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`HECHT:OPTICS
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`a
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`1
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`APPLE 1050
`Apple v. Masimo
`IPR2020-01715
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`1
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`APPLE 1050
`Apple v. Masimo
`IPR2020-01715
`
`

`

`SECONDEDITION
`
`EUGENE HECHT
`Adelphi University
`
`|
`
`With Contributions by Alfred Zajac
`
`'
`
`A"¥
`ADDISON-WESLEY PUBLISHING COMPANY
`
`Reading, Massachusetts = Menlo Park, California = Don Mills, Ontario
`Wokingham, England =» Amsterdam = Sydney = Singapore
`Tokyo » Madrid =» Bogata » Santiago » San Juan
`
`
`
`2
`
`

`

`
`
`Copyright © 1987, 1974 by Addison-Wesley Publishing Company, Inc.
`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 the prior written permission of the publisher.
`Printed in the United States of America. Published simultaneously
`in Canada.
`11 12:13 14 15 MA 96959494
`
`Sponsoring editor: Bruce Spatz
`Production supervisors: Margaret Pinetle and Lorraine Ferrier
`Text designer: Joyce Weston
`Hlustrators: Oxford [hustrators
`Art consultant: Loretia Batley
`Manufacturing supervisor: Ann DeLacey
`
`Library of Congress Cataloging-in-Publication Data
`Hecht, Eugene.
`Optics.
`Bibliography: p.
`Includes indexes.
`IL. Title.
`1, Optics.
`L. Zajac, Alfred.
`QC355.2.H42 1987
`535 86-14067
`ISBN 0-201-11609-X
`
`Reprinted with corrections May, 1990.
`
`3
`
`

`

`
`
`Contents
`
`com
`
`12
`15
`1?
`19
`2]
`
`25
`24
`27
`28
`30
`
`1 A Brief History
`t.l Prolegomenon ........-:5 fee
`1.2
`Inthe Beginning
`......2+.- a
`5
`1.3 From the Seventeenth Century
`.. 2... s
`l.4 The Nineteenth Century ....---. va
`1.56 Twentieth-Century Optics
`....-...-.
`
`2 The Mathematics of Wave Motion
`2.1 One-Dimensional Waves ........ ae
`2.2 Harmonic Waves
`
`. 2... 0.0. 04s
`2.3 Phase and Phase Velocity
`2.4 The Complex Representation .......-
`2.5 Plane Waves
`2.6 The Three-Dimensional Differential Wave
`Equation
`2.7 Spherical Waves 2... 2 ee
`2.8 Cylindrical Waves
`2.9 Scalarand Vector Waves .. 2.2... 04:
`Problems
`
`3 Electromagnetic Theory, Photons, and Light
`3.1 Basic Laws of Electromagnetic Theory
`3.2 Electromagnetic Waves
`. 2... 1. ee es
`3.3 Energy and Momentum
`S84 Radiation 2... 2. ee ee
`3.5 Light and Matter
`3.6 The Electromagnetic-Photon Spectrum
`Problems
`
`4 The Propagation of Light
`4.1
`Introduction
`
`4.2 The Laws of Reflection and Refraction
`
`4.3 The Electromagnetic Approach .......
`4.4 Familiar Aspects of the Interaction of Light and
`rr
`4.5 The Stokes Treatment of Reflection and
`Refraction
`2... 1. ee ee
`4.6 Photons and the Laws of Reflection and
`Refraction 2... 2 ee. ee ee ee
`Problems
`.. 2... 2 eee ee
`
`5 Geometrical Optics—Paraxial Theory
`5.1
`Introductory Remarks
`.......40-0585
`5.2 Lenses
`53 Stops... 0. ek ee ee
`5.40 Mirrors... 2 ek
`5.5 Prisms
`
`:
`6.6 Fiberoptics
`6.7 Optical Systems 2. 2... ee
`Problems
`
`6 More on Geometrical Optics
`6.t Thick Lenses and Lens Systems ...... -
`6.2 Analytical Ray Tracing ......- . a0 6
`6.3 Aberrations ... 1... 2 eh ee ceenn)
`Problems
`
`Te
`
`7? The Superposition of Waves
`
`The Addition of Waves of the Same Frequency
`7.1 The Algebraic Method ........2-.
`7.2 The Complex Method ........0-.,
`
`118
`
`120
`121
`
`128
`128
`129
`149
`153
`163
`170
`176
`202
`
`211
`2h
`215
`220
`240
`
`242
`
`243
`243
`
`
`
`4
`
`

`

`
`
`x
`
`Contents
`
`7.3 Phasor Addition ...........24-
`
`7A Standing Waves
`
`.. 2... ..2.-50 00585
`
`The Addition of Waves of Different Frequency
`75 Beats 2... ee ee
`7.4 Group Velocity
`7.7 Anharmonic Periadic Waves—Fourier Analysis
`7.8 Nonperiodic Waves—Fourier Integrals
`.
`.
`.
`.
`79 Pulses and Wave Packets
`. 2... .-- 50)
`
`7.10 Optical Bandwidths
`Prohlerms
`
`8 Polarization
`
`8.1 The Nature of Polarized Light
`8.2 Polarizers 2... 2... ee ee ee es
`8.3 Dichroism . 2... ee
`84 Birefringence
`. 2... 0. eee
`8.5 Scattering and Polarization
`....,.4.4.4-,
`8.6 Polarization by Reflection ........4.,
`8.7 Retarders -. 2... . 2... ee ee ee
`8.8 Circular Polarizers 2... 2.0... aa
`8.9 Polarization of Polychromatic Light
`. ... .
`8.10 Optical Activity 2... ke
`8.11 induced Optical Effects—Optical Modulators
`8.12 A Mathematical Description of Polarization
`Problems
`
`Interference
`9
`2 ee
`9.1 General Considerations . 2... 2.
`9.2 Conditions for Interference... ....4
`
`.
`.
`9.3 Wavefront-Splitting Interferometers .. .
`.
`.
`9.4 Amplitude-Splitting Interferometers
`.
`.
`.
`9.5 Types and Localization of Interference Fringes
`9.6 Multiple-Beam Interference
`9.7 Applications of Single and Multilayer Films
`9.8 Applications of Interferometry .......
`Problems
`
`10 Diffraction
`10.1 Preliminary Considerations
`10.2. Fraunhofer Diffraction
`10.3. Fresnel Diffraction
`
`ce
`
`10.4 Kirchhoff’s Scalar Diffraction Theory ... .
`10.5 Boundary Diffraction Waves 2. 2...
`Problems
`
`247
`248
`
`250
`250
`252
`254
`259
`26)
`263
`266
`
`270
`270
`277
`279
`282
`292
`296
`300
`505
`506
`309
`314
`321
`326
`
`333
`334
`337
`339
`346
`361
`563
`373
`378
`388
`
`392
`392
`401
`434
`459
`463
`465
`
`11 Fourier Optics
`11.)
`Introduction 2... 0. eee ee
`11.2 Fourier Transforms . 6.4. ee ee ee
`
`....-.-.. ane:
`11.3 Optical Applications
`Problems
`oo. ww we ee ek
`
`472
`472
`472
`483
`512
`
`12 Basics of Coherence Theory
`12.1
`Introduciion
`. 2. 02... ee ee ee
`12.2 Visibility 2... 0000.00.04
`12.3 The Mutual Coherence Theory and the
`Degree of Coherence...
`. 2... 0 .
`12,4 Coherence and Stellar Interferometry
`Problems
`
`13 Some Aspects of the Quantum Nature of
`Light
`2. 2... 2... ee
`18.1 Quantum Fields
`13.2 Blackbody Radiation—Planck’s Quantum
`Hypothesis
`.
`6 6. ee
`13.3 The Photoelectric Effect—Einstein’s Photon
`
`2 ee ee
`Concept 2... 0.
`13.4 Particlesand Waves ..........-.
`
`. 2... 0...
`18.5 Probability and Wave Optics
`13.6 Fermat, Feynman, and Photons ......
`13.7. Absorption, Emission, and Scattering
`Problems
`
`14 Sundry Topics from Contemporary Optics
`14.1
`Imagery—The Spatial Distribution of Optical
`Information ©... ee ee ee ee 7
`14.2 Lasers and Laserlight ......4..04.
`oe ee a
`14.3 Holography
`14.4 Nonlinear Optics 2.00. 2 ee ee
`Problems
`
`Appendix 1
`Appendix 2
`Table i
`Solutions to Selected Problems
`Bibliography
`Index of Tables
`Index
`
`
`
`5
`
`

`

`4)
`
`INTRODUCTION
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`THE PROPAGATION
`OF LIGHT
`
`the
`atoms. As we know from the previous “chapter,
`scattering process is responsible for the index of refrac-
`tion, as well as the resultant reflected and refracted waves.
`This atomistic description is quite satisfying concep-
`tually, even though it is not a simple matter to treat
`analytically. It should, however, be kept in mind even
`when applying macroscopic techniques, as indeed we
`shall later on.
`We now seek to determine the general principles
`governing or at least describing the propagation, reflec-
`tion, and refraction of light. In principle it should be
`possible to trace the progress of radiant energy through
`any system by applying Maxwell's equations and the
`associated boundary conditions. In practice, however,
`this is often an impractical if not an impossible task (see
`Section 10.1). So we shall take a somewhat different
`route, stopping, when appropriate, to verify that our
`results are in accord with electromagnetic theory.
`
`4.2
`
`THE LAWS OF REFLECTION AND REFRACTION
`
`4.2.1 Huygens’s Principle
`
`Recall that a wavefrontis a surface over which an optical
`disturbance has a constant phase, Asanillustration, Fig.
`4.1 shows a small portion of a spherical wavefront =
`emanating from a monochromatic point source S$ in a
`homogeneous medium. Clearly,
`if the radius of the
`wavefront as shownis 7, at somelater time it will simply
`be (r+ vé}, where v is the phase velocity of the wave.
`
`79
`
`
`
`
`
`We now consider a number of phenomena related to
`hepropagation oflight andits interaction with material
`media, In particular, we shall study the characteristics
`of lightwaves as they progress through various sub-
`stances, crossing interfaces, and being reflected and
`refracted in the process. For the most part, we shall
`envision light as a classical electromagnetic wave whose
`velocity through any medium is dependent upon that
`material's electric and magnetic properties. It is an
`intriguing fact that many of the basic principles of optics
`are predicated on the wave aspects of light but are
`completely independentof the exact nature of the wave.
`As we shall see,
`this accounts for the longevity of
`Huygens’ s principle, which has served in turn to describe
`mechanical aether waves, electromagnetic waves, and
`now, after three hundred years, applies to quantum
`Optics,
`Suppose, for the moment, that a wave impinges on
`the interface separating two different media(e.g., a
`Pléce of glass in air). As we know from oureveryday
`®xperienices, a portion of the incident flux density will
`
`be diverted back in the form of a reflected wave, while
`ibe remainder will be transmitted across the boundary
`
`m4 refracted wave. On a submicroscopic scale we can
`- an assemblage ofatomsthat scatter the incident
`Fale ic The manner in which these emitted
`
`veil tend. superimpose and combine with each other
`on the spatial distribution of the scattering
`
`6
`
`

`

`8o
`
`Chapter 4 The Propagation of Light
`
`the light passes through a
`But suppose instead that
`nonuniform sheet of glass, as in Fig. 4.2, so that the
`wavefrontitself is distorted. How can we determine its
`new form ©’? Or for that matter, what will Z' look like
`at somelater time, if it is allowed to continue unob-
`structed?
`A preliminary step toward the solution of this prob-
`lem appeared in print in 1690 in the work entitled
`Traité de la Lumiére, which had been written 12 years
`earlier by the Dutch physicist Christiaan Huygens. It
`was there that he enunciated what has since become
`known as Huygens’s principle, that every point on @
`primary wavefront serves as the source of spherical secondary
`wavelets, such that the primary wavefront at some later lime
`is the envelope of these wavelets, Moreover,
`the wavelets
`advance with a speed and frequency equal to those of the
`primary wave at each point in space. If the medium is
`homogeneous, the wavelets may be constructed with
`finite radii, whereasif it is inhomogeneous, the wavelets
`must have infinitesimal radii. Figure 4.3 should make
`this fairly clear;
`it shows a view of a wavefront %, as
`well as a numberof spherical secondary wavelets, which,
`after a time ¢, have propagated out to a radius of vt.
`The envelope of all these wavelets is then asserted to
`correspond to the advanced primary wave 2’. It is easy
`to visualize the process in termsof mechanicalvibrations
`of an elastic medium. Indeed this is
`the way that
`Huygens envisioned it within the context of anall-
`pervading aether, as is evident from this comment by
`him:
`
`We havestill to consider, in studying the spreading out
`of these waves, that each particle of matter in which a
`wave proceeds not only communicates its motion to the
`next particle to it, which is on the straight line drawn
`from the luminous point, but that it also necessarily
`gives a motiontoall the others which touchit and which
`oppose its motion. The resultis that around each particle
`there arises a wave of whichthis particle is a center.
`
`Wecan make use of these ideas in two different ways.
`On one level, a mathematical representation of the
`wavelets will serve as the basis for a valuable analytical
`technique in treating diffraction theory. One can trace
`the progress of a primary wave pastall sorts of apertures
`and obstacles by summing up the wavelet contributions
`
`——
`
`
`
`‘%
`
`\ i
`
`Figure 4.1 A segment of a spherical wave.
`
`mathematically. On anotherlevel, Fig. 4.3 represents a
`graphical application of the essential ideas and as such
`is known as Huygens’s consiruction.
`Thusfar we have merely stated Huygens’s principle,
`without any justification or proof of its validity. As we
`shall see (Chapter 10), Fresnel successfully modified
`Huygens’s principle somewhat in the 1800s. A little
`later on, Kirchhoff showed that the Huygens—Fresnel
`principle was a direct consequence of the differential
`wave equation (2.59), thereby puttingit ona firm mathe-
`matical base. That there was a need for a reformulation
`
`\E ,
`Giass
`™~,
`™~
`a
`
`ae
`,
`
`A
`
`nN
`
`S ~
`
`s
`
`Figure 4.2. Distortion of a portion of a wavefront on passing through
`a material of nonumiform thickness.
`
`
`
`7
`
`

`

`Figure 4.3 The propagation
`of a wavefront via Huygens’s
`
`principle.
`
`of the principle is evident from Fig. 4.3, where we
`deceptively only drew hemispherical wavelets.* Had we
`drawn them as spheres, there would have been a dack-
`wave moving toward the source—something that is nat
`observed. Since this difficulty was taken care of theoreti-
`cally by Fresnel and Kirchhoff, we neednot be disturbed
`by it. In fact, we shall overlook it completely when
`applying Huygens’s construction, which, in the end, is
`best thought of as a highly useful fiction.
`Sull, Huygens’s principle fits in rather nicely with our
`earlier discussion of the atomic scattering of radiant
`energy. Each atom of a material substance that interacts
`with an incident primary wavefront can be regarded as
`a point source of scattered secondary wavelets. Things
`are not quite as clear when we apply the principle to
`the propagation of light through a vacuum,Iris helpful,
`however, to keep in mindthat at any point in empty
`space on the primary wavefront there exists both a
`time-varying E-field and a time-varying B-field. These
`
`* See E. Hecht, Phys. Teach. 18, 149 (1980).
`
`4.2 The Laws of Reflection and Refraction
`
`Sr
`
`in turn create newfields that moveout from the point.
`In this sense each point on the wavefront is analogous
`to a physical scattering center.
`
`4.2.2 Snell’s Law and the Law of Reflection
`
`The fundamental laws of reflection and refraction can
`be derivedin several different ways; the first approach
`to be used here is based on Huygens’s principle. It
`should be said, however,
`that our intention at
`the
`momentis as much to elaborate on the use of the method
`as to arrive at the end results. Huygens’s principlewill
`provide a highly useful andfairly simple means of
`analyzing and visualizing some complex’ propagation
`problems,
`for example,
`those involving anisotropic
`media (p. 287) or diffraction (p. 392). Consequently,it
`is to our advantage to gain some practice in using the
`technique, even if it is not the most elegant procedure
`for deriving the desired laws.
`Figure 4.4 shows
`a monochromatic plane wave
`impinging normally down onto the smooth interface
`separating two homogeneous transparent media. When
`an incident wave comes into contact with the interface,
`it can be imagined as split into two: we observe one
`wave reflected upward and another transmitted down-
`ward. If we consider an incident wavefront Z; coin-
`cident with the interface splitting into Z, and =,, both
`also congruent with the interface, we can utilize
`Huygens’s construction (neglecting the back-waves).
`Every point on 2, serves as a source of secondary wave-
`lets, which travel more or less upward into the incident
`medium at a speed v;, At a time ! later, the front will
`advance a distance v,t and appear as &1. Similarly, every
`point on the downward-moving front =, will serve as a
`source for wavelets essentially heading down with a
`speed v,. After a time ¢ the transmitted front will appear
`a distance v,t below as Zt.
`The process is ongoing, repeating itself with the
`frequency of
`the incident wave.* The media are
`
`* This assumes the use of light whose flux density is not 50 extraor-
`dinarily high that the fields are gigantic. With this assumption the
`medium wil! behave linearly, as is most often the case. In contrast,
`observable harmonics can be generated if the fields are made large
`enough (Section 14.4}.
`
`
`
`8
`
`

`

`Now suppose the incident wave comes in at some
`other angle, as indicated in Fig. 4.5. Clearly, it sweeps
`across the interface again, essentially splitting into two
`waves: one reflected and one refracted. Let's follow the
`progress of a typical front in Fig. 4.6, envisioning the
`diagram as if it were a series of snapshots taken in
`successive intervals of time 7. Start when 2, makes
`contact with the interface at point a. At that point, both
`the reflected and transmitted wavefronts begin, so a,
`which lies on both fronts, can be taken as a source of
`both an upwardly emitted wavelet traveling at a speed
`v; and a downwardly emitted wavelet traveling at a
`speed «,. Nowfocus on another point, say, 6 on 2,.
`After atime £, the plane 2, will have moved a distance
`in the incident medium of vt), 'so that 6 then corre-
`spondsto 8’, Presumably, two wavelets will then propa-
`gate out from $' into the incident and transmitting
`media, contributing to the reflected, }, and transmit-
`ted, 2}, wavefronts. These wavelets are shown here after
`a time 4, where 7=1, + ig. The rest of the diagram
`
`Incident
`
`'
`
`
`a
`\
`
`Reflected »
`7
`
`4
`
`JO
`
`fo
`
`Refracted
`
`Figure 4.5 Reflection and transmission of plane waves.
`
`G2
`
`Chapter 4 The Propagation of Light
`
`
`
`(b)
`Figure 4.4 A monochromatic plane wave impinging down onto a
`homogeneous, isotropic medium of index m. Z;, Z,, and 2, should
`actually overlap.
`
`assumed to respondlinearly, so the reflected and trans-
`mitted waves have that same frequency (and period),
`as do all
`the secondary wavelets. Taking n, > 7;,
`it
`follows
`that
`e¢/u,> cfu,
`thus
`»,< 4,
`and
`the
`wavelengths (the distances between wavefronts drawn
`in consecutive intervals of 7} will be such that A; > A,
`and A, = A,,as shown in Fig. 4.4(b)}. The incomingplane
`wave is perpendicular to the interface, and symmetry
`produces both reflected and transmitted plane waves
`that also travel out fromthe interface perpendicularly.
`
`
`
`
`
`
`9
`
`

`

`4.2 The Laws of Reflection and Refraction
`
`83
`
`
`
`Figure 4.6 Reflection and transmission at an interface
`via Huygens’s principle.
`
`21
`
`
`
`
`should be self-explanatory. Figure 4.7 is a somewhat
`simplified version in which 0;, @,, and @,, as before, are
`the angles of incidence, reflection, and transmission (or
`refraction), respectively. Notice that
`
`1
`sin 8,
`sin@ sin ég,
`= SS (4.4)
`
`By comparison with Fig. 4.6, it should be evident that
`BD = uh
`AC = ut,
`AE = ut,
`so substituting into Eq. (4.1) and canceling #, we have
`sin @;
`sin @,
`sin 6,
`t= te,
`Uy
`ui
`U
`
`42)
`
`It follows from the first two terms that the angle of
`incidence equals the angle of reflection, that ts,
`
`o; = 0,.
`
`(4.3)
`
`Knownas the law of reflection,it first appeared in the
`book entitled Catoptrics, which was purported to have
`been written by Euclid.
`
`i‘Lae
`moll
`(+
`un
`"Al
`\'1 fu
`;
`i
`a.
`"
`Eq
`nhl ww y
`‘Noli a
`i Hey
`4
`;
`Tepe) calli Mayer
`=
`
`wi
`fos
`
`i
`
`Figure 4.7 Reflected and transmitted wavefronts at a given instant.
`
`
`
`10
`
`10
`
`

`

`84
`
`Chapter 4 The Propagation of Light
`
`The first and last terms of Eq. (4.2) yield
`sin @; vu,ae
`
`sin@,
`wu
`
`or since v,/u, = n,/7;,
`
`n; sin @, = n, sin @,.
`
`(4.4)
`
`(4.5)
`
`‘This is the very importantlaw of refraction, the physica!
`consequences of which have been studied, at least on
`record, for over eighteen hundred years. On the basis
`of some fine observations, Claudius Ptolemy of Alexan-
`dria attempted unsuccessfully to divine the expression.
`Kepler nearly succeeded in deriving the law of refrac-
`tion in his book Supplementsto Vitello in 1604, Unfortu-
`nately he was misled by some erroneous data compiled
`earlier by Vitello (ca. 1270). The correct relationship
`seems to have been arrived at first by Snell* at the
`University of Leyden and then by the French
`mathematician Descartes.7 In English-speaking coun-
`tries Eq. (4.5) is generally referred to as Snell’s law.
`Notice that it can be rewritten in the form
`
`(4.6)
`
`= thi,
`
`‘
`
` ;
`
`sin a;
`sin 6,
`
`is the ratio of the absolute indices of
`where nm, = n,/m;
`refraction. In other words,it is the relative index of refrac-
`tion of the two media, It is evident in Fig. 4.6, where
`ny > 1 (ie., n, > nj and v; > vj, that Aj > A,, whereas
`the opposite would be true if n,; < 1.
`One feature of the above treatment merits some fur-
`ther discussion. It was reasonably assumed that each
`point on the interface, such as ¢ in Fig. 4.6, coincides
`with a particular point on each of the incident, reflected,
`and transmitted waves. In other words, there is a hxed
`phase relationship between each of the waves at points
`a, 6, c, and so forth. As theincident front sweeps across
`the interface, every point on it
`im contact with the
`interface is also a point on both a correspondingreflec-
`ted front and a corresponding transmitted front. This
`situation is known as wavefront continuity, ancl it will be
`
`“This is the common spelling, although Snel
`accurate.
`
`is probably more
`
`+ For a more detailed history, see Max Herzberger, “Optics from
`Euclid to Huygens,” Appi. Opt. 5, 1383 (1966).
`
`.
`6 «a but 0 #4
`
`(cl)
`
`{e>
`
` EE —_—
`
`.
`
`.
`.*
`
`.
`8
`
`s D
`.
`.
`s
`soe #8
`
`()
`
`Figure 4.8 The reflection of a wave as the result of scattenng.
`
`
`
`11
`
`
`
`.
`
`.
`.
`
`s
`@
`
`fn.
`
`*
`
`.
`
`.
`
`.
`
`.
`*
`
`a
`
`(a)
`
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`4.2 The Laws of Reflection and Refraction
`
`85
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`medium contribute to the reflected wave, the dominant
`effect is due to a surface layer only about 3a thick, which
`is nonetheless typically several thousand atoms deep.
`Furthermore, the condition that only one beam is reflec-
`ted is true provided that A » d; it would not be the case
`with x-rays where A = d, and there several scattered
`beamsactually result; noris it the case with a diffraction
`yf
`grating, where the separation between scatterersis again
`~.|Glassoa
`comparable to A, and several reflected and transmitted
`i
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`beams are produced. A similar argument can be made
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`for the scattering process giving rise to the transmitted
`zy
`wave and Snell’s law, as Problem 4.11 establishes.
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`Figure 49 Wavetronts and rays.
`
`justified in a more mathematically rigorous treatment
`in Section 4.3.1. Interestingly, Sommerfeld* has shown
`that the laws of reflection and refraction (independent
`of the kind of wave involved) can be derived directly
`from the requirement of wavefront continuity without
`any recourse to Huygens'’s principle, and the solution
`to Problem 4.9 demonstrates as much.
`A far more physically appealing view of the whole
`process is depictedin Fig. 4,8. An electromagnetic dis-
`turbance whose wavelength (A) is several thousandtimes
`larger than the spacing betweenthe atoms (d = 0.1 nm)
`sweeps across an interface. Each atom is driven succes-
`sively and scatters a wavelet. The tilt of the incident
`wave determines the phase delay between the scattering
`of each atom in turn (see Section 10.1.3 for the details).
`The front running from C to D is composed of wavelets
`that arrive in phase, superimpose, and interfere con-
`structively. Since every point on the incident front
`(ranging from A to B in Fig. 4.7) has the same phase,
`if AC = BD,the distances traveled and therefore the
`phases of the wavelets arriving at C and D will be equal,
`as indeed they will he all across the front. From the
`geometry, this can happen only for a reflected wave-
`front propagating in the one direction such that @, = 8,.
`This picture of
`scattered interfering wavelets
`is
`essentially an atomic version of the Huygens—Fresnel
`Principle.
`Although theoretically all the dipoles throughout the
`
`*A. Sommerield, Optics, p. 151, See also J. J. Sein, Am. J. Phys. 50,
`180 (1989).
`
`4.2.3 Light Rays
`
`The concept of a light ray is one that will be of interest
`to us throughout our study of optics. A ray is a line
`drawn in space corresponding to the direction of flow of
`radiant energy. As such, it is a mathematical device rather
`than a physical entity. In practice one can produce very
`narrow beams or pencils of light (e.g., a laserbeam), and
`we might imagine a ray to be the unattainable limit on
`the narrowness of such a beam. Bear in mind that in
`an isotropic medium (i.¢., one whose properties are the
`samein all directions) rays are orthagonal trajectories af
`the wavefronis. That is to say, they are fines normal to the
`wavefronts at every point of intersection. Evidently, in suck
`a medium a ray is parallel to the propagation vector k. As
`you might suspect, this is not true in anisotropic sub-
`stances, which we will considerlater (see Section 8.4.1).
`Within homogeneous isotropic materials, rays will be straight
`lines, since by symmetry they cannot bend in any pre-
`ferred direction, there being none, Moreover, because
`the speed of propagation is identical in all directions
`within a given medium, the spatial separation between
`two wavefronts, measured along rays, must be the same
`everywhere.* Points where a single ray intersects a set
`of wavefronts are called corresponding points,
`for
`example, A, A’,and A" in Fig. 4.9. Evidently the separation
`in time between any two carresponding points on any two
`
`“When the material is inhomogencous or when there is more than
`one medium involved, it will be the optical path length (sce Section
`4.2.4) between the two wavefronts that is the same.
`
`
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`12
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`12
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`

`86
`
`Chapter 4 The Propagationof Light
`
`sequential wavefronis is identical, In other words, if wave-
`front = is transformed into 2” after atime ¢”, che distance
`between corresponding points on any andall rays will
`be traversed in that same time ¢”. This will be true even
`if the wavefronts pass from one homogeneousisotropic
`medium into another, This just means that each point
`on 3 can be imagined as following the path of a ray to
`arrive at Z" in the time ¢t”,
`[f a group of rays is such that we can find a surface
`that is orthogonal to each and every one of them, they
`are said to form a normal congruence. For example, the
`rays emanating from a point source are perpendicular
`to a sphere centered at the source and consequenily
`form a normal congruence.
`We can now briefly consider an alternative to
`Huygens’s principle thatwill also allow us to follow the
`progress of light through various isotropic media. The
`basis for this approachis the theorem of Malus and Dupin
`{introduced in 1808 by E. Malus and modified in 1816
`by CG. Bupin), according to which a group of rays will
`preserve its normal congruence after any numberof reflections
`and refractions (as in Fig. 4.9). Fromour present vantage
`point of the wavetheory, this is equivalent to the state-
`ment
`that
`rays
`remain orthogonal
`to wavefronts
`throughout
`all propagation processes
`in isotropic
`media. As shown in Problem 4.12, the theorem can be
`used to derive the law of reflection as well as Snell's
`law. It is often most convenient to carry out a ray trace
`through an optical system using the laws of reflection
`and refraction and then reconstruct the wayefronts.
`The latter can be accomplished in accord with the above
`considerations of equal transit times between corre-
`sponding points and the orthogonality of the rays and
`wavefronts,
`Figure 4.10 depicts the parallel ray formation con-
`comitant with a plane wave, where 6,, 6,., and @,, which
`have the exact same meanings as before, are now
`measured from the normal
`to the interface. The
`incident ray and the normal determine a plane known
`as the plane of incidence. Because of the symmetryof
`the situation, we must anticipate that both the reflected
`and transmitted rays will be undeflected from that
`plane. In other words, the respective unit propagation
`vectors k,, k,, and k, are coplanar.
`In summary, then, the three basic laws of reflection
`
`and refraction are:
`
`1. The incident, reflected, and refracted raysall lic in
`the plane of incidence,
`2) 0; = Ao:
`4. n; sin 6; = 7, sin 6,.
`
`{4.3}
`[4.5]
`
`These are illustrated rather nicely with a narrowlight
`beamin the photographsof Fig. 4.11. Here, the incident
`medium is air (7; = 1.0), and the transmitting medium
`is glass (rn, ~ 1.5), Consequently, n; < 7,, and it follows
`
`
`
`
`Ray representation
`
`Figure 4.10 The wave and ray representations of an incident, reflec-
`ted, and transmitted beam.
`
`
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`13
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`Figure 4.11 Refraction at various angles ofincidence. (Photos cour-
`tesy PSSC College Physics, D. C. Heath & Co., 1968.)
`
`from Snell’s law that sin 6; > sin 6,. Since both angles,
`@, and @, vary between 0° and 90°, a region over which
`the sine function is smoothly rising, it cam be concluded
`that 0, > 0,. Rays entering a higher-index mediumfrom a
`lower one refract loward the normal and vice versa. This
`much is evident in the figure. Notice that the bottom
`surface is cut circular so that the transmitted beam
`within the glass always lies along a radius andis there-
`fore normal to the lower surface in every case. If a ray
`is normal to an interface, 8; = 0 = @,, and it sails right
`through with no bending.
`The incident beam in each portion of Fig. 4.11 is
`narrow and sharp, and the reflected beam is equally
`well defined. Accordingly,
`the process is known as
`specular reflection (from the word for a common mir-
`ror alloy in ancient times, speculum). In this case, as in
`Fig. 4.12(a), the reflecting surface is smooth, or more
`precisely, any irregularities in it are small compared
`with a wavelength.” In contrast, the diffuse reflection
`
`"If the surface ridges and valleys are small compared with A, the
`Scattered wavelets will still interfere constructively in only one direc-
`tion (6, = @,),
`
`q.2 The Laws of Reflection and Refraction
`
`87
`
`in Fig. 4.12(b) occurs when the surface is relatively
`rough. For example, ‘‘nonreflecting”glass used to cover
`pictures is actually glass whose surface is roughened so
`that
`it reflects diffusely. The law of reflection holds
`exactly over any region that is small enough to be
`considered smooth. These two forms of reflection are
`extremes; a whole range of intermediate behavior is
`possible. Thus, although the paper of this page was
`manufactured deliberately to be a fairly diffuse scat-
`terer, the cover of the book reflects in a manner that is
`somewhere hetween diffuse and specular.
`Let ii,, be a unit vector normal to the interface point-
`ing in the direction from the incidentto the transmitting
`medium (Fig. 4.13). As you will have the opportunity
`to prove in Problem 4.13, the first and third basic laws
`can be combined in the form of a vector refraction
`equation:
`
`nk; XG.) = m(k, x a,)
`
`or, alternatively,
`nik, — n;k,; = (n, cos 6, — 7; cos 6,)4,.
`
`(4.7)
`
`(4.8)
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`4.2.4 Fermat's Principle
`
`The laws of reflection and refraction, and indeed the
`manner in which light propagates in general, can be
`viewed from an entirely different and intriguing per-
`spective afforded us by Fermat’s principle. The ideas
`that will unfold presently have had a tremendous
`influence on the development of physical thought in
`and beyond the study of classical optics. Apart from its
`implications in quantum optics (Section 13.6, p. 552),
`Fermat's principle provides us with an insightful and
`highly useful way of appreciating and anticipating the
`behaviorof light.
`~
`Hero of Alexandria, who lived some time between
`150 B.c. and 250 a.p., was the first to set forth what has
`since become known as a variational principle. In his
`formulation of the law of reflection, he asserted that the
`path actually taken by light in going from some point S to a
`point P via a reflecting surface was the shortest possible one.
`This can be seen rather easily in Fig. 4.14, which depicts
`a point source S emitting a number of rays that are
`
`
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`14
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`14
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`

`

`88
`
`Chapter 4 The Propagation of Light
`
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`Specular
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`Diffuse
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`Figure 4.12
`
`{2) Specular reflection. (b) Diffuse reflection. (Photos courtesy Donald Dunitz.)
`
`then “reflected” toward P. Of course, only one of these
`paths will have any physical reality. li we simply draw
`the rays as if they emanated from 5’ (the image of 5),
`noneof the distances to P will have been altered (i.e.,
`SAP =S'AP, SBP =S'BP, etc.}. But obviously the
`straight-line path S’BP, which corresponds to 6; = @,,
`is the shortest possible one. The same kind of reasoning
`(Problem 4.15) makes it evident that points S, B, and
`P must lie in what has previously been defined as the
`plane of incidence. For over fifteen hundred years
`Hero's curious observation stood alone, until in 1657
`Fermat propounded his celebrated principle of feast time,
`which encompassed both reflection and refraction.
`Obviously, a beam of light traversing an interface does
`
`not take a straightline or minimum spatial path between
`a point in the incident medium and onein the transmit-
`ting medium, Fermat
`consequently reformulated
`Hero’s statement to read:
`the actual path between iwo
`points taken by a beamof light is ihe one that is traversed in
`the least
`time. As we shall see, even this form of the
`staternent is somewhat incomplete and a bit erroneous
`at that. For the moment then, let us embrace it but not
`passionately.
`As an example of the application of the principle to
`the case of refraction, refer to Fig. 4.15, where we
`minimize #, the transit time from § to P, with respect
`to the variable x, In other words, changing x shifts point
`O, thereby changing the ray from S to P. The smallest
`
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`15
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`15
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`Figure 4.13 The ray geometry.
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`then presumably coincide with the
`transit time will
`actual path.