Color Imaging Fundamentals and Applications

iiiiiiiiColor ImagingiiiiiiiiColor ImagingFundamentals and ApplicationsErik ReinhardErum Arif KhanAhmet O guz Aky uzGarrett JohnsonA K Peters, Ltd.Wellesley, MassachusettsiiiiiiiiEditorial, Sales, and Customer Service OfceA K Peters, Ltd.888 Worcester Street, Suite 230Wellesley, MA 02482www.akpeters.comCopyright 2008 by A K Peters, Ltd.All rights reserved. No part of the material protected by this copyright notice maybe reproduced or utilized in any form, electronic or mechanical, including photo-copying, recording, or by any information storage and retrieval system, withoutwritten permission from the copyright owner.Library of Congress Cataloging-in-Publication DataReinhard, Erik, 1968Color imaging : fundamentals and applications / Erik Reinhard . . . [et al.].p. cm.Includes bibliographical references and index.ISBN: 978-1-56881-344-8 (alk. paper)1. Computer vision. 2. Image processing. 3. Color display systems. 4. Colorseparation. I. Title.TA1634.R45 2007621.367--dc222007015704Printed in India12 11 10 09 08 10 9 8 7 6 5 4 3 2 1iiiiiiiiContentsPreface xiiiI Principles 11 Introduction 31.1 Color in Nature . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Color in Society . . . . . . . . . . . . . . . . . . . . . . . . 101.3 In this Book . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 142 Physics of Light 172.1 Electromagnetic Theory . . . . . . . . . . . . . . . . . . . . 182.2 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 Polarization. . . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Spectral Irradiance . . . . . . . . . . . . . . . . . . . . . . 452.5 Reection and Refraction . . . . . . . . . . . . . . . . . . . 472.6 Birefringence . . . . . . . . . . . . . . . . . . . . . . . . . 632.7 Interference and Diffraction . . . . . . . . . . . . . . . . . . 662.8 Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . 782.9 Geometrical Optics . . . . . . . . . . . . . . . . . . . . . . 842.10 Application: Image Synthesis . . . . . . . . . . . . . . . . . 962.11 Application: Modeling the Atmosphere . . . . . . . . . . . 1042.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.13 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 1203 Chemistry of Matter 1213.1 Classical Physics . . . . . . . . . . . . . . . . . . . . . . . 1223.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 1243.3 Atoms and Ions . . . . . . . . . . . . . . . . . . . . . . . . 138viiiiiiiivi Contents3.4 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.5 Sources of Radiation . . . . . . . . . . . . . . . . . . . . . 1593.6 Polarization in Dielectric Materials . . . . . . . . . . . . . . 1823.7 Dichroism . . . . . . . . . . . . . . . . . . . . . . . . . . . 1903.8 Application: Modeling of Fire and Flames . . . . . . . . . . 1913.9 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 1974 Human Vision 1994.1 Osteology of the Skull . . . . . . . . . . . . . . . . . . . . 2004.2 Anatomy of the Eye . . . . . . . . . . . . . . . . . . . . . . 2014.3 The Retina . . . . . . . . . . . . . . . . . . . . . . . . . . . 2124.4 The Lateral Geniculate Nucleus . . . . . . . . . . . . . . . . 2284.5 The Visual Cortex . . . . . . . . . . . . . . . . . . . . . . . 2304.6 A Multi-Stage Color Model . . . . . . . . . . . . . . . . . . 2374.7 Alternative Theory of Color Vision . . . . . . . . . . . . . . 2454.8 Application: Modeling a Human Retina . . . . . . . . . . . 2474.9 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 2505 Perception 2515.1 Lightness, Brightness, and Related Denitions . . . . . . . . 2525.2 Reectance and Illumination. . . . . . . . . . . . . . . . . 2545.3 Models of Color Processing . . . . . . . . . . . . . . . . . . 2565.4 Visual Illusions . . . . . . . . . . . . . . . . . . . . . . . . 2595.5 Adaptation and Sensitivity . . . . . . . . . . . . . . . . . . 2705.6 Visual Acuity . . . . . . . . . . . . . . . . . . . . . . . . . 2795.7 Simultaneous Contrast . . . . . . . . . . . . . . . . . . . . 2825.8 Lightness Constancy . . . . . . . . . . . . . . . . . . . . . 2865.9 Color Constancy . . . . . . . . . . . . . . . . . . . . . . . . 2955.10 Category-Based Processing . . . . . . . . . . . . . . . . . . 2985.11 Color Anomalies . . . . . . . . . . . . . . . . . . . . . . . 3025.12 Application: Shadow Removal from Images . . . . . . . . . 3095.13 Application: Graphical Design. . . . . . . . . . . . . . . . 3125.14 Application: Telling Humans and Computers Apart . . . . . 3145.15 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 314II Color Models 3176 Radiometry and Photometry 3196.1 The Sensitivity of the Human Eye . . . . . . . . . . . . . . 3206.2 Radiometric and Photometric Quantities . . . . . . . . . . . 3226.3 The Efcacy of Optical Radiation . . . . . . . . . . . . . . 3376.4 Luminance, Brightness, and Contrast . . . . . . . . . . . . . 340iiiiiiiiContents vii6.5 Optical Detectors . . . . . . . . . . . . . . . . . . . . . . . 3426.6 Light Standards. . . . . . . . . . . . . . . . . . . . . . . . 3456.7 Detector Standards . . . . . . . . . . . . . . . . . . . . . . 3466.8 Measurement of Optical Radiation. . . . . . . . . . . . . . 3476.9 Visual Photometry . . . . . . . . . . . . . . . . . . . . . . . 3566.10 Application: Measuring Materials . . . . . . . . . . . . . . 3596.11 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 3627 Colorimetry 3637.1 Grassmanns Laws . . . . . . . . . . . . . . . . . . . . . . 3647.2 Visual Color Matching . . . . . . . . . . . . . . . . . . . . 3667.3 Color-Matching Functions . . . . . . . . . . . . . . . . . . 3737.4 CIE 1931 and 1964 Standard Observers . . . . . . . . . . . 3757.5 Calculating Tristimulus Values and Chromaticities . . . . . . 3787.6 Practical Applications of Colorimetry . . . . . . . . . . . . 3877.7 Application: Iso-Luminant Color Maps . . . . . . . . . . . 3977.8 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 4038 Color Spaces 4058.1 RGB Color Spaces . . . . . . . . . . . . . . . . . . . . . . 4118.2 Printers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4188.3 Luminance-Chrominance Color Spaces . . . . . . . . . . . 4278.4 Television and Video . . . . . . . . . . . . . . . . . . . . . 4308.5 Hue-Saturation-Lightness Spaces . . . . . . . . . . . . . . . 4398.6 HVS Derived Color Spaces . . . . . . . . . . . . . . . . . . 4448.7 Color Opponent Spaces . . . . . . . . . . . . . . . . . . . . 4488.8 Color Difference Metrics . . . . . . . . . . . . . . . . . . . 4598.9 Color Order Systems . . . . . . . . . . . . . . . . . . . . . 4658.10 Application: Color Transfer between Images . . . . . . . . . 4678.11 Application: Color-to-Gray Conversion . . . . . . . . . . . 4748.12 Application: Rendering . . . . . . . . . . . . . . . . . . . . 4788.13 Application: Rendering and Color-Matching Paints . . . . . 4808.14 Application: Classication of Edges . . . . . . . . . . . . . 4848.15 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 4909 Illuminants 4919.1 CIE Standard Illuminants and Sources . . . . . . . . . . . . 4919.2 Color Temperature . . . . . . . . . . . . . . . . . . . . . . 5039.3 Color-Rendering Index . . . . . . . . . . . . . . . . . . . . 5089.4 CIE Metamerism Index . . . . . . . . . . . . . . . . . . . . 5129.5 Dominant Wavelength . . . . . . . . . . . . . . . . . . . . . 5149.6 Excitation Purity . . . . . . . . . . . . . . . . . . . . . . . 5179.7 Colorimetric Purity. . . . . . . . . . . . . . . . . . . . . . 517iiiiiiiiviii Contents9.8 Application: Modeling Light-Emitting Diodes . . . . . . . . 5189.9 Application: Estimating the Illuminant in an Image . . . . . 5209.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 52410Chromatic Adaptation 52510.1 Changes in Illumination . . . . . . . . . . . . . . . . . . . . 52610.2 Measuring Chromatic Adaptation . . . . . . . . . . . . . . . 53010.3 Mechanisms of Chromatic Adaptation. . . . . . . . . . . . 53210.4 Models of Chromatic Adaptation . . . . . . . . . . . . . . . 53810.5 Application: Transforming sRGB Colors to D50 for an ICCWorkow. . . . . . . . . . . . . . . . . . . . . . . . . . . 55310.6 Application: White Balancing a Digital Camera . . . . . . . 55510.7 Application: Color-Accurate Rendering . . . . . . . . . . . 56210.8 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 56411Color and Image Appearance Models 56511.1 Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . 56611.2 Color Appearance Phenomena . . . . . . . . . . . . . . . . 58211.3 Color Appearance Modeling . . . . . . . . . . . . . . . . . 59111.4 Image Appearance Modeling . . . . . . . . . . . . . . . . . 60511.5 Applications of Color and Image Appearance Models . . . . 62011.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 629III Digital Color Imaging 63112Image Capture 63312.1 Optical Image Formation. . . . . . . . . . . . . . . . . . . 63512.2 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64912.3 Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . 65412.4 The Diaphragm. . . . . . . . . . . . . . . . . . . . . . . . 66712.5 The Shutter . . . . . . . . . . . . . . . . . . . . . . . . . . 66812.6 Filters and Coatings . . . . . . . . . . . . . . . . . . . . . . 66912.7 Solid-State Sensors . . . . . . . . . . . . . . . . . . . . . . 67212.8 In-Camera Signal Processing . . . . . . . . . . . . . . . . . 67812.9 A Camera Model . . . . . . . . . . . . . . . . . . . . . . . 68212.10 Sensor Noise Characteristics . . . . . . . . . . . . . . . . . 68312.11 Measuring Camera Noise . . . . . . . . . . . . . . . . . . . 68812.12 Radiometric Camera Calibration. . . . . . . . . . . . . . . 69412.13 Light Field Data . . . . . . . . . . . . . . . . . . . . . . . . 69712.14 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . 70112.15 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 706iiiiiiiiContents ix13High Dynamic Range Image Capture 70913.1 Multi-Exposure Techniques . . . . . . . . . . . . . . . . . . 71013.2 Response Curve Recovery . . . . . . . . . . . . . . . . . . 71513.3 Noise Removal . . . . . . . . . . . . . . . . . . . . . . . . 72213.4 Ghost Removal . . . . . . . . . . . . . . . . . . . . . . . . 72613.5 Image Alignment . . . . . . . . . . . . . . . . . . . . . . . 73313.6 Single Capture High Dynamic Range Images . . . . . . . . 73413.7 Direct High Dynamic Range Capture . . . . . . . . . . . . . 73713.8 Application: Drawing Programs . . . . . . . . . . . . . . . 73913.9 Application: Image-Based Material Editing . . . . . . . . . 74013.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 74114Display Technologies 74314.1 Cathode-Ray Tubes (CRTs) . . . . . . . . . . . . . . . . . . 74314.2 Liquid Crystal Displays (LCDs) . . . . . . . . . . . . . . . 74614.3 Transective Liquid Crystal Displays . . . . . . . . . . . . . 76714.4 Plasma Display Panels (PDPs) . . . . . . . . . . . . . . . . 76814.5 Light-Emitting Diode (LED) Displays . . . . . . . . . . . . 77014.6 Organic Light-Emitting Diode Displays . . . . . . . . . . . 77214.7 Field Emission Displays . . . . . . . . . . . . . . . . . . . 77514.8 Surface-Conduction Electron-Emitter Displays . . . . . . . 77614.9 Microcavity Plasma Devices . . . . . . . . . . . . . . . . . 77714.10 Interferometric Modulator (IMOD) Displays . . . . . . . . . 77714.11 Projection Displays. . . . . . . . . . . . . . . . . . . . . . 77914.12 Liquid Crystal Display (LCD) Projectors . . . . . . . . . . . 78114.13 Digital Light Processing (DLP R ) Projectors . . . . . . . . . 78214.14 Liquid Crystal on Silicon (LCoS) Projectors . . . . . . . . . 78514.15 Multi-Primary Display Devices . . . . . . . . . . . . . . . . 78714.16 High Dynamic Range Display Devices . . . . . . . . . . . . 79114.17 Electronic Ink . . . . . . . . . . . . . . . . . . . . . . . . . 79414.18 Display Characterization . . . . . . . . . . . . . . . . . . . 79414.19 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 80315Image Properties and Image Display 80515.1 Natural Image Statistics . . . . . . . . . . . . . . . . . . . . 80615.2 Dynamic Range . . . . . . . . . . . . . . . . . . . . . . . . 81615.3 Cross-Media Display . . . . . . . . . . . . . . . . . . . . . 82715.4 Gamut Mapping . . . . . . . . . . . . . . . . . . . . . . . . 83315.5 Gamma Correction . . . . . . . . . . . . . . . . . . . . . . 84115.6 Ambient Light . . . . . . . . . . . . . . . . . . . . . . . . . 843iiiiiiiix Contents16Color Management 84916.1 A Generic Color Management System. . . . . . . . . . . . 84916.2 ICC Color Management . . . . . . . . . . . . . . . . . . . . 85116.3 Practical Applications . . . . . . . . . . . . . . . . . . . . . 87417Dynamic Range Reduction 88117.1 Spatial Operators . . . . . . . . . . . . . . . . . . . . . . . 88517.2 Sigmoidal Compression . . . . . . . . . . . . . . . . . . . . 88817.3 Local Neighborhoods . . . . . . . . . . . . . . . . . . . . . 89217.4 Sub-Band Systems . . . . . . . . . . . . . . . . . . . . . . 89517.5 Edge-Preserving Smoothing Operators . . . . . . . . . . . . 89717.6 Gradient-Domain Operators . . . . . . . . . . . . . . . . . . 89917.7 Histogram Adjustment . . . . . . . . . . . . . . . . . . . . 90017.8 Lightness Perception . . . . . . . . . . . . . . . . . . . . . 90117.9 Counter Shading . . . . . . . . . . . . . . . . . . . . . . . 90517.10 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . 90617.11 Validation and Comparison . . . . . . . . . . . . . . . . . . 91017.12 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . 926IV Appendices 929A Vectors and Matrices 931A.1 Cross and Dot Product . . . . . . . . . . . . . . . . . . . . 931A.2 Vector Differentiation . . . . . . . . . . . . . . . . . . . . . 933A.3 Gradient of a Scalar Function . . . . . . . . . . . . . . . . . 933A.4 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . 934A.5 Gauss Theorem . . . . . . . . . . . . . . . . . . . . . . . . 934A.6 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935A.7 Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . 936A.8 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 937A.9 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . 937A.10 Homogeneous Coordinates . . . . . . . . . . . . . . . . . . 937B Trigonometry 939B.1 Sum and Difference Formulae . . . . . . . . . . . . . . . . 939B.2 Product Identities . . . . . . . . . . . . . . . . . . . . . . . 940B.3 Double-Angle Formulae . . . . . . . . . . . . . . . . . . . 941B.4 Half-Angle Formulae . . . . . . . . . . . . . . . . . . . . . 941B.5 Sum Identities . . . . . . . . . . . . . . . . . . . . . . . . . 941B.6 Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 942iiiiiiiiContents xiC Complex Numbers 945C.1 Denition. . . . . . . . . . . . . . . . . . . . . . . . . . . 945C.2 Eulers Formula . . . . . . . . . . . . . . . . . . . . . . . . 946C.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 947C.4 Time-Harmonic Quantities . . . . . . . . . . . . . . . . . . 948D Units and Constants 949E The CIE Luminous Efciency Functions 951F CIE Illuminants 955G Chromaticity Coordinates of Paints 959Bibliography 961Index 1027iiiiiiiiPrefaceColor is one of the most fascinating areas to study. Color forms an integral partof nature, and we humans are exposed to it every day. We all have an intuitiveunderstanding of what color is, but by studying the underlying physics, chemistry,optics, and human visual perception, the true beauty and complexity of color canbe appreciatedat least to some extent. Such understanding is not just importantin these areas of research, but also for elds such as color reproduction, visionscience, atmospheric modeling, image archiving, art, photography, and the like.Many of these application areas are served very well by several specicallytargeted books. These books do an excellent job of explaining in detail some as-pect of color that happens to be most important for the target audience. This isunderstandable as our knowledge of color spans many disciplines and can there-fore be difcult to fathom.It is our opinion that in application areas of computer science and computerengineering, including such exciting elds as computer graphics, computer vi-sion, high dynamic range imaging, image processing and game development, therole of color is not yet fully appreciated. We have come across several applicationsas well as research papers where color is added as an afterthought, and frequentlywrongly too. The dreaded RGB color space, which is really a collection of looselysimilar color spaces, is one of the culprits.With this book, we hope to give a deep understanding of what color is, andwhere color comes from. We also aim to show how color can be used correctlyin many different applications. Where appropriate, we include at the end of eachchapter sections on applications that exploit the material covered. While the bookis primarily aimed at computer-science and computer-engineering related areas,as mentioned above, it is suitable for any technically minded reader with an in-terest in color.In addition, the book can also be used as a text book serving agraduate-level course on color theory. In any case, we believe that to be useful inany engineering-related discipline, the theories should be presented in an intuitivemanner, while also presenting all of the mathematics in a form that allows both adeeper understanding, as well as its implementation.xiiiiiiiiiiixiv PrefaceMost of the behavior of light and color can be demonstrated with simple ex-periments that can be replicated at home. To add to the appeal of this book, wherepossible, we show how to set-up such experiments that frequently require no morethan ordinary household objects. For instance, the wave-like behavior of light iseasily demonstrated with a laser pointer and a knife. Also, several visual illusionscan be replicated at home.We have shied away from such simple experimentsonly when unavoidable.The life cycle of images starts with either photography or rendering, and in-volves image processing, storage, and display.After the introduction of digitalimaging, the imaging pipeline has remained essentially the same for more thantwo decades.The phosphors of conventional CRT devices are such that in theoperating range of the human visual system only a small number of discernibleintensity levels can be reproduced. As a result, there was never a need to captureand store images with a delity greater than can be displayed. Hence the immenselegacy of eight-bit images.High dynamic range display devices have effectively lifted this restriction, andthis has caused a rethinking of the imaging pipeline. Image capturing techniquescan and should record the full dynamic range of the scene, rather than just therestricted range that can be reproduced on older display devices.In this book,the vast majority of the photography was done in high dynamic range (HDR),with each photograph tone-mapped for reproduction on paper. In addition, highdynamic range imaging (HDRI) is integral to the writing of the text, with excep-tions only made in specic places to highlight the differences between conven-tional imaging and HDRI. Thus, the book is as future-proof as we could possiblymake it.AcknowledgmentsNumerous people have contributed to this book with their expertise and help.In particular, we would like to thank Eric van Stryland, Dean and Director ofCREOL, who has given access to many optics labs, introduced us to his col-leagues, and allowed us to photograph some of the exciting research undertakenat the School of Optics, University of Central Florida.Karen Louden, Curatorand Director of Education of the Albin Polasek Museum, Winter Park, Florida,has given us free access to photograph in the Albin Polasek collection.We have sourced many images from various researchers. In particular, weare grateful for the spectacular renderings given to us by Diego Gutierrez andhis colleagues from the University of Zaragoza.The professional photographsdonated by Kirt Witte (Savannah College of Art and Design) grace several pages,and we gratefully acknowledge his help. Several interesting weather phenomenawere photographed by Timo Kunkel, and he has kindly allowed us to reproduceiiiiiiiiPreface xvsome of them. We also thank him for carefully proofreading an early draft of themanuscript.We have had stimulating discussions with Karol Myszkowski,GrzegorzKrawczyk, Rafa Mantiuk, Kaleigh Smith, Edward Adelson and Yuanzhen Li,the results of which have become part of the chapter on tone reproduction. Thischapter also benetted from the source code of Yuanzhen Lis tone-reproductionoperator, made available by Li and her colleagues, Edward Adelson and LavanyaSharan. We are extremely grateful for the feedback received from Charles Poyn-ton, which helped improve the manuscript throughout in both form and substance.We have received a lot of help in various ways, both direct and indirect, frommany people.In no particular order, we gratefully acknowledge the help fromJanet Milliez, Vasile Rotar, Eric G Johnson, Claudiu Cirloganu, Kadi Bouatouch,Dani Lischinski, Ranaan Fattal, Alice Peters, Franz and Ineke Reinhard, GordonKindlmann, Sarah Creem-Regehr, Charles Hughes, Mark Colbert, Jared Johnson,Jaakko Konttinen, Veronica Sundstedt, Greg Ward, Mashhuda Glencross, HelgeSeetzen, Mahdi Nezamabadi, Paul Debevec, Tim Cox, Jessie Evans, MichelleWard, Denise Penrose, Tiffany Gasbarrini, Aaron Hertzmann, Kevin Suffern,Guoping Qiu, Graham Finlayson, Peter Shirley, Michael Ashikhmin, WolfgangHeidrich,Karol Myszkowski,Grzegorz Krawczyk,Rafal Mantiuk,KaleighSmith, Majid Mirmehdi, Louis Silverstein, Mark Fairchild, Nan Schaller, WaltBankes, Tom Troscianko, Heinrich B ulthoff, Roland Fleming, Bernard Riecke,Kate Devlin, David Ebert, Francisco Seron, Drew Hess, Gary McTaggart, HabibZargarpour, Peter Hall, Maureen Stone, Holly Rushmeier, Narantuja Bujantog-toch, Margarita Bratkova, Tania Pouli, Ben Long, Native Visions Art Gallery(Winter Park, Florida), the faculty, staff, and students of the Munsell Color Sci-ence Laboratory, Lawrence Taplin, Ethan Montag, Roy Berns, Val Helmink,Colleen Desimone, Sheila Brady, Angus Taggart, Ron Brinkmann, Melissa An-sweeney, Bryant Johnson, and Paul and Linda Johnson.iiiiiiiiPart IPrinciplesiiiiiiiiChapter 1IntroductionColor is a phenomenon that relates to the physics of light, chemistry of matter,geometric properties of objects, as well as to human visual perception and cog-nition. We may call sunlight yellow,and in this case we refer to a property oflight.A car may be painted red, in which case the color red is an attribute of thecar. When light enters the eye, a complex chain of events leads to the sensation ofcolor, a perceived quantity. Finally, color may be remembered, associated withevents, and reasoned about. These are cognitive aspects of color. Color meansdifferent things under different circumstances [814].At the same time, it is clear that an understanding of color will involve eachof these aspects. Thus,the study of color theory and its applications necessar-ily spans several different elds, including physics, chemistry, optics, radiometry,photometry, colorimetry, physiology, vision science, color appearance modeling,andimageprocessing. Asaresult, what onthesurfaceappearstobearela-tively simple subject, turns out to have many hidden depths. Perhaps this is thereason that in practical applications found in computer sciencecomputer vision,graphics and image processingthe use of color is often under-explored and mis-understood.In our view, to understand color with sufcient depth, and to be able to applythis knowledge to your own area of interest, it is not enough to read the literaturein any one specic discipline, be it computer graphics, computer vision, photog-raphy, art, etc. Instead, it is necessary to step outside ones own eld in order toappreciate the subtleties and complexities of color.The purpose of this book is therefore to explain color theory, its developmentandcurrentstate-of-the-art, aswellasitspracticaluseinengineering-orienteddisciplines such as computer graphics,computer vision,photography,and lm.Along the way, we delve into the physics of light, and its interaction with matter3iiiiiiii4 1.Introductionat the atomic level, such that the origins of color can be appreciated. We nd thattheintimaterelationshipbetweenenergylevels, orbitalstates, andelectromag-netic waves helps to understand why diamonds shimmer, rubies are red, and thefeathers of the blue jay are blue. Even before light enters the eye, a lot has alreadyhappened.The complexities of color multiply when perception is taken into account. Thehuman eye is not a simple light detector by any stretch of the imagination. Humanvision is able to solve an inherently under-constrained problem: it tries to makesense out of a 3D world using optical projections that are two-dimensional. Toreconstruct a three-dimensional world,the human visual system needs to makeagreatmanyassumptionsaboutthestructure ofthe world. It is quiteremark-able how well this system works, given how difcult it is to nd a computationalsolution that only partially replicates these achievements.When these assumptions are violated, the human visual system can be fooledinto perceiving the wrong thing. For instance, if a human face is lit from above, itis instantly recognizable. If the same face is lit from below, it is almost impossi-ble to determine whose face it is. It can be argued that whenever an assumption isbroken, a visual illusion emerges. Visual illusions are therefore important to learnabout how the human visual system operates. At the same time, they are impor-tant, for instance in computer graphics, to understand which image features needto be rendered correctly and which ones can be approximated while maintainingrealism.Color theory is at the heart of this book. All other topics serve to underpinthe importance of using color correctly in engineering applications. We nd thattoooftencoloristakenforgranted, andengineeringsolutions, particularlyincomputer graphics and computer vision, therefore appear suboptimal.To redressthe balance, we provide chapters detailing all important issues governing colorand its perception, along with many examples of applications.We begin this book with a brief assessment of the roles color plays in differentcontexts, including nature and society.1.1 Color in NatureLiving organisms are embedded in an environment with which they interact. Tomaximize survival, they must be in tune with this environment. Color plays animportant role in three ways:Organisms may be colored by default without this giving them a specicadvantage for survival. An example is the green color of most plants (Fig-iiiiiiii1.1.Color in Nature 5Figure 1.1.The chlorophyll in leaves causes most plants to be colored green.ure 1.1), which is due to chlorophyll, a pigment that plays a role in photo-synthesis (see Section 3.4.1).Colorhasevolvedinmanyspeciesinconjunctionwiththecolorvisionof the same or other species, for instance for camouage (Figure 1.2), forattracting partners (Figure 1.3), for attracting pollinators (Figure 1.4), or forappearing unappetizing to potential predators (Figure 1.5).Biochemical substances may be colored as a result of being optimized toserveaspecicfunctionunrelatedtocolor. Anexampleishemoglobinwhich colors blood red due to its iron content. Such functional colors arenormally found inside the body, rather than at the surface.Plants reect green and absorb all other colors of light. In particular, plantsabsorbredandbluelightandusetheenergygainedtodrivetheproductionofcarbohydrates.It has been postulated that these two colors are being absorbed asa result of two individual mechanisms that allow plants to photosynthesize withmaximum efciency under different lighting conditions [518, 769].iiiiiiii6 1.IntroductionFigure 1.2.Many animals are colored similar to their environment to evade predators.Figure 1.3. This peacock uses bright colors to attract a mate; Paignton Zoo, Devon, UK.(Photo by Brett Burridge (www.brettb.com).)iiiiiiii1.1.Color in Nature 7Figure 1.4. Many plant species grow brightly colored owers to attract pollinators suchas insects and bees; Rennes, France, June 2005.Figure 1.5.These beetles have a metallic color, presumably to discourage predators.iiiiiiii8 1.IntroductionFigure 1.6.This desert rose is colored to reect light and thereby better control itstemperature.Color in plants also aids other functions such as the regulation of temperature.In arid climates plants frequently reect light of all colors, thus appearing lightlycolored such as the desert rose shown in Figure 1.6.In humans, color vision is said to have co-evolved with the color of fruit [787,944]. In industrialized Caucasians, color deciencies occur relatively often, whilecolor vision is better developed in people who work the land. Thus, on average,color vision is diminished in people who do not depend on it for survival [216].Human skin color is largely due to pigments such as eumelanin and phaeome-lanin. The former is brown to black, whereas the latter is yellowto reddish-brown.Deeper layers contain yellow carotenoids. Some color in human skin is derivedfrom scattering, as well as the occurrence of blood vessels [520].Light scattering in combination with melanin pigmentation is also the mech-anism that determines eye color in humans and mammals. There appears to bea correlation between eye color and reactive skills. In the animal world, hunterswho stalk their prey tend to have light eye colors,whereas hunters who obtaintheir prey in a reactive manner, such as birds that catch insects in ight, have darkeyes. This difference, as yet to be fully explained, extends to humankind wheredark-eyed people tend to have faster reaction times to both visual and auditorystimuli than light-eyed people [968].iiiiiiii1.1.Color in Nature 9Figure1.7. Color plays an important role in art. An example is this photograph of theTybee Light House, taken by Kirt Witte, which won the 2006 International Color AwardsMasters of Color Photography award in the abstract category for professional photogra-phers (see also www.theothersavannah.com).iiiiiiii10 1.IntroductionFigure 1.8.Statue by Albin Polasek; Albin Polasek Museum, Winter Park, FL, 2004.1.2 Color in SocietyThe use of color by man to communicate is probably as old as mankind itself. Ar-chaeologists have found colored materials, in particular different shades of ochre,at sites occupied some 300,000 years ago by Homo erectus [519]. Approximately70,000 years ago, Homo sapiens neanderthalensis used ochre in burials, a tradi-tion followed later by Homo sapiens sapiens.Color, of course, remains an important means of communication in art [695,1257, 1291] (Figure 1.7), even in cases where color is created by subtle reections(Figure 1.8).Color also plays an important role in religion,where the staining of churchwindowsisusedtoimpresschurchgoers. Theinteriorofachurchreectsthetimesinwhichit wasbuilt, rangingfromlight andairytodarkandsomber(Figure 1.9).iiiiiiii1.2.Color in Society 11Figure 1.9. Color in different church interiors; Rennes, France, June 2005 (top); Mainau,Germany, July 2005 (bottom).With modern technology, new uses of color have come into fashion. Impor-tant areas include the reproduction of color [509], lighting design [941], photogra-phy [697], and television and video [923]. In computer graphics, computer vision,and other engineering disciplines, color also plays a crucial role, but perhaps lessso than it should. Throughout this book, examples of color use in these elds areprovided.iiiiiiii12 1.Introduction1.3 In this BookThisbookoffersanin-depthtreatmentofvarioustopicsrelatedtocolor. Ourintention is to explain only a subset of color science, but to treat each of the topicswe have chosen in depth.Our choice of topics is intended to be relevant to thosewho require a more-than-casual understanding of color for their work, which weenvisage to be in computer graphics,computer vision,animation,photography,image processing, and related disciplines.Alluseofcolorinanydisciplineiseitherexplicitlyorimplicitlybasedontheories developed to describe the physics of light, as well as the perception of itby humans.We therefore offer in the rst chapters a reasonably detailed accountof light, its propagation through vacuum as well as other media, and its interactionwith boundaries.Thephysicsoflight isgovernedbyMaxwellsequations, whichformthebasisofourthinkingaboutthewavenatureoflight. Withouttheseequations,there would not be any physical optics. Since almost everything in radiometryand geometric physics is derived from Maxwells equations, we expect that areasof study further aeld would also look very different without them. Fields affectedinclude photometry, lighting design, computer graphics, and computer vision. Assuch, webeginthisbookwithadetaileddescriptionofelectromagneticwavesand show how radiometry, geometric optics, and much of computer graphics arederived from them.While the wave nature of light constitutes a powerful model in understand-ing the properties of light, the theory of electromagnetic radiation is not able toexplain all measurable light behavior. In particular, light sometimes behaves asparticles, and this fact is not captured by electromagnetic theory. Hence, in Chap-ter 3, we briey introduce quantum mechanics as well as molecular orbital theory,as these form the basic tools for understanding how light interacts with matter atthe atomic scale. Such interaction is the cause of various behaviors such as ab-sorption, emission, diffraction, and dispersion. These theories afford insight intoquestions such as why is water pale blue and why is a ruby red and can sap-phires be blue. Thus, Chapter 3 is largely concerned with providing answers toquestions regarding the causes of color. Whereas Chapter 2 deals with the prop-agationoflight, Chapter3islargelyinformedbytheinteractionoflightwithmatter.Chapter 4 provides a brief introduction to human vision. The optics of the eye,as well as the neurophysiology of what is collectively known as the human visualsystem, aredescribed. Thescienticliteratureonthistopicisvast, andafullaccount of the neuronal processing of visual systems is beyond the scope of thisiiiiiiii1.3.In this Book 13book. However, it is clear that with the advent of more sophisticated techniques,the once relatively straightforward theories of color processing in the visual cor-tex, have progressed to be signicantly less straightforward. This trend is contin-uing to this day.The early inferences made on the basis of single-cell recordingshave been replaced with a vast amount of knowledge that is often contradictory,and every new study that becomes available poses intriguing new questions. Onthewhole, however, itappearsthatcolorisnotprocessedasaseparateimageattribute, butisprocessedtogetherwithotherattributessuchasposition, size,frequency, direction, and orientation.Color can also be surmised from a perceptual point of view. Here, the humanvisualsystemistreatedasablackbox, withoutputsthatcanbemeasured. Inpsychophysical tests, participants are set a task which must be completed in re-sponse to the presentation of visual stimuli. By correlating the task response tothe stimuli that are presented, important conclusions regarding the human visualsystem may be drawn. Chapter 5 describes some of the ndings from this eld ofstudy, as it pertains to theories of color. This includes visual illusions, adaptation,visual acuity, contrast sensitivity, and constancy.In the following chapters, we build upon the fundamentals underlying colortheory. Chapter 6 deals with radiometry and photometry, whereas Chapter 7 dis-cusses colorimetry. Much research has been devoted to color spaces that are de-signed for different purposes. Chapter 8 introduces many of the currently-usedcolor spaces and explains the strengths and weaknesses of each color space.Thepurpose of this chapter is to give transformations between existing color spacesand to enable the selection of an appropriate color space for specic tasks, realiz-ing that each task may require a different color space.Light sources, and their theoretical formulations (called illuminants), are dis-cussed in Chapter 9. Chapter 10 introduces chromatic adaptation, showing that theperception of a colored object does not only depend on the objects reectance,butalsoonitsilluminationandthestateofadaptationoftheobserver. Whilecolorimetry is sufcient for describing colors, an extended model is required toaccountfortheenvironmentinwhichthecolorisobserved. Colorappearancemodels take as input the color of a patch, as well as a parameterized descriptionof the environment. These models then compute appearance correlates that de-scribe perceived attributes of the color, given the environment. Color appearancemodels are presented in Chapter 11.In Part III, the focus is on images, and in particular their capture and display.Much of this part of the book deals with the capture of high dynamic range im-ages, as we feel that such images are gaining importance and may well becomethe de-facto norm in all applications that deal with images.iiiiiiii14 1.IntroductionChapter 12 deals with the capture of images and includes an in-depth descrip-tion of the optical processes involved in image formation, as well as issues relatedto digital sensors. This chapter also includes sections on camera characterization,and more specialized capture techniques such as holography and light eld data.Techniques for the capture of high dynamic range images are discussed in Chap-ter 13. The emphasis is on multiple exposure techniques, as these are currentlymost cost effective, requiring only a standard camera and appropriate software.DisplayhardwareisdiscussedinChapter 14, includingconventional andemerging display hardware. Here, the focus is on liquid crystal display devices, asthese currently form the dominant display technology. Further, display calibrationtechniques are discussed.Chapter 15 is devoted to a discussion on natural image statistics, a eld im-portant as a tool both to help understand the human visual system,and to helpstructure and improve image processing algorithms. This chapter also includessections on techniques to measure the dynamic range of images, and discussescross-mediadisplaytechnology, gamutmapping, gammacorrection, andalgo-rithms for correcting for light reected off display devices. Color management forimages is treated in Chapter 16, with a strong emphasis on ICC proles. Finally,Chapter 17 presents current issues in tone reproduction, a collection of algorithmsrequired to prepare a high dynamic range image for display on a conventional dis-play device.For each of the topics presented in the third part of the book,the emphasisis on color management, rather than spatial processing. As such, these chaptersaugment, rather than replace, current books on image processing.The book concludes with a set of appendices, which are designed to help clar-ify the mathematics used throughout the book (vectors and matrices, trigonome-try, and complex numbers), and to provide tables of units and constants for easyreference. We also refer to the DVD-ROMincluded with the book, which containsa large collection of images in high dynamic range format, as well as tonemappedversions of these images (in JPEG-HDR format for backward compatibility), in-cludedforexperimentation. TheDVD-ROMalsocontainsarangeofspectralfunctions, a metameric spectral image, as well as links to various resources on theInternet.1.4 Further ReadingThe history of color is described in Nassaus book [814],whereas some of thehistory of color science is collected in MacAdams books [717, 720]. A historicaliiiiiiii1.4.Further Reading 15overview of dyes and pigments is available in Colors: The Story of Dyes and Pig-ments [245]. An overview of color in art and science is presented in a collectionof papers edited by Lamb and Bourriau [642]. Finally, a history of color order,including practical applications, is collected in Rolf Kuehnis Color Space and itsDivisions: Color Order from Antiquity to Present [631].iiiiiiiiChapter 2Physics of LightLight travels through environments as electromagnetic energy that may interactwith surfaces and volumes of matter. Ultimately, some of that light reaches thehuman eye which triggers a complicated chain of events leading to the perception,cognition, and understanding of these environments.To understand the physicalaspects of light, i.e., everythingthat happenstolight before it reaches the eye, we have to study electromagnetism. To understandthe various ways by which colored light may be formed, we also need to know alittle about quantummechanics and molecular orbital theory (discussed in Chapter3). These topics are not particularly straightforward, but they are nonetheless wellworth studying. They afford insight into the foundations of many elds relatedto color theory, such as optics, radiometry (and therefore photometry), as well ascomputer graphics.The physical properties of light are well modeled by Maxwells equations. Wetherefore begin this chapter with a brief discussion of Maxwells equations. Wethen discuss various optical phenomena that may be explained by the theory ofelectromagnetic radiation. These include scattering, polarization, reection, andrefraction.There are other optical phenomena that involve the interaction between lightand materials at the atomic structure. Examples of these phenomena are diffrac-tion, interference, and dispersion, each capable of separating light into differentwavelengthsandarethereforeperceivedasproducingdifferentcolors. Withthe exceptionof interference and diffraction, the explanationof these phenom-ena requires some insight into the chemistry of matter. We therefore defer theirdescription until Chapter 3.In addition to presenting electromagnetic theory,in this chapter we also in-troduce the concept of geometrical optics,which provides a simplied view of17iiiiiiii18 2.Physics of Lightthe theory of light. It gives rise to various applications, including ray tracing inoptics. It is also the foundation for all image synthesis as practiced in the eld ofcomputer graphics. We show this by example in Section 2.10.Thus, the purpose of this chapter is to present the theory of electromagneticwaves and to show how light propagates through different media and behaves nearboundaries and obstacles. This behavior by itself gives rise to color. In Chapter 3,we explain how light interacts with matter at the atomic level, which gives rise toseveral further causes for color, including dispersion and absorption.2.1 Electromagnetic TheoryLight maybemodeledbyatransverseelectromagnetic(TEM)wavetravelingthroughamedium. Thissuggeststhatthereisaninteractionbetweenelectricand magnetic elds and their sources of charge and current. A moving electriccharge is the source of an electric eld. At the same time, electric currents pro-duce a magnetic eld. The relationship between electric and magnetic elds aregoverned by Maxwells equations which consist of four laws:Gauss law for electric elds;Gauss law for magnetic elds;Faradays law;Amperes circuital law.We present each of these laws in integral form rst, followed by their equiv-alentdifferentialform. Theintegralformhasamoreintuitivemeaningbutisrestricted to simple geometric cases, whereas the differential form is valid for anypoint in space where the vector elds are continuous.There are several systems of units and dimensions used in Maxwells equa-tions, including Gaussian units, Heaviside-Lorentz units, electrostatic units, elec-tromagnetic units, and SI units [379]. There is no specic reason to prefer onesystem over another. Since the SI system is favored in engineering-oriented dis-ciplines, we present all equations in this system. In the SI system, the basic quan-titiesarethemeter(m)forlength, thekilogram(kg)formass, thesecond(s)for time, the ampere (A) for electric current, the kelvin (K) for thermodynamictemperature, themole(mol)foramountofsubstance, andthecandela(cd)forluminous intensity. (see Table D.3 in Appendix D).iiiiiiii2.1.Electromagnetic Theory 19R-FFeQ-eQqQFigure2.1. TwochargesQandqexert equal, but oppositeforceFuponeachother(assuming the two charges have equal sign).2.1.1 Electric FieldsGiven a three-dimensional space, we may associate attributes with each point inthis space. For instance, the temperature in a room may vary by location in theroom. With heat tending to rise, the temperature near the ceiling is usually higherthan near the oor.Similarly, itispossibletoassociateotherattributestoregionsinspace. Ifthese attributes have physical meaning,we may speak of aeld,which simplyindicates that there exists a description of how these physical phenomena changewith position in space. In a time-varying eld, these phenomena also change withtime.For instance, we could place an electrical charge in space and a second chargesome distance away. These two charges exert a force on each other dependentupon the magnitude of their respective charges and their distance. Thus, if wemove the second charge around in space, the force exerted on it by the rst chargechanges (and vice versa). The two charges thus create a force eld.The forceFthat twocharges Qandqexert oneachother is givenbyCoulombs law (see Figure 2.1):F =14 0QqR2 eQ. (2.1)Inthisequation, Risthedistancebetweenthecharges, andtheconstant 0 =136 109(in Farad/meter) is called the permittivity of vacuum. The vector eQ isa unit vector pointing from the position of one charge to the position of the other.If we assume that Q is an arbitrary charge and thatq is a unit charge,thenwe can compute the electric eld intensity E by dividing the left- and right-handiiiiiiii20 2.Physics of LightRE-eQqQ1111-eQ2E2EQ2R2Figure2.2. The electric eld intensity E at the position of chargeq is due to multiplecharges located in space.sides of Equation (2.1) by q:E = Fq=Q4 0R2eQ. (2.2)Iftherearemultiplechargespresent, thentheelectriceldintensityattheposition of the test charge is given by the sum of the individual eld intensities(Figure 2.2):E =Ni=1Qi4 0R2ieQi. (2.3)Finally, if we scale the electric eld intensity E by the permittivity of vacuum(0), we obtain what is called the electric ux density, indicated by the vector Dwhich points in the same direction as E:D =0E =Ni=1Qi4 R2ieQi. (2.4)Formediaotherthanvacuum, thepermittivitywill generallyhaveadifferentvalue. In that case, we drop the subscript 0, so that, in general, we haveD =E. (2.5)This equationmay be seen as relating the electric eld intensity to the electricuxdensity, wherebythedifferencebetweenthetwovectorsisinmost casesdeterminedbyaconstant uniquetothespecicmedium. Thisconstant, , iscalled the materials permittivity or dielectric constant. Equation (2.5) is one ofiiiiiiii2.1.Electromagnetic Theory 21three so-called material equations. The remaining two material equations will bediscussed in Sections 2.1.3 and 2.1.8.If,instead of a nite number of separate charges,we have a distribution ofcharges over space, the electric ux density is governed by Gauss law for electricelds.2.1.2 Gauss Law for Electric FieldsIn a static or time-varying electric eld there is a distribution of charges. The rela-tionship between an electric eld and a charge distribution is quantied by Gausslaw for electric elds. It states that the total electric ux, D= E, emanating froma closed surface s is equal to the electric charge Q enclosed by that surface:1_sD n ds = Q. (2.6)The integral is over the closed surface s and n denotes the outward facing surfacenormal. If the charge Q is distributed over the volume v according to a chargedistribution function (also known as electric charge density), we may rewriteGauss law as follows: _sD n ds =_v dv. (2.7)Thus, a distribution of charges over a volume gives rise to an electric eld that maybe measured over a surface that bounds that volume. In other words, the electricux emanating from an enclosing surface is related to the charge containedbythat surface.2.1.3 Magnetic FieldsWhile charges may create an electric eld, electric currents may create a magneticeld. Thus, analogous to electric ux, we may speak of magnetic ux that has theability to exert a force on either a magnet or another electric current.Given that an electric current is nothing more than a ow of moving charges,it is apparent that a magnetic eld can only be produced from moving charges;stationary charges do not produce a magnetic eld. Conversely, a magnetic eldhas the ability to exert a force on moving charged particles.The magnetic ux density associated with a magnetic eld is indicated by B.A charged particle with a charge Q moving with velocity v through a magneticeld with a ux density of B is pulled by a force F which is given byF = QvB. (2.8)1SeealsoAppendixAwhichprovidesthefundamentalsofvectoralgebraandincludesfurtherdetail about the relationship between integrals over contours, surfaces, and volumes.iiiiiiii22 2.Physics of LightvQBEv BE + v BFFigure 2.3. The Lorentz force equation: F is the sum of E and the cross product of theparticles velocity v and the magnetic ux density B at the position of the particle.However,according to (2.2),a particle with chargeQ is also pulled by a forceequal to F = QE. The total force exerted on such a particle is given by the su-perposition of electric and magnetic forces, which is known as the Lorentz forceequation (Figure 2.3):F = Q (E+vB). (2.9)This equation thus provides a method to determine the motion of a charged par-ticle as it moves through a combined electric and magnetic eld.Given the cross-product between the magnetic ux density and the velocityvector in the above equation, we deduce that the direction of the magnetic force isperpendicular to both the direction of movement and the direction of the magneticux. As such, this force has the ability to change the direction of the motion, butnot the magnitude of the charged particles velocity. It also does not change theenergy associated with this particle.On the other hand, the electric eld exerts a force that is independent of themotion of the particle. As a result, energy may be transferred between the eldand the charged particle.The magnetic ux density B has a related quantity called the magnetic vector,indicatedbyH. TherelationshipbetweenBandHisgovernedbyamaterialconstant called the magnetic permeability:B = H. (2.10)Thisequationisthesecondofthreematerial equationsandwill bediscussedfurther in Section 2.1.8.2.1.4 Gauss Law for Magnetic FieldsWhile electric elds are governed by Gauss law for electric elds (the electricux emanating from a closed surface depends on the charge present in the volumeiiiiiiii2.1.Electromagnetic Theory 23enclosed by that surface), magnetic elds behave somewhat differently. The totalmagnetic ux emanating from a closed surface bounding a magnetic eld is equalto zero; this constitutes Gauss law for magnetic elds:_sB n ds = 0. (2.11)This equation implies that no free magnetic poles exist. As an example, the mag-netic ux emanating from a magnetic dipole at its north pole is matched by theux directed inward towards its south pole.2.1.5 Faradays LawAs shown earlier, currents produce a magnetic eld. Conversely, a time-varyingmagneticeldiscapableofproducingacurrent. Faradayslawstatesthattheelectric eld induced by a time-varying magnetic eld is given by_cE n dc = ddt_sB n ds. (2.12)Here, the left-hand side is an integral over the contourc that encloses an opensurface s. The quantity integrated is the component of the electric eld intensity Enormal to the contour. The right-hand side integrates the normal component of Bover the surface s. Note that the right-hand side integrates over an open surface,whereas the integral in (2.11) integrates over a closed surface.2.1.6 Amperes Circuital LawGiven a surface area s enclosed by a contour c, the magnetic ux density alongthis contour is related to the total current passing through area s. The total currentis composed of two components, namely a current as a result of moving chargedparticles and a current related to changes in electric ux density. The latter currentis also known as displacement current.Moving charged particles may be characterized by the current ux density j,whichisgiveninamperepersquaremeter(A/m2). Ifelectricchargeswithadensity of are moving with a velocity v, the current ux density j is given byj = v. (2.13)If the current ux density j is integrated over surface area, then we nd the totalcharge passing through this surface per second (coulomb/second = ampere; C/s =A). Thus, the current resulting from a ow of charges is given by_sjn ds. (2.14)iiiiiiii24 2.Physics of LightThedisplacement current dependsontheelectricuxdensity. If wein-tegratetheelectricuxdensityoverthesamesurfacearea, weobtaincharge(coulomb; C):_sD n ds. (2.15)Ifwedifferentiatethisquantitybytime, theresultisachargepassingthroughsurface s per second, i.e., current:ddt_sD n ds. (2.16)The units in (2.14) and (2.16) are now both in coulomb per second and are thusmeasures of current. Both types of current are related to the magnetic ux densityaccording to Amperes circuital law:_cH n dc =_sjn ds +ddt_sD nds. (2.17)This law states that a time-varying magnetic eld can be produced by the ow ofcharges (a current), as well as by a displacement current.2.1.7 Maxwells EquationsBoth of Gauss laws, Faradays law, and Amperes circuital law together form aset of equations that are collectively known as Maxwells equations. For conve-nience we repeat them here:_sD n ds =_v dv; Gauss law for electric elds (2.18a)_sB n ds = 0; Gauss law for magnetic elds(2.18b)_cE n dc = ddt_sB n ds; Faradays law (2.18c)_cH n dc =_sjn ds +ddt_sD n ds. Amperes circuital law (2.18d)The above equations are given in integral form. They may be rewritten in dif-ferential form,after which these equations hold for points in space where bothelectricandmagneticeldsarecontinuous. Thisfacilitatessolvingthesefoursimultaneous equations.StartingwithGausslawforelectricelds, weseethat theleft-handsideof (2.18a) is an integral over a surface, whereas the right-hand side is an integraliiiiiiii2.1.Electromagnetic Theory 25over a volume. As we are interested in a form of Maxwells equations that is validfor points in space, we would like to replace the left-hand side with an integraloverthevolumeunderconsideration. ThismaybeaccomplishedbyapplyingGauss theorem (see Appendix A), yielding_sD n ds =_vD dv. (2.19)Combining this result with (2.18a) yields_vD dv =_v dv, (2.20)and, therefore, in the limit when the volume v goes to zero, we haveD =. (2.21)A similar set of steps may be applied to Gauss law for magnetic elds, and thiswill result in a similar differential form:B = 0. (2.22)Amperes law, given in (2.18d), is stated in terms of a contour integral on the left-hand side and a surface integral on the right-hand side. Here, it is appropriate toapply Stokes theoremto bring both sides into the same domain (see Appendix A):_cH n dc =_sH n ds. (2.23)Substituting this result into (2.18d) yields a form where all integrals are over thesame surface. In the limit when this surface area s goes to zero, Equation (2.18d)becomesH=j + Dt, (2.24)which is the desired differential form.Finally, Faradays law may also be rewrit-ten by applying Stokes theorem applied to the left-hand side of (2.18c):_cE n dc =_sE n ds = _sBtn ds. (2.25)Undertheassumptionthattheareasbecomesvanishinglysmall, thisequationyieldsE = Bt. (2.26)Faradays law and Amperes law indicate that a time-varying magnetic eldhas the ability to generate an electric eld, and that a time-varying electric eldiiiiiiii26 2.Physics of Lightgenerates a magnetic eld. Thus, time-varying electric and magnetic elds cangenerate each other. This property forms the basis for wave propagation, allow-ing electric and magnetic elds to propagate away from their source. As light canbe considered to consist of waves propagating through space, Maxwells equa-tions are fundamental to all disciplines involved with the analysis, modeling, andsynthesis of light.2.1.8 Material EquationsThefourlawsthatcompriseMaxwellsequationsarenormallycomplementedwiththreematerial equations(alsoknownasconstitutiverelations). TwoofthesewerepresentedinEquations(2.5) and(2.10) andarerepeatedhereforconvenience:D =E; (2.27a)B = H. (2.27b)The third material equation relates the current ux density j to the electric eldintensity E according to a constant and is known as Ohms law. It is given herein differential form:j =E. (2.27c)Material Material Good conductorsSilver 6.17107Tungsten 1.82107Copper 5.8107Brass 1.5107Gold 4.1107Bronze 1.0107Aluminium 3.82107Iron 1.0107Poor conductorsWater (fresh) 1.0103Earth (dry) 1.0103Water (sea) 4.0100Earth (wet) 3.0102InsulatorsDiamond 2.01013Porcelain 1.01010Glass 1.01012Quartz 1.01017Polystyrene 1.01016Rubber 1.01015Table 2.1.Conductivity constants for several materials [532].iiiiiiii2.1.Electromagnetic Theory 27Theconstant is calledthespecicconductivity, andis determinedbythemedium.Values of for some materials are listed in Table 2.1.Materials with aconductivity signicantly different from 0 are called conductors. Metals, such asgold, silver, and copper are good conductors. On the other hand, materials withlow conductivity ( 0) are called insulators or dielectrics . Finally, for somematerials, (semiconductors) the conductivity increases with increasing tempera-ture. See also Section 3.3.3.Formostmaterials, themagneticpermeability willbecloseto1. Somematerials, however, have a permeability signicantly different from 1, and theseare then called magnetic.The speed v at which light travels through a medium is related to the materialconstants as follows:v =1 . (2.28)WewillderivethisresultinSection2.2.4. Thesymbol 0isreservedforthepermeability of vacuum. The value of 0 is related to both permittivity 0 and thespeed of light c in vacuum as follows:c =100. (2.29)Values for all three constants are given in Table 2.2. The permittivity and perme-ability of materials is normally given relative to those of vacuum: =0r; (2.30a) = 0r. (2.30b)Values of r and r are given for several materials in Tables 2.3 and 2.4.Normally,the three material constants, , ,and ,are independent of theeld strengths. However,this is not always the case. For some materials thesevalues also depend on past values of E or B. In this book, we will not considersuch effects of hysteresis. Similarly, unless indicated otherwise, the material con-stants are considered to be isotropic, which means that their values do not changeConstant Value Unitc 3 108m s10136109C2s2kg1m3= Fm104107kg m C2= Hm1Table 2.2.The speed of light c, permittivity 0, and permeability 0 (all in vacuum).iiiiiiii28 2.Physics of LightMaterial rMaterial rAir 1.0006 Paper 24Alcohol 25 Polystyrene 2.56Earth (dry) 7 Porcelain 6Earth (wet) 30 Quartz 3.8Glass 410 Snow 3.3Ice 4.2 Water (distilled) 81Nylon 4 Water (sea) 70Table 2.3.Dielectric constants r for several materials [532].Material rMaterial rAluminium 1.000021 Nickel 600.0Cobalt 250.0 Platinum 1.0003Copper 0.99999 Silver 0.9999976Gold 0.99996 Tungsten 1.00008Iron 5000.0 Water 0.9999901Table 2.4.Permeability constants r for several materials [532].with spatial orientation. Materials which do exhibit a variation in material con-stants with spatial orientation are called anisotropic.2.2 WavesMaxwells equations form a set of simultaneous equations that are normally dif-cult to solve. In this section, we are concerned with nding solutions to Maxwellsequations, and to accomplish this, we may apply simplifying assumptions. Theassumptions outlined in the previous section, that material constants are isotropicand independent of time, are the rst simplications. To get closer to an appro-priatesolutionwithwhichwecanmodellightandopticalphenomena, furthersimplications are necessary.Inparticular, areasonableclassofmodelsthat aresolutionstoMaxwellsequations is formed by time-harmonic plane waves. With these waves, we can ex-plain optical phenomena such as polarization, reection, and refraction. We rstderive the wave equation, which enables the decoupling of Maxwells equations,and therefore simplies the solution. We then discuss plane waves, followed bytime-harmonic elds, and time-harmonic plane waves. Each of these steps con-stitutes a further specialization of Maxwells equations.iiiiiiii2.2.Waves 292.2.1 The Wave EquationWeassumethatthesourceofanelectromagneticwaveissufcientlyfarawayfrom a given region of interest. In this case, the region is called source-free. Insuch a region,the charge and current distribution j will be 0,and as a resultMaxwells equations reduce to a simpler form:D = 0; (2.31a)B = 0; (2.31b)E = Bt; (2.31c)H=Dt. (2.31d)We may apply the material equations for D and H to yield a set of Maxwellsequations in E and B only: E = 0; (2.32a)B = 0; (2.32b)E = Bt; (2.32c)1B = Et. (2.32d)This set of equations still expresses E in terms of B and B in terms of E. Never-theless, this result may be decoupled by applying the curl operator to (2.32c) (seeAppendix A):E = t B. (2.33)Substituting (2.32d) into this equation then yieldsE = 2Et2. (2.34)To simplify this equation, we may apply identity (A.23) from Appendix A:(E) 2E = 2Et2. (2.35)From (2.32a), we know that we may set E to zero in (2.35) to yield the standardequation for wave motion of an electric eld,2E 2Et2= 0. (2.36)iiiiiiii30 2.Physics of LightFor a magnetic eld, a similar wave equation may be derived:2B 2Bt2= 0. (2.37)Thetwowaveequationsdonotdependoneachother, therebysimplifyingthesolution of Maxwells equations. This result is possible, because the charge andcurrent distributions are zero in source-free regions, and, therefore, we could sub-stitute E = 0 in (2.35) to produce (2.36) (and similarly for (2.37)). Thus, bothwave equations are valid for wave propagation problems in regions of space thatdo not generate radiation (i.e.,they are source-free). Alternatively,we may as-sume that the source of the wave is sufciently far away.Undertheseconditions, wearethereforelookingforsolutionstothewaveequations, ratherthansolutionstoMaxwellsequations. Onesuchsolutionisafforded by plane waves, which we discuss next.2.2.2 Plane WavesA plane wave may be thought of as an innitely large plane traveling in a givendirection. We will show that plane waves form a solution to the wave equationsderived in the preceding section. For a position vector r(x, y, z) and unit directionvector s = (sx, sy, sz), the solution will be of the form E =f (rs, t). A plane isthen dened asrs = constant. (2.38)For convenience, we choose a coordinate system such that one of the axes, say z,is aligned with the direction of propagation, i.e., the surface normal of the plane iss = (0, 0, 1). As a result, we have r s =z. Thus, if we consider a wave modeled byan innitely large plane propagating through space in the direction of its normalrs, wearelookingforasolutionwherebythespatialderivativesintheplaneare zero and the spatial derivatives along the surface normal are non-zero. TheLaplacian operators in (2.36) and (2.37) then simplify to22E = 2Ez2. (2.39)Substituting into (2.36) yields2Ez2 2Et2= 0. (2.40)2In this section we show results for E and note that similar results may be derived for B.iiiiiiii2.2.Waves 31A general solution to this equation is given byE =E1_rs 1 t_+E2_rs +1 t_(2.41a)=E1 (rs vt) +E2 (rs +vt), (2.41b)with E1 and E2 arbitrary functions.3The general solution to the wave equationstherefore consists of two planes propagating into opposite directions (+rs andrs).For simplicity, we consider only one of the planes and momentarily assumethat (a partial) solution is given by a single plane traveling in the +z direction:E =E1(rs vt). (2.42)We will now show that for plane waves the vectors Eand Hare both perpendicularto the direction of propagation s,and also that E and H are orthogonal to eachother. Webeginbydifferentiatingtheaboveexpressionwithrespect totime,which producesEt= vE/, (2.43)where the prime indicates differentiation with respect to the argument rs vtof E.Next, we consider the curl of E.The x-component of the curl of E is givenby(E)x = Ezy Eyz(2.44a)= E/zsyE/ysz(2.44b)=_sE/_x . (2.44c)The curl for the y- and z-components can be derived similarly. By substitutionof (2.43) and (2.44) into Maxwells equations(2.31) and applyingthe materialequations (2.5) and (2.10), we ndsH/ +vE/ = 0, (2.45a)sE/v H/ = 0. (2.45b)Using (2.29), we nd that v =_/ and v =_/. Integrating the previousequation then givesE = _ sH, (2.46a)H=_sE. (2.46b)3We will derive expressions for these functions in the following sections.iiiiiiii32 2.Physics of LightTo show that plane waves are transversal, i.e., both E and H are perpendicularto the direction of propagation s, we take the dot product between E and s and Hand s:E s =__ sH_ s, (2.47a)H s =__sE_ s. (2.47b)Using (A.6) in Appendix A, we nd that E s = H s = 0, thereby proving that Eand H are perpendicular to s. Such waves are called transversal electromagnetic(TEM) waves.These equations also imply that s = EH. The vector s is generally knownasthePoyntingvectororpowerdensityvectorandplaysanimportantroleinquantifying energy. This can be seen from the units employed in vectors E (V/m)and H(A/m), so that s is specied in watts per square meter (V/m A/m=W/m2).For planar waves in homogeneous media, the Poynting vector thus points in thedirectionofwavepropagation, anditslengthisameasureoftheenergyowinduced by the electromagnetic eld (discussed further in Section 2.4). In keepingwith general practice, we will use the symbol S when referring to the Poyntingvector in the remainder of this book.2.2.3 Time-Harmonic FieldsA reasonable approach to modeling the oscillating wave behavior of light in freespaceisbyusingsinusoidalfunctions. Thus, weassumethatthecurrentandcharge distributions, j and , at the source vary with time asa cos(t +), (2.48)where a is the amplitude and t + is the phase. The current and charge sourcesvary with time t as well as space r and are therefore written as j(r, t) and (r, t).UsingtheresultsfromAppendixC.4, wemayseparatetheseeldsintoaspatial component and a time-varying component:j(r, t) =j(r) eit, (2.49a)(r, t) =(r) eit. (2.49b)iiiiiiii2.2.Waves 33Since Maxwells equations are linear, this results in the following set of equations:Deit= eit, (2.50a)Beit= 0, (2.50b)_Eeit_= iBeit, (2.50c)_Heit_= iDeit+jeit. (2.50d)Note that all the underlined quantities are (complex) functions of space r only.In addition, the time-dependent quantity eitcancels everywhere. For a homoge-neous eld, we may set j and to zero. For a homogeneous eld in steady statewe therefore obtain the following set of equations:D = 0, (2.51a)B = 0, (2.51b)E = iB, (2.51c)H= iD. (2.51d)By a procedure similar to the one shown in Section 2.2.1, wave equations for theelectric eld E and the magnetic eld E may be derived:2E2 E = 0, (2.52a)2B2 B = 0. (2.52b)2.2.4 Harmonic Plane WavesIn this section we will nd a solution to the plane wave equations of harmonicwaves in homogeneous media, i.e., a solution to Equations (2.52). We will startby assuming that the wave is traveling through a dielectric medium, i.e., the con-ductivity of the material is close to zero ( 0). Later in this section, we willconsider conductive materials.For general plane waves we assumed that the planes are traveling in the +zand z directions.As both E and H (and thereby D and B) are transversal to thedirection or propagation, we nd thatEx= 0,Bx= 0, (2.53a)Ey= 0,By= 0. (2.53b)iiiiiiii34 2.Physics of LightFrom Faradays law, we nd identities for the partial derivatives in the z directionsfor E:Eyz= i Bx, (2.54a)Exz= i By, (2.54b)0 = i Bz. (2.54c)Similar solutions are found from Amperes law for B:Byz= i Ex, (2.54d)Bxz= i Ey, (2.54e)0 = i Ez. (2.54f)From these equations, we nd that the components of E and B in the direction ofpropagation (z) are zero.Differentiation of (2.54b) and substitution from (2.54d)leads to2Exz2= i Byz= 2 Ex. (2.55)As we assume that the eld is uniform, Exis a function of z only, and we maythereforereplacethepartial derivativeswithordinaryderivatives, yieldingthewave equation for harmonic plane waves ,d2Exdz2+2 Ex = 0. (2.56)Similar equations may be set up for Ey, Bx, and By. By letting2=2 , (2.57)a general solution for (2.56) is given byEx =E+mei z+Emei z. (2.58)For arguments sake, we will assume that the newly introduced constants E+mandEmare real, and therefore we replace them with E+mand Em.The solution of thewave equation is then the real part of Exei t:Ex(z, t) = ReExei t (2.59a)= ReE+mei(t z)+Emei(t+ z) (2.59b)=E+m cos(t z) +Em cos(t + z). (2.59c)iiiiiiii2.2.Waves 35Thus, the solution to the wave equation for harmonic plane waves in homogeneousmedia may be modeled by a pair of waves, one propagating in the +z directionand the other traveling in the z direction.Turning our attention to only one of the waves, for instance E+m cos(t z),it is clear that by keeping to a single position z, the wave produces an oscillationwith angular frequency .The frequencyf of the wave and its period Tmay bederived from as follows:f =2=1T . (2.60)At the same time, the wave E+m cos(t z) travels through space in the positivez direction,which follows from the z component of the waves phase. Thevalueofthecosinedoesnotalterifweaddorsubtractmultiplesof2tothephase. Hence, we havet +z =t +(z +) +2. (2.61)Solving for , which is called wavelength, and combining with (2.57) we get = 2=2 . (2.62)With the help of Equations (2.29) and (2.60), we nd the well-known result thatthe wavelength of a harmonic plane wave relates to its frequency by means ofthe speed of light: = vpf. (2.63)The phase velocity vp may be viewed as the speed with which the wave prop-agates. This velocity may be derived by setting the phase value to a constant:t z =C z = t C . (2.64)For a wave traveling in the z direction through vacuum, the velocity of the waveequals the time derivative in z:vp = dzdt= =1 . (2.65)Thus, we have derived the result of (2.28). In vacuum, the phase velocity is vp =c.For conductive media, the derivation is somewhat different because the elec-tric eld intensity j =E is now not zero. Thus, Amperes law is given byH=E+iE. (2.66)iiiiiiii36 2.Physics of LightRewriting this equation, we haveH= i_ i_E. (2.67)Theprevioussolutionfordielectricmaterialsmaythereforebeextendedtoin-clude conductive media by substitution of by i . As a result, the solutionof (2.58) should be rewritten asEx =E+m ei z+Emei z, (2.68)where is complex: = i_ i_. (2.69)The real and complex parts of are given by = Re = 2_1+_ _21, (2.70a) = Im = 2_1+_ _2+1. (2.70b)The complex form of the electric eld is therefore given by (compare with (2.58))Ex =E+mezei z+Em e zei z. (2.71)As before,the real part of this complex solution is of interest. By splitting thecomplex amplitude E+minto a real part and a phase part, i.e., E+m = E+m ei +, anddoing the same for Em, the real part of the solution is given byEx(z, t) = E+m e zcos_t z ++_(2.72a)+Em e zcos_t + z +_(2.72b)It is easy to observe that this result is an extension of the dielectric case discussedearlier in this section. In particular, for dielectrics with = 0, the solution is realbecause = 0. In addition, for dielectrics the value of is the same as before.For a wave propagating along the +z-axis, we see that its amplitude is mod-ulated by a factore z. This means that its amplitude becomes smaller as thewave progresses through the material. A consequence of this is that waves canonly penetrate a conductive medium by a short distance. As such, we note thatconductive materials are generally not transparent.Conversely, for dielectric materials with very low conductivity, the value ofalpha tends to zero, so that the amplitude of waves propagating through dielectricsis not signicantly attenuated. Hence, dielectric materials tend to be transparent.iiiiiiii2.2.Waves 3710 -13 10 -10 1 10 22 10 -9 1 nm 10 -6 1 m 10 5 10 15 f { { radiowavesmicrowaves infraredvisible lightultravioletX-rays-rays10 12 10 9 10 6 10 3 1 THz1 GHz1 MHz1 kHz 10 0 Figure 2.4.The electromagnetic spectrum.2.2.5 The Electromagnetic Spectrum and SuperpositionSo far, we have discussed harmonic plane waves with a specic given wavelength.Such waves are called monochromatic. The wavelength can be nearly anything,ranging for instance from 1013m for gamma rays to 105m for radio frequen-cies [447]. This range of wavelengths is known as the electromagnetic spectrumand is presented in Figure 2.4.The range of wavelengths that the human eye is sensitive to is called the visiblespectrum and ranges from roughly 380 nm to 780 nm. This range constitutes onlya very small portion of the electromagnetic spectrum and coincides with the rangeof wavelengths that interacts sensibly with materials. Longer wavelengths havelower energies associated with them, and their interaction with materials is usuallylimited to the formation of heat. At wavelengths shorter than the visible range, theenergy is strong enough to ionize atoms and permanently destroy molecules [815].As such, only waves with wavelengths in the visible range interact with the atomicstructure of matter in such a way that it produces color.So far, our discussion has centered on monochromatic light; the shape of thewave was considered to be sinusoidal. In practice, this is rarely the case. Ac-cording to the Fourier theorem, more complicated waves may be constructed bya superposition of sinusoidal waves of different wavelengths (or, equivalently, ofdifferent frequencies). This leads to the superposition principle, which is givenhere for a dielectric material (compare with (2.59c)):Ex(z, t) =_=0_E+m,cos(t z) +Em,cos(t + z)_d . (2.73)In addition, radiators tend to emit light at many different wavelengths. In Fig-ure 2.5, an example is shown of the wavelength composition of the light emittedby a tungsten radiator heated to 2000 K.iiiiiiii38 2.Physics of Light300 350 400 450 500 550 600 650 700 750 8000.410.420.430.440.450.460.470.48Wavelength (nm)EmissivityEmissivity of tungstenat 2000 (K)Figure 2.5.The relative contribution of each wavelength to the light emitted by a tungstenradiator at a temperature of 2000 K [653].2.3 PolarizationWe have already shown that harmonic waves are transversal: both E and H liein a plane perpendicular to the direction of propagation. This still leaves somedegrees of freedom. First, both vectors may be oriented in any direction in thisplane(albeitwith thecaveatthattheyareorthogonalto oneanother). Further,theorientationofthesevectorsmaychangewithtime. Third, theirmagnitudemayvarywithtime. Inall, thetime-dependentvariationofEandHleadstopolarization, as we will discuss in this section.We continue to assume without loss of generality that a harmonic plane waveis traveling along the positive z-axis. This means that the vectors E and H may bedecomposed into constituent components in the x- and y-directions:E =_Exex +Eyey_ei z. (2.74)Here, exand eyareunitnormalvectorsalongthex-andy-axes. Thecomplexamplitudes Ex and Ey are dened asEx = [Ex[ ei x, (2.75a)Ey = [Ey[ ei y. (2.75b)The phase angles are therefore given by x and y. For a given point in space z =r s, as time progresses the orientation and magnitude of the electric eld intensityvector E will generally vary. This can be seen by writing Equations (2.75) in theiriiiiiiii2.3.Polarization 39real form:Ex[Ex[ = cos(t z +x), (2.76a)Ey[Ey[ = cos(t z +y). (2.76b)It is now possible to eliminate the component of the phase that is common to bothof these equations, i.e., t z, by rewriting them in the following form (usingidentity (B.7a); see Appendix B):Ex[Ex[ = cos(t z)cos(x) sin(t z)sin(x), (2.77a)Ey[Ey[ = cos(t z)cos(y) sin(t z)sin(y). (2.77b)If we solve both equations for cos(t z) and equate them, we getEx[Ex[ cos(y) Ey[Ey[ cos(x) = sin(t z)(sin(y)cos(x) cos(y)sin(x)) (2.78a)= sin(t z)sin(yx). (2.78b)Repeating this, but now solving for sin(t z) and equating the results, we ndEx[Ex[ sin(y) Ey[Ey[ sin(x) = cos(t z)sin(yx). (2.79)By squaring and adding these equations, we obtain_Ex[Ex[_2+_Ey[Ey[_22Ex[Ex[Ey[Ey[ cos(yx) = sin2(yx). (2.80)Thisequationshowsthat thevectorErotatesaroundthez-axisdescribinganellipse. The wave is therefore elliptically polarized. The axes of the ellipse do notneed to be aligned with the x- and y-axes, but could be oriented at an angle.Two special cases exist; the rst is when the phase angles x and y are sepa-rated by multiples of :yx = m (m = 0, 1, 2, . . .). (2.81)For integer values of m, the sine operator is 0 and the cosine operator is either +1or 1 dependent on whether m is even or odd. Therefore, Equation (2.80) reducesto_Ex[Ex[_2+_Ey[Ey[_2= 2(1)mEx[Ex[Ey[Ey[. (2.82)iiiiiiii40 2.Physics of LightEBWave propagationFigure 2.6. An electromagnetic wave with wavelength is dened by electric vector Eand magnetic induction B, which are both orthogonal to the direction of propagation. Inthis case, both vectors are linearly polarized.The general formof this equation is either x2+y2=2xy or x2+y2=2xy. We areinterested in the ratio between x and y, as this determines the level of eccentricityof the ellipse. We nd this as follows:x2+y2= 2xy. (2.83a)By dividing both sides by y2we getx2y2 +1 = 2xy. (2.83b)Solving for the ratio x/y yieldsxy = 1. (2.83c)Solving x2+y2= 2xy in a similar manner yields x/y = 1. We therefore ndthat the ratio between Ex and Ey isExEy= (1)m EyEx. (2.84)As such,the ratio between x- and y-components of E are constant for xedm.This means that instead of inscribing an ellipse, this vector oscillates along a line.Thus, when the phase angles xand yare in phase, the electric eld intensityvector is linearly polarized, as shown in Figure 2.6. The same is then true for themagnetic vector H.iiiiiiii2.3.Polarization 41yxWavepropagationEFigure 2.7.For a circularly polarized wave, the electric vector E rotates around the Poynt-ing vector while propagating. Not shown is the magnetic vector, which also rotates aroundthe Poynting vector while remaining orthogonal to E.The second special case occurs when the amplitudes [Ex[ and [Ey[ are equaland the phase angles differ by either /2 2m or /2 2m. In this case,(2.80) reduces toE2x +E2y= [Ex[2. (2.85)This is the equation of a circle, and this type of polarization is therefore calledcircular polarization. If yx =/2 2m the wave is called a right-handedcircularly polarized wave. Conversely, if y x = /2 2mthe wave iscalled left-handed circularly polarized. In either case, the eld vectors inscribe acircle, as shown for E in Figure 2.7.Thecausesofpolarizationincludereectionofwavesoffsurfacesorscat-teringbyparticlessuspendedinamedium. Forinstance, sunlightenteringtheEarths atmosphere undergoes scattering by small particles, which causes the skyto be polarized.Polarization can also be induced by employing polarization lters. These l-ters are frequently used in photography to reduce glare from reecting surfaces.Such lters create linearly polarized light. As a consequence, a pair of such lterscan be stacked such that together they block all light.This is achieved if the two lters polarize light in orthogonal directions,asshown in the overlapping region of the two sheets in Figure 2.8. If the two ltersare aligned,then linearly polarized light will emerge,as if only one lter werepresent. The amount of light Ee transmitted through the pair of polarizing ltersiiiiiiii42 2.Physics of LightFigure 2.8. Two sheets of polarizing material are oriented such that together they blocklight, whereas each single sheet transmits light.Figure 2.9.A polarizing sheet in front of an LCD screen can be oriented such that all lightis blocked.iiiiiiii2.3.Polarization 43Figure 2.10.A polarizing sheet is oriented such that polarized laser light is transmitted.is a function of the angle between the two polarizers and the amount of incidentlight Ee,0:Ee = Ee,0 cos2(). (2.86)This relation is known as the Law of Malus [447, 730, 1128].The same effect is achieved by placing a single polarizing lter in front of anLCD screen,as shown in Figure 2.9 (see also Section 14.2). Here,a backlightemits non-polarized light, which is rst linearly polarized in one direction. Thenthe intensity of each pixel is adjusted by means of a second variable polarization inthe orthogonal direction. Thus, the light that is transmitted through this sequenceof lters is linearly polarized. As demonstrated in Figure 2.9, placing one furtherpolarizing lter in front of the screen thus blocks the remainder of the light.Inaddition, laserlight ispolarized. Thiscanbeshownbyusingasinglepolarization lter to block laser light. In Figure 2.10, a sheet of polarizing materialis placed in the path of a laser. The sheet is oriented such that most of the light istransmitted. Some of the light is also reected. By changing the orientation of thesheet, the light can be blocked, as shown in Figure 2.11. Here, only the reectingcomponent remains.Polarization is extensively used in photography in the form of lters that canbe attached to the camera. This procedure allows unwanted reections to be re-iiiiiiii44 2.Physics of LightFigure 2.11.A polarizing sheet is oriented such that polarized laser light is blocked.Figure 2.12.The LCD screen emits polarized white light, which undergoes further polar-ization upon reection dependent on the amount of stress in the reective object. Thus, thecolorization of the polarized reected light follows the stress patterns in the material.iiiiiiii2.4.Spectral Irradiance 45moved from scenes. For instance the glint induced by reections off water can beremoved. It can also be used to darken the sky and improve its contrast.In computer vision, polarization can be used to infer material properties us-ing a sequence of images taken with a polarization lter oriented at different an-gles [1250]. This technique is based on the fact that specular materials partiallypolarize light.Polarization also nds uses in material science, where analysis of the polar-izingpropertiesofamaterialprovidesinformationofitsinternalstresses. Anexample is shown in Figure 2.12 where the colored patterns on the CD case aredue to stresses in the material induced during fabrication. They become visibleby using the polarized light of an LCD screen.2.4 Spectral IrradianceIn Section2.2.2, wearguedthatthe Poyntingvectorofaplanewavepointsinthedirectionofpropagation. However, wedidnot discussitsmagnitude. Ingeneral, the vectors E and H vary extremely rapidly in both direction and time.For harmonic plane waves, the magnitude of the Poynting vector may thereforebe seen as giving the instantaneous energy density.In the case of a linearly polarized harmonic plane wave, the E and H eldsmay be modeled asE =E0 cos(rs t), (2.87a)H=H0 cos(rs t). (2.87b)The Poynting vector, given by S =EH, is thenS =E0H0 cos2(rs t). (2.88)The instantaneous energy density thus varies with double the angular frequency(because the cosine is squared). At this rate of change, the Poynting vector doesnot constitute a practical measure of energy ow.However, it is possible to average the magnitude of the Poynting vector overtime. The time average f (t))Tof a functionf (t) over a time interval Tis givenby f (t))T =1T_t+T/2tT/2f (t)dt. (2.89)iiiiiiii46 2.Physics of LightFor a harmonic function eit, such a time average becomeseit_T =1T_t+T/2tT/2eitdt (2.90a)=1iT eit_eiT/2eiT/2_(2.90b)=sin_T2_T2eit(2.90c)= sinc_T2_eit(2.90d)= sinc_T2_(cos(t) +i sin(t)). (2.90e)The sinc function tends to zero for suitably large time intervals,and the cosineand sine terms average to zero as well, i.e., cos(t))T = sin(t))T = 0.However, the Poynting vector for a harmonic plane wave varies with cos2. Itcan be shown that the time average of such a function is 1/2 for intervals of Tchosen to be large with respect to the period of a single oscillation:cos2(t)_T = 12T . (2.91)Applying this result to the Poynting vector of (2.88), we nd that the magnitudeof the time-averaged Poynting vector is given by|S|)T = 12 |E0H0|. (2.92)To derive an expression for the time-averaged Poynting vector, we will rst relatethe magnitude of E to the magnitude of H. From (2.46) we have that|E| =_ |sH|, (2.93a)|H| =_ |sE|. (2.93b)Thefact that all threevectorsinvolvedintheaboveequationsareorthogonalmeans we may apply Equation (A.7), yielding|E| =_ |s| |H|, (2.93c)|H| =_ |s| |E|. (2.93d)iiiiiiii2.5.Reection and Refraction 47Bysolvingbothexpressionsfor |s|andequatingthem, wendthefollowingrelation: |H| = |E|. (2.94)Using Equation (A.7) once more, the magnitude of the time-averaged Poyntingvector follows from (2.92):|S|)T = 12_00|E0|2(2.95a)= 0c2|E0|2. (2.95b)An important quantity proportional to this magnitude is called irradiance4Ee andis given by [447]Ee = 0c2|E20|_T . (2.96)ThisquantityisalsoknownasradiantuxdensityandismeasuredinW/m2.Undertheassumptionthatthematerialisdielectric, linear, homogeneous, andisotropic, this expression becomesEe = v2|E20|_T . (2.97)The irradiance is a measure of the amount of light illuminating a surface. It can beseen as the average energy reaching a surface unit per unit of time. Irradiance isone of the core concepts of the eld of radiometry, and therefore lies at the heartof all color theory. It is discussed further in Chapter 6.2.5 Reection and RefractionSo far, we have discussed harmonic plane waves traveling in free space. For manyapplications in optics, computer graphics, and for the discussion of the causes ofcolored materials following in Chapter 3, the behavior of light at the boundary oftwo different media constitutes an important and interesting case. In the followingsections, wediscussthedirectionofpropagationaswellasthestrengthofthereected and refracted waves.4In the physics literature, irradiance is often indicated with the symbol I. However, we are usingthe radiometric symbol Ee for consistency with later chapters.iiiiiiii48 2.Physics of Lightn-nCDA Bitits(i)s(i)s(t)sin ( ) = sin ( )CBAC=i isin ( ) = sin ( )ADAB=t tMedium 1Medium 2sin ( ) sin ( )=v1v2i txzz=0Figure 2.13.Geometry associated with refraction.2.5.1 Direction of PropagationFor now, we assume that a harmonic plane wave is incident upon a planar bound-arybetweentwomaterials. At thisboundary, thewavewill besplit intotwowaves, onewhichispropagatedbackintothemediumandonethatentersthesecond medium. The rst of these two is thus reected,whereas the second isrefracted (or transmitted).We will also assume that we have a coordinate system such that the boundaryislocatedatthez = 0plane, andthattheincidentplanetravelsinadirectionindicated by S(i). The direction vectors of the reected and transmitted waves arenamed S(r)and S(t), respectively.At any given location r = (x y 0) on the boundary, the time variation of eachof the three elds will be identical:t rS(i)v1=t rS(r)v1=t rS(t)v2. (2.98)As both the incident and the reected wave propagate through the same medium,their velocities will be identical. However, the refracted wave will have a differentvelocity, since the medium has different permittivity and permeability constants.The above equalities should hold for any point on the boundary. In particular, theequalities hold for locations r1 = (1 0 0) and r2 = (0 1 0). These two locationsgive us the following set of equalities:s(i)xv1= s(r)xv1= s(t)xv2, (2.99a)s(i)yv1= s(r)yv1= s(t)yv2. (2.99b)iiiiiiii2.5.Reection and Refraction 49n-nits(i)s(r)s(t)Medium 1Medium 2sin ( ) sin ( )=v1v1i rrsin ( ) sin ( )=v1v2i txzz=0Figure 2.14.Geometry associated with reection.If we assume that the Poynting vectors for the incident, reected, and trans-mitted waves lie in the x-z plane, then by referring to Figures 2.13 and 2.14, wehaveS(i)=sin(i)0cos(i), (2.100a)S(r)=sin(r)0cos(r), (2.100b)S(t)=sin(t)0cos(t). (2.100c)The z-coordinates are positive for S(i)and S(t)and negative for S(r). By combin-ing (2.99) and (2.100) we nd thatsin(i)v1= sin(r)v1= sin(t)v2. (2.101)Since sin(i) = sin(r) and the z-coordinates of S(i)and S(r)are of opposite sign,the angle of incidence and the angle of reection are related by r = i. Thisrelation is known as the law of reection.Since the speed of light in a given medium is related to the permittivity andpermeability of the material according to (2.28), we may rewrite (2.101) assin(i)sin(t) = v1v2=_2211= n2n1= n. (2.102)iiiiiiii50 2.Physics of LightFigure2.15. Refraction demonstrated by means of a laser beam making the transitionfrom smoke-lled air to water.Figure 2.16.Laser light reecting and refracting off an air-to-water boundary.iiiiiiii2.5.Reection and Refraction 51Figure 2.17.Laser light interacting with a magnifying lens. Note the displacement of thetransmitted light, as well as the secondary reections of front and back surfaces of the lens.The values n1 =11 and n2 =22 are called the absolute refractive indicesof the two media, whereas n is the relative refractive index for refraction from therstintothesecondmedium. Therelationsgivenin(2.102)constituteSnellslaw.5An example of refraction is shown in Figure 2.15, where a laser was aimedat a tank lled with water. The laser is a typical consumer-grade device normallyused as part of a light show in discotheques and clubs. A smoke machine was usedto produce smoke and allow the laser light to scatter towards the camera. For thesame reason a few drops of milk were added to the tank. Figure 2.16 shows aclose-up with a shallower angle of incidence. This gure shows that light is bothreected and refracted.AsecondexampleofSnellslawatworkisshowninFigure2.17wherealaser beam was aimed at a magnifying lens. As the beam was aimed at the centerof the lens, the transmitted light is parallel to the incident light, albeit displaced.The displacement is due to the double refraction at either boundary of the lens.Similarly, the gure shows light being reected