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.

"This book is a major text in the field, and a must-read for academicians and engineers alike." (Computing Reviews, May 1, 2008)

• Preface xviiContributors xixChapter 1. Newton, Fizeau, and Haidinger Interferometers 1 M. V. Mantravadi and D. Malacara1.1. Introduction 11.2. Newton Interferometer 11.2.1. Source and Observer’s Pupil Size Considerations 91.2.2. Some Suitable Light Sources 111.2.3. Materials for the Optical Flats 121.2.4. Simple Procedure for Estimating Peak Error 121.2.5. Measurement of Spherical Surfaces 131.2.6. Measurement of Aspheric Surfaces 151.2.7. Measurement of Flatness of Opaque Surfaces 171.3. Fizeau Interferometer 171.3.1. The Basic Fizeau Interferometer 181.3.2. Coherence Requirements for the Light Source 201.3.3. Quality of Collimation Lens Required 221.3.4. Liquid Reference Flats 231.3.5. Fizeau Interferometer with Laser Source 231.3.6. Multiple-Beam Fizeau Setup 241.3.7. Testing Nearly Parallel Plates 261.3.8. Testing the Inhomogeneity of Large Glass or Fused Quartz Samples 271.3.9. Testing the Parallelism and Flatness of the Faces of Rods, Bars and Plates 281.3.10. Testing Cube Corner and Right-Angle Prisms 281.3.11. Fizeau Interferometer for Curved Surfaces 301.3.12. Testing Concave and Convex Surfaces 321.4. Haldinger Interferometer 331.4.1. Applications of Haidinger Fringes 351.4.2. Use of Laser Source for Haidinger Interferometer 361.4.3. Other Applications of Haidinger Fringes 391.5. Absolute Testing of Flats 40Chapter 2. Twyman–Green Interferometer 46 D. Malacara2.1. Introduction 462.2. Beam-Splitter 482.2.1. Optical Path Difference Introduced by the Beam Splitter Plate 492.2.2. Required Accuracy in the Beam Splitter Plate 512.2.3. Polarizing Cube Beam Splitter 532.2.4. Nonpolarizing Cube Beam Splitter 552.3. Coherence Requirements 562.3.1. Spatial Coherence 562.3.2. Temporal Coherence 602.4. Uses of a Twyman–Green Interferometer 622.4.1. Testing of Prisms and Diffraction Rulings 642.4.2. Testing of Lenses 692.4.3. Testing of Microscope Objectives 712.5. Compensation of Intrinsic Aberrations in the Interferometer 722.6. Unequal-Path Interferometer 732.6.1. Some Special Designs 752.6.2. Improving the Fringe Stability 762.7. Open Path Interferometers 772.7.1. Mach-Zehnder Interferometers 772.7.2. Oblique Incidence Interferometers 782.8. Variations from the Twyman–Green Configuration 802.8.1. Multiple Image Interferometers 802.8.2. Interferometers with Diffractive Beam Splitters 802.8.3. Phase Conjugating Interferometer 812.9. Twyman–Green Interferograms and their Analysis 832.9.1. Analysis of Interferograms of Arbitrary Wavefronts 91Chapter 3. Common-Path Interferometers 97 S. Mallick and D. Malacara3.1. Introduction 973.2. Burch’s Interferometer Employing Two Matched Scatter Plates 983.2.1. Fresnel Zone Plate Interferometer 1023.2.2. Burch and Fresnel Zone Plate Interferometers for Aspheric Surfaces 1023.2.3. Burch and Fresnel Zone Plate Interferometers for Phase Shifting 1023.3. Birefringent Beam Splitters 1043.3.1. Savart Polariscope 1043.3.2. Wollaston Prism 1063.3.3. Double-Focus Systems 1073.4. Lateral Shearing Interferometers 1083.4.1. Use of a Savart Polariscope 1083.4.2. Use of a Wollaston Prism 1113.5. Double-Focus Interferometer 1123.6. Saunders’s Prism Interferometer 1143.7. Point Diffraction Interferometer 1163.8. Zernike Tests with Common-Path Interferometers 118Chapter 4. Lateral Shear Interferometers 122 Strojnik, G. Paez, and M. Mantravadi4.1. Introduction 1224.2. Coherence Properties of the Light Source 1234.3. Brief Theory of Lateral Shearing Interferometry 1244.3.1. Interferograms of Spherical and Flat Wavefronts 1264.3.2. Interferogams of Primary Aberrations upon Lateral Shear 1284.4. Evaluation of an Unknown Wavefront 1344.5. Lateral Shearing Interferometers in Collimated Light (White Light Compensated) 1374.5.1. Arrangements Based on the Jamin Interferometer 1374.5.2. Arrangements Based on the Michelson Interferometer 1394.5.3. Arrangements Based on a Cyclic Interferometer 1404.5.4. Arrangements Based on the Mach–Zehnder Interferometer 1424.6. Lateral Shearing Interferometers in Convergent Light (White Light Compensated) 1434.6.1. Arrangements Based on the Michelson Interferometer 1434.6.2. Arrangements Based on the Mach–Zehnder Interferometer 1464.7. Lateral Shearing Interferometers Using Lasers 1494.7.1. Other Applications of the Plane Parallel Plate Interferometer 1524.8. Other Types of Lateral Shearing Interferometers 1574.8.1. Lateral Shearing Interferometers Based on Diffraction 1584.8.2. Lateral Shearing Interferometers Based on Polarization 1624.9. Vectorial Shearing Interferometer 1644.9.1. Shearing Interferometry 1654.9.2. Directional Shearing Interferometer 1664.9.3. Simulated Interferometric Patterns 1684.9.4. Experimental Results 1734.9.5. Similarities and Differences With Other Interferometers 176Chapter 5. Radial, Rotational, and Reversal Shear Interferometer 185 D. Malacara5.1. Introduction 1855.2. Radial Shear Interferometers 1875.2.1. Wavefront Evaluation from Radial Shear Interferograms 1895.2.2. Single-Pass Radial Shear Interferometers 1905.2.3. Double-Pass Radial Shear Interferometers 1955.2.4. Laser Radial Shear Interferometers 1975.2.5. Thick-Lens Radial Shear Interferometers 2025.3. Rotational Shear Interferometers 2045.3.1. Source Size Uncompensated Rotational Shear Interferometers 2075.3.2. Source Size Compensated Rotational Shear Interferometers 2115.4. Reversal Shear Interferometers 2115.4.1. Some Reversal Shear Interferometers 213Chapter 6. Multiple-Beam Interferometers 219 C. Roychoudhuri6.1. Brief Historical Introduction 2196.2. Precision in Multiple-Beam Interferometry 2216.3. Multiple-Beam Fizeau Interferometer 2246.3.1. Conditions for Fringe Formation 2246.3.2. Fizeau Interferometry 2296.4. Fringes of Equal Chromatic Order 2326.5. Reduction of Fringe Interval in Multiple-Beam Interferometry 2356.6. Plane Parallel Fabry–Perot Interferometer 2366.6.1. Measurement of Thin-Film Thickness 2366.6.2. Surface Deviation from Planeness 2376.7. Tolansky Fringes with Fabry–Perot Interferometer 2416.8. Multiple-Beam Interferometer for Curved Surfaces 2436.9. Coupled and Series Interferometers 2446.9.1. Coupled Interferometer 2456.9.2. Series Interferometer 2466.10. Holographic Multiple-Beam Interferometers 2476.11. Temporal Evolution of FP Fringes and Its Modern Applications 2476.12. Final Comments 250Chapter 7. Multiple-Pass Interferometers 259 P. Hariharan7.1. Double-Pass Interferometers 2597.1.1. Separation of Aberrations 2597.1.2. Reduction of Coherence Requirements 2627.1.3. Double Passing for Increased Accuracy 2647.2. Multipass Interferometry 266Chapter 8. Foucault, Wire, and Phase Modulation Tests 275 J. Ojeda-Castan˜eda8.1. Introduction 2758.2. Foucault or Knife-Edge Test 2758.2.1. Description 2758.2.2. Geometrical Theory 2808.2.3. Physical Theory 2898.3. Wire Test 2938.3.1. Geometrical Theory 2978.4. Platzeck–Gaviola Test 2988.4.1. Geometrical Theory 2998.5. Phase Modulation Tests 3028.5.1. Zernike Test and its Relation to the Smart Interferometer 3028.5.2. Lyot Test 3058.5.3. Wolter Test 3078.6. Ritchey–Common Test 3108.7. Conclusions 313Chapter 9. Ronchi Test 317 A. Cornejo-Rodriguez9.1. Introduction 3179.1.1. Historical Introduction 3179.2. Geometrical Theory 3189.2.1. Ronchi Patterns for Primary Aberrations 3209.2.2. Ronchi Patterns for Aspherical Surfaces 3279.2.3. Null Ronchi Rulings 3289.3. Wavefront Shape Determination 3319.3.1. General Case 3339.3.2. Surfaces with Rotational Symmetry 3359.4. Physical Theory 3379.4.1. Mathematical Treatment 3379.4.2. Fringe Contrast and Sharpness 3409.4.3. Physical versus Geometrical Theory 3439.5. Practical Aspects of the Ronchi Test 3449.6. Some Related Tests 3479.6.1. Concentric Circular Grid 3479.6.2. Phase Shifting Ronchi Test 3489.6.3. Sideband Ronchi Test 3489.6.4. Lower Test 3499.6.5. Ronchi–Hartmann and Null Hartmann Tests 350Chapter 10. Hartmann, Hartmann–Shack, and Other Screen Tests 361 D. Malacara-Doblado and I. Ghozeil10.1. Introduction 36110.2. Some Practical Aspects 36310.3. Hartmann Test Using a Rectangular Screen 36610.4. Wavefront Retrieval 36810.4.1. Tilt and Defocus Removal 36810.4.2. Trapezoidal Integration 37010.4.3. Southwell Algorithm 37310.4.4. Polynomial Fitting 37410.4.5. Other Methods 37510.5. Hartmann Test Using a Screen with Four Holes 37610.5.1. Four Holes in Cross 37710.5.2. Four Holes in X 37810.6. Hartmann Test of Ophthalmic Lenses 37910.7. Hartmann Test Using Nonrectangular Screens 37910.7.1. Radial Screen 38010.7.2. Helical Screen 38210.8. Hartmann–Shack Test 38310.9. Crossed Cylinder Test 38610.10. Testing with an Array of Light Sources or Printed Screens 38710.10.1. Testing Convergent Lenses 38810.10.2. Testing Concave and Convex Surfaces 38910.11. Michelson–Gardner–Bennett Tests 39310.12. Other Developments 394Chapter 11. Star Tests 398 D. Malacara and W. T. Welford11.1. Introduction 39811.2. Star Test with Small Aberrations 39911.2.1. The Aberration Free Airy Pattern 40011.2.2. The Defocused Airy Pattern 40311.2.3. Polychromatic Light 40511.2.4. Systems with Central Obstructions 40711.2.5. Effects of Small Aberrations 40811.2.6. Gaussian Beams 40911.2.7. Very Small Convergence Angles (Low Fresnel Numbers) 40911.3. Practical Aspects with Small Aberrations 41011.3.1. Effects of Visual Star Testing 41011.3.2. The Light Source for Star Testing 41211.3.3. The Arrangement of the Optical System for Star Testing 41311.3.4. Microscope Objectives 41511.4. The Star Test with Large Aberrations 41611.4.1. Spherical Aberration 41711.4.2. Longitudinal Chromatic Aberration 41811.4.3. Axial Symmetry 41811.4.4. Astigmatism and Coma 41911.4.5. Distortion 41911.4.6. Non-Null Tests 42011.5. Wavefront Retrieval with Slope and Curvature Measurements 42111.5.1. The Laplacian and Local Average Curvatures 42111.5.2. Wavefront Determination with Iterative Fourier Transforms 42211.5.3. Irradiance Transport Equation 42511.6. Wavefront Determination with Two Images Using the Irradiance Transport Equation 42611.7. Wavefront Determination with a Single Defocused Image Using Fourier Transform Iterations 42911.8. Wavefront Determination with Two or Three Defocused Images Using Fresnel Transform Iterations 430Chapter 12. Testing of Aspheric Wavefronts and Surfaces 435 D. Malacara, K. Creath, J. Schmit and J. C. Wyant12.1. Introduction 43512.2 Some Methods to Test Aspheric Wavefronts 43712.3. Imaging of the Interference Pattern in Non-Null Tests 43912.4. Some Null Testing Configurations 44212.4.1. Flat and Concave Spherical Surfaces 44212.4.2. Telescope Refracting Objectives 44212.4.3. Concave Paraboloidal Surfaces 44312.4.4. Concave Ellipsoidal or Spheroidal Surfaces 44412.5. Testing of Convex Hyperboloidal Surfaces 44512.5.1. Hindle Type Tests 44512.5.2. Testing by Refraction 44912.6. Testing of Cylindrical Surfaces 45312.7. Early Compensators 45412.7.1. Couder, Burch, and Ross Compensators 45612.7.2. Dall Compensator 45812.8. Refractive Compensators 46112.8.1. Refractive Offner Compensator 46212.8.2. Shafer Compensator 46412.8.3. General Comments about Refracting Compensators 46512.9. Reflecting Compensators 46612.9.1. Reflective Offner Compensators 46812.9.2. Reflective Adaptive Compensator 47112.10. Other Compensators for Concave Conicoids 47112.11. Interferometers Using Real Holograms 47412.11.1. Holographic Wavefront Storage 47612.11.2. Holographic Test Plate 47612.12. Interferometers Using Synthetic Holograms 47712.12.1. Fabrication of Computer-Generated Holograms (CGHs) 47812.12.2. Using a CGH in an Interferometer 48012.12.3. Off-Axis CGH Aspheric Compensator 48312.12.4. In-Line CGH Aspheric Compensator 48512.12.5. Combination of CGH with Null Optics 48612.13. Aspheric Testing with Two-Wavelength Holography 48812.14. Wavefront Stitching 49112.14.1. Annular Zones 49112.14.2. Circular Zones 49312.14.3. Dynamic Tilt Switching 493Chapter 13. Zernike Polynomial and Wavefront Fitting 498 Virendra N. Mahajan13.1. Introduction 49813.2. Aberrations of a Rotationally Symmetric System with a Circular Pupil 49913.2.1. Power Series Expansion 49913.2.2. Primary or Seidel Aberration Function 50113.2.3. Secondary or Schwarzschild Aberration Function 50413.2.4. Zernike Circle Polynomial Expansion 50513.2.5. Zernike Circle Polynomials as Balanced Aberrations for Minimum Wave Aberration Variance 50813.2.6. Relationships Between Coefficients of Power-Series and Zernike-Polynomial Expansions 51013.2.7. Conversion of Seidel Aberrations into Zernike Aberrations 51313.2.8. Conversion of Zernike Aberrations into Seidel Aberrations 51513.3. Aberration Function of a System with a Circular Pupil, but Without an Axis of Rotational Symmetry 51613.3.1. Zernike Circle Polynomial Expansion 51613.3.2. Relationships Among the Indices n, m, and j 51813.3.3. Isometric, Interferometric, and PSF Plots for a Zernike Circle Polynomial Aberration 52013.3.4. Primary Zernike Aberrations and Their Relationships with Seidel Aberrations 52113.4. Zernike Annular Polynomials as Balanced Aberrations for Systems with Annular Pupils 52513.4.1. Balanced Aberrations 52513.4.2. Zernike Annular Polynomials 52513.4.3. Isometric, Interferometric, and PSF Plots for a Zernike Annular Polynomial Aberration 52913.5. Determination of Zernike Coefficients From Discrete Wavefront Error Data 53013.5.1. Introduction 53013.5.2. Orthonormal Coefficients and Aberration Variance 53513.5.3. Orthonormal Polynomials 53713.5.4. Zernike Coefficients 53813.5.5. Numerical Example 53913.6. Summary 543Chapter 14. Phase Shifting Interferometry 547 Horst Schreiber and John H. Bruning14.1. Introduction 54714.2. Fundamental Concepts 54814.3. Advantages of PSI 55014.4. Methods of Phase Shifting 55214.5. Detecting the Wavefront Phase 55714.6. Data Collection 56014.6.1. Temporal Methods 56014.6.2. Spatial Methods 56414.7. PSI Algorithms 56814.7.1. Three Step Algorithms 56914.7.2. Least-Squares Algorithms 57114.7.3. Carre´ Algorithm 57414.7.4. Family of Averaging Algorithms 57614.7.5. Hariharan Algorithm 57714.7.6. 2 þ 1 Algorithm 58014.7.7. Methods to Generate Algorithms 58214.7.8. Methods to Evaluate Algorithms 58614.7.9. Summary of Algorithms 59114.8. Phase Shift Calibration 59614.9. Error Sources 59914.9.1. Phase Shift Errors 60014.9.2. Detector Nonlinearities 60214.9.3. Source Stability 60514.9.4. Quantization Errors 60614.9.5. Vibration Errors 60714.9.6. Air Turbulence 61014.9.7. Extraneous Fringes and Other Coherent Effects 61014.9.8. Interferometer Optical Errors 61114.10. Detectors and Spatial Sampling 61314.10.1. Solid State Sensors 61314.10.2. Spatial Sampling 61414.11. Quality Functions 61714.11.1. Modulation 61814.11.2. Residues 61914.11.3. Filtering 62214.12. Phase Unwrapping 62314.12.1. Unwrapping in One Dimension 62314.12.2. 2-D Phase Unwrapping 62514.12.3. Path-Following Algorithms 62614.12.4. Path Independent Methods 62814.13. Aspheres and Extended Range PSI Techniques 62914.13.1. Aliasing 63014.13.2. Sub-Nyquist Interferometry 63114.13.3. Two Wavelength PSI 63514.13.4. Subaperture Stitching 63714.14. Other Analysis Methods 63814.14.1. Zero Crossing Analysis 63814.14.2. Synchronous Detection 63914.14.3. Heterodyne Interferometry 64014.14.4. Phase Lock Interferometry 64114.14.5. Spatial Synchronous and Fourier Methods 64214.15. Computer Processing and Output 64414.16. Implementation and Applications 64714.16.1. Commercial Instrumentation 64714.16.2. Interferometer Configurations 65014.16.3. Absolute Calibration 65114.16.4. Sources 65414.16.5. Alignment Fiducials 65514.17. Future Trends for PSI 655Chapter 15. Surface Profilers, Multiple Wavelength, and White Light Intereferometry 667 J. Schmit, K. Creath, and J. C. Wyant15.1. Introduction to Surface Profilers 66715.1.1. Contact Profilometers 66815.1.2. Optical Profilometers 66815.1.3. Interferometric Optical Profilometers 66815.1.4. Terms and Issues in Determining System Performance 66915.2. Contact Profilometers 67015.2.1. Stylus Profilers 67015.2.2. Scanning Probe Microscopes 67415.2.3. Comparison of AFM and Stylus Profiler 68315.3. Optical Profilers 68515.3.1. Optical Focus Sensors 68715.3.2. Confocal Microscopy 68915.4. Interferometric Optical Profilers 69515.4.1. Common Features 69615.5. Two Wavelength and Multiple Wavelength Techniques 70215.5.1. Two-wavelengths Phase Measurement 70415.5.2. Multiple-wavelength Phase Measurement 70715.5.3. Reducing Measurement Time 71015.6. White Light Interference Optical Profilers 71115.6.1. White Light Interference 71115.6.2. Image Buildup 71215.6.3. Signal Processing of White Light Interferograms 71315.6.4. Light Sources 71615.6.5. Dispersion in White Light Fringes 71615.6.6. Other Names for Interferometric Optical Profilers 72315.7. Wavelength Scanning Interferometer 72415.7.1. Wavelength Tunable Light Sources 72415.7.2. Image Buildup 72515.7.3. Signal Analysis 72815.7.4. Film and Plate Thickness Measurement 72915.8. Spectrally Resolved White Light Interferometry (SRWLI) 73115.8.1. Image Buildup 73115.8.2. Signal Analysis 73215.8.3. Other Names for Spectral Interferometry 73515.9. Polarization Interferometers 73515.9.1. Differential Interference Contrast Microscope (Nomarski) 73615.9.2. Geometric Phase Shifting 73815.10. Optical Ranging Methods 74115.10.1. Interferometric Ranging 74115.10.2. Optical Triangulation 74215.10.3. Time of Flight (TOF) 74215.11. Summary 742Chapter 16. Optical Metrology of Diffuse Surfaces 756 K. Creath, J. Schmit, and J. C Wyant16.1. Moire´ and Fringe Projection Techniques 75616.1.1. Introduction 75616.1.2. What is Moire´? 75716.1.3. Moire´ and Interferograms 76216.1.4. Historical Review 76816.1.5. Fringe Projection 76916.1.6. Shadow Moire´ 77316.1.7. Projection Moire´ 77716.1.8. Two-angle Holography 77816.1.9. Common Features 77916.1.10. Comparison to Conventional Interferometry 77916.1.11. Coded and Structured Light Projection 78016.1.12. Applications 78116.1.13. Summary 78316.2. Holographic and Speckle Tests 78316.2.1. Introduction 78316.2.2. Holographic Interferometry for Nondestructive Testing 78416.2.3. Speckle Interferometry and Digital Holography 791Chapter 17. Angle, Prisms, Curvature, and Focal Length Measurements 808 Z. Malacara17.2.1. Divided Circles and Goniometers 80817.2.2. Autocollimator 81017.2.3. Interferometric Measurements of Angles 81217.3. Testing of Prisms 81217.4. Radius of Curvature Measurements 81717.4.1. Mechanical Measurement of Radius of Curvature 81717.4.2. Optical Measurement of Radius of Curvature 82017.5. Focal Length Measurements 82317.5.1. Nodal Slide Bench 82317.5.2. Focimeters 82417.5.3. Other Focal Length Measurements 825Chapter 18. Mathematical Representation of an Optical Surface and Its Characteristics 832 D. Malacara18.1. Definition of an Optical Surface 83218.1.1. Parameters for Conic Surfaces 83518.1.2. Some Useful Expansions of z 83518.1.3. Aberration of the Normals to the Surface 83618.2. Caustic Produced by an Aspheric Surface 83718.3. Primary Aberrations of Spherical Surfaces 83918.3.1. Spherical Aberration of and Aspherical Surface 83918.3.2. Coma of a Concave Mirror 84018.3.3. Astigmatism of a Concave Mirror 84118.4. Astigmatic Surfaces 84118.4.1. Toroidal Surface 84218.4.2. Astigmatic Ellipsoidal and Oblate Spheroidal Surfaces 84218.4.3. Sphero-Cylindrical Surface 84418.4.4. Testing Astigmatic Surfaces and Reference Astigmatic Surface 84618.4.5. Comparison Between Astigmatic Surfaces 84718.5. Off-Axis Conicoids 84918.5.1. Off-Axis Paraboloids 850Appendix. Optical Testing Programs 852Index 855