Physics of Light and Optics Justin Peatross and Harold Stokes Brigham Young University Department of Physics and Astronomy All Publication Rights Reserved (2001) Revised April 2002 This project is supported in part by the National Science Foundation Division of Undergraduate Education (Grant number DUE9952773). i
CONTENTS Chapter 0 Chapter 1 Chapter 2 Chapter 3 Mathematical Tools 0.1 Introduction 1 0.2 Complex Numbers 1 0.3 Vector Calculus 5 0.4 Fourier Theory 9 Appendix 0.A Sylvester s Theorem 14 Appendix 0.B Integrals and Formulas 15 Electromagnetic Phenomena 1.1 Introduction 17 1.2 Coulomb s and Gauss Laws 17 1.3 BiotSavart and Ampere s Laws 20 1.4 Faraday s Law 22 1.5 Continuity Equation 22 1.6 Maxwell s Equations 24 1.7 The Wave Equation 25 1.8 Wave Equation in Isotropic Media 26 Plane Waves and Refractive Index 2.1 Introduction 29 2.2 Plane Wave Solutions to the Wave Equation 30 2.3 Dielectric Model of Refractive Index and Absorption 32 2.4 Conductor Model of Refractive Index and Absorption 37 2.5 Poynting s Theorem 39 2.6 Irradiance of a Plane Wave 41 Appendix 2.A Energy Density of Electric Fields 43 Appendix 2.B Energy Density of Magnetic Fields 44 Appendix 2.C Radiometry Versus Photometry 45 Reflection and Refraction 3.1 Introduction 47 3.2 Refraction at an Interface 47 3.3 The Fresnel Coefficients 51 3.4 Reflection and Transmission 52 3.5 Brewster s Angle 55 3.6 Total Internal Reflection 55 Appendix 3.A Boundary Conditions for Fields at an Interface 56 ii
Chapter 4 Chapter 5 Polarization 4.1 Introduction Linear, Circular, and Elliptical Polarization 59 4.2 Jones Vectors for Representing Polarization 60 4.3 Jones Matrices 64 4.4 Jones Matrix for Polarizers at Arbitrary Angle 65 4.5 Jones Matrices for Wave plates 69 4.6 Polarization Effects of Reflection and Transmission 72 4.7 Reflection from Metallic or other Absorptive Surfaces 73 4.8 Polarization and Total Internal Reflection 75 4.9 Ellipsometry 78 Light Propagation in Crystals 5.1 Introduction 79 5.2 Wave Propagation in Nonisotropic Media 80 5.3 Fresnel s Equation 82 5.4 Uniaxial Crystal 84 5.5 Poynting Vector in a Uniaxial Crystal 88 Appendix 5.A Rotation of Coordinates 90 Appendix 5.A Huygens' Elliptical Construct for a Uniaxial Crystal 92 Review Problems Chapters 15 95 Chapter 6 Chapter 7 Multiple Boundary Problem 6.1 Introduction 101 6.2 Double Boundary Problem Solved Using Fresnel Coefficients 101 6.3 Double Boundary Problem at Sub Critical Angles 106 6.4 Beyond Critical Angle: Tunneling of Evanescent Waves 109 6.5 FabryPerot Etalon 112 6.6 Distinguishing Nearby Wavelengths in a FabryPerot Instrument 114 6.7 Setup of a FabryPerot Instrument 120 6.8 Multilayer Coatings 122 6.9 Repeated Multilayer Stacks 126 Superposition of QuasiParallel, Uniformly Polarized Plane Waves 7.1 Introduction 129 7.2 Intensity 130 7.3 Group Versus Phase Velocity: Superposition of Two Plane Waves 133 7.4 Frequency Spectrum of Light 135 7.5 Phase Delay and Group Delay 140 7.6 Quadratic Dispersion 141 7.7 Generalized Context for Group Delay 144 Appendix 7.A Causality and Exchange of Energy with the Medium 149 iii
Chapter 8 Coherence Theory 8.1 Introduction 157 8.2 Michelson Interferometer 158 8.3 Temporal Coherence 160 8.4 Fringe Visibility and Coherence Length 162 8.5 Fourier Spectroscopy 164 8.6 Young s TwoSlit Setup and Spatial Coherence 167 Appendix 8.A Spatial Coherence with a Continuous Source 173 Appendix 8.B Non Random Phase and the van CittertZernike Theorem 174 Review Problems Chapters 68 177 Chapter 9 Chapter 10 Light as Rays 9.1 Introduction 181 9.2 The Eikonal Equation 182 9.3 Fermat s Principle 186 9.4 Paraxial Rays and ABCD Matrices 190 9.5 Paraxial Reflections and Transmissions at Curved Surfaces 192 9.6 Image Formation by a Mirrors and Lenses 197 9.7 Image Formation by a Complex Optical Systems 199 9.8 Stability of Laser Cavities 202 Appendix 9.A Ray Tracing To be written Diffraction 10.1 Introduction 207 10.2 FresnelKirchhoff Diffraction Formula 209 10.3 Babinet's principle 213 10.4 Fresnel Approximation 215 10.5 Fraunhofer Approximation 216 10.6 Diffraction with Cylindrical Symmetry 218 Appendix 10.A Significance of the Scalar Wave Approximation 219 Appendix 10.B Green s Theorem 220 Chapter 11 Diffraction Applications 11.1 Introduction 223 11.2 Diffraction of a Gaussian Field Profile 223 11.3 Gaussian Laser Beams 225 11.4 Fraunhofer Diffraction Through a Lens 230 11.5 Resolving Power of a Telescope 235 11.6 The Array Theorem 240 11.7 Diffraction Grating 241 11.8 Spectrometers 244 11.9 ABCD law for Gaussian beams 249 Review Problems Chapters 911 253 iv
Chapter 12 Chapter 13 Chapter 14 Interferograms and Holography To Be Written 12.1 Introduction 12.2 Generating Interferograms 12.3 Testing Optical Surfaces 12.4 Creating Holograms 12.5 Holographic Wave Front Reconstruction Blackbody Radiation and Lasers 13.1 Introduction 259 13.2 Failure of Equipartition Principle 261 13.3 Planck's Formula 263 13.4 Einstein's A and B Coefficients 266 Appendix 13.A Thermodynamic Derivation of StefanBoltzman Law 267 Introduction to Quantum Optics To Be Written 14.1 Introduction 14.2 Photon Statistics 14.3 Coincidence Counting 14.4 Parametric Down Conversion 14.5 Interference of Correlated Photon Pairs Appendix 14.A Field Operators Appendix 14.B Analysis of Correlated Photon Pairs v
Preface This book is currently in a very preliminary form and will likely take several years to complete. It is used for a seniorlevel optics course for physics majors at Brigham Young University. Constructive feedback is welcome. The book is available at no cost at optics.byu.edu. In addition to completing the writing and editing of this material, we are developing a series of film clips and animations to enhance it. This curriculum benefits from a CCLI grant from the National Science Foundation Division of Undergraduate Education (DUE9952773). The grant has enabled the development of a number of laboratory exercises that are being integrated into the project. vi
Constants e o = 8 854 10 12 2. C N m 2 Permittivity mo = 4p 10 7 T m A Permeability c = 2. 9979 10 8 ms Speed of light in vacuum q e m e k B = 1. 602 10 19 C Charge of an electron = 9. 108 10 31 kg Mass of an electron = 1. 380 10 23 JK Boltzmann constant h = 6. 626 10 34 J s Planck s constant h = h 2p = 1. 054 10 34 J s s = 5. 670 10 8 Wm K 2 4 StefanBoltzmann constant vii