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Modern Optics Notes H. M. Van Driel Department of Physics University of Toronto Version Modern Optics Notes H. M. van Driel Department of Physics University of Toronto Version: 2.2 Contents Introduction 9 0.1. History of optics 9 0.2. Outline of the notes 10 References 11 Part 1. Light as an Electromagnetic Phenomenon 12 Chapter 1. Propagation of Light 13 1.1. Maxwell’s Equations and the Constitutive Relations 13 1.2. Wave Solutions to Maxwell’s Equations in Homogeneous, Dielectric Media 14 1.3. Plane, Spherical and Cylindrical Waves 15 1.4. Phasor Representation of Waves 18 1.5. Complex Form of Maxwell’s Equations in Dielectric and Conducting Media 18 1.6. Dispersion Relation for Plane Waves 19 1.7. Classical Model for the Dielectric Function 21 1.7.1. Bound electron systems 21 1.7.2. Free electron systems 26 1.8. Pulses 28 Appendix 2.1: Fourier Transforms 29 References 30 Problems 30 Special Note on Phase Conventions 31 Chapter 2. Energy and Linear Momentum in an Electromagnetic Wave. 32 2.1. Transport of Energy in an Electromagnetic Wave 32 2 CONTENTS 3 2.2. Time Average of Sinusoidal Quantities 33 2.3. Poynting’s Theorem for Dispersive Media 33 2.4. Optical Units 34 2.5. Linear Momentum and Radiation Pressure of Light 34 References 35 Problems 35 Chapter 3. The Vector Nature of Light: Polarization Effects 36 3.1. Introduction 36 3.2. Description of Polarization States 36 3.3. Anisotropic Optical Media 40 3.4. Matrix Representation of Polarization–The Jones Calculus 41 3.5. Optical Activity 44 3.6. Magneto- and Electro-optic Effects 46 i) Faraday Rotation in Solids 46 ii) Voigt Effect 46 iii) Pockels Effect 46 iv) Kerr Effect 46 v) The Cotton-Mouton Effect; 47 References 47 Problems 48 Special Note on Phase Conventions 48 Chapter 4. Reflection and Refraction at an Interface 50 4.1. Reflection and Refraction of a Plane Wave at a Planar Interface 50 4.2. Amplitudes of Reflected, Transmitted and Refracted Waves 52 4.3. Total Internal Reflection 57 4.4. Reflection from Rough Surfaces 60 References 63 Problems 63 Part 2. Ray Optics 65 CONTENTS 4 Chapter 5. Geometrical Optics 66 5.1. Geometrical Optics Approximation 66 5.2. Rays at a Plane Interface 66 5.3. Prisms 68 5.4. Reflection at a Curved Interface: Spherical Mirrors 71 5.5. Refraction at a Curved Interface 72 5.6. Thin Lenses 73 5.7. Lens Aberrations 74 5.7.1. Spherical Aberration 74 5.7.2. Coma 74 5.7.3. Chromatic Aberration 74 5.7.4. Astigmatism and curvature of the field 75 5.7.5. Distortion 75 References 75 Problems 75 Chapter 6. Matrix Methods in Paraxial Optics 76 6.1. Optical Rays and Transformations 76 6.2. Ray Propagation Through Cascaded Elements 78 6.3. Telescopes and Microscopes 79 References 81 Problems 81 Part 3. Wave Optics 82 Chapter 7. Superposition of Optical Waves 83 7.1. Interference of Two Beams 83 7.2. Partial Coherence 85 7.3. Coherence Time and Coherence Length 87 7.4. The Wiener-Khintchine Theorem 89 7.5. Temporal and Spatial Coherence 91 7.6. Interferometer: wavefront-splitting 93 7.6.1. The Michelson Interferometer 93 7.6.2. The Fabry-Perot Interferometer 95 CONTENTS 5 7.7. Multilayer Thin Films 100 References 105 Problems 105 Chapter 8. Diffraction Phenomena 107 8.1. Fresnel-Kirchoff Theory of Diffraction 108 8.2. Babinet’s Principle 111 8.3. Fresnel and Fraunhofer Diffraction 112 8.4. Fresnel Approximation 113 References 122 Problems 122 Chapter 9. Fraunhofer Diffraction, Fourier Optics and Holography 123 9.1. The Fraunhofer Approximation 123 9.2. Aperture Functions Made Simple 126 9.3. Properties of Fourier Transforms 129 9.4. Fourier Optics and Gratings 130 9.5. Transmission Characteristics of a Lens 133 9.6. The Lens as a Fourier Transforming System. 136 9.7. Holography 138 References 143 Problems 143 Chapter 10. Gaussian Beams 146 10.1. Paraxial Optics 146 10.2. Gaussian Beams 147 10.3. Gaussian Beams in Resonators with Curved Mirrors: Form of the Modes 150 10.4. Hermite-Gaussian Modes 153 10.5. Hermite-Gaussian Beams in Resonators—Allowed Frequencies 155 10.6. Transformation of Gaussian Beams 156 References 160 Problems 160 CONTENTS 6 Chapter 11. Optical Waveguides 162 11.1. Waveguides and Ray Optics 162 11.2. Modes of Planar Dielectric Waveguides 163 11.3. Modal Dispersion 170 11.4. Optical Fibres and Waveguide Coupling 171 References 172 Problems 173 Part 4. Quantum Optics 174 Chapter 12. Introduction to Quantum Optics and Spectroscopy 175 12.1. Black body Radiation and the Onset of Quantum Optics 175 12.2. Two Level Atoms 179 12.3. Optical Sources 181 12.4. Line shape Functions 183 12.5. Macroscopic Aspects of the Interaction of a Monochromatic Wave with Two-level Atoms 184 12.6. Derivation of the Dielectric Function for a Collection of Two-level Atoms 186 References 188 Problems 188 Chapter 13. The Laser 189 13.1. Threshold Condition for CW Laser Oscillation 189 13.2. Threshold Operation of a Laser— Amplitude Condition 190 13.3. Threshold Operation of a Laser—Frequency Condition 192 13.4. Three and Four Level Lasers 194 13.5. Effects of Spontaneous Emission 199 13.6. Pulsed lasers 199 References 199 Problems 199 Chapter 14. Specific Laser Systems 201 14.1. Ruby Laser 201 CONTENTS 7 14.2. The Nd:YAG laser 203 14.3. The He-Ne laser 203 14.4. The CO2 laser 204 14.5. The Semiconductor Laser 206 14.6. Dye Lasers 208 14.7. Titanium Sapphire laser 208 References 209 Index 210 List of Constants c = speed of light; in vacuum = 2:9979 × 108ms−1 = 0:29979m(ns)−1 −12 −1 0 = electric permittivity of free space = 8:854 × 10 F m −7 −1 µ0 = magnetic permeability of free space = 4π × 10 Hm e = electric charge = 1:602 × 10−19C h = Planck’s constant = 6:625 × 10−34Js ~ = h=2π = 1:054 × 10−34J s m = mass of electron = 9:109 × 10−31 kg −23 −1 kB = Boltzmann’s constant= 1.38 ×10 JK Simple conversions for photon energy/wavelength: 1 µm ≡ 1.24 eV ≡ 1:89 × 1015rads−1 ≡ 3.00×1014 Hz Some notes on typeface conventions used A : real-valued scalar; A : complex-valued scalar A~ : real-valued vector; A~ : complex-valued vector Introduction And God said: "Let there be light. And there was light. And God saw the light, that it was good; And God divided the light from the darkness.” The Torah 0.1. History of optics Light, as a phenomenon of nature, has been of fundamental interest to man since the dawn of time. To the ancients light from the sun was the source of all life, since it helped to produce crops for food. It was also regarded as a conveyor of knowledge since it allowed man to have information about his environment, while its absence created doubt, uncertainty and, consequently, fear. The sun was regarded as an object that had control over mankind, and for many peoples it was elevated to the status of a god. Because of the central role that it played in ancient lives, it is perhaps not surprising that the study of optical phenomena, or Optics, is one of the oldest fields of natural science. Nonetheless, many aspects of the nature of light, paricularly quantum properties, remain a mystery today. Early natural philosophers such as the Greeks, Thales, Plato, Democritus and Aristotle were deeply interested in the phenomenon of vision and are known to have proposed several theories to explain it. These were centred around the observations that light travels in a straight line and that the angle of incidence is equal to the angle of reflection for a light beam striking a mirror surface. The most popular theories stated that vision took place as a two step process. In the first step a stream of "particles” or “rays” emanates from the eye in the direction it is looking. An object struck by these rays releases particles back towards the eye and forms an impression. More sophisticated versions of the theory stated that the released particles or chips from the objects were miniature versions of the object itself and therefore each carried full information about that object. However, the Greeks recognized that these theories could not be complete, since they failed to account for such simple phenomena as the refraction of light and the process of inversion in the development of mirror images. Although the true nature of light and its propagation properties had to await the discovery of the origins of electromagnetic and quantum phenomena in the 19th and 20th centuries it was nevertheless possible for an operational or phenomenological knowledge of light to develop. Archimedes, perhaps the greatest scientist of antiquity, is known to have discovered the focusing properties of lenses and mirrors. Legend has it that he used this knowledge to focus sunbeams to set fire to Roman ships about to invade Syracuse in what is now Sicily. Around the year 1000 A.D., the Arab Alhazen made sweeping discoveries in optics and offered explanations for the phenomena of reflection from non-planar surfaces and refraction at the boundary between two different media. During the Renaissance, the Italian, Leonardo da Vinci showed tremendous insight into how light scatters in the atmosphere and applied much of this knowledge to his paintings. He is also known for his development of the pinhole camera working inside a room-sized camera obscura to make perspective and landscape studies. His countryman, Galileo Galilei is credited with the development of the telescope, following suggestions by the Dutch lens grinders Lipperschey and DeWard.
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