6419–6425 (2005).
9. J. L. Rayces, “Exact Relation between Wave Aberration and Ray Aberration,” Opt. Acta. 11, 85–88 (1964).
10. P. Mercere, P. Zeitoun, M. Idir, S. Le Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Boucourt, K. A. Goldberg,
P. P. Naullleau and S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with λEUV /120 accuracy”, Opt.
Lett. 28, 1534–1536 (2003).
11. A. Poteomkin, N. Andreev, I. Ivanov, E. Khazanov, A. Shaykin, and V. Zelenogorsky, “Use of a scanning Hartmann sensor for measurement of thermal lensing in TGG crystal,” Proc. SPIE 4970, 10–21 (2003).
12. M. Kasper, D. Looze, S. Hippler, R. Davies and A. Glindemann, “Increasing the sensitivity of a Shack-Hartmann
sensor,” in Proceedings of the Canterbury Conference on Wavefront sensing and its applications, Canterbury(1999), http://mpia-hd.mpg.de/ALFA/PAPERS/canterbury99MEK.pdf.
13. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006(1980).

1.

Introduction

Hartmann wavefront sensors, first described in 1904 [1], and particularly the Shack-Hartmannderivatives, have enjoyed a recent surge of popularity due to improvements in the speed, quality
and size of CCD arrays. They are currently used for a variety of applications, including lensmetrology [2], ophthalmology [3], adaptive optics [4] and the measurement of thermal lensing

212

APPENDIX C. RELEVANT PAPERS[5] [6]. The measurement of wavefront distortion induced by absorption in the optics of advanced gravitational wave interferometers is a particularly demanding application, requiring asensitivity equivalent to at least λ /600 at λ = 820 nm [7].
The type of Hartmann sensor used for an application is usually dictated by the intensity of theavailable light source. Shack-Hartmann sensors use micro-lens arrays to sample the wavefront
and thus optimize the light collection efficiency but they are susceptible to imperfections in
the micro-lens array [8]. A Hartmann wavefront sensor, by contrast, samples the wavefront, W,
using an opaque plate containing an array of holes, the Hartmann plate, as shown schematicallyin part of Fig. 1. It is therefore less light efficient but it is simple to optimize and, as shown
here, capable of superior performance.
Interferometerread−out
Optical fibre
from SLDL

z0
W

Hartmann plate

HeNe laser
Translation stage

CCD

Hartmann sensor

Fig. 1. A schematic of the Hartmann wavefront sensor and the system used to test it. The
sensor consists of a Hartmann plate mounted a distance L from a CCD. It was illuminatedby a wavefront W emitted from a fiber-coupled super luminescent diode (SLD), the free
end of the which was mounted on a micrometer-controlled translation stage.

The rays created by the Hartmann plate propagate normal to the incident wavefront W, to theactive surface of a CCD where they produce an array of spots. If the local slope of the wavefront changes then the positions of the spots will change. Dividing the transverse displacementof each spot by the ’lever-arm’ propagation length L yields the gradient of the wavefront change
at each hole (see [9] for example), and the wavefront change, ∆W , can be determined by numerically integrating this gradient field. While knowledge of the initial wavefront would thenenable the new wavefront to be calculated, in this paper we do not make any assumptions about
the initial wavefront but rather consider only the change, ∆W , in the wavefront as required forthe measurement of thermal lensing in gravitational wave interferometers.
The position of each spot is specified by its centroid, the precision of which is maximized byensuring that each spot consists of a large number of pixels and that each pixel acquires a large
number of photoelectrons in the available integration time. This minimizes the effects of darkcurrent, CCD read-out noise, non-uniformity in pixel response and photoelectron shot noise.
Since Hartmann wavefront sensors naturally provide large spot sizes, they should be able toprovide high sensitivity measurements of wavefront changes. Indeed, a Hartmann sensor thathas a reproducibility (sensitivity) of λ /1500 at λ = 13.4 nm [10] and a scanning Hartmann
sensor that has a sensitivity of λ /500 at λ = 1064 nm [11] have been reported. The accuracyof the scanning Hartmann sensor was estimated to be λ /50 at λ = 1064 nm. In this paper, we
describe a detailed investigation of the optimization of a Hartmann wavefront sensor (H-WFS)
and demonstrate large improvements in the sensitivity and accuracy of this type of sensor.

C.2. OPT. EXPRESS 15 (16), 10370-10375 (2007)

2.

213

Hartmann wavefront sensor

The camera used in our sensor is a 12-bit digitized, 1024 × 1024 pixel CCD camera that has anominal dynamic range of 66 dB (11 bits) and a nominal pixel spacing of 12 µ m. The actual
average pixel spacing was measured by translating the camera sideways using a micrometercontrolled translation stage and observing the displacement on the CCD of a fixed diffraction
pattern, yielding a value of 11.975 ± 0.005 µ m.
The Hartmann plate was made from 50 µ m thick brass plate into which 150 µ m diameterholes in a uniform hexagonal-close-packed array spaced 430 µ m apart were laser drilled. It was
mounted on the body of the camera and the distance between the plate and the active surfaceof the CCD, the lever arm L, was calibrated by illuminating the sensor with two laser beams
separated by a precisely measured angle and measuring the average displacement between thetwo spot patterns, giving L = 10.43 ± 0.02 mm. The hole diameter, pitch, pattern and valueof L were chosen to ensure that cross-talk between neighbouring spots was negligible while
maintaining sensitivity.
A weighted centroiding algorithm was used to determine the spot positions [12]:xc = ∑i p2i xi / ∑i p2i and yc = ∑i p2i yi / ∑i p2i where pi is the digital number, directly proportional
to the number of photoelectrons, and xi and yi are the coordinates of the ith pixel. The summation range of the algorithm was adjusted to minimize the variance in the centroids. In practice,
this meant using only pixels within a box that was 15 pixels square, which is 10% larger thanthe hole diameter. Simulations indicate that cross-talk due to diffraction should introduce asystematic error of less than 0.1% for the defocus measurement discussed in this paper.
All measurements reported here were recorded after a 3 hour warm-up period to reduce theeffects of thermal expansion of the sensor. However, there was a residual 0.5 mHz oscillationin the output of the H-WFS due to a periodic variation in the temperature of the sensor with
an amplitude of 150 mK, caused by the cycling of the room temperature. The magnitude of
the oscillation is consistent with thermal expansion of the Hartmann plate. To reduce the effectof this variation, we used data recorded at the turning points of the oscillation for the analysis
reported below.
The H-WFS was tested using the system shown in Fig. 1, in which it was illuminated bylight emitted from an optical fiber with a 50 µ m core and 0.36 NA that was coupled to an 820
nm super-luminescent diode, which had a full-width-half-maximum coherence length of about
5 µ m. A Michelson interferometer was used to measure changes in the distance, z0 , between
the light source and the H-WFS with a precision of 50 nm.
3.

Results

The statistics of the noise in the H-WFS were investigated by illuminating the CCD with theoutput from the fiber at three different intensity levels. A short integration time was used toensure that the dark current was negligible. We observed that the fluctuation, ∆p, in the digital
number in a pixel, p, was well described by ∆p ∝ p0.5 , and thus the noise has Poissonian
statistics. A relative fluctuation of 0.25% was obtained for the maximum digital number (4095),
indicating a maximum photoelectron count of approximately 1.6 × 105, which is similar to thespecified electron well depth of each pixel, confirming that the dominant noise is photoelectron
shot noise.
Numerical simulation predicts that the RMS uncertainty in the centroid position due to photoelectron shot noise in a single Hartmann image should be about 0.25% of a pixel, or about
30 nm, if the brightest pixel in each spot is full. If the noise in sequential Hartmann images
is uncorrelated then the RMS error in the displacement of a centroid, σ∆y , will be a factor of 21/2 larger. The RMS wavefront difference between adjacent holes, σ∆W , is given by

214

APPENDIX C. RELEVANT PAPERSσ∆W = σ∆y h p /L, where h p is the spacing between adjacent holes in the Hartmann plate. The
RMS wavefront error for a zonal reconstruction across all holes, assuming uncorrelated noise,
1/2
is given by Cpd × σ∆W , where Cpd is the noise coefficient developed by Southwell [13], which
is dependent on the number, Nholes , and arrangement of holes.
The wavefront error was measured using the system shown in Fig. 1 while keeping z0 constant. Spot centroids were calculated for consecutive Hartmann images separated in time by 15s and the average prism in each image was removed. These zero-prism centroids were used to
calculate the error in the wavefront change. A typical map of the wavefront change, which has
an RMS error of λ /1450, is shown in Fig. 2. The RMS error for these maps varied between
λ /1000 and λ /2000 which is consistent with the shot noise limit and Cpd = 0.2, calculated for
Nholes = 263 used in this example.

Fig. 2. Measured single-frame wavefront error map over a 7.2 mm × 7.2 mm region.

If the statistical characteristics of the noise do not vary with time then the wavefront error
should be reduced by averaging over multiple Hartmann images. To test this, we recorded asequence of 2000 Hartmann images at 30 images/second and removed the global prism fromeach image. A set of reference centroids was then calculated by averaging over Nref = 1000
images, consisting of the first and last 500 images. The central 1000 images were used to
calculate sets of centroids averaged over Navg images, where Navg = 1, . . . , 990. This processensured that the reference and average centroids were statistically independent. The σ∆W isplotted in Fig. 3, showing that σ∆W < λ /15, 500.
The σ∆W is plotted in Fig. 3. Also plotted is the result of a numerical simulation that assumes
stationary random noise and in which the only free parameter is the Navg = 1 error. For small
1/2Navg , the error decreases as Navg , as expected, and it asymptotically approaches a value that is afactor Nref times smaller than the Navg = 1, the limit due to the noise in the reference centroids.
Note the good agreement between the measurement and the numerical simulation except forNavg > 200 where the measured error is slightly larger than that predicted.
The RMS wavefront error across all holes for Navg = 990 was λ /15, 500, which is larger than
1/2
the value Cpd × σ∆W = λ /21, 000 predicted using the simulated data and the Southwell noise
coefficient appropriate for this measurement. These discrepancies are probably due to the effectof the low-frequency temperature fluctuation described above.
The ability of the H-WFS to measure a small (modal) change in the wavefront, ∆W , wasdemonstrated by translating the fiber light source. As shown in Fig. 4, translating the source a
distance ∆z from an initial position z0 displaces the spot on the CCD by ∆y (h), assuming thatthe change in the slope of each wavefront across the hole is small. The expected local gradientC.2. OPT. EXPRESS 15 (16), 10370-10375 (2007)

215

Fig. 3. The improvement in H-WFS sensitivity due to averaging over Navg Hartmann images. The solid curve shows the improvement predicted by a numerical simulation assuming only random, stationary noise in the spot centroids.

of the wavefront change can then be calculated using
∆z y0
∆z h
∂ (∆W ) ∆y
=
=
=
= Sh
∂h
L
(z0 − ∆z) (z0 + L) (z0 − ∆z) z0(1)

where h is the position of the hole in the Hartmann plate, S is the primary aberration defocus
and we have considered only one dimension for clarity.

Fig. 4. Schematic diagram showing the displacement of the Hartmann spot on the CCD due
to a change in the distance between the fiber end and the H-WFS.

Unfortunately, the 0.5 mHz oscillation in the output of the H-WFS resulted in a synchronous
oscillation in the calculated defocus. We therefore continously recorded Hartmann images at 40
Hz and translated the fibre by about 10 µ m every 5-10 minutes. The translation that occurred
nearest a turning point of the 0.5 mHz oscillation was then selected for analysis, as this ensured
that the average temperature of the H-WFS was the same before and after the translation. Plotsof the local gradient of the wavefront change for ∆z = 9.60 ± 0.05 µ m, versus the transverse
position, y0 , of each spot are shown in Fig. 5. They show a linear relationship between the local
gradient and transverse position as predicted by Equation 1, and the improvement in sensitivity
due to averaging.
The defocus due to the source translation can be calculated using the slope, m, of the line-ofbest-fit to this data and S = m (z0 + L)/z0 . The defocus for the 1-image and 5000-image averages are −1.3 ± 0.2 × 10−3 m−1 and −1.159 ± 0.007 × 10−3 m−1 at 95% confidence level. The
uncertainty in S when averaged over 5000 images is equivalent to an uncertainty in the wavefront sag of 0.1 nm (λ /9300) over the CCD aperture (≈ 10 mm), which is roughly twice the216

APPENDIX C. RELEVANT PAPERS

Fig. 5. Measured local gradient of the wavefront change versus spot position at the CCD, y0 ,due to translation of the fiber light source, averaged over (left) 1 and (right) 5000 Hartmann
images.

previously measured λ /15, 500 due probably to the non-stationary noise during the extended
acquistion time.
The accuracy of the defocus measurement can be determined by comparing the measured defocus with that predicted using Equation 1, but this comparison requires an accurate measurement of z0 . The distance between the source and the H-WFS was determined by exploiting
the non-linear dependence of the local gradient on ∆z: analyzing Hartmann images recordedwith large ∆z and determining the value of z0 that would produce the best agreement between
the measured local gradient and that predicted by Equation 1. With this approach, we foundz0 = 91.7 ± 0.2 mm, giving an expected defocus of −1.14 ± 0.01 × 10−3 m−1 , which differsfrom the measurement by about 1.7%. This error is equivalent to λ /3, 300 and could be explained by a change in average temperature of the H-WFS of order 10 mK.
4.

Conclusion

We have demonstrated that the H-WFS can measure changes in a wavefront with a single-framesensitivity of λ /1450, which is primarily limited by shot noise. We have also shown that thesensitivity can be improved to λ /15, 500 by averaging multiple Hartmann images. Finally, wedemonstrated its application to the measurement of a small wavefront change due to defocus
with a precision of 7 × 10−6 m−1 and established that the sensor is accurate to within about
2.0 × 10−5 m−1 .
Together with the simplicity of the H-WFS, these results show that it is ideal for high precision and high accuracy measurement of wavefront changes, and represent a large improvementin the state-of-the-art. The H-WFS clearly exceeds the requirement for the measurement ofabsorption-induced wavefront distortion in advanced gravitational wave interferometers.
Acknowledgements
We acknowledge support from the Australian Research Council.

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