Phase estimation in optical interferometry

364 Pages·2014·12.027 MB·English

by  Rastogi, Pramod

Physics Rastogi
Hack
Phase Estimation in
Optical Interferometry
P
h
a
s
Phase Estimation in Optical Interferometry covers the essentials of e
phase-stepping algorithms used in interferometry and pseudointerferometric
E
techniques. It presents the basic concepts and mathematics needed for
s
understanding the phase estimation methods in use today.
t
i
m
The first four chapters focus on phase retrieval from image transforms using
a single frame. Chapters 5 through 8 examine the local environment of a a
fringe pattern, give a broad picture of the phase estimation approach based t
i
on local polynomial phase modeling, cover temporal high-resolution phase o
evaluation methods, and present methods of phase unwrapping. The final n
chapter discusses experimental imperfections that are liable to adversely

influence the accuracy of phase measurements. i
n

Responding to the push for the deployment of novel technologies and fast- O
evolving techniques, this book provides a framework for understanding
p
various modern phase estimation methods. It also helps readers get a
t
comparative view of the performance and limitations of the approaches.
i
c
a
Features
l
• Discusses important aspects that must be taken into account in
I
both spatial and temporal phase measurements n
• Covers a range of algorithms, including the vortex operator, phase- t
e
unwrapping, linear phase-stepping, Carré-type, and single-frame
r
algorithms f
e
• Compares the performance and limitations of various approaches
r
• Explains how the spiral phase transform allows retrieving phase o
from fringe patterns containing closed fringes m
• Describes how to minimize several systematic and random errors
e
that arise during phase measurements
t
r
y
Cover illustration by courtesy of Prof Mitsuo Takeda.
Phase unwrapping constitutes an essential part of phase estimation in optical interferometry. The wrapped
phase map in the right with 2\pi phase jumps is unwrapped into the continuous phase in the left.
K20882
ISBN: 978-1-4665-9831-7
90000
9 781466 598317
K20882_COVER_final.indd 1 9/29/14 12:29 PM

Phase Estimation
in
Optical
Interferometry

Phase Estimation
in
Optical
Interferometry
Edited by
Pramod Rastogi
EPFL - Swiss Federal Institute of Technology
Lausanne, Switzerland
Erwin Hack
Empa - Swiss Federal Laboratories for Materials Science and Technology
Dubendorf, Switzerland
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2015 by Taylor & Francis Group, LLC
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Version Date: 20140922
International Standard Book Number-13: 978-1-4665-9832-4 (eBook - PDF)
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Contents
Preface, xiii
About the Editors, xvii
List of Contributors, xix
Abbreviations, xxi
Chapter 1 ◾ F ourier Fringe Demodulation 1
Mitsuo takeda
1.1 INTRODUCTION 1
1.2 PRINCIPLE OF THE GENERIC FTM FOR FRINGE
DEMODULATION 2
1.3 GENERAL FEATURES OF THE FTM 11
1.4 APPLICATIONS OF FOURIER FRINGE DEMODULATION 15
1.4.1 Vibration Mode Measurement 15
1.4.2 Imaging Polarimetry 17
1.4.3 Plasma Diagnosis 21
1.4.4 X-ray Phase Tomography 23
1.4.5 Measurement of Ultrashort Optical Pulses 24
1.5 CONCLUSION 27
REFERENCES 28
Chapter 2 ◾ W indowed Fourier Transforms 31
Jingang Zhong and Jiawen weng
2.1 INTRODUCTION 31
2.2 PHASE DEMODULATION BASED ON FOURIER
TRANSFORM 33
v
vi ◾ Contents
2.3 PHASE DEMODULATION BASED ON WINDOWED
FOURIER TRANSFORM 41
2.3.1 Principle of Windowed Fourier Transform for
Phase Demodulation 41
2.3.2 Deficiency of Windowed Fourier Transform with
an Invariable Window Size 42
2.4 PHASE DEMODULATION BASED ON ADAPTIVE
WINDOWED FOURIER TRANSFORM 45
2.4.1 Principle of Adaptive Windowed Fourier Transform 45
2.4.2 Principle of the Determination of the Scale Factor
for AWFT 46
2.5 NUMERICAL ANALYSIS 49
2.5.1 Numerical Analysis by FT 49
2.5.2 Numerical Analysis by WFT with an Invariable
Window Size 51
2.5.3 Numerical Analysis by AWFT 54
2.6 EXPERIMENTAL ANALYSIS EXAMPLE 60
2.7 CONCLUSION 64
REFERENCES 65
Chapter 3 ◾ C ontinuous Wavelet Transforms 69
LioneL r. watkins
3.1 INTRODUCTION 69
3.2 THE ONE-DIMENSIONAL CONTINUOUS WAVELET
TRANSFORM 70
3.3 WAVELET CENTERS AND BANDWIDTHS 78
3.3.1 Heisenberg Principle 84
3.4 SCALOGRAMS 84
3.5 RIDGE OF THE CONTINUOUS WAVELET TRANSFORM 85
3.6 THE GRADIENT METHOD 89
3.6.1 Correcting the Instantaneous Frequency 90
3.7 THE PHASE METHOD 92
Contents ◾ vii
3.8 FOURIER APPROACH TO CWT 93
3.9 EFFECT OF DISCONTINUITIES AT THE SIGNAL EDGE 95
3.10 ONE-DIMENSIONAL WAVELET FUNCTIONS 97
3.11 TWO-DIMENSIONAL CONTINUOUS WAVELET
TRANSFORM 103
3.12 CONCLUSIONS 116
APPENDIX A 118
RIDGE OF THE TWO-DIMENSIONAL CWT 118
REFERENCES 118
Chapter 4 ◾ T he Spiral Phase Transform 121
kieran g. Larkin
4.1 INTRODUCTION 121
4.2 THEORY 121
4.2.1 Demodulation in One and Two Dimensions 121
4.2.2 Quadrature Signals 124
4.2.3 Intrinsically 1-D Structure of 2-D Fringe Patterns 125
4.2.4 SIGNUM Returns 127
4.3 IMPLEMENTATION 128
4.3.1 Vortex Operator Algorithm 128
4.3.2 Orientation and Direction Estimation 131
4.4 WHEN TO USE THE SPIRAL PHASE TRANSFORM 133
4.4.1 Single Frame: Open or Closed Fringes 133
4.4.2 Amplitude Demodulation and Fringe Normalization 133
4.4.3 Multiframe Sequences with Arbitrary (and
Unknown) Phase Shifts 134
4.4.4 Other Fringe-like Patterns 134
4.5 PRACTICAL DEMODULATION EXAMPLE 134
4.6 SUMMARY 138
REFERENCES 138
viii ◾ Contents
Chapter 5 ◾ R egularized Phase Estimation Methods in
Interferometry 141
Moises padiLLa and ManueL servin
5.1 INTRODUCTION 141
5.2 REGULARIZED LOW-PASS LINEAR FILTERING 144
5.2.1 Frequency Response of Low-Pass Regularizing
Filters 148
5.3 CONVOLUTION-BASED TEMPORAL PHASE-
SHIFTING INTERFEROMETRY 153
5.4 SPATIALLY REGULARIZED TEMPORAL LINEAR
CARRIER INTERFEROMETRY 158
5.5 CONVOLUTION-BASED SPATIAL-CARRIER
INTERFEROMETRY 161
5.6 REGULARIZATION IN GENERAL SPATIAL CARRIER
INTERFEROMETRY 163
5.7 TEMPORAL REGULARIZATION IN PHASE-SHIFTING
INTERFEROMETRY 167
5.8 REGULARIZED PHASE ESTIMATION OF SINGLE-
IMAGE CLOSED-FRINGES INTERFEROGRAMS 170
5.9 REGULARIZED SPATIAL INTERPOLATION-
EXTRAPOLATION IN INTERFEROMETRY 173
5.10 REGULARIZATION IN LATERAL SHEARING
INTERFEROMETRY 174
5.10.1 Standard Method for Wavefront Estimation in
Lateral Shearing Interferometry 176
5.10.2 Regularized Methods for Wavefront Estimation in
Lateral Shearing Interferometry 179
5.11 CONCLUSIONS 182
REFERENCES 183
Chapter 6 ◾ L ocal Polynomial Phase Modeling and Estimation 187
gannavarpu raJshekhar, sai siva gorthi, and praMod rastogi
6.1 INTRODUCTION 187
6.2 DIGITAL HOLOGRAPHIC INTERFEROMETRY 188
6.3 PRINCIPLE 193