Optical Shop Testing

Herausgeber: Malacara, Daniel

• Broschiertes Buch

Produktbeschreibung

This updated Third Edition of the classic textbook is an essential reference for specialists and nonspecialists in the field of optical testing Since the publication of the Second Edition of this book, many advances have taken place in the field of optical testing. Taking into account the changes in telecommunications, including the various forms of digital networks and their testing aspects, this Third Edition compiles the vast amount of research being conducted in optical engineering into one easily accessible source. Optical Shop Testing, Third Edition brings together descriptions of all tests carried out in the optical shop that are applicable to optical components and systems. In addition to new chapters and modified material, this revised edition also includes information on: * Testing of aspheric wavefronts, compensators, and null configurations * Zernike polynomial and wavefront fitting * Optical metrology of diffuse objects * Angle, prism, curvature, and focal length measurements * Mathematical representation of an optical surface and its characteristics Intended for anyone engaged in optical shop testing, this essential textbook also includes a CD-ROM with an exhaustive list of resources and two programs for Windows(R), which will be useful when teaching or working in optical testing.

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Produktdetails

• Verlag: Wiley
• 3rd edition
• Seitenzahl: 888
• Erscheinungstermin: 1. Mai 2007
• Englisch
• Abmessung: 240mm x 161mm x 52mm
• Gewicht: 1478g
• ISBN-13: 9780471484042
• ISBN-10: 0471484040
• Artikelnr.: 22725557

Autorenporträt

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.

Inhaltsangabe

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