Optical Metrology Third Edition Optical Metrology. Kjell J. G ˚ asvik Copyright  2002 John Wiley & Sons, Ltd. ISBN: 0-470-84300-4 Optical Metrology Third Edition Kjell J. G ˚ asvik Spectra Vision AS, Trondheim, Norway Copyright  2002 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. 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Contents Preface to the Third Edition xi 1Basics 1 1.1 Introduction 1 1.2 Wave Motion. The Electromagnetic Spectrum 1 1.3 The Plane Wave. Light Rays 3 1.4 Phase Difference 4 1.5 Complex Notation. Complex Amplitude 5 1.6 Oblique Incidence of A Plane Wave 5 1.7 The Spherical Wave 7 1.8 The Intensity 8 1.9 Geometrical Optics 8 1.10 The Simple Convex (Positive) Lens 10 1.11 A Plane-Wave Set-Up 11 2 Gaussian Optics 15 2.1 Introduction 15 2.2 Refraction at a Spherical Surface 15 2.2.1 Examples 19 2.3 The General Image-Forming System 19 2.4 The Image-Formation Process 21 2.5 Reflection at a Spherical Surface 23 2.6 Aspheric Lenses 25 2.7 Stops and Apertures 26 2.8 Lens Aberrations. Computer Lens Design 28 2.9 Imaging and The Lens Formula 29 2.10 Standard Optical Systems 30 2.10.1 Afocal Systems. The Telescope 30 2.10.2 The Simple Magnifier 32 2.10.3 The Microscope 34 vi CONTENTS 3 Interference 37 3.1 Introduction 37 3.2 General Description 37 3.3 Coherence 38 3.4 Interference between two Plane Waves 41 3.4.1 Laser Doppler Velocimetry (LDV) 45 3.5 Interference between other Waves 46 3.6 Interferometry 49 3.6.1 Wavefront Division 50 3.6.2 Amplitude Division 51 3.6.3 The Dual-Frequency Michelson Interferometer 54 3.6.4 Heterodyne (Homodyne) Detection 55 3.7 Spatial and Temporal Coherence 56 3.8 Optical Coherence Tomography 61 4 Diffraction 67 4.1 Introduction 67 4.2 Diffraction from a Single Slit 67 4.3 Diffraction from a Grating 70 4.3.1 The Grating Equation. Amplitude Transmittance 70 4.3.2 The Spatial Frequency S pectrum 73 4.4 Fourier Optics 75 4.5 Optical Filtering 76 4.5.1 Practical Filtering Set-Ups 78 4.6 Physical Optics Description of Image Formation 81 4.6.1 The Coherent Transfer Function 83 4.6.2 The Incoherent Transfer Function 85 4.6.3 The Depth of Focus 88 4.7 The Phase-Modulated Sinusoidal Grating 89 5 Light Sources and Detectors 99 5.1 Introduction 99 5.2 Radiometry. Photometry 99 5.2.1 Lambertian Surface 102 5.2.2 Blackbody Radiator 103 5.2.3 Examples 105 5.3 Incoherent Light Sources 108 5.4 Coherent Light Sources 109 5.4.1 Stimulated Emission 109 5.4.2 Gas Lasers 112 5.4.3 Liquid Lasers 114 5.4.4 Semiconductor Diode Lasers. Light Emitting Diodes 114 5.4.5 Solid-State Lasers 117 5.4.6 Other Lasers 119 CONTENTS vii 5.4.7 Enhancements of Laser Operation 119 5.4.8 Applications 122 5.4.9 The Coherence Length of a Laser 123 5.5 Hologram Recording Media 125 5.5.1 Silver Halide Emulsions 125 5.5.2 Thermoplastic Film 126 5.5.3 Photopolymer Materials 127 5.6 Photoelectric Detectors 127 5.6.1 Photoconductors 128 5.6.2 Photodiodes 129 5.7 The CCD Camera 131 5.7.1 Operating Principles 131 5.7.2 Responsitivity 134 5.8 Sampling 135 5.8.1 Ideal Sampling 135 5.8.2 Non-Ideal Sampling 138 5.8.3 Aliasing 139 5.9 Signal Transfer 139 6 Holography 147 6.1 Introduction 147 6.2 The Holographic Process 147 6.3 An Alternative Description 150 6.4 Uncollimated Reference and Reconstruction Waves 150 6.5 Diffraction Efficiency. The Phase Hologram 153 6.6 Volume Holograms 154 6.7 Stability Requirements 156 6.8 Holographic I nterferometry 157 6.8.1 Double-Exposure I nterferometry 157 6.8.2 Real-Time Interferometry 157 6.8.3 Analysis of Interferograms 158 6.8.4 Localization of Interference Fringes 161 6.9 Holographic Vibration Analysis 165 6.10 Holographic I nterferometry of Transparent Objects 168 7Moir ´ e M ethods. Triangulation 173 7.1 Introduction 173 7.2 Sinusoidal Gratings 173 7.3 Moir ´ e Between Two Angularly Displaced Gratings 175 7.4 Measurement of In-Plane Deformation and Strains 175 7.4.1 Methods for Increasing the Sensitivity 177 viii CONTENTS 7.5 Measurement of Out-Of-Plane Deformations. Contouring 179 7.5.1 Shadow Moir ´ e 179 7.5.2 Projected Fringes 180 7.5.3 Vibration Analysis 186 7.5.4 Moir ´ e Technique by Means of Digital Image Processing 188 7.6 Reflection Moir ´ e 189 7.7 Triangulation 190 8 Speckle Methods 193 8.1 Introduction 193 8.2 The Speckle Effect 193 8.3 Speckle Size 195 8.4 Speckle Photography 197 8.4.1 The Fourier Fringe Method 197 8.4.2 The Young Fringe Method 201 8.5 Speckle Correlation 203 8.6 Speckle-Shearing Interferometry 208 8.7 White-Light Speckle Photography 212 9 Photoelasticity and Polarized Light 217 9.1 Introduction 217 9.2 Polarized Light 217 9.3 Polarizing Filters 219 9.3.1 The Linear Polarizer 219 9.3.2 Retarders 221 9.4 Unpolarized Light 223 9.5 Reflection and Refraction at an Interface 223 9.6 The Jones Matrix Formalism of Polarized Light 227 9.7 Photoelasticity 230 9.7.1 Introduction 230 9.7.2 The Plane Polariscope 231 9.7.3 The Circular Polariscope 232 9.7.4 Detection of Isochromatics of Fractional Order. Compensation 234 9.8 Holographic Photoelasticity 237 9.9 Three-Dimensional Photoelasticity 239 9.9.1 Introduction 239 9.9.2 The Frozen Stress Method 241 9.9.3 The Scattered Light Method 242 9.10 Ellipsometry 245 10 Digital Image Processing 249 10.1 Introduction 249 10.2 The Frame Grabber 249 CONTENTS ix 10.3 Digital Image Representation 251 10.4 Camera Calibration 251 10.4.1 Lens Distortion 252 10.4.2 Perspective Transformations 254 10.5 Image Processing 254 10.5.1 Contrast Stretching 255 10.5.2 Neighbourhood Operations. Convolution 256 10.5.3 Noise Suppression 257 10.5.4 Edge Detection 259 10.6 The Discrete Fourier Transform (DFT) and the FFT 262 11 Fringe Analysis 269 11.1 Introduction 269 11.2 Intensity-Based Analysis Methods 269 11.2.1 Introduction 269 11.2.2 Prior Knowledge 270 11.2.3 Fringe Tracking and Thinning 270 11.2.4 Fringe Location by Sub-Pixel Accuracy 273 11.3 Phase-Measurement Interferometry 276 11.3.1 Introduction 276 11.3.2 Principles of TPMI 276 11.3.3 Means of Phase Modulation 279 11.3.4 Different Techniques 279 11.3.5 Errors in TPMI Measurements 281 11.4 Spatial Phase-Measurement Methods 282 11.4.1 Multichannel Interferometer 282 11.4.2 Errors in Multichannel Interferometers 285 11.4.3 Spatial-Carrier Phase-Measurement Method 285 11.4.4 Errors in the Fourier Transform Method 287 11.4.5 Space Domain 289 11.5 Phase Unwrapping 290 11.5.1 Introduction 290 11.5.2 Phase-Unwrapping Techniques 292 11.5.3 Path-Dependent Methods 292 11.5.4 Path-Independent Methods 293 11.5.5 Temporal Phase Unwrapping 295 12 Computerized Optical Processes 297 12.1 Introduction 297 12.2 TV Holography (ESPI) 298 12.3 Digital Holography 301 12.4 Digital Speckle Photography 305 13 Fibre Optics in Metrology 307 13.1 Introduction 307 13.2 Light Propagation through Optical Fibres 307 13.3 Attenuation and Dispersion 310 x CONTENTS 13.4 Different Types of Fibres 313 13.5 Fibre-Optic Sensors 315 13.6 Fibre-Bragg Sensors 318 Appendices A. Complex Numbers 325 B. Fourier Optics 327 B.1 The Fourier Transform 327 B.2 Some Functions and Their Transforms 329 B.3 Some Implications 332 C. Fourier Series 335 D. The Least-Squares Error Method 339 E. Semiconductor Devices 343 References and Further Reading 347 Index 355 Preface to the Third Edition This edition of Optical Metrology contains a new chapter about computerized optical processes, including digital holography and digital speckle photography. Chapter 2, on Gaussian optics, and Chapter 5, on light sources and detectors, are greatly expanded to include descriptions of standard imaging systems, light-emitting diodes and solid-state detectors. Separate new sections on optical coherence tomography, speckle correlation, the Fast Fourier Transform, temporal phase unwrapping and fibre Bragg sensors are included. Finally, a new appendix about Fourier series is given. Solutions to the end-of-chapter problems can be found at http://www.wiley.co.uk/opticalmetrology. Since the previous edition, the electronic camera has taken over more and more as the recording medium. The word ‘digital’ is becoming a prefix to an increasing number of techniques. I think this new edition reflects this trend. It gives me great pleasure to acknowledge the many stimulating discussions with Pro- fessor H.M. Pedersen at The Norwegian University of Science a nd Technology. Thanks also to John Petter G ˚ asvik for designing many of the new figures. [...].. .Optical Metrology Kjell J G˚ svik a Copyright  2002 John Wiley & Sons, Ltd ISBN: 0-4 7 0-8 430 0-4 1 Basics 1.1 INTRODUCTION Before entering into the different techniques of optical metrology some basic terms and definitions have to be established Optical metrology is about light and therefore we must develop a mathematical description... propagates along the optical axis In Figure 1.13(c) the point source is displaced along the focal plane a distance h from the optical axis We then get a plane wave propagating in a direction that makes an angle θ to the optical axis where tan θ = h/f (1.20) 1.11 A PLANE-WAVE SET-UP Finally, we refer to Figure 1.14 which shows a commonly applied set-up to form a uniform, expanded plane wave from a laser... normal pointing from the incident to the transmitting medium (b) In the same way, derive a vector expression equivalent to the law of reflection Optical Metrology Kjell J G˚ svik a Copyright  2002 John Wiley & Sons, Ltd ISBN: 0-4 7 0-8 430 0-4 2 Gaussian Optics 2.1 INTRODUCTION Lenses are an important part of most optical systems Good results in optical measurements often rely on the best selection of... As can be realized, a ray is completely determined at any plane normal to the z-axis by specifying x, its height above the z-axis in that plane, and its angle α relative to the z-axis A ray therefore can be speci ed by a column matrix x α The two components of this matrix will be altered as the ray propagates through an optical system At the point A in Figure 2.1 the height is unaltered, and this fact... fixed planes in space Let us consider the simple case sketched in Figure 1.6 where a plane wave falls obliquely on to a plane parallel to the xy-plane a distance z from it The wave propagates along the unit vector n which is lying in the xz-plane (defined as the plane of incidence) and makes an angle θ to the z-axis The components of the n- and r-vectors are therefore n = (sin θ, 0, cos θ ) r = (x, y,... and 2.17 Reprinted with permission.) y x q n z Figure 1.6 THE SPHERICAL WAVE 7 These expressions put into Equation (1.6) (Re and temporal part omitted) give u = U eik(x sin θ+z cos θ ) (1.9a) For z = 0 (the xy-plane) this reduces to u = U eikx sin θ (1.9b) 1.7 THE SPHERICAL WAVE A spherical wave, illustrated in Figure 1.5(b), is a wave emitted by a point source It should be easily realized that the complex... magnification m= In Figure 1.13(a), the case of a point source lying on the optical axis forming a spherical diverging wave that is converted to a converging wave and focuses onto a point on the optical axis is illustrated In Figure 1.13(b) the point source is lying on-axis at a distance Po D f f a Pi b Figure 1.11 ho hi Figure 1.12 A PLANE-WAVE SET-UP 11 (a) (b) h q (c) Figure 1.13 from the lens equal to the focal... should have considered the ray to lie in an arbitrary plane, taken its components in the xz- and yz-planes and introduced the component angles α and β relative to the z-axis We then would have found that x and α at a given point depend only on x and α at other points, not on y and β In other words, the pairs of variables (x, α) and (y, β) are decoupled from one another and may be treated independently... We do the calculations on the projection in the xz-plane and the answers will also apply for the yz-plane with the substitutions x → y and α → β The xz projections behave as though y and β were zero Such rays, which lie in a single plane containing the z-axis are called meridional rays In this theory we have assumed that an optical axis can be defined and that all light rays and all normals to refracting... Equation (1.6) is commonly adopted and ‘Re’ is omitted because it is silently understood that the field is described by the real part One advantage of such complex representation of the field is that the spatial and temporal parts factorize: ψ(x, y, z, t) = U ei(φ−2πνt) = U eiφ e−i2πvt (1.7) In optical metrology (and in other branches of optics) one is most often interested in the spatial distribution . Optical Metrology Third Edition Optical Metrology. Kjell J. G ˚ asvik Copyright  2002 John Wiley & Sons, Ltd. ISBN: 0-4 7 0-8 430 0-4 Optical Metrology Third. have v = c = 3 × 10 8 m/s Optical Metrology. Kjell J. G ˚ asvik Copyright  2002 John Wiley & Sons, Ltd. ISBN: 0-4 7 0-8 430 0-4 2 BASICS z y( z , t ) dl/2p l U Figure