Interferogram Analysis

for Optical Testing

Second Edition

Daniel Malacara Centro de Investigaciones de Optica

Leon, Mexico

Manuel Servin Centro de Investigaciones de Optica

Leon, Mexico

Zacarias Malacara Centro de Investigaciones de Optica

Leon, Mexico

i^\ Taylor &. Francis w J Taylor & Francis Group

Boca Raton London New York Singapore

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Croup, the academic division of T&F Informa pic.

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Contents

Chapter 1 Review and Comparison of the Main Interferometric Systems 1

1.1 Two-Wave Interferometers and Configurations Used in Optical Testing 1

1.2 Twyman-Green Interferometer 5 1.3 Fizeau Interferometers 8 1.4 Typical Interferograms in Twyman-Green

and Fizeau Interferometers 11 1.5 Lateral Shear Interferometers 14

1.5.1 Primary Aberrations 16 1.5.1.1 Defocus 17 1.5.1.2 Spherical Aberration 17 1.5.1.3 Coma 17 1.5.1.4 Primary Astigmatism 17

1.5.2 Rimmer-Wyant Method To Evaluate Wavefronts.... 18 1.5.3 Saunders Method To Evaluate Interferograms 20 1.5.4 Spatial Frequency Response of Lateral

Shear Interferometers. 21 1.5.5 Regularization Method To Obtain Wavefronts 23

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viii Malacara, Servi'n, and Malacara

1.6 RonchiTest 26 1.7 Har tmann Test 30 1.8 Fringe Projection 34 1.9 Talbot Interferometry and Moire Deflectometry 37 1.10 Common Light Sources Used in Interferometry 39 1.11 Aspherical Compensators and Aspheric Wavefronts 41 1.12 Imaging of the Pupil on the Observation Plane 41

1.12.1 Imaging the Pupil Back on Itself 42 1.12.2 Imaging the Pupil on the Observing Screen 43 1.12.3 Requirements on the Imaging Lens 46

1.13 Multiple-Wavelength Interferometry 51 References 54

Chapter 2 Fourier Theory Review 63

2.1 Introduction 63 2.1.1 Complex Functions 63

2.2 Fourier Series 66 2.3 Fourier Transforms 68

2.3.1 Parseval Theorem 71 2.3.2 Central Ordinate Theorem 71 2.3.3 Translation Property 72 2.3.4 Derivative Theorem 72 2.3.5 Symmetry Properties of Fourier Transforms 73

2.4 The Convolution of Two Functions 75 2.4.1 Filtering by Convolution 78

2.5 The Cross-Correlation of Two Functions 79 2.6 Sampling Theorem 80 2.7 Sampling of a Periodical Function 83

2.7.1 Sampling of a Periodical Function with Interval Averaging 85

2.8 Fast Fourier Transform 89 References 94

Chapter 3 Digital Image Processing 95

3.1 Introduction 95 3.2 Histogram and Gray-Scale Transformations 96 3.3 Space and Frequency Domain of Interferograms 98 3.4 Digital Processing of Images 100

3.4.1 Point and Line Detection 102 3.4.2 Derivative and Laplacian Operators 102

Contents ix

3.4.3 Spatial Filtering by Convolution Masks 103 3.4.4 Edge Detection 109 3.4.5 Smoothing by Regularizing Filters 110

3.5 Some Useful Spatial Filters 112 3.5.1 Square Window Filter 112 3.5.2 Hamming and Hanning Window Filters 114 3.5.3 Cosinusoidal and Sinusoidal

Window Filters 115 3.6 Extrapolation of Fringes Outside of the Pupil 116 3.7 Light Detectors Used To Digitize Images 118

3.7.1 Image Detectors and Television Cameras 119 3.7.2 Frame Grabbers 123

References 125

Chapter 4 Fringe Contouring and Polynomial Fitting 127

4.1 Fringe Detection Using Manual Digitizers 127 4.2 Fringe Tracking and Fringe Skeletonizing 129

4.2.1 Spatial Filtering of the Image 131 4.2.2 Identification of Fringe Maxima 131 4.2.3 Assignment of Order Number to Fringes 133

4.3 Global Polynomial Interpolation 135 4.3.1 Zernike Polynomials 137 4.3.2 Properties of Zernike Polynomials 140 4.3.3 Least-Squares Fit to Zernike Polynomials 141 4.3.4 Gram-Schmidt Orthogonalization 143

4.4 Local Interpolation by Segments 144 4.5 Wavefront Representation by an Array of Gaussians 148 References 150

Chapter 5 Periodic Signal Phase Detection and Algorithm Analysis 159 *

5.1 Least-Squares Phase Detection of a Sinusoidal Signal 159 5.2 Quadrature Phase Detection of a Sinusoidal Signal 165

5.2.1 Low-Pass Filtering in Phase Detection 168 5.3 Discrete Low-Pass Filtering Functions 173

5.3.1 Examples of Discrete Filtering Functions 176 5.3.1.1 Wyant's Three-Step Algorithm 176 5.3.1.2 Four-Steps-in-Cross Algorithm 177 5.3.1.3 Schwider-Hariharan Five-Step

( 4 + 1) Algorithm 178

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5.4 Fourier Description of Synchronous Phase Detection 179 5.5 Synchronous Detection Using a Few Sampling Points 188

5.5.1 General Discrete Sampling 190 5.5.2 Equally Spaced and Uniform Sampling 194 5.5.3 Applications of Graphical Vector Representation.... 196 5.5.4 Graphic Method To Design

Phase-Shifting Algorithms 198 5.6 Signal Amplitude Measurement 201 5.7 Characteristic Polynomial of a Sampling Algorithm 203 5.8 General Error Analysis of Synchronous

Phase-Detection Algorithms 206 5.8.1 Exact Phase-Error Analysis 207 5.8.2 Phase-Error Approximation

in Two Particular Cases 210 5.9 Some Sources of Phase Error 212

5.9.1 Phase-Shifter Miscalibration and Nonlinearities.... 214 5.9.1.1 Error in the Sampling

Reference Functions 215 5.9.1.2 Error in the Measured Signal 216

5.9.2 Measurement and Compensation of Phase-Shift Errors 217

5.9.3 Linear or Detuning Phase-Shift Error 220 5.9.4 Quadratic Phase-Shift Errors 221 5.9.5 High-Order, Nonlinear, Phase-Shift

Errors with a Sinusoidal Signal 224 5.9.6 High-Order, Nonlinear, Phase-Shift

Errors with a Distorted Signal 226 5.9.7 Nonuniform Phase-Shifting Errors 229 5.9.8 Phase Detection of a Harmonically

Distorted Signal 231 5.9.9 Light-Detector Nonlinearities 234 5.9.10 Random Phase Error 235

5.10 Shifting Algorithms with Respect to the Phase Origin 239 5.10.1 Shifting the Algorithm by ±JI /2 242 5.10.2 Shifting the Algorithm by +JI/4 243

5.11 Optimization of Phase-Detection Algorithms 247 5.12 Influence of Window Function of Sampling Algorithms 249 5.13 Conclusions 252 Appendix. Derivative of the Amplitude of the Fourier

Transform of the Reference Sampling Functions 253 References 254

Contents xi

Chapter 6 Phase-Detection Algorithms 259

6.1 General Properties of Synchronous Phase-Detection Algorithms 259

6.2 Three-Step Algorithms To Measure the Phase 260 6.2.1 120° Three-Step Algorithm 261 6.2.2 Inverted T Three-Step Algorithm 266 6.2.3 Wyant's Tilted T Three-Step Algorithm 268 6.2.4 Two-Steps-Plus-One Algorithm 270

6.3 Four-Step Algorithms To Measure the Phase 274 6.3.1 Four Steps in the Cross Algorithm 275 6.3.2 Algorithm for Four Steps in X 278

6.4 Five-Step Algorithm 281 6.5 Algorithms with Symmetrical N + 1 Phase Steps 284

6.5.1 Symmetrical Four-Step (3 + 1) Algorithm 290 6.5.2 Schwider-Hariharan Five-Step

(4 + 1) Algorithm 294 6.5.3 Symmetrical Six-Step (5 + 1) Algorithm 298 6.5.4 Symmetrical Seven-Step (6 + 1) Algorithm 301

6.6 Combined Algorithms in Quadrature 304 6.6.1 Schwider Algorithm 308 6.6.2 Schmit and Creath Algorithm 315 6.6.3 Other Detuning-Insensitive Algorithms 319

6.7 Detuning-Insensitive Algorithms for Distorted Signals 321 6.7.1 Zhao and Surrel Algorithm 322 6.7.2 Hibino Algorithm 326 6.7.3 Six-Sample, Detuning-Insensitive Algorithm 328

6.8 Algorithms Corrected for Nonlinear Phase-Shifting Error 330

6.9 Continuous Sampling in a Finite Interval 334 6.10 Asynchronous Phase-Detection Algorithms 339

6.10.1 Carre Algorithm 340 6.10.2 Schwider Asynchronous Algorithm 346 6.10.3 Two Algorithms in Quadrature 349 6.10.4 An Algorithm for Zero Bias

and Three Sampling Points 349 6.10.5 Correlation with Two Sinusoidal

Signals in Quadrature 351 6.11 Algorithm Summary 352

6.11.1 Detuning Sensitivity 352 6.11.2 Harmonie Sensitivity 355

References 355

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XII Malacara, Servin, and Malacara.

Chapter 7 Phase-Shifting Interferometry 359

7.1 Phase-Shifting Basic Principles 359 7.2 An Introduction to Phase Shifting 360

7.2.1 Moving Mirror with a Linear Transducer 360 7.2.2 Rotating Glass Plate 361 7.2.3 Moving Diffraction Gräting 362 7.2.4 Rotating Phase Plate 363 7.2.5 Moire in an Interferogram

with a Linear Carrier 365 7.2.6 Frequency Changes in the Laser Light Source 365 7.2.7 Simultaneous Phase-Shift Interferometry 366

7.3 Phase-Shifting Schemes and Phase Measurement 366 7.4 Heterodyne Interferometry 368 7.5 Phase-Lock Detection 370 7.6 Sinusoidal Phase Oscillation Detection 373 7.7 Practical Sources of Phase Error 376

7.7.1 Vibration and Air Turbulence 376 7.7.2 Multiple-Beam Interference

and Frequency Mixing 378 7.7.3 Spherical Reference Wavefronts 381 7.7.4 Quantization Noise 382 7.7.5 Photon Noise Phase Errors 382 7.7.6 Laser Diode Intensity Modulation 382

7.8 Selection of the Reference Sphere in Phase-Shifting Interferometry 383 7.8.1 Paraxial Focus 385

i 7.8.2 Best Focus 385 7.8.3 Marginal Focus 386 7.8.4 Optimum Tut and Defocusing in

Phase-Shifting Interferometry 387 7.8.4.1 Temporal Phase-Shifting Techniques 389 7.8.4.2 Spatial Linear Carrier Demodulation 390 7.8.4.3 Spatial Circular Carrier Demodulation.... 391

References 392

Chapter 8 Spatial Linear and Circular Carrier Analysis 399

8.1 Spatial Linear Carrier Analysis 399 8.1.1 Introduction of a Linear Carrier 400 8.1.2 Holographie Interpretation of the Interferogram.... 403 8.1.3 Fourier Spectrum of the Interferogram

and Filtering 407 8.1.4 Pupil Diffraction Effects 411

Contents XIII

8.2 Space-Domain Phase Demodulation with a Linear Carrier 414 8.2.1 Basic Space-Domain Phase

Demodulation Theory 414 8.2.2 Phase Demodulation with

an Aspherical Reference 416 8.2.3 Analog and Digital Implementations

of Phase Demodulation 418 8.2.4 Spatial Low-Pass Filtering 419 8.2.5 Sinusoidal Window Filter Demodulation 422 8.2.6 Spatial Carrier Phase-Shifting Method 424 8.2.7 Phase-Locked Loop Demodulation 428

8.3 Circular Spatial Carrier Analysis 432 8.4 Phase Demodulation with a Circular Carrier 433

8.4.1 Phase Demodulation with a Spherical Reference Wavefront 433

8.4.2 Phase Demodulation with a Tilted-Plane Reference Wavefront 436

8.5 Fourier Transform Phase Demodulation with a Linear Carrier 440 8.5.1 Sources of Error in the Fourier

Transform Method 444 8.5.2 Spatial Carrier Frequency, Spectrum Width,

and Interferogram Domain Determination 446 8.6 Fourier Transform Phase Demodulation

with a Circular Carrier 447 References 449

Chapter 9 Interferogram Analysis with Moire Methods 455

9.1 Moire Techniques 455 9.2 Moire Formed by Two Interferograms

with a Linear Carrier 456 9.2.1 Moire with Interferograms

of Spherical Wavefronts 458 9.2.2 Moire with Interferograms

of Aspherical Wavefronts 462 9.3 Moire Formed by Two Interferograms

with a Circular Carrier 465 9.3.1 Moire with Interferograms

of Spherical Wavefronts 467 9.3.2 Moire with Interferograms

of Aspherical Wavefronts 468

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9.4 Summary of Moire Effects 470 9.5 Holographie Interpretation of Moire Patterns 470 9.6 Conclusion 472 References 473

Chapter 10 Interferogram Analysis without a Carrier 475

10.1 Introduction 475 10.2 Mathematical Model of the Fringes 476 10.3 The Phase Tracker 481 10.4 The Af-Dimensional Quadrature Transform 485

10.4.1 Using the Fourier Transform To Calculate the Isotropie Hubert Transform 487

10.4.2 The Fringe Orientation Term 488 10.5 Conclusion 490 References 491

Chapter 11 Phase Unwrapping 493

11.1 The Phase Unwrapping Problem 493 11.2 Unwrapping Consistent Phase Maps 500

11.2.1 Unwrapping Full-Field Consistent Phase Maps.... 500 11.2.2 Unwrapping Consistent Phase Maps

within a Simple Connected Region 502 11.3 Unwrapping Noisy Phase Maps 504

11.3.1 Unwrapping Using Least-Squares Integration 504 11.3.2 The Regularized Phase Tracking Unwrapper 507

11.4 Unwrapping Subsampled Phase Maps 511 11.4.1 Greivenkamp's Method 513 11.4.2 Null Fringe Analysis of Subsampled Phase

Maps Using a Computer-Stored Compensator 516 11.4.3 Unwrapping of Smooth Continuous

Subsampled Phase Maps 518 11.4.4 Unwrapping the Partial Derivative

of the Wavefront 520 11.5 Conclusions 521 References 522

Chapter 12 Wavefront Curvature Sensing 525

12.1 Wavefront Determination by Slope Sensing 525 12.2 Wavefront Curvature Sensing 525

Contents xv

12.2.1 The Laplacian and Local Average Curvatures 526 12.2.2 Irradiance Transport Equation 527 12.2.3 Laplacian Determination with

Irradiance Transport Equation 529 12.2.4 Wavefront Determination with

Iterative Fourier Transforms 533 12.3 Wavefront Determination with Defocused Images 537 12.4 Conclusions 538 References 538

Index 541