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SIMG(455 Physical Optics SPSP(455 Optical Physics SIMG-455 Physical Optics SPSP-455 Optical Physics Roger L. Easton, Jr. Chester F. Carlson Center for Imaging Science Rochester Institute of Technology 54 Lomb Memorial Drive Rochester, NY 14623-5604 585-475-5969 585-475-5988 (FAX) [email protected] 29 April 2008 Contents Preface ix 0.1 Sources: . 1 0.1.1 Optics Texts: . 2 0.1.2 Physics Texts: . 2 1 Wave Optics = Physical Optics 1 1.1 The Wave Equation . 2 1.1.1 Traveling Waves . 2 1.1.2 Notation and Dimensions for Traveling Sinusoidal Waves in a Medium . 4 1.2 Electric and Magnetic Fields . 6 1.2.1 A Note on Units . 6 1.2.2 Magnetic Fields . 7 1.3 Maxwell’sEquations . 9 1.3.1 Vector Operations . 9 1.3.2 Vector Calculus . 15 1.3.3 Curl of Curl . 22 1.4 1. Gauss’Law for Electric Fields . 22 1.5 2. Gauss’Law for Magnetic Fields . 23 1.6 3. Faraday’sLaw of Magnetic Induction . 23 1.7 4. Ampère’sLaw . 23 1.8 Solution of Maxwell’sEquations . 24 1.8.1 Wave Equation for EM Waves . 25 1.8.2 Electromagnetic Waves from Maxwell’sEquations . 28 1.8.3 Phase Velocity of Electromagnetic Waves . 30 1.9 Consequences of Maxwell’sEquations . 31 1.9.1 “Shape”of the Wavefronts . 32 1.10 Optical Frequencies –Detector Response . 33 1.11 Dual Nature of Light: Photons . 34 1.11.1 Momentum of Photons . 36 2 Index of Refraction 37 2.0.2 Refractive Indices of Di¤erent Materials: . 39 2.0.3 Optical Path Length . 39 2.1 Lorentz’sModel of Refractive Index . 40 2.1.1 Refractive Index in terms of Phase . 41 2.1.2 Refractive Index in Terms of Forces . 43 2.1.3 Damping Forces and Complex Refractive Index . 47 2.1.4 Dispersion Curves . 51 2.1.5 Empirical Expressions for Dispersion . 53 2.1.6 Negative Refractive Index . 56 v vi CONTENTS 3 Polarization 57 3.1 Plane Polarization = Linear Polarization . 57 3.2 Circular Polarization . 58 3.2.1 Nomenclature for Circular Polarization . 59 3.2.2 Elliptical Polarization, Re‡ections . 59 3.2.3 Change of Handedness on Re‡ection . 59 3.2.4 Natural Light . 60 3.3 Description of Polarization States . 60 3.3.1 Jones Vector . 60 3.3.2 Actions of Optical Elements on Jones Vectors . 61 3.3.3 Coherency Matrix J ................................ 64 3.4 Generation of Polarized Light . 69 3.4.1 Selective Emission: . 69 3.4.2 Selective Transmission or Absorption . 69 3.4.3 Generating Polarized Light by Re‡ection –Brewster’sAngle . 70 3.4.4 Polarization by Scattering . 70 3.5 Birefringence –Double Refraction . 71 3.5.1 Examples: . 71 3.5.2 Phase Delays in Birefringent Materials; Wave Plates . 71 3.5.3 Critical Angle –Total Internal Re‡ection . 73 4 Electromagnetic Waves at an Interface 75 4.1 Fresnel Equations . 75 4.1.1 De…nitions of Vectors . 75 4.1.2 Snell’sLaw for Re‡ection and Refraction of Waves . 77 4.1.3 Boundary Conditions for Electric and Magnetic Fields . 78 4.1.4 Transverse Electric (TE) Waves = “s”= “ ”Polarization . 81 4.1.5 Transverse Magnetic (TM) Waves = “p”=? “ ”polarization . 84 4.1.6 Comparison of Coe¢ cients for TE and TM Wavesk . 85 4.1.7 Angular Dependence of Amplitude Coe¢ cients at “Rare-to-Dense”Interface 87 4.1.8 Re‡ectance and Transmittance at “Rare-to-Dense”Interface . 88 4.1.9 Re‡ection and Transmittance at “Dense-to-Rare”Interface, Critical Angle . 91 4.1.10 Practical Applications for Fresnel’sEquations . 92 5 Optical Interference and Coherence 93 5.1 Addition of Oscillations . 93 5.1.1 Addition of 1-D Traveling Waves . 94 5.2 Division-of-Wavefront Interference . 97 5.2.1 Addition of 3-D Traveling Waves with same ! . 97 5.2.2 Superposition of Two Plane Waves with !1 = !2 . 103 5.3 Fringe Visibility –Temporal Coherence . .6 . 106 5.3.1 Coherence of Light . 111 5.3.2 Coherence Time and Coherence Length . 112 5.3.3 Polarization and Fringe Visibility . 114 5.4 Van Cittert - Zernike Theorem . 115 5.5 Division-of-Amplitude Interferometers . 115 5.5.1 Fizeau Interferometer . 116 5.5.2 Michelson Interferometer . 118 5.5.3 Mach-Zehnder Interferometer . 125 5.5.4 Sagnac Interferometer . 126 5.6 Interference by Multiple Re‡ections . 127 5.6.1 Thin Films . 127 5.6.2 Fabry-Perot Interferometer . 135 CONTENTS vii 6 Optical Di¤raction and Imaging 139 6.1 Huygens’Principle . 140 6.2 Di¤raction Integrals . 141 6.3 Fresnel Di¤raction; the Near Field ............................ 143 6.3.1 Fresnel Di¤raction as a Convolution Integral . 145 6.3.2 Characteristics of Fresnel Di¤raction . 154 6.3.3 Transfer Function of Fresnel Di¤raction . 155 6.4 Fraunhofer Di¤raction . 156 6.4.1 Examples of Fraunhofer Di¤raction . 158 6.5 Coherence of Light and Imaging . 160 6.6 Transmissive Optical Elements . 161 6.6.1 Optical Elements with Constant or Linear Phase . 162 6.6.2 Lenses with Spherical Surfaces . 163 6.7 Fraunhofer Di¤raction and Imaging . 166 6.7.1 “Di¤raction-Limited”and “Detector-Limited”Imaging Systems . 171 6.8 Depth of Field and Depth of Focus . 171 Preface ix 0.1 SOURCES: 1 The science of optics is often divided into three classi…cations based on the scale of the phenomena considered. I. Geometrical Optics (Ray Optics): Macroscopic-scale Phenomena (a) considers light to be a RAY that travels in a straight line until it encounters an interface between media. The wavelength and temporal frequency of the light are assumed to be zero and in…nity, respectively: 0; ; ! !1 (b) explains re‡ection and refraction; (c) calculations are simple; (d) useful for designing imaging systems (locate the images and their magni…cations) (e) more di¢ cult to assess the quality of the resulting image II. Physical Optics (Wave Optics): Microscopic-scale Phenomena (a) considers light (electromagnetic radiation) to be a WAVE; (b) action of light is described by Maxwell’s equations; (c) “complicated”mathematical calculations; (d) light has wavelength , frequency , velocity c; (e) leads to explanations of re‡ection, refraction, di¤raction, interference, polarization, dis- persion; (f) Useful for assessing the quality of the images. III. Quantum Optics: Atomic-scale Phenomena (a) light is a PHOTON with both wave-like and particle-like characteristics; (b) used to analyze the interaction of light and matter on a sub-microscopic level; (c) explains the photoelectric e¤ect, lasers. Phenomena in the …rst two categories are most relevant to imaging. You may have already explored the concepts of ray optics in an earlier course. This course considers the principles within the second category. The …rst several chapters are reviews of physical principles that are most relevant to wave optics, starting with oscillations, complex numbers, traveling waves, and the Doppler e¤ect. From there, we will consider electromagnetic waves and Maxwell’s equations, leading to the interaction of light within matter as demonstrated by the phenomenon of dispersion. The interaction of light at an interface between two media is considered next, including some discussion of Fresnel’s equations for re‡ectivity and transmittance. Then follows a discussion of polarization of light and optical birefringence. Finally we will consider the aspects of wave optics that are most relevant to optical imaging systems: interference and di¤raction. As we shall see, these are really just two aspects of the same phenomenon that arises from interactions of light from di¤erent locations as the light propagates. This will lead to the discussion of the fundamental limitation to the image created by an optical system; the constraint to the ability of the image to distinguish point sources that are close together. 0.1 Sources: Many references exist for the subject of wave optics, some from the point of view of physics and many others from the subdiscipline of optics. Unfortunately, relatively few.
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