Fringe Pattern Analysis for Optical Metrology

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 XIList of Symbols and Acronyms XV1 Digital Linear Systems 11.1 Introduction to Digital Phase Demodulation in Optical Metrology 11.1.1 Fringe Pattern Demodulation as an Ill-Posed Inverse Problem 11.1.2 Adding a priori Information to the Fringe Pattern: Carriers 31.1.3 Classification of Phase Demodulation Methods in Digital Interferometry 71.2 Digital Sampling 91.2.1 Signal Classification 91.2.2 Commonly Used Functions 111.2.3 Impulse Sampling 131.2.4 Nyquist–Shannon Sampling Theorem 141.3 Linear Time-Invariant (LTI) Systems 141.3.1 Definition and Properties 151.3.2 Impulse Response of LTI Systems 151.3.3 Stability Criterion: Bounded-Input Bounded-Output 171.4 Z-Transform Analysis of Digital Linear Systems 181.4.1 Definition and Properties 181.4.2 Region of Convergence (ROC) 191.4.3 Poles and Zeros of a Z-Transform 201.4.4 Inverse Z-Transform 211.4.5 Transfer Function of an LTI System in the Z-Domain 221.4.6 Stability Evaluation by Means of the Z-Transform 231.5 Fourier Analysis of Digital LTI Systems 241.5.1 Definition and Properties of the Fourier Transform 251.5.2 Discrete-Time Fourier Transform (DTFT) 251.5.3 Relation Between the DTFT and the Z-Transform 261.5.4 Spectral Interpretation of the Sampling Theorem 271.5.5 Aliasing: Sub-Nyquist Sampling 291.5.6 Frequency Transfer Function (FTF) of an LTI System 311.5.7 Stability Evaluation in the Fourier Domain 331.6 Convolution-Based One-Dimensional (1D) Linear Filters 341.6.1 One-Dimensional Finite Impulse Response (FIR) Filters 341.6.2 One-Dimensional Infinite Impulse Response (IIR) Filters 371.7 Convolution-Based two-dimensional (2D) Linear Filters 391.7.1 Two-Dimensional (2D) Fourier and Z-Transforms 391.7.2 Stability Analysis of 2D Linear Filters 401.8 Regularized Spatial Linear Filtering Techniques 421.8.1 Classical Regularization for Low-Pass Filtering 421.8.2 Spectral Response of 2D Regularized Low-Pass Filters 461.9 Stochastic Processes 481.9.1 Definitions and Basic Concepts 481.9.2 Ergodic Stochastic Processes 511.9.3 LTI System Response to Stochastic Signals 521.9.4 Power Spectral Density (PSD) of a Stochastic Signal 521.10 Summary and Conclusions 542 Synchronous Temporal Interferometry 572.1 Introduction 572.1.1 Historical Review of the Theory of Phase-Shifting Algorithms (PSAs) 572.2 Temporal Carrier Interferometric Signal 602.3 Quadrature Linear Filters for Temporal Phase Estimation 622.3.1 Linear PSAs Using Real-Valued Low-Pass Filtering 642.4 The Minimum Three-Step PSA 682.4.1 Algebraic Derivation of the Minimum Three-Step PSA 682.4.2 Spectral FTF Analysis of the Minimum Three-Step PSA 692.5 Least-Squares PSAs 712.5.1 Temporal-to-Spatial Carrier Conversion: Squeezing Interferometry 732.6 Detuning Analysis in Phase-Shifting Interferometry (PSI) 742.7 Noise in Temporal PSI 802.7.1 Phase Estimation with Additive Random Noise 822.7.2 Noise Rejection in N-Step Least-Squares (LS) PSAs 852.7.3 Noise Rejection of Linear Tunable PSAs 862.8 Harmonics in Temporal Interferometry 872.8.1 Interferometric Data with Harmonic Distortion and Aliasing 882.8.2 PSA Response to Intensity-Distorted Interferograms 912.9 PSA Design Using First-Order Building Blocks 952.9.1 Minimum Three-Step PSA Design by First-Order FTF Building Blocks 972.9.2 Tunable Four-Step PSAs with Detuning Robustness at ;; = −;;0 1002.9.3 Tunable Four-Step PSAs with Robust Background Illumination Rejection 1012.9.4 Tunable Four-Step PSA with Fixed Spectral Zero at ;; = π 1022.10 Summary and Conclusions 1043 Asynchronous Temporal Interferometry 1073.1 Introduction 1073.2 Classification of Temporal PSAs 1083.2.1 Fixed-Coefficients (Linear) PSAs 1083.2.2 Tunable (Linear) PSAs 1083.2.3 Self-Tunable (Nonlinear) PSAs 1093.3 Spectral Analysis of the Carré PSA 1103.3.1 Frequency Transfer Function of the Carré PSA 1123.3.2 Meta-Frequency Response of the Carré PSA 1133.3.3 Harmonic-Rejection Capabilities of the Carré PSA 1143.3.4 Phase-Step Estimation in the Carré PSA 1163.3.5 Improvement of the Phase-Step Estimation in Self-Tunable PSAs 1183.3.6 Computer Simulations with the Carré PSA with Noisy Interferograms 1203.4 Spectral Analysis of Other Self-Tunable PSAs 1223.4.1 Self-Tunable Four-Step PSA with Detuning-Error Robustness 1233.4.2 Self-Tunable Five-Step PSA by Stoilov and Dragostinov 1263.4.3 Self-Tunable Five-Step PSA with Detuning-Error Robustness 1283.4.4 Self-Tunable Five-Step PSA with Double Zeroes at the Origin and the Tuning Frequency 1303.4.5 Self-Tunable Five-Step PSA with Three Tunable Single Zeros 1313.4.6 Self-Tunable Five-Step PSA with Second-Harmonic Rejection 1333.5 Self-Calibrating PSAs 1363.5.1 Iterative Least-Squares, the Advanced Iterative Algorithm 1373.5.2 Principal Component Analysis 1403.6 Summary and Conclusions 1454 Spatial Methods with Carrier 1494.1 Introduction 1494.2 Linear Spatial Carrier 1494.2.1 The Linear Carrier Interferogram 1494.2.2 Instantaneous Spatial Frequency 1524.2.3 Synchronous Detection with a Linear Carrier 1554.2.4 Linear and Nonlinear Spatial PSAs 1594.2.5 Fourier Transform Analysis 1644.2.6 Space–Frequency Analysis 1704.3 Circular Spatial Carrier 1734.3.1 The Circular Carrier Interferogram 1734.3.2 Synchronous Detection with a Circular Carrier 1744.4 2D Pixelated Spatial Carrier 1774.4.1 The Pixelated Carrier Interferogram 1774.4.2 Synchronous Detection with a Pixelated Carrier 1804.5 Regularized Quadrature Filters 1864.6 Relation Between Temporal and Spatial Analysis 1984.7 Summary and Conclusions 1985 Spatial Methods Without Carrier 2015.1 Introduction 2015.2 Phase Demodulation of Closed-Fringe Interferograms 2015.3 The Regularized Phase Tracker (RPT) 2045.4 Local Robust Quadrature Filters 2155.5 2D Fringe Direction 2165.5.1 Fringe Orientation in Interferogram Processing 2165.5.2 Fringe Orientation and Fringe Direction 2195.5.3 Orientation Estimation 2225.5.4 Fringe Direction Computation 2255.6 2D Vortex Filter 2295.6.1 The Hilbert Transform in Phase Demodulation 2295.6.2 The Vortex Transform 2305.6.3 Two Applications of the Vortex Transform 2335.7 The General Quadrature Transform 2355.8 Summary and Conclusions 2396 Phase Unwrapping 2416.1 Introduction 2416.1.1 The Phase Unwrapping Problem 2416.2 Phase Unwrapping by 1D Line Integration 2446.2.1 Line Integration Unwrapping Formula 2446.2.2 Noise Tolerance of the Line Integration Unwrapping Formula 2466.3 Phase Unwrapping with 1D Recursive Dynamic System 2506.4 1D Phase Unwrapping with Linear Prediction 2516.5 2D Phase Unwrapping with Linear Prediction 2556.6 Least-Squares Method for Phase Unwrapping 2576.7 Phase Unwrapping Through Demodulation Using a Phase Tracker 2586.8 Smooth Unwrapping by Masking out 2D Phase Inconsistencies 2626.9 Summary and Conclusions 266Appendix A List of Linear Phase-Shifting Algorithms (PSAs) 271References 315Index 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)