Optics, Global Edition 5th Edition (PDF)

Global Global edition edition For these Global Editions, th...

Global Global edition edition For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range Optics of subjects and requirements, equipping students with the best possible Optics learning tools. This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version. fifth edition FIFTH edition Eugene Hecht This is a special edition of an established title widely Hecht used by colleges and universities throughout the world. Pearson published this exclusive edition for the benefit of students outside the United States and Canada. If you edition G LO Ba l purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author. Pearson Global Edition Hecht_05_1292096934_Final.indd 1 07/09/16 4:41 PM  5 Optics ed Global Edition Eugene Hecht Adelphi University Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo A01_HECH6933_05_GE_FM.indd 1 06/09/16 10:34 AM  Program Management Team Lead: Kristen Flathman Assistant Acquisitions Editor, Global Editions: Murchana Borthakur Assistant Project Editor, Global Editions: Vikash Tiwari Senior Manufacturing Controller, Global Editions: Trudy Kimber Media Production Manager, Global Editions: Vikram Kumar Compositor: Cenveo® Publisher Services Cover Designer: Lumina Datamatics Illustrators: Jim Atherton Rights & Permissions Project Manager: Rachel Youdelman Manufacturing Buyer: Maura Zaldivar-Garcia Marketing Manager: Elizabeth Elsworth Cover Photo Credit: © krugloff/Shutterstock.com Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on appropriate page within text. Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2017 The right of Eugene Hecht to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Authorized adaptation from the United States edition, entitled Optics, Fifth Edition, ISBN 9780133977226 by Eugene Hecht, published by Pearson Education © 2017. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 9 8 7 6 5 4 3 2 1 ISBN 10: 1-292-09693-4 ISBN 13: 978-1-292-09693-3 Typeset by Cenveo Publisher Services Printed and bound in Malaysia A01_HECH6933_05_GE_FM.indd 2 12/09/16 4:29 PM  Preface To Ca, b. w. l. T conditions, more on evanescent waves, subsections on Refraction of Light From a Point Source, Negative Refraction, Huygens’s he creation of this 5th edition was guided by three overarch- Ray Construction, and The Goos-Hänchen Shift; in Chapter 5 ing imperatives: wherever possible, to improve the pedagogy; to (Geometrical Optics), lots of new art illustrating the behavior of continue to modernize the treatment (e.g., with a bit more on pho- lenses and mirrors, along with additional remarks on fiberoptics, tons, phasors, and Fourier); and to update the content to keep pace as well as subsections on Virtual Objects, Focal-Plane Ray Trac- with technological advances (e.g., the book now discusses atomic ing, and Holey/Microstructured Fibers; in Chapter 6 (More on interferometers, and metamaterials). Optics is a fast-evolving Geometrical Optics), there is a fresh look at simple ray tracing field and this edition strives to provide an up-to-date approach to through a thick lens; in Chapter 7 (The Superposition of Waves), the discipline, all the while focusing mainly on pedagogy. one can find a new subsection on Negative Phase Velocity, a To that end there are several goals: (1) to sustain an apprecia- much extended treatment of Fourier analysis with lots of dia- tion of the central role played by atomic scattering in almost every grams showing—without calculus—how the process actually aspect of Optics; (2) to establish from the outset, the underlying works, and a discussion of the optical frequency comb (which quantum-mechanical nature of light (indeed, of all quantum par- was recognized by a 2005 Nobel Prize); in Chapter 8 (Polariza- ticles), even as the book is grounded in traditional methodology. tion), a powerful technique is developed using phasors to analyze Thus the reader will find electron and neutron diffraction patterns polarized light; there is also a new discussion of the transmittance pictured alongside the customary photon images; (3) to provide an of polarizers, and a subsection on Wavefronts and Rays in Uni- early introduction to the powerful perspective of Fourier theory, axial Crystals; Chapter 9 (Interference), begins with a brief which has come to be so prevalent in modern-day analysis. Ac- conceptual discussion of diffraction and coherence as it relates to cordingly, the concepts of spatial frequency and spatial period are Young’s Experiment. There are several new subsections, among introduced and graphically illustrated as early as Chapter 2, right which are Near Field/Far Field, Electric Field Amplitude via along with temporal frequency and period. Phasors, Manifestations of Diffraction, Particle Interference, At the request of student users, I have dispersed throughout Establishing The Wave Theory of Light, and Measuring Coher- the text over one-hundred completely worked-out EXAMPLES ence Length. Chapter 10 (Diffraction), contains a new subsec- that make use of the principles explored in each Section. More tion called Phasors and the Electric-Field Amplitude. Dozens of than two hundred problems, sans solutions, have been added to newly created diagrams and photographs extensively illustrate a the ends of the chapters to increase the available selection of variety of diffraction phenomena. Chapter 11 (Fourier Optics), fresh homework questions. A complete teacher’s solutions now has a subsection, Two-Dimensional Images, which contains manual is available upon request. Inasmuch as “a picture is a remarkable series of illustrations depicting how spatial frequency worth a thousand words,” many new diagrams and photographs components combine to create images. Chapter 12 (Basics of further enhance the text. The book’s pedagogical strength lies Coherence Theory), contains several new introductory subsec- in its emphasis on actually explaining what is being discussed. tions among which are Fringes and Coherence, and Diffraction This edition furthers that approach. and the Vanishing Fringes. There are also a number of additional Having taught Optics every year since the 4th edition was highly supportive illustrations. Chapter 13 (Modern Optics: Lasers published, I became aware of places in the book where things and Other Topics), contains an enriched and updated treatment of could be further clarified for the benefit of today’s students. lasers accompanied by tables and illustrations as well as several Accordingly, this revision addresses dozens of little sticking new subsections, including Optoelectronic Image Reconstruction. points, and fills in lots of missing steps in derivations. Every piece This 5th edition offers a substantial amount of new material of art has been scrutinized for accuracy, and altered where appro- that will be of special interest to teachers of Optics. For example: priate to improve readability and pedagogical effectiveness. in addition to plane, spherical, and cylindrical waves, we can Substantial additions of new materials can be found: in Chap- now generate helical waves for which the surface of constant ter 2 (Wave Motion), namely, a subsection on Twisted Light; in phase spirals as it advances through space (Section 2.11, p. 39). Chapter 3 (Electromagnetic Theory, Photons, and Light), an Beyond the mathematics, students often have trouble under- elementary treatment of divergence and curl, additional discus- standing what the operations of divergence and curl correspond sion of photons, as well as subsections on Squeezed Light, and to physically. Accordingly, the present revision contains a sec- Negative Refraction; in Chapter 4 (The Propagation of Light), a tion exploring what those operators actually do, in fairly simple short commentary on optical density, a piece on EM boundary terms (Section 3.1.5, p. 51). 3 A01_HECH6933_05_GE_FM.indd 3 06/09/16 10:34 AM  4 Preface The phenomenon of negative refraction is an active area of probability-amplitude approach to quantum mechanics. In any contemporary research and a brief introduction to the basic event, it provides students with a complementary means of physics involved can now be found in Chapter 4 (p. 114). apprehending diffraction that is essentially free of calculus. Huygens devised a method for constructing refracted rays The reader interested in Fourier optics can now find a wonder- (p. 116), which is lovely in and of itself, but it also allows for a ful series of illustrations showing how sinusoidal spatial frequency convenient way to appreciate refraction in anisotropic crystals contributions can come together to generate a recognizable two- (p. 358). dimensional image; in this case of a young Einstein (p. 555). This When studying the interaction of electromagnetic waves extraordinary sequence of figures should be discussed, even in an with material media (e.g., in the derivation of the Fresnel Equa- introductory class where the material in Chapter 11 might other- tions), one utilizes the boundary conditions. Since some student wise be beyond the level of the course—it’s fundamental to mod- readers may have little familiarity with E&M, the 5th edition ern image theory, and conceptually beautiful. contains a brief discussion of the physical origins of those To make the advanced treatment of coherence in Chapter 12 conditions (Section 4.6.1, p. 122). more accessible to a wider readership, this edition now contains The book now contains a brief discussion of the Goos-Hänchen an essentially non-mathematical introduction (p. 590); it sets shift which occurs in total internal reflection, It’s a piece of inter- the stage for the traditional presentation. esting physics that is often overlooked in introductory treatments Finally, the material on lasers, though only introductory, has (Section 4.7.1, p. 137). been extended (p. 619) and brought more into line with the con- Focal-plane ray tracing is a straightforward way to track rays temporary state of affairs. through complicated lens systems. This simple yet powerful Over the years since the 4th edition dozens of colleagues technique, which is new to this edition, works nicely in the class- around the world have provided comments, advice, sugges- room and is well worth a few minutes of lecture time (p. 177). tions, articles, and photographs for this new edition; I sincerely Several fresh diagrams now make clear the nature of virtual thank them all. I am especially grateful to Professor Chris Mack images and, more subtly, virtual objects arising via lens systems of the University of Texas at Austin, and Dr. Andreas Karpf of (p. 176–177). Adelphi University. I’m also indebted to my many students The widespread use of fiberoptics has necessitated an up-to- who have blind tested all the new expositive material, worked date exposition of certain aspects of the subject (p. 208–212). the new problems (often on exams), and helped take some of Among the new material the reader can now find a discussion the new photos. Regarding the latter I particularly thank Tanya of microstructured fibers and, more generally, photonic crys- Spellman, George Harrison, and Irina Ostrozhnyuk for the tals, both entailing significant physics (p. 212–214). hours spent, cameras in hand. In addition to the usual somewhat formulaic, and alas, “dry” I am most appreciative of the support provided by the team mathematical treatment of Fourier series, the book now con- at Addison Wesley, especially by Program Manager Katie tains a fascinating graphical analysis that conceptually shows Conley who has ably and thoughtfully guided the creation of this what those several integrals are actually doing. This is great 5th edition from start to finish. The manuscript was scrupulously stuff for undergraduates (Section 7.3.1, p. 309–313). and gracefully copy edited by Joanne Boehme who did a remark- Phasors are utilized extensively to help students visualize able job. Hundreds of complex diagrams were artfully drawn by the addition of harmonic waves. The technique is very useful in Jim Atherton of Atherton Customs; his work is extraordinary and treating the orthogonal field components that constitute the speaks for itself. This edition of Optics was developed under the various polarization states (p. 344). Moreover, the method ever-present guidance of John Orr of Orr Book Services. His ­provides a nice graphical means to analyze the behavior of abiding commitment to producing an accurate, beautiful book wave plates (p. 371). deserves special praise. In an era when traditional publishing is Young’s Experiment and double-beam interference in gen- undergoing radical change, he uncompromisingly maintained the eral, are central to both classical and quantum Optics. Yet the very highest standards, for which I am most grateful. It was truly usual introduction to this material is far too simplistic in that it a pleasure and a privilege working with such a consummate overlooks the limitations imposed by the phenomena of diffrac- professional. tion and coherence. The analysis now briefly explores those Lastly I thank my dear friend, proofreader extraordinaire, concerns early on (Section 9.1.1, p. 402). my wife, Carolyn Eisen Hecht who patiently coped with the The traditional discussion of interference is extended using travails of one more edition of one more book. Her good hu- phasors to graphically represent electric-field amplitudes, giv- mor, forbearance, emotional generosity, and wise counsel were ing students an alternative way to visualize what’s happening essential. (Section 9.3.1, p. 409). Anyone wishing to offer comments or suggestions concern- Diffraction can also be conveniently appreciated via electric- ing this edition, or to provide contributions to a future edition, field phasors (p. 470–471). That methodology leads naturally to can reach me at Adelphi University, Physics Department, the classical vibration curve, which brings to mind Feynman’s Garden City, NY, 11530 or better yet, at [email protected]. A01_HECH6933_05_GE_FM.indd 4 06/09/16 10:34 AM  Preface   5 Acknowledgments for the Global Edition Pearson would like to thank and acknowledge the following people for their work on the Global Edition. Contributors D. K. Bhattacharya Defence Research and Development Organization Nikhil Marriwalla University Institute of Engineering and Technology, Kurukshetra University Samrat Mukherjee National Institute of Technology Patna Stephan Nikolov Plovdiv University Reviewers D. K. Bhattacharya Defence Research and Development Organization Maruthi Manoj Brundavanam Indian Institute of Technology Kharagpur Shailendra Kumar Varshney Indian Institute of Technology Kharagpur Sushil Kumar Hansraj College, Delhi University A01_HECH6933_05_GE_FM.indd 5 06/09/16 10:34 AM  This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 2 24/11/14 5:26 PM  Contents 1 A Brief History   9 4.5 Fermat’s Principle  117 1.1 Prolegomenon  9 4.6 The Electromagnetic Approach   121 1.2 In the Beginning   9 4.7 Total Internal Reflection   133 1.3 From the Seventeenth Century   10 4.8 Optical Properties of Metals   139 1.4 The Nineteenth Century   12 4.9 Familiar Aspects of the Inter­action of Light 1.5 Twentieth-Century Optics  15 and Matter  142 4.10 The Stokes Treatment of Reflection and 2 Wave Motion  18 Refraction  147 4.11 Photons, Waves, and Probability   148 2.1 One-Dimensional Waves  18 Problems  151 2.2 Harmonic Waves   22 2.3 Phase and Phase Velocity   26 5 Geometrical Optics  159 2.4 The Superposition Principle   28 5.1 Introductory Remarks  159 2.5 The Complex Representation   30 5.2 Lenses  159 2.6 Phasors and the Addition of Waves   31 5.3 Stops  183 2.7 Plane Waves  32 5.4 Mirrors  188 2.8 The Three-Dimensional Differential 5.5 Prisms  199 Wave Equation   36 5.6 Fiberoptics  204 2.9 Spherical Waves  37 5.7 Optical Systems  215 2.10 Cylindrical Waves  39 5.8 Wavefront Shaping  239 2.11 Twisted Light  39 5.9 Gravitational Lensing  244 Problems  41 Problems  246 3 Electromagnetic Theory, Photons, 6 More on Geometrical Optics   255 and Light  45 6.1 Thick Lenses and Lens Systems   255 3.1 Basic Laws of Electromagnetic Theory   46 6.2 Analytical Ray Tracing   259 3.2 Electromagnetic Waves  54 6.3 Aberrations  266 3.3 Energy and Momentum   57 6.4 Grin Systems  284 3.4 Radiation  69 6.5 Concluding Remarks  286 3.5 Light in Bulk Matter   76 Problems  286 3.6 The Electromagnetic-Photon Spectrum   83 3.7 Quantum Field Theory   90 7 The Superposition of Waves   290 Problems  92 7.1 The Addition of Waves of the Same Frequency  291 4 The Propagation of Light   96 7.2 The Addition of Waves of 4.1 Introduction  96 Different Frequency  302 4.2 Rayleigh Scattering  96 7.3 Anharmonic Periodic Waves   308 4.3 Reflection  104 7.4 Nonperiodic Waves  318 4.4 Refraction  108 Problems  332 7 A01_HECH6933_05_GE_FM.indd 7 06/09/16 10:34 AM  8 Contents 8 Polarization  338 11 Fourier Optics   542 8.1 The Nature of Polarized Light   338 11.1 Introduction  542 8.2 Polarizers  346 11.2 Fourier Transforms  542 8.3 Dichroism  347 11.3 Optical Applications  552 8.4 Birefringence  351 Problems  583 8.5 Scattering and Polarization   361 8.6 Polarization by Reflection   363 12 Basics of Coherence Theory   588 8.7 Retarders  366 12.1 Introduction  588 8.8 Circular Polarizers  373 12.2 Fringes and Coherence   590 8.9 Polarization of Polychromatic Light   374 12.3 Visibility  594 8.10 Optical Activity  375 12.4 The Mutual Coherence Function and the 8.11  Induced Optical Effects—Optical Degree of Coherence   597 Modulators  380 12.5 Coherence and Stellar Interferometry   603 8.12 Liquid Crystals  384 Problems  609 8.13  A Mathematical Description of Polarization  387 13 Modern Optics: Lasers and Problems  392 Other Topics  612 13.1 Lasers and Laserlight   612 9 Interference  398 13.2 Imagery—The Spatial Distribution of Optical 9.1 General Considerations  398 Information  638 9.2 Conditions for Interference   402 13.3 Holography  652 9.3 Wavefront-Splitting Interferometers  405 13.4 Nonlinear Optics  667 9.4 Amplitude-Splitting Interferometers  416 Problems  672 9.5 Types and Localization Appendix 1  677 of Interference Fringes   432 Appendix 2  680 9.6 Multiple-Beam Interference  433 9.7 Applications of Single and Multilayer Films   441 Table 1  681 9.8 Applications of Interferometry   446 Solutions to Selected Problems   685 Problems  452 Bibliography  708 Index  712 10 Diffraction  457 List of Tables   722 10.1 Preliminary Considerat ions  457 10.2 Fraunhofer Diffraction  465 10.3 Fresnel Diffraction  505 10.4 Kirchhoff’s Scalar Diffraction Theory   532 10.5 Boundary Diffraction Waves   535 Problems  536 A01_HECH6933_05_GE_FM.indd 8 06/09/16 10:34 AM  1 A Brief History 1.1 Prolegomenon pointed out that a glass globe filled with water could be used for magnifying purposes. And it is certainly possible that some In chapters to come we will evolve a formal treatment of much Roman artisans may have used magnifying glasses to facilitate of the science of Optics, with particular emphasis on aspects of very fine detailed work. contemporary interest. The subject embraces a vast body of After the fall of the Western Roman Empire (475 c.e.), which knowledge accumulated over roughly three thousand years of the roughly marks the start of the Dark Ages, little or no scientific human scene. Before embarking on a study of the modern view progress was made in Europe for a great while. The dominance of things optical, let’s briefly trace the road that led us there, if of the Greco-Roman-Christian culture in the lands embracing the for no other reason than to put it all in perspective. Mediterranean soon gave way by conquest to the rule of Allah. The center of scholarship shifted to the Arab world. Refraction was studied by Abu Sa`d al-`Ala’ Ibn Sahl (940– 1000 c.e.), who worked at the Abbasid court in Baghdad, where 1.2 In the Beginning he wrote On the Burning Instruments in 984. His accurate dia- grammatical illustration of refraction, the first ever, appears in The origins of optical technology date back to remote antiqui- that book. Ibn Sahl described both parabolic and ellipsoidal burn- ty. Exodus 38:8 (ca. 1200 b.c.e.) recounts how Bezaleel, while ing mirrors and analyzed the hyperbolic plano-convex lens, as preparing the ark and tabernacle, recast “the looking-glasses of well as the hyperbolic biconvex lens. The scholar Abu Ali al- the women” into a brass laver (a ceremonial basin). Early mir- Hasan ibn al-Haytham (965–1039), known in the Western world rors were made of polished copper, bronze, and later on of as Alhazen, was a prolific writer on a variety of topics, including speculum, a copper alloy rich in tin. Specimens have survived 14 books on Optics alone. He elaborated on the Law of Reflec- from ancient Egypt—a mirror in perfect condition was un- tion, putting the angles of incidence and reflection in the same earthed along with some tools from the workers’ quarters near plane normal to the interface (p. 107); he studied spherical and the pyramid of Sesostris II (ca. 1900 b.c.e.) in the Nile valley. parabolic mirrors and gave a detailed description of the human The Greek philosophers Pythagoras, Democritus, Empedocles, eye (p. 215). Anticipating Fermat, Alhazen suggested that light Plato, Aristotle, and others developed several theories of the travels the fastest path through a medium. nature of light. The rectilinear propagation of light (p. 99) was By the latter part of the thirteenth century, Europe was only known, as was the Law of Reflection (p. 105) enunciated by beginning to rouse from its intellectual stupor. Alhazen’s work Euclid (300 b.c.e.) in his book Catoptrics. Hero of Alexandria was translated into Latin, and it had a great effect on the writings attempted to explain both these phenomena by asserting that of Robert Grosseteste (1175–1253), Bishop of Lincoln, and on the light traverses the shortest allowed path between two points. Polish mathematician Vitello (or Witelo), both of whom were in- The burning glass (a positive lens used to start fires) was fluential in rekindling the study of Optics. Their works were alluded to by Aristophanes in his comic play The Clouds known to the Franciscan Roger Bacon (1215–1294), who is con- (424 b.c.e.). The apparent bending of objects partly immersed sidered by many to be the first scientist in the modern sense. He in water (p. 113) is mentioned in Plato’s Republic. Refraction seems to have initiated the idea of using lenses for correcting was studied by Cleomedes (50 c.e.) and later by Claudius Ptol- vision and even hinted at the possibility of combining lenses to emy (130 c.e.) of Alexandria, who tabulated fairly precise form a telescope. Bacon also had some understanding of the way measurements of the angles of incidence and refraction for in which rays traverse a lens. After his death, Optics again lan- several media (p. 108). It is clear from the accounts of the his- guished. Even so, by the mid-1300s, European paintings were de- torian Pliny (23–79 c.e.) that the Romans also possessed burn- picting monks wearing eyeglasses. And alchemists had come up ing glasses. Several glass and crystal spheres have been found with a liquid amalgam of tin and mercury that was rubbed onto the among Roman ruins, and a planar convex lens was recovered in back of glass plates to make mirrors. Leonardo da Vinci (1452– Pompeii. The Roman philosopher Seneca (3 b.c.e.–65 c.e.) 1519) described the camera obscura (p. 228), later popularized by 9 M01_HECH6933_05_GE_C01.indd 9 26/08/16 11:00 AM  10 Chapter 1 A Brief History 1.3 From the Seventeenth Century It is not clear who actually invented the refracting telescope, but records in the archives at The Hague show that on October 2, 1608, Hans Lippershey (1587–1619), a Dutch spectacle maker, applied for a patent on the device. Galileo Galilei (1564–1642), in Padua, heard about the invention and within several months had built his own instrument (p. 235), grinding the lenses by hand. The compound microscope was invented at just about the same time, possibly by the Dutchman Zacha- rias Janssen (1588–1632). The microscope’s concave eye- piece was replaced with a convex lens by Francisco Fontana (1580–1656) of Naples, and a similar change in the telescope was introduced by Johannes Kepler (1571–1630). In 1611, Kepler published his Dioptrice. He had discovered total inter- nal reflection (p. 133) and arrived at the small angle approxi- mation to the Law of Refraction, in which case the incident and transmission angles are proportional. He evolved a treat- Giovanni Battista Della Porta (1535–1615). (US National Library of Medicine) ment of first-order Optics for thin-lens systems and in his book describes the detailed operation of both the Keplerian (positive eyepiece) and Galilean (negative eyepiece) tele- scopes. Willebrord Snel (1591–1626), whose name is usually the work of Giovanni Battista Della Porta (1535–1615), who dis- inexplicably spelled Snell, professor at Leyden, empirically cussed multiple mirrors and combinations of positive and negative discovered the long-hidden Law of Refraction (p. 108) in lenses in his Magia naturalis (1589). 1621—this was one of the great moments in Optics. By learn- This, for the most part, modest array of events constitutes ing precisely how rays of light are redirected on traversing a what might be called the first period of Optics. It was undoubt- boundary between two media, Snell in one swoop swung open edly a beginning—but on the whole a humble one. The whirl- the door to modern applied Optics. René Descartes (1596–1650) wind of accomplishment and excitement was to come later, in was the first to publish the now familiar formulation of the the seventeenth century. Law of Refraction in terms of sines. Descartes deduced the A very early picture of an outdoor European village scene. The man on the left is selling eyeglasses. (INTERFOTO/Alamy) M01_HECH6933_05_GE_C01.indd 10 26/08/16 11:00 AM  1.3 From the Seventeenth Century 11 Johannes Kepler (1571–1630). (Nickolae/Fotolia) Sir Isaac Newton (1642–1727). (Georgios Kollidas/Fotolia) law using a model in which light was viewed as a pressure transmitted by an elastic medium; as he put it in his La Diop- trique (1637) also observed diffraction effects. He was the first to study the colored interference patterns (p. 416) generated by thin films recall the nature that I have attributed to light, when I said that it (Micrographia, 1665). He proposed the idea that light was a is nothing other than a certain motion or an action conceived in a rapid vibratory motion of the medium propagating at a very very subtle matter, which fills the pores of all other bodies.... great speed. Moreover, “every pulse or vibration of the lumi- The universe was a plenum. Pierre de Fermat (1601–1665), tak- nous body will generate a sphere”—this was the beginning of ing exception to Descartes’s assumptions, rederived the Law the wave theory. Within a year of Galileo’s death, Isaac New- of Reflection (p. 117) from his own Principle of Least Time ton (1642–1727) was born. The thrust of Newton’s scientific (1657). effort was to build on direct observation and avoid speculative The phenomenon of diffraction, that is, the deviation from hypotheses. Thus he remained ambivalent for a long while rectilinear propagation that occurs when light advances beyond about the actual nature of light. Was it corpuscular—a stream an obstruction (p. 457), was first noted by Professor Francesco of particles, as some maintained? Or was light a wave in an Maria Grimaldi (1618–1663) at the Jesuit College in Bologna. all-pervading medium, the aether? At the age of 23, he began He had observed bands of light within the shadow of a rod his now famous experiments on dispersion. illuminated by a small source. Robert Hooke (1635–1703), I procured me a triangular glass prism to try therewith the cele- curator of experiments for the Royal Society, London, later brated phenomena of colours. Newton concluded that white light was composed of a mix- ture of a whole range of independent colors (p. 201). He main- tained that the corpuscles of light associated with the various colors excited the aether into characteristic vibrations. Even though his work simultaneously embraced both the wave and emission (corpuscular) theories, he did become more commit- ted to the latter as he grew older. His main reason for rejecting the wave theory as it stood then was the daunting problem of explaining rectilinear propagation in terms of waves that spread out in all directions. After some all-too-limited experiments, Newton gave up try- ing to remove chromatic aberration from refracting telescope lenses. Erroneously concluding that it could not be done, he turned to the design of reflectors. Sir Isaac’s first reflecting telescope, completed in 1668, was only 6 inches long and 1 inch in diameter, but it magnified some 30 times. At about the same time that Newton was emphasizing the René Descartes by Frans Hals (1596–1650). (Georgios Kollidas/Shutterstock) emission theory in England, Christiaan Huygens (1629–1695), M01_HECH6933_05_GE_C01.indd 11 26/08/16 11:00 AM  12 Chapter 1 A Brief History The great weight of Newton’s opinion hung like a shroud over the wave theory during the eighteenth century, all but sti- fling its advocates. Despite this, the prominent mathematician Leonhard Euler (1707–1783) was a devotee of the wave theory, even if an unheeded one. Euler proposed that the undesirable color effects seen in a lens were absent in the eye (which is an erroneous assumption) because the different media present ne- gated dispersion. He suggested that achromatic lenses (p. 280) might be constructed in a similar way. Inspired by this work, Samuel Klingenstjerna (1698–1765), a professor at Uppsala, reperformed Newton’s experiments on achromatism and deter- mined them to be in error. Klingenstjerna was in communica- tion with a London optician, John Dollond (1706–1761), who was observing similar results. Dollond finally, in 1758, com- bined two elements, one of crown and the other of flint glass, to form a single achromatic lens. Incidentally, Dollond’s invention was actually preceded by the unpublished work of the amateur scientist Chester Moor Hall (1703–1771) in Essex. Christiaan Huygens (1629–1695). (Portrait of Christiaan Huygens (ca. 1680), Abraham Bloteling. Engraving. Rijksmuseum [Object number RP-P-1896-A-19320].) on the continent, was greatly extending the wave theory. Unlike 1.4 The Nineteenth Century Descartes, Hooke, and Newton, Huygens correctly concluded that light effectively slowed down on entering more dense me- The wave theory of light was reborn at the hands of Dr. Thomas dia. He was able to derive the Laws of Reflection and Refrac- Young (1773–1829), one of the truly great minds of the century. tion and even explained the double refraction of calcite (p. 352), In 1801, 1802, and 1803, he read papers before the Royal Society, using his wave theory. And it was while working with calcite extolling the wave theory and adding to it a new fundamental that he discovered the phenomenon of polarization (p. 338). concept, the so-called Principle of Interference (p. 398): As there are two different refractions, I conceived also that there When two undulations, from different origins, coincide either are two different emanations of the waves of light.... perfectly or very nearly in direction, their joint effect is a com- Thus light was either a stream of particles or a rapid undula- bination of the motions belonging to each. tion of aethereal matter. In any case, it was generally agreed that its speed was exceedingly large. Indeed, many believed that light propagated instantaneously, a notion that went back at least as far as Aristotle. The fact that it was finite was deter- mined by the Dane Ole Christensen Römer (1644–1710). Jupi- ter’s nearest moon, Io, has an orbit about that planet that is nearly in the plane of Jupiter’s own orbit around the Sun. Römer made a careful study of the eclipses of Io as it moved through the shadow behind Jupiter. In 1676 he predicted that on November 9 Io would emerge from the dark some 10 min- utes later than would have been expected on the basis of its yearly averaged motion. Precisely on schedule, Io performed as predicted, a phenomenon Römer correctly ex­plained as aris- ing from the finite speed of light. He was able to determine that light took about 22 minutes to traverse the diameter of the Earth’s orbit around the Sun—a distance of about 186 million miles. Huygens and Newton, among others, were quite convinced of the validity of Römer’s work. Independently estimating the Earth’s orbital diameter, they assigned values to c equivalent to 2.3 * 108 m>s and 2.4 * 108 m>s, respectively.* *A. Wróblewski, Am. J. Phys. 53, 620 (1985). Thomas Young (1773–1829). (Smithsonian Institution) M01_HECH6933_05_GE_C01.indd 12 26/08/16 11:00 AM  1.4 The Nineteenth Century 13 Huygens was aware of the phenomenon of polarization aris- ing in calcite crystals, as was Newton. Indeed, the latter in his Opticks stated, Every Ray of Light has therefore two opposite Sides.... It was not until 1808 that Étienne Louis Malus (1775–1812) discovered that this two-sidedness of light also arose upon reflection (p. 363); the phenomenon was not inherent to crys- talline media. Fresnel and Dominique François Arago (1786– 1853) then conducted a series of experiments to determine the effect of polarization on interference, but the results were utterly inexplicable within the framework of their longitudi- nal wave picture. This was a dark hour indeed. For several years Young, Arago, and Fresnel wrestled with the problem until finally Young suggested that the aethereal vibration might be transverse, as is a wave on a string. The two-sidedness of light was then simply a manifestation of the two orthogo- nal vibrations of the aether, transverse to the ray direc­tion. Fresnel went on to evolve a mechanistic description of aether Augustin Jean Fresnel (1788–1827). (US National Library of Medicine) oscillations, which led to his now famous formulas for the amplitudes of reflected and transmitted light (p. 123). By 1825 the emission (or corpuscular) theory had only a few te- He was able to explain the colored fringes of thin films and nacious advocates. determined wavelengths of various colors using Newton’s The first terrestrial determination of the speed of light was per- data. Even though Young, time and again, maintained that his formed by Armand Hippolyte Louis Fizeau (1819–1896) in 1849. conceptions had their very origins in the research of Newton, His apparatus, consisting of a rotating toothed wheel and a distant he was severely attacked. In a series of articles, probably writ- mirror (8633 m), was set up in the suburbs of Paris from Suresnes ten by Lord Brougham, in the Edinburgh Review, Young’s pa- to Montmartre. A pulse of light leaving an opening in the wheel pers were said to be “destitute of every species of merit.” struck the mirror and returned. By adjusting the known rotational Augustin Jean Fresnel (1788–1827), born in Broglie, Nor- speed of the wheel, the returning pulse could be made either to mandy, began his brilliant revival of the wave theory in France, pass through an opening and be seen or to be obstructed by a unaware of the efforts of Young some 13 years earlier. Fresnel tooth. Fizeau arrived at a value of the speed of light equal to synthesized the concepts of Huygens’s wave description and 315 300 km>s. His colleague Jean Bernard Léon Foucault (1819– the interference principle. The mode of propagation of a pri- 1868) was also involved in research on the speed of light. In 1834 mary wave was viewed as a succession of spherical secondary Charles Wheatstone (1802–1875) had designed a rotating-mirror wavelets, which overlapped and interfered to re-form the ad- arrangement in order to measure the duration of an electric spark. vancing primary wave as it would appear an instant later. In Using this scheme, Arago had proposed to measure the speed of Fresnel’s words: light in dense media but was never able to carry out the experi- ment. Foucault took up the work, which was later to provide mate- The vibrations of a luminous wave in any one of its points may rial for his doctoral thesis. On May 6, 1850, he reported to the be considered as the sum of the elementary movements con- Academy of Sciences that the speed of light in water was less than veyed to it at the same moment, from the separate action of all that in air. This result was in direct conflict with Newton’s formu- the portions of the unobstructed wave considered in any one of lation of the emission theory and a hard blow to its few remaining its anterior positions. devotees. These waves were presumed to be longitudinal, in analogy with While all of this was happening in Optics, quite indepen- sound waves in air. Fresnel was able to calculate the diffraction dently, the study of electricity and magnetism was also patterns arising from various obstacles and apertures and satis- bearing fruit. In 1845 the master experimentalist Michael factorily accounted for rectilinear propagation in homogeneous Faraday (1791–1867) established an interrelationship be- isotropic media, thus dispelling Newton’s main objection to the tween electromagnetism and light when he found that the undulatory theory. When finally apprised of Young’s priority to polarization direction of a beam could be altered by a strong the interference principle, a somewhat disappointed Fresnel magnetic field applied to the medium. James Clerk Maxwell nonetheless wrote to Young, telling him that he was consoled by (1831–1879) brilliantly summarized and extended all the finding himself in such good company—the two great men be- empirical knowledge on the subject in a single set of math- came allies. ematical equations. Beginning with this remarkably succinct M01_HECH6933_05_GE_C01.indd 13 26/08/16 11:00 AM  14 Chapter 1 A Brief History research, evolving quietly on its own, that ultimately led to the next great turning point. In 1725 James Bradley (1693–1762), then Savilian Professor of Astronomy at Oxford, attempted to measure the distance to a star by observing its orientation at two different times of the year. The position of the Earth changed as it orbited around the Sun and thereby provided a large baseline for triangulation on the star. To his surprise, Bradley found that the “fixed” stars displayed an apparent sys- tematic movement related to the direction of motion of the Earth in orbit and not dependent, as had been anticipated, on the Earth’s position in space. This so-called stellar aberration is analogous to the well-known falling-raindrop situation. A raindrop, although traveling vertically with respect to an ob- server at rest on the Earth, will appear to change its incident angle when the observer is in motion. Thus a corpuscular James Clerk Maxwell (1831–1879). (E.H.) and beautifully symmetrical synthesis, he was able to show, purely theoretically, that the electromagnetic field could propagate as a transverse wave in the luminiferous aether (p. 54). Solving for the speed of the wave, Maxwell arrived at an ex- pression in terms of electric and magnetic properties of the me- dium (c = 1> 1P0m0). Upon substituting known empirically determined values for these quantities, he obtained a numerical result equal to the measured speed of light! The conclusion was inescapable—light was “an electromagnetic disturbance in the form of waves” propagated through the aether. Maxwell died at the age of 48, eight years too soon to see the experimental con- firmation of his insights and far too soon for physics. Heinrich Rudolf Hertz (1857–1894) verified the existence of long electro- magnetic waves by generating and detecting them in an exten- sive series of experiments published in 1888. The acceptance of the wave theory of light seemed to necessitate an equal acceptance of the existence of an all- pervading substratum, the luminiferous aether. If there were waves, it seemed obvious that there must be a supporting me- dium. Quite naturally, a great deal of scientific effort went into determining the physical nature of the aether, yet it would have to possess some rather strange properties. It had to be so tenuous as to allow an apparently unimpeded motion of celestial bodies. At the same time, it could support the ex- ceedingly high-frequency (∼1015 Hz) oscillations of light traveling at 186 000 miles per second. That implied remark- ably strong restoring forces within the aethereal substance. The speed at which a wave advances through a medium is dependent on the characteristics of the disturbed substratum and not on any motion of the source. This is in contrast to the behavior of a stream of particles whose speed with respect to the source is the essential parameter. Table of Opticks from Volume 2 of the Cyclopedia: or, An Universal Dictionary Certain aspects of the nature of aether intrude when study- of Arts and Sciences, edited by Ephraim Chambers, published in London by ing the optics of moving objects, and it was this area of James and John Knapton in 1728. (University of Wisconsin Digital Collections) M01_HECH6933_05_GE_C01.indd 14 26/08/16 11:00 AM  1.5 Twentieth-Century Optics 15 model of light could explain stellar aberration rather handily. 1.5 Twentieth-Century Optics Alternatively, the wave theory also offers a satisfactory expla- nation provided that the aether remains totally undisturbed as Jules Henri Poincaré (1854–1912) was perhaps the first to grasp the Earth plows through it. the significance of the experimental inability to observe any ef- In response to speculation as to whether the Earth’s motion fects of motion relative to the aether. In 1899 he began to make through the aether might result in an observable difference be- his views known, and in 1900 he said: tween light from terrestrial and extraterrestrial sources, Arago set out to examine the problem experimentally. He found that Our aether, does it really exist? I do not believe that more pre- cise observations could ever reveal anything more than relative there were no such observable differences. Light behaved just displacements. as if the Earth were at rest with respect to the aether. To ex- plain these results, Fresnel suggested in effect that light was In 1905 Albert Einstein (1879–1955) introduced his Special partially dragged along as it traversed a transparent medium in Theory of Relativity, in which he too, quite independently, re- motion. Experiments by Fizeau, in which light beams passed jected the aether hypothesis. down moving columns of water, and by Sir George Biddell The introduction of a “luminiferous aether” will prove to be su- Airy (1801–1892), who used a water-filled telescope in 1871 perfluous inasmuch as the view here to be developed will not to examine stellar aberration, both seemed to confirm Fres- require an “absolutely stationary space.” nel’s drag hypothesis. Assuming an aether at absolute rest, Hendrik Antoon Lorentz (1853–1928) derived a theory that He further postulated: encompassed Fresnel’s ideas. light is always propagated in empty space with a definite velocity In 1879 in a letter to D. P. Todd of the U.S. Nautical Almanac c which is independent of the state of motion of the emitting body. Office, Maxwell suggested a scheme for measuring the speed at which the solar system moved with respect to the lumi- The experiments of Fizeau, Airy, and Michelson–Morley niferous aether. The American physicist Albert Abraham were then explained quite naturally within the framework of Michelson (1852–1931), then a naval instructor, took up the Einstein’s relativistic kinematics.* Deprived of the aether, idea. Michelson, at the tender age of 26, had already estab- physicists simply had to get used to the idea that electromag- lished a favorable reputation by performing an extremely pre- netic waves could propagate through free space—there was no cise determination of the speed of light. A few years later, he alternative. Light was now envisaged as a self-sustaining wave began an experiment to measure the effect of the Earth’s mo- with the conceptual emphasis passing from aether to field. The tion through the aether. Since the speed of light in aether is electromagnetic wave became an entity in itself. constant and the Earth, in turn, presumably moves in relation On October 19, 1900, Max Karl Ernst Ludwig Planck (1858– to the aether (orbital speed of 67 000 mi>h), the speed of light 1947) read a paper before the German Physical Society in which measured with respect to the Earth should be affected by the he introduced the hesitant beginnings of what was to become yet planet’s motion. In 1881 he published his findings. There was no detectable motion of the Earth with respect to the aether— the aether was stationary. But the decisiveness of this surprising result was blunted somewhat when Lorentz pointed out an oversight in the calculation. Several years later Michelson, then professor of physics at Case School of Applied Science in Cleveland, Ohio, joined with Edward Williams Morley (1838– 1923), a well-known professor of chemistry at Western Reserve, to redo the experiment with considerably greater precision. Amazingly enough, their results, published in 1887, once again were negative: It appears from all that precedes reasonably certain that if there be any relative motion between the earth and the luminiferous aether, it must be small; quite small enough entirely to refute Fresnel’s explanation of aberration. Thus, whereas an explanation of stellar aberration within the context of the wave theory required the existence of a relative motion between Earth and aether, the Michelson–Morley ex- Albert Einstein (1879–1955). (Orren Jack Turner/Library of Congress Prints and Photographs Division [LC-USZ62-60242]) periment refuted that possibility. Moreover, the findings of Fizeau and Airy necessitated the inclusion of a partial drag of light due to motion of the medium. *See, for example, Special Relativity by French, Chapter 5. M01_HECH6933_05_GE_C01.indd 15 26/08/16 11:00 AM  16 Chapter 1 A Brief History (a) (b) (c) Figure 1.1   A rather convincing illustration of the particle nature of light. This sequence of photos was made using a position-sensing photomultiplier tube illuminated by an (8.5 * 103 count-per-second) image of a bar chart. The exposure times were (a) 8 ms, (b) 125 ms, (c) 1 s, (d) 10 s, and (e) 100 s. Each dot can be interpreted as the arrival of a single photon. (ITT Electro-Optical Products Division) (d) (e) another great revolution in scientific thought—Quantum of momentum p had an associated wavelength l, such that Mechanics, a theory embracing submicroscopic phenomena p = h>l. The easy images of submicroscopic specks of matter (p. 61). In 1905, boldly building on these ideas, Einstein pro- became untenable, and the wave-particle dichotomy dissolved posed a new form of corpuscular theory in which he asserted that into a duality. light consisted of globs or “particles” of energy. Each such quan- Quantum Mechanics also treats the manner in which light is tum of radiant energy or photon,† as it came to be called, had an absorbed and emitted by atoms (p. 74). Suppose we cause a gas energy proportional to its frequency n, that is, ℰ = hn, where h to glow by heating it or passing an electrical discharge through is known as Planck’s constant (Fig. 1.1). By the end of the 1920s, it. The light emitted is characteristic of the very structure of the through the efforts of Bohr, Born, Heisenberg, Schrödinger, atoms constituting the gas. Spectroscopy, which is the branch of De Broglie, Pauli, Dirac, and others, Quantum Mechanics had Optics dealing with spectrum analysis (p. 83), developed from become a well-verified theory. It gradually became evident that the research of Newton. William Hyde Wollaston (1766–1828) the concepts of particle and wave, which in the macroscopic made the earliest observations of the dark lines in the solar spec- world seem so obviously mutually exclusive, must be merged in trum (1802). Because of the slit-shaped aperture generally used the submicroscopic domain. The mental image of an atomic par- in spectroscopes, the output consisted of narrow colored bands ticle (e.g., electrons and neutrons) as a minute localized lump of of light, the so-called spectral lines. Working independently, matter would no longer suffice. Indeed, it was found that these Joseph Fraunhofer (1787–1826) greatly extended the subject. “particles” could generate interference and diffraction patterns After accidentally discovering the double line of sodium (p. 144), in precisely the same way as would light (p. 412). Thus photons, he went on to study sunlight and made the first wavelength de- protons, electrons, neutrons, and so forth—the whole lot—have terminations using diffraction gratings (p. 496). Gustav Robert both particle and wave manifestations. Still, the matter was by Kirchhoff (1824–1887) and Robert Wilhelm Bunsen (1811–1899), no means settled. “Every physicist thinks that he knows what a working together at Heidelberg, established that each kind of photon is,” wrote Einstein. “I spent my life to find out what a atom had its own signature in a characteristic array of spectral photon is and I still don’t know it.” lines. And in 1913 Niels Henrik David Bohr (1885–1962) set Relativity liberated light from the aether and showed the kin- forth a precursory quantum theory of the hydrogen atom, which ship between mass and energy (via ℰ0 = mc2). What seemed to was able to predict the wavelengths of its emission spectrum. be two almost antithetical quantities now became interchange- The light emitted by an atom is now understood to arise from its able. Quantum Mechanics went on to establish that a particle‡ outermost electrons (p. 74). The process is the domain of mod- ern quantum theory, which describes the most minute details with incredible precision and beauty. † The word photon was coined by G. N. Lewis, Nature, December 18, 1926. The flourishing of applied Optics in the second half of the ‡ Perhaps it might help if we just called them all wavicles. twentieth century represents a renaissance in itself. In the 1950s M01_HECH6933_05_GE_C01.indd 16 26/08/16 11:00 AM  1.5 Twentieth-Century Optics 17 several workers began to inculcate Optics with the mathemati- new devices. The technology needed to produce a practicable cal techniques and insights of communications theory. Just as optical communications system developed rapidly. The sophis- the idea of momentum provides another dimension in which to ticated use of crystals in devices such as second-harmonic gen- visualize aspects of mechanics, the concept of spatial frequency erators (p. 668), electro-optic and acousto-optic modulators, offers a rich new way of appreciating a broad range of optical and the like spurred a great deal of contemporary research in phenomena. Bound together by the mathematical formalism of crystal optics. The wavefront reconstruction technique known Fourier analysis (p. 308), the outgrowths of this contemporary as holography (p. 652), which produces magnificent three- emphasis have been far-reaching. Of particular interest are the dimensional images, was found to have numerous additional theory of image formation and evaluation (p. 552), the transfer applications (nondestructive testing, data storage, etc.). functions (p. 578), and the idea of spatial filtering (p. 328). The military orientation of much of the developmental work The advent of the high-speed digital computer brought with in the 1960s continued into the 2000s with added vigor. Today it a vast improvement in the design of complex optical systems. that technological interest in Optics ranges across the spectrum Aspherical lens elements (p. 160) took on renewed practical from “smart bombs” and spy satellites to “death rays” and infra- significance, and the diffraction-limited system with an appre- red gadgets that see in the dark. But economic considerations ciable field of view became a reality. The technique of ion bom- coupled with the need to improve the quality of life have brought bardment polishing, in which one atom at a time is chipped products of the discipline into the consumer marketplace as away, was introduced to meet the need for extreme precision in never before. Lasers are in use everywhere: reading videodiscs the preparation of optical elements. The use of single and mul- in living rooms, cutting steel in factories, scanning labels in tilayer thin-film coatings (reflecting, antireflecting, etc.) be- supermarkets, and performing surgery in hospitals. Millions of came commonplace (p. 443). Fiberoptics evolved into a practi- optical display systems on clocks and calculators and comput- cal communications tool (p. 204), and thin-film light guides ers are blinking all around the world. The almost exclusive use, continued to be studied. A great deal of attention was paid to the for the last one hundred years, of electrical signals to handle infrared end of the spectrum (surveillance systems, missile and transmit data is now rapidly giving way to more efficient guidance, etc.), and this in turn stimulated the development of optical techniques. A far-reaching revolution in the methods of infrared materials. Plastics began to be used extensively in processing and communicating information is quietly taking Optics (lens elements, replica gratings, fibers, aspherics, etc.). place, a revolution that will continue to change our lives in the A new class of partially vitrified glass ceramics with exceed- years ahead. ingly low thermal expansion was developed. A resurgence in Profound insights are slow in coming. What few we have took the construction of astronomical observatories (both terrestrial over three thousand years to glean, even though the pace is ever and extraterrestrial) operating across the whole spectrum was quickening. It is marvelous indeed to watch the answer subtly well under way by the end of the 1960s and vigorously sus- change while the question immutably remains—what is light?* tained into the twenty-first century (p. 236). The first laser was built in 1960, and within a decade laser- beams spanned the range from infrared to ultraviolet. The *For more reading on the history of optics, see F. Cajori, A History of Physics, availability of high-power coherent sources led to the discov- and V. Ronchi, The Nature of Light. Excerpts from a number of original papers ery of a number of new optical effects (harmonic generation, can conveniently be found in W. F. Magie, A Source Book in Physics, and in frequency mixing, etc.) and thence to a panorama of marvelous M. H. Shamos, Great Experiments in Physics. M01_HECH6933_05_GE_C01.indd 17 02/09/16 1:57 PM  2 Wave Motion The issue of the actual nature of light is central to a complete domain, the classical concept of a physical wave is an illusion. treatment of Optics, and we will struggle with it throughout this Still, in the large-scale regime in which we ordinarily work, work. The straightforward question “Is light a wave phenome- electromagnetic waves seem real enough and classical theory non or a particle phenomenon?” is far more complicated than it applies superbly well. might at first seem. For example, the essential feature of a par- Because both the classical and quantum-mechanical treat- ticle is its localization; it exists in a well-defined, “small” region ments of light make use of the mathematical description of of space. Practically, we tend to take something familiar like a waves, this chapter lays out the basics of what both formal­isms ball or a pebble and shrink it down in imagination until it be- will need. the ideas we develop here will apply to all physical comes vanishingly small, and that’s a “particle,” or at least the waves, from a surface tension ripple in a cup of tea to a pulse of basis for the concept of “particle.” But a ball interacts with its light reaching us from some distant galaxy. environment; it has a gravitational field that interacts with the Earth (and the Moon, and Sun, etc.). This field, which spreads out into space—whatever it is—cannot be separated from the ball; it is an inextricable part of the ball just as it is an inextri- 2.1 One-Dimensional Waves cable part of the definition of “particle.” Real particles interact via fields, and, in a sense, the field is the particle and the particle An essential aspect of a traveling wave is that it is a self- is the field. That little conundrum is the domain of Quantum sustaining disturbance of the medium through which it propa- Field Theory, a discipline we’ll talk more about later (p. 148). gates. The most familiar waves, and the easiest to visualize Suffice it to say now that if light is a stream of submicroscopic (Fig. 2.1), are the mechanical waves, among which are waves particles (photons), they are by no means “ordinary” miniball on strings, surface waves on liquids, sound waves in the air, classical particles. and compression waves in both solids and fluids. Sound waves On the other hand, the essential feature of a wave is its non- are longitudinal—the medium is displaced in the direction of localization. A classical traveling wave is a self-sustaining dis- motion of the wave. Waves on a string (and electromagnetic turbance of a medium, which moves through space transporting waves) are transverse—the medium is displaced in a direction energy and momentum. We tend to think of the ideal wave as a perpendicular to that of the motion of the wave. In all cases, continuous entity that exists over an extended region. But when although the energy-carrying disturbance advances through the we look closely at real waves (such as waves on strings), we see medium, the individual participating atoms remain in the vi- composite phenomena comprising vast numbers of particles cinity of their equilibrium positions: the disturbance advances, moving in concert. The media supporting these waves are atomic not the material medium. That’s one of several crucial features (i.e., particulate), and so the waves are not continuous entities in of a wave that distinguishes it from a stream of particles. The and of themselves. The only possible exception might be the wind blowing across a field sets up “waves of grain” that sweep electromagnetic wave. Conceptually, the classical electromag- by, even though each stalk only sways in place. Leonardo da netic wave (p. 54) is supposed to be a continuous entity, and it Vinci seems to have been the first person to recognize that a serves as the model for the very notion of wave as distinct from wave does not transport the medium through which it travels, particle. But in the past century we found that the energy of and it is precisely this property that allows waves to propagate an electromagnetic wave is not distributed continuously. The at very great speeds. classical formulation of the electromagnetic theory of light, What we want to do now is figure out the form the wave equa- however wonderful it is on a macroscopic level, is profoundly tion must have. To that end, envision some such disturbance c wanting on a microscopic level. Einstein was the first to suggest moving in the positive x-direction with a constant speed v. The that the electromagnetic wave, which we perceive macroscopi- specific nature of the disturbance is at the moment unimportant. cally, is the statistical manifestation of a fundamentally granular It might be the vertical displacement of the string in Fig. 2.2 or underlying microscopic phenomenon (p. 61). In the subatomic the magnitude of an electric or magnetic field associated with an 18 M02_HECH6933_05_GE_C02.indd 18 26/08/16 11:14 AM  2.1 One-Dimensional Waves 19 v (a) Figure 2.2   A wave on a string. For the moment we limit ourselves to a wave that does not change its shape as it progresses through space. After a time t the pulse has moved along the x-axis a distance vt, but in all other respects it remains unaltered. We now introduce a coordinate sys- tem S′, that travels along with the pulse (Fig. 2.3b) at the speed v. In this system c is no longer a function of time, and as we move along with S′, we see a stationary constant profile described by Eq. (2.2). Here, the coordinate is x′ rather than x, so that (b) c = ƒ(x′)(2.3) Figure 2.1 (a) A longitudinal wave in a spring. (b) A transverse wave in a spring. (a) S c = f (x,t) v electromagnetic wave (or even the quantum-mechanical proba- bility amplitude of a matter wave). 0 x Since the disturbance is moving, it must be a function of both position and time; (b) S′ c(x, t) = ƒ(x, t)(2.1) c = f(x′) where ƒ(x, t) corresponds to some specific function or wave shape. This is represented in Fig. 2.3a, which shows a pulse 0′ x′ traveling in the stationary coordinate system S at a speed v. The shape of the disturbance at any instant, say, t = 0, can be found (c) S S′ by holding time constant at that value. In this case, c = f (x–vt) c(x, t) 0 t = 0 = ƒ(x, 0) = ƒ(x)(2.2) represents the 2 profile of the wave at that time. For example, if 0 0′ x′ x ƒ(x) = e-ax , where a is a constant, the profile has the shape of vt x′ a bell; that is, it is a Gaussian function. (Squaring the x makes x it symmetrical around the x = 0 axis.) Setting t = 0 is analo- gous to taking a “photograph” of the pulse as it travels by. Figure 2.3   Moving reference frame. M02_HECH6933_05_GE_C02.indd 19 26/08/16 11:14 AM  20 Chapter 2 Wave Motion The disturbance looks the same at any value of t in S′ as it did (a) c (x, 0) = f(x) at t = 0 in S when S and S′ had a common origin (Fig. 2.3c). 3.0 We now want to rewrite Eq. (2.3) in terms of x to get the 2.5 wave as it would be described by someone at rest in S. It follows from Fig. 2.3c that 2.0 x′ = x - vt(2.4) 1.5 and substituting into Eq. (2.3) 1.0 c(x, t) = ƒ(x - vt)(2.5) 0.5 This then represents the most general form of the one-dimensional wavefunction. To be more specific, we have only to choose a x shape, Eq. (2.2), and then substitute (x - vt) for x in ƒ(x). The –6 –4 –2 0 2 4 6 resulting expression describes a wave having the desired pro- 11 1 10 2 9 3 8 4 7 6 5 file, moving in the positive x-direction with a speed v. Thus, t=0 2 c(x, t) = e-a(x - vt) is a bell-shaped wave, a pulse. (b) c (x, t) To see how this all works in a bit more detail, let’s unfold 3.0 the analysis for a specific pulse, for example, c(x) = 2.5 3>[10x2 + 1] = ƒ(x). That profile is plotted in Fig. 2.4a, and if it was a wave on a rope, c would be the vertical displacement 2.0 v = 1.0 ms and we might even replace it by the symbol y. Whether c rep- resents displacement or pressure or electric field, we now have 1.5 the profile of the disturbance. To turn ƒ(x) into c(x, t), that is, 1.0 to turn it i