Optical Shop Testing

Daniel Malacara, PhD, is a Professor at the Centro de Investigaciones en Optica, Leon, Gto, Mexico. A designer and constructor of optical instruments, including telescopes, he is well known for his books, including Optical Shop Testing, which has been translated into several languages. Dr. Malacara is a Fellow of the Optical Society of America and of SPIE, the International Society of Optical Engineering.

Preface xvii

Contributors xix

Chapter 1. Newton, Fizeau, and Haidinger Interferometers 1 
M. V. Mantravadi and D. Malacara

1.1. Introduction 1

1.2. Newton Interferometer 1

1.2.1. Source and Observer’s Pupil Size Considerations 9

1.2.2. Some Suitable Light Sources 11

1.2.3. Materials for the Optical Flats 12

1.2.4. Simple Procedure for Estimating Peak Error 12

1.2.5. Measurement of Spherical Surfaces 13

1.2.6. Measurement of Aspheric Surfaces 15

1.2.7. Measurement of Flatness of Opaque Surfaces 17

1.3. Fizeau Interferometer 17

1.3.1. The Basic Fizeau Interferometer 18

1.3.2. Coherence Requirements for the Light Source 20

1.3.3. Quality of Collimation Lens Required 22

1.3.4. Liquid Reference Flats 23

1.3.5. Fizeau Interferometer with Laser Source 23

1.3.6. Multiple-Beam Fizeau Setup 24

1.3.7. Testing Nearly Parallel Plates 26

1.3.8. Testing the Inhomogeneity of Large Glass or Fused Quartz Samples 27

1.3.9. Testing the Parallelism and Flatness of the Faces of Rods, Bars and Plates 28

1.3.10. Testing Cube Corner and Right-Angle Prisms 28

1.3.11. Fizeau Interferometer for Curved Surfaces 30

1.3.12. Testing Concave and Convex Surfaces 32

1.4. Haldinger Interferometer 33

1.4.1. Applications of Haidinger Fringes 35

1.4.2. Use of Laser Source for Haidinger Interferometer 36

1.4.3. Other Applications of Haidinger Fringes 39

1.5. Absolute Testing of Flats 40

Chapter 2. Twyman–Green Interferometer 46 
D. Malacara

2.1. Introduction 46

2.2. Beam-Splitter 48

2.2.1. Optical Path Difference Introduced by the Beam Splitter Plate 49

2.2.2. Required Accuracy in the Beam Splitter Plate 51

2.2.3. Polarizing Cube Beam Splitter 53

2.2.4. Nonpolarizing Cube Beam Splitter 55

2.3. Coherence Requirements 56

2.3.1. Spatial Coherence 56

2.3.2. Temporal Coherence 60

2.4. Uses of a Twyman–Green Interferometer 62

2.4.1. Testing of Prisms and Diffraction Rulings 64

2.4.2. Testing of Lenses 69

2.4.3. Testing of Microscope Objectives 71

2.5. Compensation of Intrinsic Aberrations in the Interferometer 72

2.6. Unequal-Path Interferometer 73

2.6.1. Some Special Designs 75

2.6.2. Improving the Fringe Stability 76

2.7. Open Path Interferometers 77

2.7.1. Mach-Zehnder Interferometers 77

2.7.2. Oblique Incidence Interferometers 78

2.8. Variations from the Twyman–Green Configuration 80

2.8.1. Multiple Image Interferometers 80

2.8.2. Interferometers with Diffractive Beam Splitters 80

2.8.3. Phase Conjugating Interferometer 81

2.9. Twyman–Green Interferograms and their Analysis 83

2.9.1. Analysis of Interferograms of Arbitrary Wavefronts 91

Chapter 3. Common-Path Interferometers 97 
S. Mallick and D. Malacara

3.1. Introduction 97

3.2. Burch’s Interferometer Employing Two Matched Scatter Plates 98

3.2.1. Fresnel Zone Plate Interferometer 102

3.2.2. Burch and Fresnel Zone Plate Interferometers for Aspheric Surfaces 102

3.2.3. Burch and Fresnel Zone Plate Interferometers for Phase Shifting 102

3.3. Birefringent Beam Splitters 104

3.3.1. Savart Polariscope 104

3.3.2. Wollaston Prism 106

3.3.3. Double-Focus Systems 107

3.4. Lateral Shearing Interferometers 108

3.4.1. Use of a Savart Polariscope 108

3.4.2. Use of a Wollaston Prism 111

3.5. Double-Focus Interferometer 112

3.6. Saunders’s Prism Interferometer 114

3.7. Point Diffraction Interferometer 116

3.8. Zernike Tests with Common-Path Interferometers 118

Chapter 4. Lateral Shear Interferometers 122 
Strojnik, G. Paez, and M. Mantravadi

4.1. Introduction 122

4.2. Coherence Properties of the Light Source 123

4.3. Brief Theory of Lateral Shearing Interferometry 124

4.3.1. Interferograms of Spherical and Flat Wavefronts 126

4.3.2. Interferogams of Primary Aberrations upon Lateral Shear 128

4.4. Evaluation of an Unknown Wavefront 134

4.5. Lateral Shearing Interferometers in Collimated Light (White Light Compensated) 137

4.5.1. Arrangements Based on the Jamin Interferometer 137

4.5.2. Arrangements Based on the Michelson Interferometer 139

4.5.3. Arrangements Based on a Cyclic Interferometer 140

4.5.4. Arrangements Based on the Mach–Zehnder Interferometer 142

4.6. Lateral Shearing Interferometers in Convergent Light (White Light Compensated) 143

4.6.1. Arrangements Based on the Michelson Interferometer 143

4.6.2. Arrangements Based on the Mach–Zehnder Interferometer 146

4.7. Lateral Shearing Interferometers Using Lasers 149

4.7.1. Other Applications of the Plane Parallel Plate Interferometer 152

4.8. Other Types of Lateral Shearing Interferometers 157

4.8.1. Lateral Shearing Interferometers Based on Diffraction 158

4.8.2. Lateral Shearing Interferometers Based on Polarization 162

4.9. Vectorial Shearing Interferometer 164

4.9.1. Shearing Interferometry 165

4.9.2. Directional Shearing Interferometer 166

4.9.3. Simulated Interferometric Patterns 168

4.9.4. Experimental Results 173

4.9.5. Similarities and Differences With Other Interferometers 176

Chapter 5. Radial, Rotational, and Reversal Shear Interferometer 185 
D. Malacara

5.1. Introduction 185

5.2. Radial Shear Interferometers 187

5.2.1. Wavefront Evaluation from Radial Shear Interferograms 189

5.2.2. Single-Pass Radial Shear Interferometers 190

5.2.3. Double-Pass Radial Shear Interferometers 195

5.2.4. Laser Radial Shear Interferometers 197

5.2.5. Thick-Lens Radial Shear Interferometers 202

5.3. Rotational Shear Interferometers 204

5.3.1. Source Size Uncompensated Rotational Shear Interferometers 207

5.3.2. Source Size Compensated Rotational Shear Interferometers 211

5.4. Reversal Shear Interferometers 211

5.4.1. Some Reversal Shear Interferometers 213

Chapter 6. Multiple-Beam Interferometers 219 
C. Roychoudhuri

6.1. Brief Historical Introduction 219

6.2. Precision in Multiple-Beam Interferometry 221

6.3. Multiple-Beam Fizeau Interferometer 224

6.3.1. Conditions for Fringe Formation 224

6.3.2. Fizeau Interferometry 229

6.4. Fringes of Equal Chromatic Order 232

6.5. Reduction of Fringe Interval in Multiple-Beam Interferometry 235

6.6. Plane Parallel Fabry–Perot Interferometer 236

6.6.1. Measurement of Thin-Film Thickness 236

6.6.2. Surface Deviation from Planeness 237

6.7. Tolansky Fringes with Fabry–Perot Interferometer 241

6.8. Multiple-Beam Interferometer for Curved Surfaces 243

6.9. Coupled and Series Interferometers 244

6.9.1. Coupled Interferometer 245

6.9.2. Series Interferometer 246

6.10. Holographic Multiple-Beam Interferometers 247

6.11. Temporal Evolution of FP Fringes and Its Modern Applications 247

6.12. Final Comments 250

Chapter 7. Multiple-Pass Interferometers 259 
P. Hariharan

7.1. Double-Pass Interferometers 259

7.1.1. Separation of Aberrations 259

7.1.2. Reduction of Coherence Requirements 262

7.1.3. Double Passing for Increased Accuracy 264

7.2. Multipass Interferometry 266

Chapter 8. Foucault, Wire, and Phase Modulation Tests 275 
J. Ojeda-Castan˜eda

8.1. Introduction 275

8.2. Foucault or Knife-Edge Test 275

8.2.1. Description 275

8.2.2. Geometrical Theory 280

8.2.3. Physical Theory 289

8.3. Wire Test 293

8.3.1. Geometrical Theory 297

8.4. Platzeck–Gaviola Test 298

8.4.1. Geometrical Theory 299

8.5. Phase Modulation Tests 302

8.5.1. Zernike Test and its Relation to the Smart Interferometer 302

8.5.2. Lyot Test 305

8.5.3. Wolter Test 307

8.6. Ritchey–Common Test 310

8.7. Conclusions 313

Chapter 9. Ronchi Test 317 
A. Cornejo-Rodriguez

9.1. Introduction 317

9.1.1. Historical Introduction 317

9.2. Geometrical Theory 318

9.2.1. Ronchi Patterns for Primary Aberrations 320

9.2.2. Ronchi Patterns for Aspherical Surfaces 327

9.2.3. Null Ronchi Rulings 328

9.3. Wavefront Shape Determination 331

9.3.1. General Case 333

9.3.2. Surfaces with Rotational Symmetry 335

9.4. Physical Theory 337

9.4.1. Mathematical Treatment 337

9.4.2. Fringe Contrast and Sharpness 340

9.4.3. Physical versus Geometrical Theory 343

9.5. Practical Aspects of the Ronchi Test 344

9.6. Some Related Tests 347

9.6.1. Concentric Circular Grid 347

9.6.2. Phase Shifting Ronchi Test 348

9.6.3. Sideband Ronchi Test 348

9.6.4. Lower Test 349

9.6.5. Ronchi–Hartmann and Null Hartmann Tests 350

Chapter 10. Hartmann, Hartmann–Shack, and Other Screen Tests 361 
D. Malacara-Doblado and I. Ghozeil

10.1. Introduction 361

10.2. Some Practical Aspects 363

10.3. Hartmann Test Using a Rectangular Screen 366

10.4. Wavefront Retrieval 368

10.4.1. Tilt and Defocus Removal 368

10.4.2. Trapezoidal Integration 370

10.4.3. Southwell Algorithm 373

10.4.4. Polynomial Fitting 374

10.4.5. Other Methods 375

10.5. Hartmann Test Using a Screen with Four Holes 376

10.5.1. Four Holes in Cross 377

10.5.2. Four Holes in X 378

10.6. Hartmann Test of Ophthalmic Lenses 379

10.7. Hartmann Test Using Nonrectangular Screens 379

10.7.1. Radial Screen 380

10.7.2. Helical Screen 382

10.8. Hartmann–Shack Test 383

10.9. Crossed Cylinder Test 386

10.10. Testing with an Array of Light Sources or Printed Screens 387

10.10.1. Testing Convergent Lenses 388

10.10.2. Testing Concave and Convex Surfaces 389

10.11. Michelson–Gardner–Bennett Tests 393

10.12. Other Developments 394

Chapter 11. Star Tests 398 
D. Malacara and W. T. Welford

11.1. Introduction 398

11.2. Star Test with Small Aberrations 399

11.2.1. The Aberration Free Airy Pattern 400

11.2.2. The Defocused Airy Pattern 403

11.2.3. Polychromatic Light 405

11.2.4. Systems with Central Obstructions 407

11.2.5. Effects of Small Aberrations 408

11.2.6. Gaussian Beams 409

11.2.7. Very Small Convergence Angles (Low Fresnel Numbers) 409

11.3. Practical Aspects with Small Aberrations 410

11.3.1. Effects of Visual Star Testing 410

11.3.2. The Light Source for Star Testing 412

11.3.3. The Arrangement of the Optical System for Star Testing 413

11.3.4. Microscope Objectives 415

11.4. The Star Test with Large Aberrations 416

11.4.1. Spherical Aberration 417

11.4.2. Longitudinal Chromatic Aberration 418

11.4.3. Axial Symmetry 418

11.4.4. Astigmatism and Coma 419

11.4.5. Distortion 419

11.4.6. Non-Null Tests 420

11.5. Wavefront Retrieval with Slope and Curvature Measurements 421

11.5.1. The Laplacian and Local Average Curvatures 421

11.5.2. Wavefront Determination with Iterative Fourier Transforms 422

11.5.3. Irradiance Transport Equation 425

11.6. Wavefront Determination with Two Images Using the Irradiance Transport Equation 426

11.7. Wavefront Determination with a Single Defocused Image Using Fourier Transform Iterations 429

11.8. Wavefront Determination with Two or Three Defocused Images Using Fresnel Transform Iterations 430

Chapter 12. Testing of Aspheric Wavefronts and Surfaces 435 
D. Malacara, K. Creath, J. Schmit and J. C. Wyant

12.1. Introduction 435

12.2 Some Methods to Test Aspheric Wavefronts 437

12.3. Imaging of the Interference Pattern in Non-Null Tests 439

12.4. Some Null Testing Configurations 442

12.4.1. Flat and Concave Spherical Surfaces 442

12.4.2. Telescope Refracting Objectives 442

12.4.3. Concave Paraboloidal Surfaces 443

12.4.4. Concave Ellipsoidal or Spheroidal Surfaces 444

12.5. Testing of Convex Hyperboloidal Surfaces 445

12.5.1. Hindle Type Tests 445

12.5.2. Testing by Refraction 449

12.6. Testing of Cylindrical Surfaces 453

12.7. Early Compensators 454

12.7.1. Couder, Burch, and Ross Compensators 456

12.7.2. Dall Compensator 458

12.8. Refractive Compensators 461

12.8.1. Refractive Offner Compensator 462

12.8.2. Shafer Compensator 464

12.8.3. General Comments about Refracting Compensators 465

12.9. Reflecting Compensators 466

12.9.1. Reflective Offner Compensators 468

12.9.2. Reflective Adaptive Compensator 471

12.10. Other Compensators for Concave Conicoids 471

12.11. Interferometers Using Real Holograms 474

12.11.1. Holographic Wavefront Storage 476

12.11.2. Holographic Test Plate 476

12.12. Interferometers Using Synthetic Holograms 477

12.12.1. Fabrication of Computer-Generated Holograms (CGHs) 478

12.12.2. Using a CGH in an Interferometer 480

12.12.3. Off-Axis CGH Aspheric Compensator 483

12.12.4. In-Line CGH Aspheric Compensator 485

12.12.5. Combination of CGH with Null Optics 486

12.13. Aspheric Testing with Two-Wavelength Holography 488

12.14. Wavefront Stitching 491

12.14.1. Annular Zones 491

12.14.2. Circular Zones 493

12.14.3. Dynamic Tilt Switching 493

Chapter 13. Zernike Polynomial and Wavefront Fitting 498 
Virendra N. Mahajan

13.1. Introduction 498

13.2. Aberrations of a Rotationally Symmetric System with a Circular Pupil 499

13.2.1. Power Series Expansion 499

13.2.2. Primary or Seidel Aberration Function 501

13.2.3. Secondary or Schwarzschild Aberration Function 504

13.2.4. Zernike Circle Polynomial Expansion 505

13.2.5. Zernike Circle Polynomials as Balanced Aberrations for Minimum Wave Aberration Variance 508

13.2.6. Relationships Between Coefficients of Power-Series and Zernike-Polynomial Expansions 510

13.2.7. Conversion of Seidel Aberrations into Zernike Aberrations 513

13.2.8. Conversion of Zernike Aberrations into Seidel Aberrations 515

13.3. Aberration Function of a System with a Circular Pupil, but Without an Axis of Rotational Symmetry 516

13.3.1. Zernike Circle Polynomial Expansion 516

13.3.2. Relationships Among the Indices n, m, and j 518

13.3.3. Isometric, Interferometric, and PSF Plots for a Zernike Circle Polynomial Aberration 520

13.3.4. Primary Zernike Aberrations and Their Relationships with Seidel Aberrations 521

13.4. Zernike Annular Polynomials as Balanced Aberrations for Systems with Annular Pupils 525

13.4.1. Balanced Aberrations 525

13.4.2. Zernike Annular Polynomials 525

13.4.3. Isometric, Interferometric, and PSF Plots for a Zernike Annular Polynomial Aberration 529

13.5. Determination of Zernike Coefficients From Discrete Wavefront Error Data 530

13.5.1. Introduction 530

13.5.2. Orthonormal Coefficients and Aberration Variance 535

13.5.3. Orthonormal Polynomials 537

13.5.4. Zernike Coefficients 538

13.5.5. Numerical Example 539

13.6. Summary 543

Chapter 14. Phase Shifting Interferometry 547 
Horst Schreiber and John H. Bruning

14.1. Introduction 547

14.2. Fundamental Concepts 548

14.3. Advantages of PSI 550

14.4. Methods of Phase Shifting 552

14.5. Detecting the Wavefront Phase 557

14.6. Data Collection 560

14.6.1. Temporal Methods 560

14.6.2. Spatial Methods 564

14.7. PSI Algorithms 568

14.7.1. Three Step Algorithms 569

14.7.2. Least-Squares Algorithms 571

14.7.3. Carre´ Algorithm 574

14.7.4. Family of Averaging Algorithms 576

14.7.5. Hariharan Algorithm 577

14.7.6. 2 þ 1 Algorithm 580

14.7.7. Methods to Generate Algorithms 582

14.7.8. Methods to Evaluate Algorithms 586

14.7.9. Summary of Algorithms 591

14.8. Phase Shift Calibration 596

14.9. Error Sources 599

14.9.1. Phase Shift Errors 600

14.9.2. Detector Nonlinearities 602

14.9.3. Source Stability 605

14.9.4. Quantization Errors 606

14.9.5. Vibration Errors 607

14.9.6. Air Turbulence 610

14.9.7. Extraneous Fringes and Other Coherent Effects 610

14.9.8. Interferometer Optical Errors 611

14.10. Detectors and Spatial Sampling 613

14.10.1. Solid State Sensors 613

14.10.2. Spatial Sampling 614

14.11. Quality Functions 617

14.11.1. Modulation 618

14.11.2. Residues 619

14.11.3. Filtering 622

14.12. Phase Unwrapping 623

14.12.1. Unwrapping in One Dimension 623

14.12.2. 2-D Phase Unwrapping 625

14.12.3. Path-Following Algorithms 626

14.12.4. Path Independent Methods 628

14.13. Aspheres and Extended Range PSI Techniques 629

14.13.1. Aliasing 630

14.13.2. Sub-Nyquist Interferometry 631

14.13.3. Two Wavelength PSI 635

14.13.4. Subaperture Stitching 637

14.14. Other Analysis Methods 638

14.14.1. Zero Crossing Analysis 638

14.14.2. Synchronous Detection 639

14.14.3. Heterodyne Interferometry 640

14.14.4. Phase Lock Interferometry 641

14.14.5. Spatial Synchronous and Fourier Methods 642

14.15. Computer Processing and Output 644

14.16. Implementation and Applications 647

14.16.1. Commercial Instrumentation 647

14.16.2. Interferometer Configurations 650

14.16.3. Absolute Calibration 651

14.16.4. Sources 654

14.16.5. Alignment Fiducials 655

14.17. Future Trends for PSI 655

Chapter 15. Surface Profilers, Multiple Wavelength, and White Light Intereferometry 667 
J. Schmit, K. Creath, and J. C. Wyant

15.1. Introduction to Surface Profilers 667

15.1.1. Contact Profilometers 668

15.1.2. Optical Profilometers 668

15.1.3. Interferometric Optical Profilometers 668

15.1.4. Terms and Issues in Determining System Performance 669

15.2. Contact Profilometers 670

15.2.1. Stylus Profilers 670

15.2.2. Scanning Probe Microscopes 674

15.2.3. Comparison of AFM and Stylus Profiler 683

15.3. Optical Profilers 685

15.3.1. Optical Focus Sensors 687

15.3.2. Confocal Microscopy 689

15.4. Interferometric Optical Profilers 695

15.4.1. Common Features 696

15.5. Two Wavelength and Multiple Wavelength Techniques 702

15.5.1. Two-wavelengths Phase Measurement 704

15.5.2. Multiple-wavelength Phase Measurement 707

15.5.3. Reducing Measurement Time 710

15.6. White Light Interference Optical Profilers 711

15.6.1. White Light Interference 711

15.6.2. Image Buildup 712

15.6.3. Signal Processing of White Light Interferograms 713

15.6.4. Light Sources 716

15.6.5. Dispersion in White Light Fringes 716

15.6.6. Other Names for Interferometric Optical Profilers 723

15.7. Wavelength Scanning Interferometer 724

15.7.1. Wavelength Tunable Light Sources 724

15.7.2. Image Buildup 725

15.7.3. Signal Analysis 728

15.7.4. Film and Plate Thickness Measurement 729

15.8. Spectrally Resolved White Light Interferometry (SRWLI) 731

15.8.1. Image Buildup 731

15.8.2. Signal Analysis 732

15.8.3. Other Names for Spectral Interferometry 735

15.9. Polarization Interferometers 735

15.9.1. Differential Interference Contrast Microscope (Nomarski) 736

15.9.2. Geometric Phase Shifting 738

15.10. Optical Ranging Methods 741

15.10.1. Interferometric Ranging 741

15.10.2. Optical Triangulation 742

15.10.3. Time of Flight (TOF) 742

15.11. Summary 742

Chapter 16. Optical Metrology of Diffuse Surfaces 756 
K. Creath, J. Schmit, and J. C Wyant

16.1. Moire´ and Fringe Projection Techniques 756

16.1.1. Introduction 756

16.1.2. What is Moire´? 757

16.1.3. Moire´ and Interferograms 762

16.1.4. Historical Review 768

16.1.5. Fringe Projection 769

16.1.6. Shadow Moire´ 773

16.1.7. Projection Moire´ 777

16.1.8. Two-angle Holography 778

16.1.9. Common Features 779

16.1.10. Comparison to Conventional Interferometry 779

16.1.11. Coded and Structured Light Projection 780

16.1.12. Applications 781

16.1.13. Summary 783

16.2. Holographic and Speckle Tests 783

16.2.1. Introduction 783

16.2.2. Holographic Interferometry for Nondestructive Testing 784

16.2.3. Speckle Interferometry and Digital Holography 791

Chapter 17. Angle, Prisms, Curvature, and Focal Length Measurements 808 
Z. Malacara

17.2.1. Divided Circles and Goniometers 808

17.2.2. Autocollimator 810

17.2.3. Interferometric Measurements of Angles 812

17.3. Testing of Prisms 812

17.4. Radius of Curvature Measurements 817

17.4.1. Mechanical Measurement of Radius of Curvature 817

17.4.2. Optical Measurement of Radius of Curvature 820

17.5. Focal Length Measurements 823

17.5.1. Nodal Slide Bench 823

17.5.2. Focimeters 824

17.5.3. Other Focal Length Measurements 825

Chapter 18. Mathematical Representation of an Optical Surface and Its Characteristics 832 
D. Malacara

18.1. Definition of an Optical Surface 832

18.1.1. Parameters for Conic Surfaces 835

18.1.2. Some Useful Expansions of z 835

18.1.3. Aberration of the Normals to the Surface 836

18.2. Caustic Produced by an Aspheric Surface 837

18.3. Primary Aberrations of Spherical Surfaces 839

18.3.1. Spherical Aberration of and Aspherical Surface 839

18.3.2. Coma of a Concave Mirror 840

18.3.3. Astigmatism of a Concave Mirror 841

18.4. Astigmatic Surfaces 841

18.4.1. Toroidal Surface 842

18.4.2. Astigmatic Ellipsoidal and Oblate Spheroidal Surfaces 842

18.4.3. Sphero-Cylindrical Surface 844

18.4.4. Testing Astigmatic Surfaces and Reference Astigmatic Surface 846

18.4.5. Comparison Between Astigmatic Surfaces 847

18.5. Off-Axis Conicoids 849

18.5.1. Off-Axis Paraboloids 850

Appendix. Optical Testing Programs 852

Index 855