Phase estimation in optical interferometry

Phase Estimation in Optical Interferometry covers the essentials of phase-stepping algorithms used in interferometry and pseudointerferometric techniques. It presents the basic concepts and mathematics needed for understanding the phase estimation methods in use today.

The first four chapters focus on phase retrieval from image transforms using a single frame. The next several chapters examine the local environment of a fringe pattern, give a broad picture of the phase estimation approach based on local polynomial phase modeling, cover temporal high-resolution phase evaluation methods, and present methods of phase unwrapping. The final chapter discusses experimental imperfections that are liable to adversely influence the accuracy of phase measurements.

Responding to the push for the deployment of novel technologies and fast-evolving techniques, this book provides a framework for understanding various modern phase estimation methods. It also helps readers get a comparative view of the performance and limitations of the approaches.

Author Notes

Professor Pramod Rastogi is the author or coauthor of over 150 scientific papers in peer-reviewed archival journals, the author of encyclopedia articles, and editor of several books in the field of optical metrology. Professor Rastogi is also the co-editor-in-chief of the International Journal of Optics and Lasers in Engineering . A recipient of the 2014 SPIE Dennis Gabor Award, he is a member of the Swiss Academy of Engineering Sciences and a fellow of the Society of the Photo-Optical Instrumentation Engineers and the Optical Society of America. He received a PhD from the University of Franche Comté.

Dr. Erwin Hack is a senior scientist at EMPA, lecturer at ETH Zurich, associate editor of Optics and Lasers in Engineering , vice chair of CEN WS71 on validation of computational solid mechanics models using strain fields from calibrated measurement (VANESSA), vice president of the Swiss Society for Non-Destructive Testing, and a member of EOS and OSA. Dr. Hack has authored or coauthored more than 80 papers in peer-reviewed journals and conferences and coedited the book Optical Methods in Solid Mechanics . He received a PhD in physical chemistry from the University of Zurich. His research interests include THz imaging, digital speckle pattern interferometry, and thermography.

Mitsuo TakedaJingang Zhong and Jiawen WengLionfl R. WatkinsKitran G. LarkinMoises Papilla and Manuel SeryinGannavarpu Rajshekhar and Sai Siva Gorthi and Pramou RastogiAbhijit Patil and Rajesh Lancoju and Pramod RastogiDavid R. BurtonErwin Hack

Preface	p. xiii
About the Editors	p. xvii
List of Contributors	p. xix
Abbreviations	p. xxi
Chapter 1 Fourier Fringe Demodulation	p. 1
1.1 Introduction	p. 1
1.2 Principle of the Generic FTM for Fringe Demodulation	p. 2
1.3 General Features of the FTM	p. 11
1.4 Applications of Fourier Fringe Demodulation	p. 15
1.4.1 Vibration Mode Measurement	p. 15
1.4.2 Imaging Polarimetry	p. 17
1.4.3 Plasma Diagnosis	p. 21
1.4.4 X-ray Phase Tomography	p. 23
1.4.5 Measurement of Ultrashort Optical Pulses	p. 24
1.5 Conclusion	p. 27
References	p. 28
Chapter 2 Windowed Fourier Transforms	p. 31
2.1 Introduction	p. 31
2.2 Phase Demodulation Based on Fourier Transform	p. 33
2.3 Phase Demodulation Based on Windowed Fourier Transform	p. 41
2.3.1 Principle of Windowed Fourier Transform for Phase Demodulation	p. 41
2.3.2 Deficiency of Windowed Fourier Transform with an Invariable Window Size	p. 42
2.4 Phase Demodulation Based on Adaptive Windowed Fourier Transform	p. 45
2.4.1 Principle of Adaptive Windowed Fourier Transform	p. 45
2.4.2 Principle of the Determination of the Scale Factor for AWFT	p. 46
2.5 Numerical Analysis	p. 49
2.5.1 Numerical Analysis by FT	p. 49
2.5.2 Numerical Analysis by WFT with an Invariable Window Size	p. 51
2.5.3 Numerical Analysis by AWFT	p. 54
2.6 Experimental Analysis Example	p. 60
2.7 Conclusion	p. 64
References	p. 65
Chapter 3 Continuous Wavelet Transforms	p. 69
3.1 Introduction	p. 69
3.2 The One-Dimensional Continuous Wavelet Transform	p. 70
3.3 Wavelet Centers and Bandwidths	p. 78
3.3.1 Heisenberg Principle	p. 84
3.4 Scalograms	p. 84
3.5 Ridge of the Continuous Wavelet Transform	p. 85
3.6 The Gradient Method	p. 89
3.6.1 Correcting the Instantaneous Frequency	p. 90
3.7 The Phase Method	p. 92
3.8 Fourier Approach to CWT	p. 93
3.9 Effect of Discontinuities at the Signal Edge	p. 95
3.10 One-Dimensional Wavelet Functions	p. 97
3.11 Two-Dimensional Continuous Wavelet Transform	p. 103
3.12 Conclusions	p. 116
Appendix A p. 118
Ridge of the Two-Dimensional CWT	p. 118
References	p. 118
Chapter 4 The Spiral Phase Transform	p. 121
4.1 Introduction	p. 121
4.2 Theory	p. 121
4.2.1 Demodulation in One and Two Dimensions	p. 121
4.2.2 Quadrature Signals	p. 124
4.2.3 Intrinsically 1-D Structure of 2-D Fringe Patterns	p. 125
4.2.4 SIGNUM Returns	p. 127
4.3 Implementation	p. 128
4.3.1 Vortex Operator Algorithm	p. 128
4.3.2 Orientation and Direction Estimation	p. 131
4.4 When to Use the Spiral Phase Transform	p. 133
4.4.1 Single Frame: Open or Closed Fringes	p. 133
4.4.2 Amplitude Demodulation and Fringe Normalization	p. 133
4.4.3 Multiframe Sequences with Arbitrary (and Unknown) Phase Shifts	p. 134
4.4.4 Other Fringe-like Patterns	p. 134
4.5 Practical Demodulation Example	p. 134
4.6 Summary	p. 138
References	p. 138
Chapter 5 Regularized Phase Estimation Methods in Interferometry	p. 141
5.1 Introduction	p. 141
5.2 Regularized Low-Pass Linear Filtering	p. 144
5.2.1 Frequency Response of Low-Pass Regularizing Filters	p. 148
5.3 Convolution-Based Temporal Phase-Shifting Interferometry	p. 153
5.4 Spatially Regularized Temporal Linear Carrier Interferometry	p. 158
5.5 Convolution-Based Spatial-Carrier Interferometry	p. 161
5.6 Regularization in General Spatial Carrier Interferometry	p. 163
5.7 Temporal Regularization in Phase-Shifting Interferometry	p. 167
5.8 Regularized Phase Estimation of Single-Image Closed-Fringes Interferograms	p. 170
5.9 Regularized Spatial Interpolation-Extrapolation in Interferometry	p. 173
5.10 Regularization in Lateral Shearing Interferometry	p. 174
5.10.1 Standard Method for Wavefront Estimation in Lateral Shearing Interferometry	p. 176
5.10.2 Regularized Methods for Wavefront Estimation in Lateral Shearing Interferometry	p. 179
5.11 Conclusions	p. 182
References	p. 183
Chapter 6 Local Polynomiae Phase Modeling and Estimation	p. 187
6.1 Introduction	p. 187
6.2 Digital Holographic Interferometry	p. 188
6.3 Principle	p. 193
6.4 Maximum Likelihood Estimation	p. 199
6.5 Cubic Phase Function	p. 206
6.6 High-Order Ambiguity Function	p. 213
6.7 Phase-Differencing Operator	p. 222
6.8 Conclusions	p. 230
References	p. 231
Chapter 7 Signal-Processing Methods in Phase-Shifting Interferometry	p. 235
7.1 Introduction	p. 235
7.2 Temporal Techniques	p. 237
7.3 Linear Phase Step Estimation Methods	p. 241
7.3.1 Multiple Signal Classification Method: root-MUSIC	p. 241
7.3.2 Multiple Signal Classification Method: spectral-MUSIC	p. 243
7.3.3 Estimation of Signal Parameter via Rotational Invariance Technique	p. 248
7.4 Evaluation of Phase Distribution	p. 251
7.4.1 Evaluation of Linear Phase Step Estimation Methods	p. 252
7.4.2 Phase Extraction Using ESPRIT: Experimental Results	p. 253
7.5 Dual PZT in Holographic Moiré	p. 255
7.6 Evaluation of Phase Distribution In Holographic Moiré	p. 256
7.6.1 Holographic Moiré Experiments	p. 257
7.7 Nonlinear Phase Step Estimation Method	p. 259
7.7.1 Nonlinear Maximum Likelihood Estimation Method for Holographic Interferometry	p. 262
7.7.2 Evaluation of Nonlinear Phase Step Estimation Method	p. 264
7.7.3 Nonlinear Maximum Likelihood Estimation Method for Holographic Moiré	p. 266
7.7.4 Evaluation of Nonlinear Phase Step Estimation Method for Holographic Moiré	p. 268
7.8 Summary of Signal-Processing Methods	p. 269
References	p. 271
Chapter 8 Phase Unwrapping	p. 273
8.1 Introduction	p. 273
8.2 The Basic Operation of Phase Unwrapping	p. 276
8.3 Phase Unwrapping: The Practical Issues and Challenges	p. 279
8.4 Phase Unwrapping and Defensive Programming	p. 280
8.5 Phase-Unwrapping Algorithms	p. 281
8.5.1 Path-Guiding Unwrapping Algorithms	p. 281
8.5.2 Area-Based Unwrapping Algorithms	p. 287
8.5.3 Other Methods of Phase Unwrapping	p. 289
8.6 Online Sources of Unwrapping Codes	p. 290
8.7 Conclusion	p. 290
References	p. 291
Chapter 9 Uncertainty in Phase Measurements	p. 293
9.1 Introduction	p. 293
9.2 Influence Quantities	p. 296
9.2.1 Test Object and Environment	p. 296
9.2.2 Illumination and Image Acquisition	p. 296
9.2.3 Phase Retrieval and Image Processing	p. 296
9.3 Quantification of Uncertainty Contributions	p. 298
9.4 Uncertainty Contributions for Imaging	p. 300
9.4.1 Lateral and Temporal Image Resolution	p. 300
9.4.2 Signal-Independent Contributions	p. 301
9.4.3 Signal-Dependent Contributions	p. 303
9.5 Uncertainty Contributions for Linear Phase-Stepping Algorithms	p. 304
9.5.1 Combined Uncertainty	p. 304
9.5.2 Uncertainty from Uncorrelated Influences	p. 306
9.5.3 Uncertainty from Phase Stepping	p. 307
9.5.4 Example of Combined Uncertainty	p. 312
9.6 Phase Measurement Uncertainty for Carré-Type Algorithms	p. 316
9.7 Phase Measurement Uncertainty for Single-Frame Algorithms	p. 317
9.7.1 Relation to Linear Phase-Stepping Algorithms	p. 317
9.7.2 Combined Phase Measurement Uncertainty	p. 321
9.7.3 Uncertainty from Uncorrelated Influences	p. 323
9.7.4 Uncertainty from Correlated Influences	p. 324
9.8 Summary	p. 326
References	p. 326
Index	p. 331

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