McGRAW-HILL
`BOOK COMPANY
`New York
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`
`FRANCIS A. JENKINS
`Late Professor of Physics
`University of California, Berkeley
`
`HARVEY E. WHITE
`Professor of Physics, Emeritus
`Director of the Lawrence Hall
`of Science, Emeritus
`University of California, Berkeley
`
`Fundamentals
`of Optics
`
`FOURTH EDITION
`
`LG Electronics, Inc. et al.
`EXHIBIT 1009
`IPR Petition for
`U.S. Patent No. RE43,106
`
`

`
`FUNDAMENTALS
`
`FUNDAMENTALSFUNDAMENTALS
`OF OPTICS
`
`OF OPTICSOF OPTICS
`
`

`
`This book was set in Times Roman.
`The editors were Robert A. Fry and Anne T. Vinnicombe;
`the cover was designed by Pencils Portfolio, Inc.;
`the production supervisor was Dennis J. Conroy.
`The new drawings were done by ANCO Technical Services.
`Kingsport Press, Inc., was printer and binder.
`
`Library of Congress Cataloging in Publication Data
`
`Jenkins, Francis Arthur, dates
`Fundamentals of optics.
`
`First ed. published in 1937 under title: Funda-
`mentals of physical optics.
`Includes index.
`1. Optics. I. White, Harvey Elliott, date
`joint author. II. Title.
`QC355.2.J46 1976
`ISBN 0-07-032330-5
`
`75-26989
`
`535
`
`FUNDAMENTALS
`OF OPTICS
`
`Copyright © 1957, 1976 by McGraw-Hill, Inc. All rights reserved.
`Copyright 1950 by McGraw-Hill, Inc. All rights reserved.
`Formerly published under the title of FUNDAMENTALS OF
`PHYSICAL OPTICS, copyright 1937 by McGraw-Hill, Inc.
`Copyright renewed 1965 by Francis A. Jenkins and Harvey E. White.
`Printed in the United States of America. 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.
`16 KPKP 89
`
`

`
`CONTENTS
`
`Preface to the Fourth Edition
`
`Preface to the Third Edition
`
`Part One
`
`Geometrical Optics
`
`1
`
`Properties of Light
`
`1.1
`1.2
`1.3
`1.4
`1.5
`1.6
`1.7
`1.8
`1.9
`1.10
`
`The Rectilinear Propagation of Light
`The Speed of Light
`The Speed of Light in Stationary Matter
`The Refractive Index
`Optical Path
`Laws of Reflection and Refraction
`Graphical Construction for Refraction
`The Principle of Reversibility
`Fermat’s Principle
`Color Dispersion
`
`xvii
`
`xix
`
`3
`5
`6
`8
`9
`10
`11
`13
`14
`14
`18
`
`

`
`v[ CONTENTS
`
`2 Plane Surfaces and Prisms
`
`2.1
`2.2
`2.3
`2.4
`2.5
`2.6
`2.7
`2.8
`2.9
`2.10
`2.11
`2.12
`2.13
`
`Parallel Beam
`The Critical Angle and Total Reflection
`Plane-Parallel Plate
`Refraction by a Prism
`Minimum Deviation
`Thin Prisms
`Combinations of Thin Prisms
`Graphical Method of Ray Tracing
`Direct-Vision Prisms
`Reflection of Divergent Rays
`Refraction of Divergent Rays
`Images Formed by Paraxial Rays
`Fiber Optics
`
`3 Spherical Surfaces
`
`3.1
`3.2
`3.3
`3.4
`3.5
`3.6
`3.7
`3.8
`3.9
`3.10
`3.11
`
`Focal Points and Focal Lengths
`Image Formation
`Virtual Images
`Conjugate Points and Planes
`Convention of Signs
`Graphical Constructions. The Parallel-Ray Method
`Oblique-Ray Methods
`Magnification
`Reduced Vergence
`Derivation of the Gaussian Formula
`Nomography
`
`4 Thin Lenses
`
`4.1
`4.2
`4.3
`4.4
`4.5
`4.6
`4.7
`4.8
`4.9
`4.10
`4.11
`4.12
`4.13
`4.14
`4.15
`
`Focal Points and Focal Lengths
`Image Formation
`Conjugate Points and Planes
`The Parallel-Ray Method
`The Oblique-Ray Method
`Use of the Lens Formula
`Lateral Magnification
`Virtual Images
`Lens Makers’ Formula
`Thin-Lens Combinations
`Object Space and Image Space
`The Power of a Thin Lens
`Thin Lenses in Contact
`Derivation of the Lens Formula
`Derivation of the Lens Makers’ Formula
`
`24
`
`24
`25
`28
`29
`30
`32
`32
`33
`34
`36
`36
`38
`40
`
`44
`
`45
`46
`47
`47
`50
`50
`52
`54
`54
`56
`57
`
`6O
`
`60
`62
`62
`62
`63
`64
`64
`65
`67
`68
`70
`70
`71
`72
`73
`
`

`
`5 Thick Lenses
`
`5.1
`5.2
`5.3
`5.4
`5.5
`5.6
`5.7
`5.8
`5.9
`5.10
`5.11
`5.12
`
`Two Spherical Surfaces
`The Parallel-Ray Method
`Focal Points and Principal Points
`Conjugate Relations
`The Oblique-Ray Method
`General Thick-Lens Formulas
`Special Thick Lenses
`Nodal Points and Optical Center
`Other Cardinal Points
`Thin-Lens Combination as a Thick Lens
`Thick-Lens Combinations
`Nodal Slide
`
`6 Spherical Mirrors
`
`6.1 Focal Point and Focal Length
`6.2 Graphical Constructions
`6.3 Mirror Formulas
`6.4 Power of Mirrors
`6.5 Thick Mirrors
`6.6 Thick-Mirror Formulas
`6.7 Other Thick Mirrors
`6.8 Spherical Aberration
`6.9 Astigmatism
`
`7 The Effects of Stops
`
`7.1
`7.2
`7.3
`7.4
`7.5
`7.6
`7.7
`7.8
`7.9
`7.10
`7.11
`
`Field Stop and Aperture Stop
`Entrance and Exit Pupils
`Chief Ray
`Front Stop
`Stop between Two Lenses
`Two Lenses with No Stop
`Determination of the Aperture Stop
`Field of View
`Field of a Plane Mirror
`Field of a Convex Mirror
`Field of a Positive Lens
`
`8 Ray Tracing
`
`8.1 Oblique Rays
`8.2 Graphical Method for Ray Tracing
`8.3 Ray-tracing Formulas
`8.4 Sample Ray-tracing Calculations
`
`78
`
`78
`79
`81
`82
`82
`84
`88
`88
`90
`91
`93
`93
`
`98
`
`98
`99
`102
`104
`105
`107
`109
`109
`111
`
`115
`
`115
`116
`117
`117
`118
`120
`121
`122
`122
`124
`124
`
`130
`
`130
`131
`134
`135
`
`

`
`viii commtcrs
`
`9
`
`9.1
`9.2
`9.3
`9.4
`9..5
`9.6
`9.7
`9.8
`9.9
`9.10
`9.11
`9.12
`9.13
`9.14
`
`Lens Aberrations
`Expansion of the Sine. First-Order Theory
`Third-Order Theory of Aberrations
`Spherical Aberration of a Single Surface
`Spherical Aberration of a Thin Lens
`Results of Third-Order Theory
`Fifth-Order Spherical Aberration
`Coma
`Aplanatic Points of a Spherical Surface
`Astigmatism
`Curvature of Field
`Distortion
`The Sine Theorem and Abbe’s Sine Condition
`Chromatic Aberration
`Separated Doublet
`
`10 Optical Instruments
`The Human Eye
`10.1
`Cameras and Photographic Objectives
`10.2
`Speed of Lenses
`10.3
`Meniscus Lenses
`10.4
`Symmetrical Lenses
`10.5
`Triplet Anastigmats
`10.6
`Telephoto Lenses
`10.7
`Magnifiers
`10.8
`Types of Magnifiers
`10.9
`Spectacle Lenses
`10.10
`Microscopes
`10.11
`Microscope Objectives
`10.12
`Astronomical Telescopes
`10.13
`10 Id
`Oeular.~ and Eyepieces
`Huygens Eyepiece
`10.15
`Ramsden Eyepiece
`10.16
`Kellner or Achromatized Ramsden Eyepiece
`10.17
`Special Eyepieces
`10.18
`Prism Binoculars
`10.19
`The Kellner-Schmidt Optical System
`10.20
`Concentric Optical Systems
`10.21
`
`Part
`
`Two Wave Optics
`
`11 Vibrations and Waves
`
`11.1
`11.2
`11.3
`11.4
`
`Simple Harmonic Motion
`The Theory of Simple Harmonic Motion
`Stretching of a Coiled Spring
`Vibrating Spring
`
`149
`150
`151
`152
`153
`157
`160
`162
`166
`167
`170
`171
`173
`176
`182
`
`188
`188
`191
`191
`193
`193
`194
`195
`195
`198
`198
`200
`201
`202
`205
`205
`206
`206
`206
`207
`208
`209
`
`214
`
`216
`217
`218
`221
`
`

`
`CONTENTS iX
`
`11.5
`11.6
`11.7
`11.8
`11.9
`11.10
`11.11
`
`Transverse Waves
`Sine Waves
`Phase Angles
`Phase Velocity and Wave Velocity
`Amplitude and Intensity
`Frequency and Wavelength
`Wave Packets
`
`12 The Superposition of Waves
`
`12.1
`12.2
`12.3
`12.4
`12.5
`12.6
`12.7
`12.8
`12.9
`
`Addition of Simple Harmonic Motions along the Same Line
`Vector Addition of Amplitudes
`Superposition of Two Wave Trains of the Same Frequency
`Superposition of Many Waves with Random Phases
`Complex Waves
`Fourier Analysis
`Group Velocity
`Graphical Relation between Wave and Group Velocity
`Addition of Simple Harmonic Motions at Right Angles
`
`13
`
`Interference of Two Beams of Light
`
`13.1
`13.2
`13.3
`13.4
`13.5
`13.6
`13.7
`13.8
`13.9
`13.10
`13.11
`13.12
`13.13
`13.14
`13.15
`
`Huygens’ Principle
`Young’s Experiment
`Interference Fringes from a Double Source
`Intensity Distribution in the Fringe System
`Fresnel’s Biprism
`Other Apparatus Depending on Division of the Wave Front
`Coherent Sources
`Division of Amplitude. Michelson Interferometer
`Circular Fringes
`Localized Fringes
`White-Light Fringes
`Visibility of the Fringes
`Interferometric Measurements of Length
`Twyman and Green Interferometer
`Index of Refraction by Interference Methods
`
`14 Interference Involving Multiple Reflections
`Reflection from a Plane-Parallel Film
`14.1
`Fringes of Equal Inclination
`14.2
`Interference in the Transmitted Light
`14.3
`Fringes of Equal Thickness
`14.4
`Newton’s Rings
`14.5
`Nonreflecting Films
`14.6
`Sharpness of the Fringes
`14.7
`Method of Complex Amplitudes
`14.8
`Derivation of the Intensity Function
`14.9
`
`223
`224
`225
`228
`229
`232
`235
`
`238
`
`239
`240
`242
`244
`246
`248
`250
`252
`253
`
`259
`
`260
`261
`263
`265
`266
`268
`270
`271
`273
`275
`276
`277
`279
`281
`282
`
`286
`288
`291
`292
`293
`294
`295
`297
`299
`300
`
`

`
`CONTENTS
`
`14.10
`14.11
`14.12
`14.13
`14.14
`14.15
`14.16
`
`Fabry-Perot Interferometer
`Brewster’s Fringes
`Chromatic Resolving Power
`Comparison of Wavelengths with the Interferometer
`Study of Hyperfine Structure and of Line Shape
`Other Interference Spectroscopes
`Channeled Spectra. Interference Filter
`
`15
`
`Fraunhofer Diffraction by a Single Opening
`
`15.1
`15.2
`15.3
`15.4
`15.5
`15.6
`15.7
`15.8
`15.9
`15.10
`15.11
`
`Fresnel and Fraunhofer Diffraction
`Diffraction by a Single Slit
`Further Investigation of the Single-Slit Diffraction Pattern
`Graphical Treatment of Amplitudes. The Vibration Curve
`Rectangular Aperture
`Resolving Power with a Rectangular Aperture
`Chromatic Resolving Power of a Prism
`Circular Aperture
`Resolving Power of a Telescope
`Resolving Power of a Microscope
`Diffraction Patterns with Sound and Microwaves
`
`16 The Double SHt
`Qualitative Aspects of the Pattern
`Derivation of the Equation for the Intensity
`Comparison of the Single-Slit and Double-Slit Patterns
`Distinction between Interference and Diffraction
`Position of the Maxima and Minima. Missing Orders
`Vibration Curve
`Effect of Finite Width of Source Slit
`Michelson’s Stellar Interferometer
`Correlation Interferometer
`Wide-Angle Interference
`
`16.1
`16.2
`16.3
`16.4
`16.5
`16.6
`16.7
`16.8
`16.9
`16.10
`
`17
`
`The Diffraction Grating
`
`17.1
`17.2
`17.3
`17.4
`17.5
`17.6
`17.7
`17.8
`1.7.9
`17.10
`17.11
`17.12
`
`Effect of Increasing the Number of Slits
`Intensity Distribution from an Ideal Grating
`Principal Maxima
`Minima and Secondary Maxima
`Formation of Spectra by a Grating
`Dispersion
`Overlapping of Orders
`Width of the Principal Maxima
`Resolving Power
`Vibration Curve
`Production of Ruled Gratings
`Ghosts
`
`301
`302
`303
`3O5
`308
`310
`311
`
`315
`
`315
`316
`319
`322
`324
`325
`327
`329
`330
`332
`334
`
`338
`
`338
`339
`341
`341
`342
`346
`347
`349
`351
`352
`
`355
`
`355
`357
`358
`358
`359
`362
`362
`363
`364
`365
`368
`370
`
`

`
`17.13
`17.14
`17.15
`17.16
`
`18
`
`18.1
`18.2
`18.3
`18.4
`18.5
`18.6
`18.7
`18,8
`18.9
`18.10
`18.11
`18.12
`18.13
`18,14
`18.15
`
`Control of the Intensity Distribution among Orders
`Measurement of Wavelength with the Grating
`Concave Grating
`Grating Spectrographs
`
`Fresnel Diffraction
`Shadows
`Fresnel’s Half-Period Zones
`Diffraction by a Circular Aperture
`Diffraction by a Circular Obstacle
`Zone Plate
`Vibration Curve for Circular Division of the Wave Front
`Apertures and Obstacles with Straight Edges
`Strip Division of the Wave Front
`Vibration Curve for Strip Division. Cornu’s Spiral
`Fresnel’s Integrals
`The Straight Edge
`Rectilinear Propagation of Light
`Single Slit
`Use of Fresnel’s Integrals in Solving Diffraction Problems
`Diffraction by an Opaque Strip
`
`19
`
`The Speed of Light
`
`19.1
`19.2
`19.3
`19.4
`19.5
`19.6
`19.7
`19.8
`19.9
`19.10
`19.11
`19.12
`19.13
`19.14
`19.15
`
`20
`
`20.1
`20.2
`20.3
`20.4
`20.5
`20.6
`
`R6mer’s Method
`Bradley’s Method. The Aberration of Light
`Michelson’s Experiments
`Measurements in a Vacuum
`Kerr-Cell Method
`Speed of Radio Waves
`Ratio of the Electrical Units
`The Speed of Light in Stationary Matter
`Speed of Light in Moving Matter
`Fresnel’s Dragging Coefficient
`Airy’s Experiment
`Effect of Motion of the Observer
`The Michelson-Morley Experiment
`Principle of Relativity
`The Three First-Order Relativity Effects
`
`The Electromagnetic Character of Light
`Transverse Nature of Light Vibrations
`Maxwell’s Equations for a Vacuum
`Displacement Current
`The Equations for Plane Electromagnetic Waves
`Pictorial Representation of an Electromagnetic Wave
`Light Vector in an Electromagnetic Wave
`
`370
`373
`373
`374
`
`378
`
`378
`380
`383
`384
`385
`386
`388
`389
`389
`390
`393
`395
`397
`399
`400
`
`4O3
`
`403
`4O5
`406
`408
`408
`410
`411
`411
`412
`413
`414
`414
`416
`418
`419
`
`423
`
`424
`424
`425
`427
`428
`429
`
`

`
`20.7
`20.8
`20.9
`20.10
`20.11
`20.12
`
`Energy and Intensity of the Electromagnetic Wave
`Radiation from an Accelerated Charge
`Radiation From a Charge in Periodic Motion
`Hertz’s Verification of the Existence of Electromagnetic Waves
`Speed of Electromagnetic Waves in Free ’Space
`~erenkov Radiation
`
`21 Sources of Light and Their Spectra
`Classification of Sources
`Solids at High Temperature
`Metallic Arcs
`Bunsen Flame
`Spark
`Vacuum Tube
`Classification of Spectra
`Emittanee and Absorptance
`Continuous Spectra
`Line Spectra
`Series of Spectral Lines
`Band Spectra
`
`21.1
`21.2
`21.3
`21.4
`21.5
`21.6
`21.7
`21.8
`21.9
`21.10
`21.11
`21.12
`
`22
`
`Absorption and Scattering
`
`22.1
`22.2
`22.3
`22.4
`22.5
`22.6
`22.7
`22.8
`22.9
`22.10
`22.11
`22.12
`22.13
`
`23
`23.1
`23.2
`23.3
`23.4
`23.5
`23.6
`23.7
`23.8
`23.9
`
`General and Selective Absorption
`Distinction between Absorption and Scattering
`Absorption by Solids and Liquids
`Absorption by Gases
`Resonance and Fluorescence of Gases
`Fluorescence of Solids and Liquids
`Selective Reflection. Residual Rays
`Theory of the Connection between Absorption and Reflection
`Scattering by Small Particles
`Molecular Scattering
`Raman Effect
`Theory of Scattering
`Scattering and Refractive Index
`
`Dispersion
`
`Dispersion of a Prism
`Normal Dispersion
`Cauchy’s Equation
`Anomalous Dispersion
`Sellmeier’s Equation
`Effect of Absorption on Dispersion
`Wave and Group Velocity in the Medium
`The Complete Dispersion Curve of a Substance
`The Electromagnetic Equations for Transparent Media
`
`429
`430
`432
`432
`434
`434
`
`438
`
`438
`439
`439
`442
`442
`443
`445
`445
`447
`450
`452
`453
`
`457
`
`457
`458
`459
`461
`461
`464
`464
`465
`466
`468
`469
`470
`471
`
`474
`
`474
`475
`479
`479
`482
`485
`487
`488
`489
`
`

`
`CONTENTS xiii
`
`23.10
`23.11
`
`Theory of Dispersion
`Nature of the Vibrating Particles and Frictional Forces
`
`24 The Polarization of Light
`
`24.1
`24.2
`24.3
`24.4
`24.5
`24.6
`24.7
`24.8
`24.9
`24.10
`24.11
`24.12
`24.13
`24.14
`24.15
`24.16
`24.17
`24.18
`
`Polarization by Reflection
`Representation of the Vibrations in Light
`Polarizing Angle and Brewster’s Law
`Polarization by a Pile of Plates
`Law of Malus
`Polarization by Dichroie Crystals
`Double Refraction
`Optic Axis
`Principal Sections and Principal Planes
`Polarization by Double Refraction
`Nicol Prism
`Parallel and Crossed Polarizers
`Refraction by Calcite Prisms
`Rochon and Wollaston Prisms
`Scattering of Light and the Blue Sky
`The Red Sunset
`Polarization by Scattering
`The Optical Properties of Gemstones
`
`25
`
`Reflection
`
`25.1
`25.2
`25.3
`25.4
`25.5
`25.6
`25.7
`25.8
`25.9
`25.10
`25.11
`
`25.12
`
`Reflection from Dielectrics
`Intensities of the Transmitted Light
`Internal Reflection
`Phase Changes on Reflection
`Reflection of Plane-polarized Light from Dielectrics
`Elliptically Polarized Light by Internal Reflection
`Penetration into the Rare Medium
`Metallic Reflection
`Optical Constants of Metals
`Description of the Light Reflected from Metals
`Measurement of the Principal Angle of Incidence and Principal
`Azimuth
`Wiener’s Experiments
`
`26 Double Refraction
`Wave Surfaces for Uniaxial Crystals
`Propagation of Plane Waves in Uniaxial Crystals
`Plane Waves at Oblique Incidence
`Direction of the Vibrations
`Indices of Refraction for Uniaxial Crystals
`Wave Surfaces in Biaxial Crystals
`Internal Conical Refraction
`
`26.1
`26.2
`26.3
`26.4
`26.5
`26.6
`26.7
`
`491
`494
`
`497
`
`498
`499
`500
`501
`503
`504
`505
`507
`507
`508
`510
`511
`511
`513
`514
`515
`516
`518
`
`523
`
`523
`526
`527
`527
`529
`531
`533
`
`536
`538
`
`540
`541
`
`544
`
`544
`546
`549
`550
`551
`553
`556
`
`

`
`26.8
`26.9
`
`External Conical Refraction
`Theory of Double Refraction
`
`27
`
`Interference of Polarized Light
`
`2Z1
`27.2
`27.3
`27.4
`27.5
`27.6
`27.7
`27.8
`27.9
`
`EUiptically and Circularly Polarized Light
`Quarter- and Half-Wave Plates
`Crystal Hates between Crossed Polarizers
`Babinet Compensator
`Analysis of Polarized Light
`Interference with White Light
`Polarizing Monochromatic Filter
`Applications of Interference in Parallel Light
`Interference in Highly Convergent Light
`
`28
`
`Optical Actifity and Modern Wave Optics
`
`28.1
`28.2
`28.3
`28.4
`28.5
`28.6
`28.7
`28.8
`28.9
`28.10
`28.11
`28.12
`28.13
`28.14
`
`Rotation of the Plane of Polarization
`Rotary Dispersion
`Fresnel’s Explanation of Rotation
`Double Refraction in Optically Active Crystals
`Shape of the Wave Surfaces in Quartz
`Fresnel’s Multiple Prism
`Comu Prism
`Vibration Forms an~i Intensities in Active Crystals
`Theory of Optical Activity
`Rotationin Liquids
`Modern Wave Optics
`Spatial Filtering
`Phase-Contrast Microscope
`Schlieren Optics
`
`Part Three
`
`Quantum Optics
`
`29
`
`29.1
`29.2
`29.3
`29.4
`29.5
`29.6
`29.7
`29.8
`29.9
`
`Light Quanta and Their Origin
`
`The Bohr Atom
`Energy Levels
`Bohr-Stoner Scheme for Building Up Atoms
`Elliptical Orbits, or Penetrating Orbitals
`Wave Mechanics
`The Spectrum of Sodium
`Resonance Radiation
`Metastable States
`Optical Pumping
`
`30
`
`Lasers
`
`30.1
`30.2
`
`Stimulated Emission
`Laser Design
`
`557
`559
`
`564
`
`564
`567
`568
`569
`571
`572
`575
`576
`576
`
`581
`
`581
`582
`584
`586
`588
`589
`590
`591
`593
`594
`596
`597
`602
`604
`
`611
`
`612
`616
`617
`619
`622
`625
`626
`629
`630
`
`632
`
`633
`634
`
`

`
`30.3
`30.4
`30.5
`30.6
`30.7
`30.8
`30.9
`30.10
`30.11
`30.12
`30.13
`
`The Ruby Laser
`The Helium-Neon Gas Laser
`Concave Mirrors and Brewster’s Windows
`The Carbon Dioxide Laser
`Resonant Cavities
`Coherence Length
`Frequency Doubling
`Other Lasers
`Laser Safety
`The Speckle Effect
`Laser Applications
`
`31 . Holography.
`
`31.1
`31.2
`31.3
`31.4
`31.5
`31.6
`31.7
`
`The Basic Principles of Holography
`Viewing a Hologram
`The Thick, or Volume, Hologram
`Multiplex Holograms
`White-Light-Reflection Holograms
`Other Holograms
`Student Laboratory Holography
`
`32
`
`Magneto-Optics and Electro-Optics
`
`32.1
`32.2
`32.3
`32.4
`32.5
`32.6
`32.7
`32.8
`32.9
`32.10
`32.11
`
`33
`
`33.1
`33.2
`33.3
`33.4
`33.5
`33.6
`33.7
`33.8
`33.9
`33.10
`
`Zeeman Effect
`Inverse Zeeman Effect
`Faraday Effect
`Voigt Effect, or Magnetic Double Refraction
`Cotton-Mouton Effect
`Kerr Magneto-optic Effect
`Stark Effect
`Inverse Stark Effect
`Electric Double Refraction
`Kerr Electro-optic Effect
`Pockels Electro-optic Effect
`
`The Dual, Nature of Light
`Shortcomings of the Wave Theory
`Evidence for Light Quanta
`Energy, Momentum, and Velocity of Photons
`Development of Quantum Mechanics
`Principle of Indeterminacy
`Diffraction by a Slit
`Complementarity
`Double Slit
`Determination of Position with a Microscope
`Use of a Shutter
`
`CONTENTS X¥
`
`635
`636
`642
`643
`646
`650
`652
`653
`653
`653
`654
`
`658
`
`659
`664
`665
`669
`670
`672
`675
`
`678
`
`679
`685
`686
`688
`
`691
`691
`692
`693
`693
`695
`
`698
`
`699
`700
`703
`704
`705
`705
`707
`707
`709
`710
`
`

`
`Interpretation of the Dual Character of Light
`Realms of Applicability of Waves and Photons
`
`Appendixes
`I The Physical Constants
`H Electron Subshells
`III Refractive Indices and Dispersions for Optical Glasses
`1V Refractive Indices and Dispersions of Optical Crystals
`V The Most Intense Fraunhofer Lines
`II1 Abbreviated Number System
`VII Significant Figures
`
`Index
`
`711
`712
`
`715
`716
`717
`720
`721
`722
`723
`724
`
`727
`
`

`
`15
`
`FRAUNHOFER DIFFRACTION
`BY A SINGLE OPENING
`
`When a beam of light passes through a narrow slit, it spreads out to a certain extent
`into the region of the geometrical shadow. This effect, already noted and illustrated
`at the beginning of Chap. 13, Fig. 13B, is one of the simplest examples of diffraction,
`¯ ; o ,-,,.F÷I~o "f~;l’nr,~ ,,",~l’;rrI~f f,"., "l",-’o’~r,M ;n ee.,-.:,;,-,-h÷ l;n,:,(cid:128), T÷ ,,-,~,n 1-,,~ e.:,÷;o~,~,-.÷,,-,,.;lw ,~v,-,lM,,.,,~A
`only by assuming a wave character for light, and in this chapter we shall investigate
`quantitatively the diffraction pattern, or distribution of intensity of the light behind
`the aperture, using the principles of wave motion already discussed.
`
`15.1 FRESNEL AND FRAUNHOFER DIFFRACTION
`
`Diffraction phenomena are conveniently divided into two general classes, (1) those in
`which the source of light and the screen on which the pattern is observed are effec-
`tively at infinite distances from the aperture causing the diffraction and (2) those in
`which either the source or the screen, or both, are at finite distances from the aperture.
`The phenomena coming under class (1) are called, for historical reasons, Fraunhofer
`diffraction, and those coming under class (2) Fresnel diffraction. Fraunhofer diffrac-
`tion is much simpler to treat theoretically. It is easily observed in practice by rendering
`
`

`
`16 FUNDAMENTALS OF OPTICS
`
`Oiffrocting
`slit
`
`Screer
`
`FIGURE 15A
`Experimental arrangement for obtaining the diffraction pattern of a single slit;
`Fraunhofer diffraction.
`
`the light from a source para!le! with a lens and focusing it on a screen with another
`lens placed behind the aperture, an arrangement which effectively removes the source
`and screen to infinity. In the observation of Fresnel diffraction, on the other hand,
`no lenses are necessary, but here the wave fronts are divergent instead of plane, and
`the theoretical treatment is consequently more complex. Only Fraunhofer diffraction
`will be considered in this chapter, and Fresnel diffraction in Chap. 18.
`
`15.2 DIFFRACTION BY A SINGLE SLIT
`
`A slit is a rectangular aperture of length large compared to its breadth. Consider a
`slit S to be set up as in Fig. 15A, with its long dimension perpendicular to the plane
`of the page, and to be illuminated by parallel monochromatic light from the narrow
`slit S’, at the principal focus of the lens L1. The light focused by another lens L2
`on a screen or photographic plate P at its principal focus will form a diffraction pat-
`tern, as indicated schematically. Figure 15B(b) and (c) shows two actual photographs,
`taken with "" .................... " .....m~ ~u,~" a .......... : .... -~,~÷ 1;~. ,,~’~" ....... 1=,,,+~,
`v.tv.twL ~txS.t,t~.
`Qllle£elIt I~ApUSUIe
`4358/~. The distance S’Lt was 25.0 cm, and L2P was 100 cm. The width of the slit
`S was 0.090 mm, and of S’, 0.10 mm. If S’ was widened to more than about 0.3 mm,
`the details of the pattern began to be lost. On the original plate, the half width d
`of the central maximum was 4.84 mm. It is important to notice that the width of the
`central maximum is twice as great as that of the fainter side maxima. That this effect
`comes under the heading of diffraction as previously defined is clear when we note
`that the strip drawn in Fig. 15B(a) is the width of the geometical image of the slit S’,
`or practically that which would be obtained by removing the second slit and using
`the whole aperture of the lens. This pattern can easily be observed by ruling a single
`transparent line on a photographic plate and using it in front of the eye as explained
`in See. 13.2, Fig. 13E.
`The explanation of the single-slit pattern lies in the interference of the Huygens
`secondary wavelets which can be thought of as sent out from every point on the wave
`
`

`
`FRAI.FNHOFER DIFFRACTION BY A SINGLE OPENING 317
`
`~)
`
`(b)
`
`I !
`FIGURE 15B d
`Photographs of the single-slit diffraction pattern.
`
`front at the instant that it occupies the plane of the slit. To a first approximation,
`one may consider these wavelets to be uniform spherical waves, the emission of
`which stops abruptly at the edges of the slit. The results obtained in this way, although
`they give a fairly accurate account of the observed facts, are subject to certain modifi-
`cations in the light of the more rigorous theory.
`Figure 15C represents a section of a slit of width b, illuminated by parallel
`light from the left. Let ds be an element of width of the wave front in the plane of
`the slit, at a distance s from the center O, which we shall call the origin. The parts
`of each secondary wave which travel normal to the plane of the slit will be focused
`at Po, while those which travel at any angle 0 will reach P. Considering first the wave-
`let emitted by the element ds situated at the origin, its amplitude will be directly
`proportional to the length ds and inversely proportional to the distance x. At P it
`will produce an infinitesimal displacement which, for a spherical wave, may be ex-
`pressed as
`
`dyo -
`
`ads
`
`X
`
`sin (cot - kx)
`
`As the position of ds is varied, the displacement it produces will vary in phase because
`of the different path length to P. When it is at a distance s below the origin, the con-
`tribution will be
`
`dy~ -
`
`ads
`
`X
`
`sin [wt- k(x + A)]
`
`ads
`- ~ sin (ogt - kx - ks sin 0)
`X
`
`(15a)
`
`

`
`318 FUNDAMENTALS OF OPTICS
`
`FIGURE 15C
`Geometrical construction for investigating the intensity in the single-slit diffrac-
`tion pattern.
`
`1
`
`We now wish to sum the effects of all elements from one edge of the slit to the other.
`This can be done by integrating Eq. (i 5a) from s = - t)/2 to b/2. The simplest way*
`is to integrate the contributions from pairs of elements symmetrically placed at s
`and -s, each contribution being
`ay=dy_,+dys
`
`_ a ds [sin (cot - kx - ks sin 0) + sin (cot - kx + ks sin 0)]
`x
`By the identity sin ~ + sin/3 = 2 cos ½(~ - /~) sin ½(~ + fl), we have
`
`which must be integrated from s = 0 to b/2. In doing so, x may be regarded as con-
`stant, insofar as it affects the amplitude. Thus
`
`L .... k ........ / .... \ ....
`
`2a
`y = -- sin (cot - kx)
`
`cos (ks sin 0) ds
`
`[b/2
`,do
`0) sin (cot - kx)
`
`=
`
`x L k sin 0
`
`17
`
`= ab sin (½kb sin 0) sin (cot - kx)
`x ½kb sin 0
`
`(15b)
`
`* The method of complex amplitudes (See. 14.8) starts with (ab/x) ~ exp (iks sin 0) ds,
`and yields the real amplitude upon multiplication of the result by its complex
`conjugate. No simplification results from using the method here.
`
`

`
`FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 319
`
`The resultant vibration will therefore be a simple harmonic one, the amplitude of
`which varies with the position of P, since the latter is determined by 0. We may
`represent its amplitude as
`
`¯
`
`A = A0 sin___.._~
`B
`where fl = ½kb sin 0 = (rob sin 0)/2 and Ao = ab/x. The quantity/~ is a convenient
`variable, which signifies one-half the phase difference between the contributions
`coming from opposite edges of the slit. The intensity on the screen is then
`
`(15c)
`
`¯
`
`I ~ A2 = Ao2 sin2 fl
`
`(15d)
`
`If the light, instead of being incident on the slit perpendicular to its plane, makes an
`angle i, a little consideration will show that it is merely necessary to replace the above
`expression for fl by the more general expression
`
`¯
`
`fl = 7rb(sin i + sin 0)
`2
`
`(15e)
`
`15.3 FURTHER INVESTIGATION OF THE SINGLE-SLIT
`DIFFRACTION PATTERN
`
`In Fig. 15D(a) graphs are shown of Eq. (15c) for the amplitude (dotted curve) and
`Eq. (15d) for the intensity, taking the constant A0 in each case as unity. The intensity
`curve will be seen to have the form required by the experimental result in Fig. 15B.
`The maximum intensity of the strong central band comes at the point P0 of Fig. 15C,
`where evidently all the secondary wavelets will arrive in phase because the path differ-
`ence A = 0. For this point fl = 0, and although the quotient (sin fl)/fl becomes
`indeterminate for fl = 0, it will be remembered that sin fl approaches fl for small
`angles and is equal to it when fl vanishes. Hence for fl = 0, (sin fl)/fl = 1. We now
`see the significance of the constant Ao. Since for fl = 0, A = Ao, it represents the
`amplitude when all the wavelets arrive in phase. Ao2 is then the value of the maximum
`intensity, at the center of the pattern. From this principal maximum the intensity
`falls to zero at fl = -re, then passes through several secondary maxima, with equally
`spaced points of zero intensity at fl = ---7r, + 27t, + 3zt,..., or in general fl = mzt.
`The secondary maxima do not fall halfway between these points, but are displaced
`toward the center of the pattern by an amount which decreases with increasing m.
`The exact values of fl for these maxima can be found by differentiating Eq. (15c) with
`respect to fl and equating to zero. This yields the condition
`
`tan fl = fl
`
`The values of fl satisfying this relation are easily found graphically as the intersections
`of the curve y = tan fl and the straight line y = t- In Fig. 15D(b) these points of inter-
`section lie directly below the corresponding secondary maxima.
`The intensities of the secondary maxima can be calculated to a very close
`approximation by finding the values of (sinz fl)/fl2 at the halfway positions, i.e.,
`
`

`
`320 FUNDAMENTALS OF OPTICS
`
`\
`\
`\
`
`(b)
`
`f
`Y
`t
`
`FIGURE 15D
`Amplitude and intensity contours for Fraunhofer diffraction of a single slit,
`showing positions of maxima and minima.
`
`where fl = 3n/2, 5nI2, 77t/2, .... This gives 4/9rc2, 4/25rc2, 4/4992,..., or 1/22.2,
`1/61.7, 1/121,..., of the intensity of the principal maximum. Reference to Table 15A
`ahead are the exact values of the intensity for every 15° intervals for the central
`maximum. These values are useful in plotting graphs. The first secondary maximum
`is only 4.72 percent the intensity of the central maximum, while the second and third
`secondary maxima are only 1.65 and 0.83 percent respectively.
`A very clear idea of the origin of the single-slit pattern is obtained by the follow-
`ut~
`
`light ......... -" ........ " ..... ’- - oi
`i J~ culmug
`pi
`irom
`LU
`ing sim e treatment. Consider the
`[Ile SIlL
`point P~ on the screen, this point being just one wavelength farther from the upper
`
`Table 15A VALUES FOR CENTRAL MAXIMUM FOR FRAUNHOFER
`DIFFRACTION OF A SINGLE SLIT
`
`deg
`
`rad
`
`0
`15
`30
`45
`60
`75
`90
`
`0
`0.2618
`0.5236
`0.7854
`1.0472
`1.3090
`1.5708
`
`sin fl
`
`0
`0.2588
`0.5000
`0.7071
`0.8660
`0.9659
`1.0000
`
`Az
`
`1
`0.9774
`0.9119
`0.8106
`0.6839
`0.5445
`0.4053
`
`#
`deg tad
`
`105
`120
`135
`150
`165
`180
`195
`
`1.8326
`2.0944
`2.3562
`2.6180
`2.8798
`3.1416
`3.4034
`
`sin fl
`
`0.9659
`0.8660
`0.7071
`0.5000
`0.2588
`0
`0.2588
`
`A2
`
`0.2778
`0.1710
`0.0901
`0.0365
`0.0081
`0
`0.0058
`
`

`
`FRAUNtIOFER DIFFRACTION BY A SINGLE OPENING 321
`
`FIGURE 15E
`Angle of the first minimum of the single-
`slit diffraction pattern.
`
`I
`
`(
`Intensity
`
`edge of the slit than from the lower. The secondary wavelet from the point in the slit
`adjacent to the upper edge will travel approximately 2/2 farther than that from the
`point at the center, and so these two will produce vibrations with a phase difference
`of ~t and will give a resultant displacement of zero at Pi. Similarly the wavelet from
`the next point below the upper edge will cancel that from the next point below the
`center, and we can continue this pairing off to include all points in the wave front,
`so that the resultant effect at P1 is zero. At Pa the path difference is 24, and if we
`divide the slit into four parts, the pairing of points again gives zero resultant, since
`the parts cancel in pairs. For the point P2, on the other hand, the path difference
`is 31/2, and we divide the slit into thirds, two of which will cancel, leaving one third
`to account for the intensity at this point. The resultant amplitude at P2 is, of course,
`not even approximately one-third that at Po, because the phases of the wavelets from
`the remaining third are not by any means equal.
`The above method, though instructive, is not exact if the screen is at a finite
`distance from the slit. As Fig. 15E is drawn, the shorter broken line is drawn to cut
`off equal distances on the rays to P1. It will be seen from this that the path difference
`to P1 between the light coming from the upper edge and that from the center is slightly
`greater than ,l/2 and that between the center and lower edge slightly less than 1/2.
`Hence the resultant intensity will not be zero at P1 and Pa, but it will be more nearly
`so the greater the distance between slit and screen or the narrower the slit. This
`corresponds to the transition from Fresnel diffraction to Fraunhofer diffraction.
`Obviously, with the relative dimensions shown in the figure, the geometrical shadow
`of the slit would considerably widen the central maximum as drawn. Just as was true
`with Young’s experiment (Sec. 13.3), when the screen is at infinity, the relations be-
`come simpler. Then the t