Title Details

Manuel Servin received his engineering diploma from the École Nationale Supérieure des Télécommunications in France (1982), and his Ph.D. from the Centro de Investigaciones en Óptica A. C. (CIO) at Leon Mexico in 1994. He is co-author of the book `Interferogram Analysis for Optical Testing?. Dr. Servin has published more than 100 papers in scientific peer-reviewed journals on Digital Interferometry and Fringe Analysis.

Juan Antonio Quiroga received his Ph.D. in physics in 1994 from the Universidad Complutense de Madrid, Spain. He is now teaching there at the Physics Faculty. His current principal areas of interest are Digital image processing applied to Optical Metrology and applied optics

Moises Padilla is a Ph.D. student in optical sciences at the Centro de Investigaciones en Óptica (CIO) at León Mexico. He is associated with the optical metrology division of the CIO. His research activities are in digital signal processing and electrical communication engineering applied to processing and analysis of optical interferogram images.

Preface XI

List of Symbols and Acronyms XV

1 Digital Linear Systems 1

1.1 Introduction to Digital Phase Demodulation in Optical Metrology 1

1.1.1 Fringe Pattern Demodulation as an I11-Posed Inverse Problem 1

1.1.2 Adding a priori Information to the Fringe Pattern: Carriers 3

1.1.3 Classification of Phase Demodulation Methods in Digital Interferometry 7

1.2 Digital Sampling 9

1.2.1 Signal Classification 9

1.2.2 Commonly Used Functions 11

1.2.3 Impulse Sampling 13

1.2.4 Nyquist–Shannon Sampling Theorem 14

1.3 Linear Time-Invariant (LTI) Systems 14

1.3.1 Definition and Properties 15

1.3.2 Impulse Response of LTI Systems 15

1.3.3 Stability Criterion: Bounded-Input Bounded-Output 17

1.4 Z-Transform Analysis of Digital Linear Systems 18

1.4.1 Definition and Properties 18

1.4.2 Region of Convergence (ROC) 19

1.4.3 Poles and Zeros of a Z-Transform 20

1.4.4 Inverse Z-Transform 21

1.4.5 Transfer Function of an LTI System in the Z-Domain 22

1.4.6 Stability Evaluation by Means of the Z-Transform 23

1.5 Fourier Analysis of Digital LTI Systems 24

1.5.1 Definition and Properties of the Fourier Transform 25

1.5.2 Discrete-Time Fourier Transform (DTFT) 25

1.5.3 Relation Between the DTFT and the Z-Transform 26

1.5.4 Spectral Interpretation of the Sampling Theorem 27

1.5.5 Aliasing: Sub-Nyquist Sampling 29

1.5.6 Frequency Transfer Function (FTF) of an LTI System 31

1.5.7 Stability Evaluation in the Fourier Domain 33

1.6 Convolution-Based One-Dimensional (1D) Linear Filters 34

1.6.1 One-Dimensional Finite Impulse Response (FIR) Filters 34

1.6.2 One-Dimensional Infinite Impulse Response (IIR) Filters 37

1.7 Convolution-Based two-dimensional (2D) Linear Filters 39

1.7.1 Two-Dimensional (2D) Fourier and Z-Transforms 39

1.7.2 Stability Analysis of 2D Linear Filters 40

1.8 Regularized Spatial Linear Filtering Techniques 42

1.8.1 Classical Regularization for Low-Pass Filtering 42

1.8.2 Spectral Response of 2D Regularized Low-Pass Filters 46

1.9 Stochastic Processes 48

1.9.1 Definitions and Basic Concepts 48

1.9.2 Ergodic Stochastic Processes 51

1.9.3 LTI System Response to Stochastic Signals 52

1.9.4 Power Spectral Density (PSD) of a Stochastic Signal 52

1.10 Summary and Conclusions 54

2 Synchronous Temporal Interferometry 57

2.1 Introduction 57

2.1.1 Historical Review of the Theory of Phase-Shifting Algorithms (PSAs) 57

2.2 Temporal Carrier Interferometric Signal 60

2.3 Quadrature Linear Filters for Temporal Phase Estimation 62

2.3.1 Linear PSAs Using Real-Valued Low-Pass Filtering 64

2.4 The Minimum Three-Step PSA 68

2.4.1 Algebraic Derivation of the Minimum Three-Step PSA 68

2.4.2 Spectral FTF Analysis of the Minimum Three-Step PSA 69

2.5 Least-Squares PSAs 71

2.5.1 Temporal-to-Spatial Carrier Conversion: Squeezing Interferometry 73

2.6 Detuning Analysis in Phase-Shifting Interferometry (PSI) 74

2.7 Noise in Temporal PSI 80

2.7.1 Phase Estimation with Additive Random Noise 82

2.7.2 Noise Rejection in N-Step Least-Squares (LS) PSAs 85

2.7.3 Noise Rejection of Linear Tunable PSAs 86

2.8 Harmonics in Temporal Interferometry 87

2.8.1 Interferometric Data with Harmonic Distortion and Aliasing 88

2.8.2 PSA Response to Intensity-Distorted Interferograms 91

2.9 PSA Design Using First-Order Building Blocks 95

2.9.1 Minimum Three-Step PSA Design by First-Order FTF Building Blocks 97

2.9.2 Tunable Four-Step PSAs with Detuning Robustness at 𝜔 = −𝜔₀ 100

2.9.3 Tunable Four-Step PSAs with Robust Background Illumination Rejection 101

2.9.4 Tunable Four-Step PSA with Fixed Spectral Zero at 𝜔 = π 102

2.10 Summary and Conclusions 104

3 Asynchronous Temporal Interferometry 107

3.1 Introduction 107

3.2 Classification of Temporal PSAs 108

3.2.1 Fixed-Coefficients (Linear) PSAs 108

3.2.2 Tunable (Linear) PSAs 108

3.2.3 Self-Tunable (Nonlinear) PSAs 109

3.3 Spectral Analysis of the Carré PSA 110

3.3.1 Frequency Transfer Function of the Carré PSA 112

3.3.2 Meta-Frequency Response of the Carré PSA 113

3.3.3 Harmonic-Rejection Capabilities of the Carré PSA 114

3.3.4 Phase-Step Estimation in the Carré PSA 116

3.3.5 Improvement of the Phase-Step Estimation in Self-Tunable PSAs 118

3.3.6 Computer Simulations with the Carré PSA with Noisy Interferograms 120

3.4 Spectral Analysis of Other Self-Tunable PSAs 122

3.4.1 Self-Tunable Four-Step PSA with Detuning-Error Robustness 123

3.4.2 Self-Tunable Five-Step PSA by Stoilov and Dragostinov 126

3.4.3 Self-Tunable Five-Step PSA with Detuning-Error Robustness 128

3.4.4 Self-Tunable Five-Step PSA with Double Zeroes at the Origin and the Tuning Frequency 130

3.4.5 Self-Tunable Five-Step PSA with Three Tunable Single Zeros 131

3.4.6 Self-Tunable Five-Step PSA with Second-Harmonic Rejection 133

3.5 Self-Calibrating PSAs 136

3.5.1 Iterative Least-Squares, the Advanced Iterative Algorithm 137

3.5.2 Principal Component Analysis 140

3.6 Summary and Conclusions 145

4 Spatial Methods with Carrier 149

4.1 Introduction 149

4.2 Linear Spatial Carrier 149

4.2.1 The Linear Carrier Interferogram 149

4.2.2 Instantaneous Spatial Frequency 152

4.2.3 Synchronous Detection with a Linear Carrier 155

4.2.4 Linear and Nonlinear Spatial PSAs 159

4.2.5 Fourier Transform Analysis 164

4.2.6 Space–Frequency Analysis 170

4.3 Circular Spatial Carrier 173

4.3.1 The Circular Carrier Interferogram 173

4.3.2 Synchronous Detection with a Circular Carrier 174

4.4 2D Pixelated Spatial Carrier 177

4.4.1 The Pixelated Carrier Interferogram 177

4.4.2 Synchronous Detection with a Pixelated Carrier 180

4.5 Regularized Quadrature Filters 186

4.6 Relation Between Temporal and Spatial Analysis 198

4.7 Summary and Conclusions 198

5 Spatial Methods Without Carrier 201

5.1 Introduction 201

5.2 Phase Demodulation of Closed-Fringe Interferograms 201

5.3 The Regularized Phase Tracker (RPT) 204

5.4 Local Robust Quadrature Filters 215

5.5 2D Fringe Direction 216

5.5.1 Fringe Orientation in Interferogram Processing 216

5.5.2 Fringe Orientation and Fringe Direction 219

5.5.3 Orientation Estimation 222

5.5.4 Fringe Direction Computation 225

5.6 2D Vortex Filter 229

5.6.1 The Hilbert Transform in Phase Demodulation 229

5.6.2 The Vortex Transform 230

5.6.3 Two Applications of the Vortex Transform 233

5.7 The General Quadrature Transform 235

5.8 Summary and Conclusions 239

6 Phase Unwrapping 241

6.1 Introduction 241

6.1.1 The Phase Unwrapping Problem 241

6.2 Phase Unwrapping by 1D Line Integration 244

6.2.1 Line Integration Unwrapping Formula 244

6.2.2 Noise Tolerance of the Line Integration Unwrapping Formula 246

6.3 Phase Unwrapping with 1D Recursive Dynamic System 250

6.4 1D Phase Unwrapping with Linear Prediction 251

6.5 2D Phase Unwrapping with Linear Prediction 255

6.6 Least-Squares Method for Phase Unwrapping 257

6.7 Phase Unwrapping Through Demodulation Using a Phase Tracker 258

6.8 Smooth Unwrapping by Masking out 2D Phase Inconsistencies 262

6.9 Summary and Conclusions 266

Appendix A List of Linear Phase-Shifting Algorithms (PSAs) 271

A.1 Brief Review of the PSAs Theory 271

A.2 Two-Step Linear PSAs 274

A.2.1 Two-Step PSA with a First-Order Zero at −𝜔0 (𝜔0 = π∕2) 274

A.3 Three-Step Linear PSAs 275

A.3.1 Three-Step Least-Squares PSA (𝜔0 = 2π∕3) 275

A.3.2 Three-Step PSA with First-Order Zeros at 𝜔 = {0,−𝜔0} (𝜔0 = π∕2) 276

A.4 Four-Step Linear PSAs 277

A.4.1 Four-Step Least-Squares PSA (𝜔0 = 2π∕4) 277

A.4.2 Four-Step PSA with a First-Order Zero at 𝜔 = 0 and a Second-Order Zero at −𝜔0 (𝜔0 = π∕2) 278

A.4.3 Four-Step PSA with First-Order Zeros at 𝜔 = {0,−𝜔0∕2,−𝜔0} (𝜔0 = π∕2) 279

A.4.4 Four-Step PSA with a First-Order Zero at −𝜔0 and a Second-Order Zero at 𝜔 = 0 (𝜔0 = π∕2) 280

A.4.5 Four-Step PSA with a First-Order Zero at 𝜔 = 0 and a Second-Order Zero at −𝜔0 (𝜔0 = 2π∕3) 281

A.5 Five-Step Linear PSAs 282

A.5.1 Five-Step Least-Squares PSA (𝜔0 = 2π∕5) 282

A.5.2 Five-Step PSA with First-Order Zeros at 𝜔 = {0,±2𝜔0} and a Second-Order Zero at −𝜔0 (𝜔0 = π∕2) 283

A.5.3 Five-Step PSA with Second-Order Zeros at 𝜔 = {0,−𝜔0} (𝜔0 = 2π∕3) 284

A.5.4 Five-Step PSA with Second-Order Zeros at 𝜔 = {0,−𝜔0} (𝜔0 = π∕2) 285

A.5.5 Five-Step PSA with a First-Order Zero at 𝜔 = 0 and a Third-Order Zero at −𝜔0 (𝜔0 = π∕2) 286

A.5.6 Five-Step PSA with a First-Order Zero at 𝜔 = 0 and a Third-Order Zero at −𝜔0 (𝜔0 = 2π∕3) 287

A.6 Six-Step Linear PSAs 288

A.6.1 Six-Step Least-Squares PSA (𝜔0 = 2π∕6) 288

A.6.2 Six-Step PSA with First-Order Zeros at {0,±2𝜔0} and a Third-Order Zero at −𝜔0 (𝜔0 = π∕2) 289

A.6.3 Six-Step PSA with a First-Order Zero at 𝜔 = 0 and a Fourth-Order Zero at −𝜔0 (𝜔0 = π∕2) 290

A.6.4 Six-Step PSA with a First-Order Zero at 𝜔 = 0 and Second-OrderZeros at {−𝜔0,±2𝜔0} (𝜔0 = π∕2) 291

A.6.5 Six-Step (5LS + 1) PSA with a Second-Order Zero at −𝜔0 (𝜔0 = 2π∕5) 292

A.7 Seven-Step Linear PSAs 293

A.7.1 Seven-Step Least-Squares PSA (𝜔0 = 2π∕7) 293

A.7.2 Seven-Step PSA with First-Order Zeros at {0,−𝜔0, 2𝜔0,±3𝜔0} and a Second-Order Zero at −2𝜔0 (𝜔0 = 2π∕6) 294

A.7.3 Seven-Step PSA with First-Order Zeros at {0,−𝜔0, 2𝜔0} and a Second-Order Zero at ±3𝜔0 (𝜔0 = 2π∕6) 295

A.7.4 Seven-Step PSA with First-Order Zeros at {0,±2𝜔0} and a Fourth-Order Zero at −𝜔0 (𝜔0 = π∕2) 296

A.7.5 Seven-Step PSA with Second-Order Zeros at {0,−𝜔0,±2𝜔0} (𝜔0 = π∕2) 297

A.7.6 Seven-Step PSA with a First-Order Zero at 𝜔 = 0 and a Fifth-Order Zero at −𝜔0 (𝜔0 = π∕2) 298

A.7.7 Seven-Step (6LS + 1) PSA with a Second-Order Zero at −𝜔0 (𝜔0 = 2π∕6) 299

A.8 Eight-Step Linear PSAs 300

A.8.1 Eight-Step Least-Squares PSA (𝜔0 = 2π∕8) 300

A.8.2 Eight-Step Frequency-Shifted LS-PSA (𝜔0 = 2 × 2π∕8) 301

A.8.3 Eight-Step PSA with First-Order Zeros at {0,−𝜔0, ±2𝜔0, π∕10, −3π∕10, −7π∕10, 9π∕10} 302

A.8.4 Eight-Step PSA with Second-Order Zeros at {0,±2𝜔0} and a Third-Order Zero at −𝜔0 (𝜔0 = π∕2) 303

A.8.5 Eight-Step PSA with First-Order Zeros at {0, −π∕6,−5π∕6, ±2𝜔0} and a Fourth-Order Zero at −𝜔0 (𝜔0 = π∕2) 304

A.8.6 Eight-Step PSA with First-Order Zeros at {0,±2𝜔0} and a Fifth-Order Zero at −𝜔0 (𝜔0 = π∕2) 305

A.9 Nine-Step Linear PSAs 306

A.9.1 Nine-Step Least-Squares PSA (𝜔0 = 2π∕9) 306

A.9.2 Nine-Step PSA with First-Order Zeros at {0,±2𝜔0} and Second-Order Zeros at {−𝜔0, −π∕4,−3π∕4} (𝜔0 = π∕2) 307

A.9.3 Nine-Step (8LS + 1) PSA (𝜔0 = 2π∕8) 308

A.10 Ten-Step Linear PSAs 309

A.10.1 Ten-Step Least-Squares PSA (𝜔0 = 2π∕10) 309

A.10.2 Ten-Step PSA with a First-Order Zero at 𝜔 = 0 and Second-Order Zeros at {−𝜔0,±2𝜔0,±3𝜔0} (𝜔0 = π∕3) 310

A.11 Eleven-Step Linear PSAs 311

A.11.1 Eleven-Step Least-Squares PSA (𝜔0 = 2π∕11) 311

A.11.2 Eleven-Step PSA with Second-Order Zeros at {0,−𝜔0,±2𝜔0, ±3𝜔0} (𝜔0 = π∕3) 312

A.11.3 Eleven-Step Frequency-Shifted LS-PSA (𝜔0 = 3 × 2π∕11) 313

A.12 Twelve-step linear PSAs 314

A.12.1 Twelve-step frequency-shifted LS-PSA (𝜔0 = 5 × 2π∕12) 314

References 315

Index 325

“I recommend this book for several reasons: it provides great insights into the principles and practical applications of classical and advanced interferometry in optical metrology, and it presents the main algorithms for recovering the modulating phase from single or multiple patterns.”  (Optics & Photonics, 8 October 2014)