Insight into Optics

O. S. Heavens

Department of Physics, University of York

and the late

R. W. Ditchburn

JOHN WILEY & SONS

Chichester • New York • Brisbane • Toronto • Singapore

Contents

Preface xvii 1. General Background 1

1.1 Why study optics? 1 1.2 Historical background 1 1.3 The scope of the subject 3 1.4 Characteristics of optical radiation 3 1.5 Waves and rays 4 1.6 Where to? 5

2. Ray Optics (1): Reflection and Refraction 6

2.1 Introduction 6 2.2 Principle of least time 6 2.3 Sign convention 7 2.4 Reflection and refraction 7 2.5 Refraction by a prism 8 2.6 Refraction by a thin prism 8 2.7 Dispersion 8 2.8 Total reflection 9 2.9 Fibre optics 10 2.10 Numerical aperture of a coated fibre 10 2.11 Imaging through fibre bundles 10 2.12 Intrascopes 11 Appendix 2.A Fermat's principle 11

3. Ray Optics (2): Paraxial Rays 13

3.1 Introduction: coaxial systems 13 3.2 Paraxial rays 14 3.3 Refraction at a single spherical surface 14 3.4 Magnification 15 3.5 Reflection at a spherical mirror 15

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vi Contents

3.6 Reflection and refraction at a plane surface 16 3.7 Thin lenses in air 16 3.8 Positive and negative lenses 17 3.9 Real and virtual images 17 3.10 General treatment of coaxial systems 17 3.11 Cardinal points 18 3.12 Magnification 19 3.13 Newton's formula 19 3.14 Catadioptric systems 19 3.15 Calculation of the power of a system 20 3.16 Two thin lenses in air 20 3.17 Thick lens 20 3.18 Nodal points y 21 3.19 Experimental location of the cardinal points 21 3.20 Telescopic systems 22 3.21 Longitudinal magnification 23 3.22 More on telescopic systems 23 3.23 Matrix methods 23 3.24 System matrices 24 3.25 Going through the system 25 3.26 Analysis of the thin lens by matrix methods 26 3.27 Was it worth it? 26 3.28 Electron- and charged-particle optics 26

4. Wave Theory (1): Interference 28

4.1 Periodicity in time and in space ^ 28 4.2 Progressive waves 28 4.3 We cannot follow 29 4.4 Phase velocity of a progressive wave 29 4.5 Waves in three dimensions 29 4.6 Superposition of light waves <- 30 4.7 Temporal beats 31 4.8 Spatial beats: interference fringes 31 4.9 Interference in thin films 32 4.10 Newton's rings 32 4.11 Optical path 32 4.12 Contour fringes 33 4.13 Young's experiment • 33 4.14 Fresnel's biprism and Lloyd's mirror 34 4.15 The Michelson interferometer 35 4.16 Circular fringes 36 4.17 Interference with a number of waves 36 4.18 Grating spectra 36 4.19 Spectrum lines 38 4.20 Why spectrum lines? 38 4.21 Free spectral range 38 4.22 Interference of multiple beams by repeated reflections 39 4.23 Fabry-Perot etalon 40 4.24 Method of exact fractions 41 4.25 Analysis of hyperfine structure 41 4.26 Comparison of two etalons 42 4.27 Channelled spectrum 43

Contents vii

4.28 Multiple-beam contour fringes 43 4.29 Where are all these fringes? 44 4.30 Multilayer dielectric films 44 4.31 Antireflection coatings 44 4.32 High-efficiency reflection films 45 4.33 Beam-splitters 46 4.34 Interference filters 46 Appendix 4.A The wave equation 47 Appendix 4.B Representation of waves by complex quantities 47

5. Wave Theory (2): Wave Groups 49 5.1 Finite wave trains 49 5.2 Fourier transforms 50 5.3 Damped harmonic waves 51 5.4 Gaussian wave groups 52 5.5 Limited wave trains 52 5.6 Very short pulses • 52 5.7 Quasi-monochromaticity 53 5.8 Propagation of a wave group in a dispersive medium 53 5.9 Coherence 54 5.10 Photometric summation 55 5.11 Visibility of interference fringes 55 5.12 Mutual coherence function 56 5.13 Complex degree of coherence 56 5.14 Coherence of a single beam 57 5.15 Spatial coherence , 57 5.16 Coherence—slit-width dependence 58 5.17 Coherence and the Michelson interferometer 58 5.18 , Measurement of degree of coherence 59 Appendix 5.A Gaussian wave group in a dispersive medium 59 Appendix 5.B Calculation of Гi2(0) and 712(0) 60

6. Diffraction . 62 6.1 Introduction 62 6.2 Calculating diffraction patterns 62 6.3 Huygens' principle 63 6.4 Fresnel's equation 63 6.5 Near-field and far-field diffraction 64 6.6 Fraunhofer diffraction 64 6.7 Fourier transforms 66 6.8 Far-field (Fraunhofer) diffraction by a rectangular aperture 66 6.9 Far-field diffraction by a circular aperture 67 6.10 Apodised circular aperture 68 6.11 Diffraction by a number of similar apertures / 68 6.12 Random and regular arrays 69 6.13 Diffraction gratings 69 6.14 Blazed gratings 70 6.15 Near-field (Fresnel) diffraction by a circular aperture 71 6.16 Diffraction by a circular obstacle 73 6.17 Fresnel zone plate 74 6.18 Multiple foci of the zone plate 74 6.19 Fresnel integrals 74

viii Contents

6.20 Cornu's spiral 75 6.21 Diffraction by a straight edge 76 6.22 Babinet's principle 76

7. Polarised Light 79

7.1 Introduction 79 7.2 Types of polarisation 80 7.3 Double refraction / 80 7.4 Methods of producing plane-polarised light 80 7.5 Interference with polarised light 81 7.6 Wave surfaces in crystals 81 7.7 Where is the plane of polarisation? 82 7.8 Quarter- and half-wave plates 83 7.9 Elliptically polarised light 83 7.10 Stokes parameters 84 7.11 Matrix treatment of polarisation problems 85 7.12 Jones vectors and Jones matrices 85 7.13 Analysis of polarised light 86 7.14 Circularly and elliptically polarised light 86 7.15 Mixtures 86 7.16 Ellipsometers 87 7.17 Compensators 87 7.18 Analysers 88 7.19 Colours of thin crystalline plates 88 7.20 The Lyot filter 89 7.21 Savart-Francon plate 89 7.22 Interference microscopy 89 7.23 Rotatory polarisation 90 7.24 Allogyric birefringence 90 7.25 Dispersion 91 7.26 Photoelasticity 91 Appendix 7.A Plane and circularly polarised light 91 Appendix 7.B Elliptical polarisation—the most general case 92

8. Electromagnetic Theory of Dielectric Media , 93

8.1 The nature of light waves 93 8.2 The basis of electromagnetic theory 93 8.3 Electric and magnetic properties of materials 94 8.4 Conductivity 95 8.5 The laws of electromagnetism 95 8.6 Maxwell's equations 95 8.7 Electromagnetic waves 96 8.8 Propagation in vacuum 96 8.9 Propagation in a perfect dielectric 96 8.10 Properties of electromagnetic waves 97 8.11 Polarisation of light 98 8.12 Energy of the electromagnetic field 99 8.13 Power flow: the Poynting vector 99 8.14 Reflection and refraction of light at an interface 99 8.15 Boundary conditions 100 8.16 Reflected and transmitted amplitudes 101 8.17 Normal incidence 103

Contents ix

8.18 Energy reflection and transmission coefficients 103 8.19 Degree of polarisation 104 8.20 Total reflection 104 8.21 Total reflection: the region beyond the interface 105 8.22 Is it right? 106 Appendix 8.A Optical multilayers 106

9. Electromagnetic Theory of Absorptive Materials 108

9.1 Preamble 108 9.2 Light incident on an absorbing medium 109 9.3 State of polarisation of reflected light 110 9.4 Optical constants of conducting materials 111 9.5 Why the theory fails for the visible region 111 9.6 Optical behaviour of non-conductors 112 9.7 Dispersion in gases at low pressure 112 9.8 Dispersion in condensed matter 113 9.9 Measurement of oscillator strength 114 9.10 Optical behaviour of metals 114 9.11 Non-local effects 115

10. Scattering of Light : 117

10.1 How does scattering arise? 117 10.2 Scattering cross-section 117 10.3 Elastic and inelastic scattering 118 10.4 Rayleigh scattering 118 10.5 Rayleigh scattering in quantum mechanical terms 120 10.6 Mie scattering 120 10.7 Raman scattering 121 10.8 Scattering in solids , 122 10.9 Brillouin scattering 123 10.10 Raman-Nath and Bragg scattering 124 10.11 Scattering by free electrons 125 Appendix 10.A Radiation from an oscillating dipole 125 Appendix 10.B Scattering by free electrons 126

11. Electro- and Magneto-optics 128

11.1 General considerations 128 11.2 Light propagation in crystals 128 11.3 The wave surface 130 11.4 Electro-optic effects 130 11.5 Magneto-optic effects 133

12. Interaction of Radiation and Matter 137

12.1 Limitation of classical ideas 137 12.2 The need for quanta 138 12.3 The photoelectric effect 1 138 12.4 Photon momentum 139 12.5 Characteristics of radiation sources 139 12.6 Power and line width 139 12.7 A digression into electronics 140 12.8 Radiation/matter interaction 140

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12.9 Relative rates of stimulated and spontaneous emission 141 12.10 Amplifying material in a resonant cavity 141 12.11 Establishing a population inversion 142 12.12 Laser systems 142 12.13 Characteristics of laser output 144

13. Holography ^ 146

13.1 Background 146 13.2 Gabor's proposal 147 13.3 Enter the laser 148 13.4 Side-band holograms 148 13.5 The reconstruction is truly three dimensional 149 13.6 Analysis of the side-band hologram 149 13.7 How good is the reconstruction? 150 13.8 Fourier transform holograms 151 13.9 Interference holography 153 13.10 Holographic diffraction gratings 153

14. Waveguides, Fibres and Optical Communications 154

14.1 Introduction 154 14.2 The metallic waveguide 154 14.3 The dielectric waveguide ; 155 14.4 Field strengths in the guides and adjacent media 156 14.5 Dielectric waveguides in integrated optical systems 157 14.6 Propagation along optical fibres 157 14.7 Modes in cylindrical fibres 158 14.8 Optical fibres in communications 158 14.9 Optical signal processing 160 14.10 Fibre-optic sensors 160 14.11 Integrated optics 161 14.12 A few of the problems 161

15. Non-linear Optics , 164

15.1 Polarisation at high field strengths 164 15.2 Consequences of non-linear susceptibility 164 15.3 Phase-matching in second harmonic generation 166 15.4 A practical SHG system 167 15.5 Optical mixing and parametric amplification 168 15.6 Phase conjugation 168 15.7 Stimulated Brillouin scattering 169 15.8 Degenerate four-wave mixing 170 15.9 Application of phase conjugation 171 15.10 Self-focusing 171 15.11 Optical bistability 172 15.12 Propagation of pulses along optical fibres 173 15.13 Solitons 175

16. Radiometry and Photometry 178

16.1 Symbols and definitions 178 16.2 Photometric units 178 16.3 Luminous intensity 180

16.4 The relation between exitance and luminance 16.5 Noise 16.6 Detectivity 16.7 Sources of noise 16.8 Combination of sources of noise 16.9 Responsivity 16.10 Thermal detectors 16.11 Bolometers 16.12 Pyroelectric detectors 16.13 Photoemissive devices 16.14 Semiconductor photoconductive devices 16.15 PIN diodes 16.16 Avalanche photodiodes 16.17 Photovoltaic devices 16.18 Intense sources 16.19 Giant-pulse lasers 16.20 Photographic integration of light 16.21 Absorption spectrophotometry 16.22 Light amplifier 16.23 Infrared imaging by scanning 16.24 Semiconductor devices in photometry 16.25 The eye as a system of radiation detectors 16.26 Colour 16.27 Trichromatism 16.28 Psychometric and psychophysical terms 16.29 Chromaticity 16.30 Defective colour vision

7. Tnterferometry

17.1 Preamble 17.2 Sensitivity and stability 17.3 Classification of interferometers 17.4 Lasers in relation to interferometry 17.5 Location of fringe position 17.6 Laser long-path interferometers 17.7 Adjacent-path interferometers 17.8 Common-path interferometers __ 17.9 Scatter interferometer 17.10 Polarising interferometers 17.11 Compensated interferometer 17.12 Spherical Fabry-Perot interferometer 17.13 Fourier transform spectroscopy 17.14 Fourier transform system 17.15 Resolving power 17.16 Laser speckle 17.17 The Burch-Tokarski experiment 17.18 Astronomical applications of speckle 17.19 Looking at stars

8. Optical Instruments F 1

18.1 Instruments and systems 18.2 Use of detectors

Contents xi

180 180 181 181 182 182 182 183 183 184 185 186 186 186 187 187 187 188 189 189 189 189 190 190 191 191 192

193

193 193 193 194 194 195 195 196 196 197 197 197 198 199 200 200 200 201 201

?ПЗ

203 203

xii Contents

18.3 Need to consider whole system 203 18.4 Use and limitation of ray theory 204 18.5 Aperture 204 18.6 The exit pupil I 204 18.7 Field 205 18.8 Relay systems 206 18.9 Modern endoscopes 206 18.10 Increasing the light throughput 207 18.11 Optical fibre system 207 18.12 The visual system 208 18.13 The eye 208 18.14 Aberrations of the eye 208 18.15 The eye's receptors 208 18.16 Interpreting the visual signals 209 18.17 Accommodation 209 18.18 Advancing geriatry 209 18.19 Correction of vision defects 210 18.20 Convergence 210 18.21 Bifocals 210 18.22 Astigmatism 211 18.23 Contact lenses 211 18.24 Hand magnifiers 211 18.25 Oculars and eyepieces 211 18.26 More eyepieces 212 18.27 Plane mirrors and prisms 212 18.28 Deflection and inversion 213 18.29 Use of prisms 213 18.30 And more prisms 214 18.31 Dispersing prisms . 215 18.32 Small telescopes and binoculars 216 18.33 Goniometers 216 18.34 Cameras 216 18.35 Telephoto lenses 216 18.36 Zoom lenses 217 18.37 Soft focus lenses 217 18.38 High-speed photography 217 18.39 Repetitive high-speed photography 218 18.40 Freezing the image '218 18.41 The streak camera 219 18.42 Astronomical telescopes 219 18.43 Where to put them? 220 18.44 How are telescopes used? 220 18.45 Problems of the paraboloid 220 18.46 Enter the hyperbola and the Schmidt plate 220 18.47 Making mirrors on a lathe 221 18.48 Plate-measuring machines 222 18.49 Next generation telescope (NGT) 222 18.50 The multimirror telescope 222 18.51 The large steerable dish 222 18.52 The rotating shoe 222 18.53 The multitelescope designs 223 18.54 The 108-mirror system 223 18.55 What next? 224

• P

Contents xiii

18.56 Microscopes 224 18.57 Resolution 224 18.58 Common types of microscope 225 18.59 The standard microscope 225 18.60 The research microscope 225 18.61 Reflecting microscopes 225 18.62 Flux collectors—solar energy 226 18.63 Imaging systems 227 18.64 Non-imaging systems 227

19. Assessment of Optical Images 229

19.1 Introduction 229 19.2 Images formed with coherent light 229 19.3 Amplitude objects and phase objects 230 19.4 Sinusoidal gratings 230 19.5 Oblique illumination 230 19.6 Contrast 231 19.7 Image formed with non-coherent light 231 19.8 Optical transfer function (OTF) 232 19.9 Line-spread function 232 19.10 Measurement of OTF or of the spread function 233 19.11 Rayleigh limit of resolution for a telescope 234 19.12 Limit of resolution of the eye 234 19.13 Useful and empty magnification 235 19.14 Resolving power of a spectroscope 235 19.15 Resolving power of a Fabry-Perot etalon 235 19.16 Reproduction of detail by a microscope 236 19.17 Phase-contrast microscopy 237 19.18 Dark-ground illumination 238 19.19 Spatial filtering 238 19.20 Schlieren method 240 19.21 Aberrations 240 19.22 Effect of aberrations on the OTF 241 19.23 Diffraction and image formation 242

20. Lasers 244

20.1 Conditions for laser action 244 20.2 Parallel mirror system 244 20.3 A closer look at cavities 245 20.4 The cavity length 245 20.5 The field distribution in the plane mirror cavity 246 20.6 The closed cavity 246 20.7 The confocal laser 248 20.8 The field distribution in the confocal cavity 249 20.9 Unstable resonators 250 20.10 Practical laser systems 250 20.11 Q-switching and mode-locking 253 20.12 Laser characteristics 254 20.13 Laser applications 254 20.14 Absorption spectroscopy with dye lasers 255 20.15 Laser Raman spectroscopy 255 20.16 Stimulated Raman scattering 256

xiv Contents

20.17 Brillouin scattering 256 20.18 Doppler-free spectroscopy 257 20.19 The laser and fusion 258 20.20 The laser gyroscope 258

21. Temporal Analysis—Photon Correlation 260

21.1 Fluctuation with time of a light signal 260 21.2 What are real light sources like? 261 21.3 Determination of spectral composition 261 21.4 Temporal analysis of wave trains 262 21.5 Temporal correlation and spectral density 262 21.6 Spatial correlations 263 21.7 Shot noise 263 21.8 Light-beating spectroscopy 263 21.9 Velocimeter 263 21.10 Molecular weight determination --p 264 21.11 Photon correlations 264 21.12 Experimental difficulties 264 21.13 Computational difficulties 265 21.14 Cross-correlation: the Hanbury-Brown and Twiss experiment 266 21.15 Interference and photon correlation 266

22. The Velocity of Light and Relativisitic Optics 267

22.1 Preamble 267 22.2 Early measurements • 267 22.3. Michelson's measurements 267 22.4. The electro-optical shutter 268 22.5. Measurements of X and v 268 22.6. Standards of length and time 268 22.7. Variation of velocity with refractive index 269 22.8. Relativity in relation to optics 269 22.9 Constancy of the velocity of light 269 22.10 The Michelson-Morley experiment 270 22.11 Further such experiments 270 22.12 The laser method 271 • 22.13 Transfer between axes in relative motion 271 22.14 Einstein's theory—addition of velocities 271 22.15 Velocity in a moving medium 272 22.16 Light received when source and observer are in relative motion 272 22.17 Aberration 273 22.18 The Doppler effect 273 22.19 Reflection from a moving mirror 274 22.20 Rotating interferometers 274 22.21 Momentum and energy of the photon 275 22.22 Light pressure 276 22.23 Interaction of gravitation and light 276 22.24 Nebular red shift 277 22.25 Synchrotron radiation 277 22.26 Cherenköv radiation 278

23. The Quantum Theory of Light 280

23.1 Introduction 280

23.2 Photons in relation to modes 23.3 Definition of a photon 23.4 The uncertainty principle 23.5 Planck's law 23.6 Photon statistics 23.7 Interaction with atoms: perturbation theory 23.8 Field quantisation 23.9 When do photoelectrons appear? 23.10 Taylor's experiment 23.11 Conclusion

The Limitations of Optical Experiments

24.1 Optics in relation to information theory 24.2 Sampling theorem 24.3 Information in an optical picture 24.4 Transmission of optical information 24.5 Noise 24.6 Fidelity 24.7 Ultimate limits 24.8 Thermal noise 24.9 Wave and particle properties of light 24.10 Squeezed light 24.11 Interferometry 24.12 Storage of optical information 24.13 Limitations in laser optics 24.14 High-power systems 24.15 The future Appendix 24.A The sampling theorem Answers, Hints and Solutions

Contents xv

280 281 281 281 282 283 283 284 284 285

287

287 288 288 288 289 289 289 289 290 290 291 291 291 292 292 293 295

Index 303