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Fundamentals of optics
Fundamentals of optics
Francis Jenkins, Harvey White
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Jenkins F.A., White H.E., Jenkins F., White H. Fundamentals of Optics (MGH Science Engineering Math, 2001)(ISBN 0072561912)(766s)
Год:
2001
Издание:
4
Издательство:
McGrawHill Science/Engineering/Math
Язык:
english
Страницы:
766
ISBN 10:
0072561912
ISBN 13:
9780072561913
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PDF, 35,25 MB
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angle^{641}
plane^{629}
ray^{567}
rays^{562}
waves^{551}
focal^{507}
axis^{501}
optics^{485}
parallel^{458}
intensity^{430}
slit^{416}
sin^{415}
wavelength^{409}
phase^{379}
index^{362}
diffraction^{351}
optical^{344}
fringes^{337}
fundamentals^{333}
lenses^{332}
spherical^{323}
fundamentals of optics^{320}
velocity^{320}
beam^{319}
interference^{306}
refraction^{303}
incident^{300}
spectrum^{286}
sec^{283}
principal^{281}
reflection^{281}
mirror^{277}
polarized^{277}
prism^{270}
surfaces^{258}
plate^{256}
amplitude^{253}
frequency^{251}
absorption^{251}
shown in fig^{249}
reflected^{246}
crystal^{239}
angles^{233}
dispersion^{224}
method^{217}
refractive^{212}
equation^{208}
wavelengths^{207}
incidence^{207}
aberration^{205}
focal length^{204}
vibrations^{201}
grating^{192}
vibration^{189}
maximum^{187}
values^{187}
cos^{184}
experiment^{183}
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FUNDAMENTALS OF OPTICS FourthEdition 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 ~ McGrawHili Primls WfiJIiI Custom Publishing New York Stl.oois SmFzarrisco Auckland Bogota Caracas lisbon London Madrid Mexia:> Milan Montreal NewDeIhi Pari5 SmJuan Singapore Sydney Tokyo Toronto McGrawHill Higher Education ~ A Division of The McGrawHill Companies This book was set in Times Roman. The editors were Robert A. Fry and Anne T. Vinnicombe; the production supervisor was Dennis J. Conroy. The new drawings were done by ANCO Technical Services. FUNDAMENTALS OF OPTICS Fourth Edition Copyright@ 2001 by The McGrawHill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base retrieval system, without prior written permission of the publisher. This book contains all material from Fundamentals of Optics, Fourth Edition by Francis A. Jenkins and Harvey E. White. Copyright@1976, 1957, 1950 by The McGrawHill Companies, Inc. Formerly published under the title of Fundamentals of Physical Optics. Copyright@ 1937 by The McGrawHill Companies, Inc. Copyright renewed 1965 by Francis A. Jenkins and Harvey E. White. Reprinted with permission of the publisher. 3 4 5 6 7 8 9 0 QSR QSR 0 9 8 7 6 5 4 3 2 ISBN 0072561912 Editor: Shirley Grall Printer/Binder: Quebecor World CONTENTS Part One Preface to the Fourth Edition xvii Preface to the Third Edition xix Geometrical Optics 1 Properties of Light 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 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 Pri; nciple of Reversibility Fermat's Principle 1.10 Color Dispersion 3 5 6 8 9 10 11 13 14 14 18 vi CONTENTS 2 Plane Surfaces and Prisms 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Parallel Beam The Critical Angle and Total Reflection PlaneParallel Plate Refraction by a Prism Minimum Deviation Thin Prisms Combinations of Thin Prisms Graphical Method of Ray Tracing DirectVision Prisms 2.10 Reflection of Divergent Rays 2.11 Refraction of Divergent Rays 2.12 Images Formed by Paraxial Rays 2.13 Fiber Optics 3 Spherical Surfaces 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Focal Points and Focal Lengths Image Formation Virtual Images Conjugate Points and Planes Convention of Signs Graphical Constructions. The ParallelRay Method ObliqueRay Methods Magnification Reduced Vergence 3.10 Derivation of the Gaussian Formula 3.11 Nomography 4 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 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 Thin Lenses 60 Focal Points and Focal Lengths Image Formation Conjugate Points and Planes The ParallelRay Method The ObliqueRay Method Use of the Lens Formula Lateral Magnification Virtual Images Lens Makers' Formula ThinLens 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 60 62 62 62 63 64 64 65 67 68 70 70 71 72 73 CONTENTS 5 Thick Lenses 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Two Spherical Surfaces The ParallelRay Method Focal Points and Principal Points Conjugate Relations The ObliqueRay Method General ThickLens Formulas Special Thick Lenses Nodal Points and Optical Center, Other Cardinal Points 5.10 ThinLens Combination as a Thick Lens 5.11 ThickLens Combinations 5.12 Nodal Slide 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Spherical Mirrors Focal Point and Focal Length Graphical Constructions Mirror Formulas Power of Mirrors Thick Mirrors ThickMirror Formulas Other Thick Mirrors Spherical Aberration Astigmatism 7 The Effects of Stops 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 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 7.10 Field of a Convex Mirror 7.11 Field of a Positive Lens 8 8.1 8.2 8.3 8.4 vii 78 78 79 81 82 82 84 88 88 90 91 93 93 98 98 99 102 104 105 107 109 109 1I1 115 lIS 1I6 1I7 1I7 1I8 120 121 122 122 124 124 Ray Tracing 130 Oblique Rays Graphical Method for Ray Tracing Raytracing Formulas Sample Raytracing Calculations 130 131 134 135 viii CONTENTS 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 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 10.21 Part Two 11 11.1 11.2 11.3 11.4 Lens Aberrations 149 Expansion of the Sine. FirstOrder Theory ThirdOrder Theory of Aberrations Spherical Aberration of a Single Surface Spherical Aberration of a Thin Lens Results of ThirdOrder Theory FifthOrder 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 150 151 152 153 157 160 162 166 167 170 17l 173 176 182 Optical Instruments 188 The Human Eye Cameras and Photographic Objectives Speed of Lenses Meniscus Lenses Symmetrical Lenses Triplet Anastigmats Telephoto Lenses Magnifiers Types of Magnifiers Spectacle Lenses Microscopes Microscope Objectives Astronomical Telescopes Oculars and Eyepieces Huygens Eyepiece Ramsden Eyepiece Kellner or Achromatized Ramsden Eyepiece Special Eyepieces Prism Binoculars The KellnerSchmidt Optical System Concentric Optical Systems 188 191 191 193 193 194 195 195 198 198 200 201 202 205 205 206 206 206 207 208 209 Wave Optics Vibrations and Waves 215 Simple Harmonic Motion The Theory of Simple Harmonic Motion Stretching of a Coiled Spring Vibrating Spring 216 217 218 221 CONTENTS 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 WhiteLight Fringes Visibility of the Fringes Interferometric Measurements of Length Twyman and Green Interferometer Index of Refraction by Interference Methods 14 Interference Involving Multiple Reflections 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 Reflection from a PlaneParallel Film Fringes of Equal Inclination Interference in the Transmitted Light Fringes of Equal Thickness Newton's Rings Nonreflecting Films Sharpness of the Fringes Method of Complex Amplitudes Derivation of the Intensity Function Ix 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 ~ X ~ CONTENTS 14.10 14.11 14.12 14.13 14.14 14.15 14.16 FabryPerot 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 Fresnel and Fraunhofer Diffraction Diffraction by a Single Slit Further Investigation of the SingleSlit 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 15.10 Resolving Power of a Microscope 15.11 Diffraction Patterns with Sound and Microwaves 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 16 The Double Slit 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 Qualitative Aspects of the Pattern Derivation of the Equation for the Intensity Comparison of the SingleSlit and DoubleSlit 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 WideAngle Interference 17 The Diffraction Grating 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 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 17.10 Vibration Curve 17.11 Production of Ruled Gratings 17.12 Ghosts 301 302 303 305 308 310 311 315 315 316 319 322 324 325 327 329 330 332 334 338 338 339 341 341 342 346 347 349 351 352 355 355 357 358 358 359 362 362 363 364 365 368 370 CONTENTS 17.13 17.14 17.15 17.16 Control of the Intensity Distribution among Orders Measurement of Wavelength with the Grating Concave Grating Grating Spectrographs 18 Fresnel Diffraction 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 Shadows Fresnel's HalfPeriod 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 xl 370 373 373 374 378 378 380 383 384 385 386 388 389 389 390 393 395 397 399 400 403 Romer's Method Bradley's Method. The Aberration of Light Michelson's Experiments Measurements in a Vacuum KerrCell 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 MichelsonMorleyExperiment Principle of Relativity The Three FirstOrder Relativity Effects 403 405 406 408 408 410 411 411 412 413 414 414 416 418 419 20 The Electromagnetic Character of Light 423 19.10 19.11 19.12 19.13 19.14 19.15 20.1 20.2 20.3 20.4 20.5 20.6 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 424 424 425 427 428 429 xii CONTllNTS 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 Cerenkov Radiation 21 Sources of Light and Their Spectra Classification of Sources Solids at High Temperature Metallic Arcs Bunsen Flame Spark 21.6 Vacuum Tube 21.7 Classification of Spectra 21.8 Emittance and Absorptance 21.9 Continuous Spectra 21.10 Line Spectra 21.11 Series of Spectral Lines 21.12 Band Spectra 21.1 21.2 21.3 21.4 21.5 22 Absorption and Scattering General and Selective Absorption Distinction between Absorption and Scattering Absorption by Solids and Liquids Absorption by Gases Resonance and Fluorescence of Gases 22.6 Fluorescence of Solids and Liquids 22.7 Selective Reflection. Residual Rays 22.8 Theory of the Connection between Absorption and Reflection 22.9 Scattering by Small Particles 22.10 Molecular Scattering 22.11 Raman Effect 22.12 Theory of Scattering 22.13 Scattering and Refractive Index 22.1 22.2 22.3 22.4 22.5 23 Dispersion 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 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 23.10 23.11 Theory of Dispersion Nature of the Vibrating Particles and Frictional 24 The Polarization 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 Forces of Light 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 Dichroic 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 2S 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 Planepolarized 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 26.1 26.2 26.3 26.4 26.5 26.6 26.7 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 xlli 491 494 497 498 499 500 501 503 504 505 507 507 508 510 511 511 513 514 515 516 518 S23 523 526 527 527 529 531 533 534 536 538 540 541 S44 544 546 549 550 551 553 556 xiv CONTENTS 26.8 26.9 External Conical Refraction. Theory of Double Refraction 27 Interference 27.1 27.2 27.3 27.4 27.5 27.6 27.7 27.8 27.9 of Polarized 557 559 Light Elliptically and Circularly Polarized Light Quarter and HalfWave Plates Crystal Plates 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 Activity and Modern 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 Part Three 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 Cornu Prism Vibration Forms ana Intensities in Active Crystals Theory of Optical Activity Rotation in Liquids Modem Wave Optics Spatial Filtering PhaseContrast Microscope Schlieren Optics Quantum and Their Origin The Bohr Atom Energy Levels BohrStoner 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 30.2 581 581 582 584 586 588 589 590 591 593 594 596 597 602 604 Optics 29 Light Quanta 29.1 29.2 29.3 29.4 29.5 29.6 29.7 29.8 29.9 Wave Optics 564 564 567 568 569 571 572 575 576 576 Stimulated Emission Laser Design 611 612 616 617 619 622 625 626 629 630 632 633 634 CONTENTS 30.3 30.4 30.5 30.6 30.7 30.8 30.9 30.10 30.11 30.12 30.13 31 31.1 31.2 31.3 31.4 31.5 31.6 31.7 32 32.1 32.2 32.3 32.4 32.5 32.6 32.7 32.8 32.9 32.10 32.11 The Ruby Laser The HeliumNeon 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 635 636 642 643 646 650 652 653 653 653 654 Holography 658 The Basic Principles of Holography Viewing a Hologram The Thick, or Volume, Hologram Multiplex Holograms White Light Reflection Holograms Other Holograms Student Laboratory Holography 659 664 665 669 670 672 675 MagnetoOptics and ElectroOptics 678 Zeeman Effect Inverse Zeeman Effect Faraday Effect Voigt Effect, or Magnetic Double Refraction CottonMouton Effect Kerr Magnetooptic Effect Stark Effect Inverse Stark Effect Electric Double Refraction Kerr Electrooptic Effect Pockels Electrooptic Effect 679 685 686 688 33 The Dual Nature of Light 33.1 33.2 33.3 33.4 33.5 33.6 33.7 33.8 33.9 33.10 XV 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 691 691 692 693 693 695 698 699 700 703 704 705 705 707 707 709 710 xvi CONTENTS 33.11 33.12 Interpretation of the Dual Character of Light Realms of Applicability of Waves and Photons 712 Appendixes 715 I The Physical Constants II Electron Subshells III IV V VI VII 711 716 717 Refractive Indices and Dispersions for Optical Glasses Refractive Indices and Dispersions of Optical Crystals The Most Intense Fraunhofer Lines Abbreviated Number System Significant Figures 722 723 724 Index 727 720 721 .. PREFACE TO THE FOURTH EDITION This fourth edition is written primarily to be used as a textbook by college students majoring in one of the physical sciences. The first, second, and third editions were written by Francis A. Jenkins and Harvey E. White while teaching optics in the physics department at the University of California, Berkeley. With the passing of Professor Jenkins in 1960 this fourth edition has been revised by Harvey E. White. A considerable number of innovative: ideas and new concepts have been developed in the field of ()ptics since the third .edition was published in 1957, thereby requiring a sizable amount of new material. Three new chapters, a number of new sections on modern optics, a number of new references, and all new problems at the ends of all chapters have been added to bring the fourth edition up to date. Fizeau's experiments on the speed of light in air and Foucault's experiments on the speed of light in stationary matter have beeh moved to Chapter 1. This serves as a better introduction to the important concept of refractive index and leaves the rest of Chapter 19 relatively unchanged. In Part One, Geometrical Optics, the long and tedious calculations of ray tracing, using logarithms, has been replaced by direct calculations using the relatively new electronic calculators, thereby permitting lens design engineers to program larger computers. xviii PREFACETO THEFOURm EDmON In Part Two, Wave Optics, Chapter 11 has been modified to give a better approach to the subject of wave motion. In Chapter 16 a section has been added on the correlation interferometer. Some of the major features of recent developments have been added at the end of Chapter 28: modern wave optics, spatial filtering, the phasecontrast microscope, and schlieren optics. In Part Three, Quantum Optics, three new chapters have been added as important new developments: Chapter 29, Light Quanta and Their Origin; Chapter 30, Lasers; and Chapter 31, Holography. I wish to take this opportunity of thanking Dr. Donald H. White for his assistance in gathering much of the new material used in this the fourth edition. HARVEY E. WHITE PREFACE TO THE THIRD EDITION The chief objectives in preparing this new edition have been simplification and modernization. Experience on the part of the authors and of the many other users of the book over the last two decades has shown that many passages and mathematical derivations were overly cumbersome, thereby losing the emphasis they should have had. As an example of the steps taken to rectify this defect, the chapter on reflection has been entirely rewritten in simpler form and placed ahead of the more difficult aspects of polarized light. Furthermore, by expressing frequency and wavelength in circular measure, and by introducing the complex notation in a few places, it has been possible to abbreviate the derivations in wave theory to make room for new material. In any branch of physics fashions change as they are influenced by the development of the field as a whole. Thus, in optics the notions of wave packet, line width, and coherence length are given more prominence because of their importance in quantum mechanics. For the same reason, our students now usually learn to deal with complex quantities at an earlier stage, and we have felt justified in giving some examples of how helpful these can be. Because of the increasing use of concentric optics, as well as graphical methods of ray tracing, these subjects have been introduced in the chapters on geometrical optics. The elegant relationships between geometrical optics and particle mechanics, as in the electron microscope and quadrupole lenses, XX PREFACETO THB TIIIRD EDmON could not be developed because of lack of space; the instructor may wish to supplement the text in this direction. The same may be true of the rather too brief treatments of some subjects where old principles have recently come into prominence, as in Cerenkov radiation, the echelle grating, and multilayer films. A difficulty that must present itself to the authors of all textbooks at this level is that of avoiding the impression that the subject is a definitive, closed body of knowledge. If the student can be persuaded to read the original literature to any extent, this impression soon fades. To encourage such reading, we have inserted many references, to original papers as well as to books, throughout the text. An entirely new set of problems, representing a rather greater spread of difficulty than heretofore, is included. It is not possible to mention all those who have assisted us by suggestions for improvement. Specific errors or omissions have been pointed out by L. W. Alvarez, W. A. Bowers, J. E. Mack, W. C. Price, R. S. Shankland, and J. M. Stone, while H. S. Coleman, J. W. Ellis, F. S. Harris, Jr., R. Kingslake, C. F. J. Overhage, and R. E. Worley have each contributed several valuable ideas. We wish to express our gratitude to all of these, as well as to T. L. Jenkins, who suggested the simplification of certain derivations and checked the answers to many of the problems. FRANCIS A. JENKINS HARVEY E. WHITE FUNDAMENTALS OF OPTICS PART ONE Geometrical Optics 1 PROPERTIES OF LIGHT All the known properties of light are described in terms of the experiments by which they were discovered and the many and varied demonstrations by which they are frequently illustrated. Numerous though these properties are, their demonstrations can be grouped together and classified under one of three heads: geometrical optics, wave optics, and quantum optics, each of which may be subdivided as follows: Geometrical optics Rectilinear propagation Finite speed Reflection Refraction Dispersion Wave optics Interference Diffraction Electromagnetic character Polarization Double refraction Quantum optics Atomic orbits Probability densities Energy levels Quanta Lasers 4 FUNDAMENTALS OF OPTICS Screen Screen FIGURE lA A demonstration experiment illustrating the principle that light rays travel in straight lines. The rectilinear propagation of light. The first group of phenomena classified as geometrical optics are treated in the first 10 chapters of this text and are most easily described in terms of straight lines and plane geometry. The second group, wave optics, deals with the wave nature of light, and is treated in Chaps. 11 to 28. The third group, quantum optics, deals with light as made up of tiny bundles of energy called quanta, and is treated from the optical standpoint in Chaps. 29 to 33. 1.1 THE RECTILINEAR PROPAGATION OF LIGHT The rectilinear propagation of light is the technical terminology applied to the principle that "light travels in straight lines." The fact that objects can be made to cast fairly sharp shadows may be considered a good demonstration of this principle. Another illustration is found in the pinhole camera. In this simple and inexpensive device the image of a stationary object is formed on a photographic film or plate by light passing through a small opening, as diagramed in Fig. lAo In this figure the object is an ornamental light bulb emitting white light. To see how an image is formed, consider the rays of light emanating from a single point a near the top of the bulb. Of the many rays of light radiating in many directions the ray that travels in the exact direction of the hole passes through to the point a' near the bottom of the image screen. Similarly, a ray leaving b near the bottom of the bulb and passing through the hole will arrive at b', near the top of the image screen. Thus it can be seen how an inverted image of the entire bulb is formed. If the image screen is moved closer to the pinhole screen, the image will be proportionately smaller, whereas if it is moved farther away, the image will be proportionately larger. Excellent sharp photographs of stationary objects can be made with this arrangement. By making a pinhole in one end of a small box and placing a photographic film or plate at the other end, taking several time exposures as trial runs, good pictures are attainable. For good, sharp photographs the hole "' ". it., ...., ...~.. l t> "~'" "' ::i 51;.'. ~ :a.J~.i]d LJ~d~~ ~ FIGURE IB Photograph of the University of California Hospital, San Francisco, taken with a pinhole camera. exposure 3.0 min; square hole = 0.33 mm. Plate distance 9.5 em; Panchromatic film; 6 FUNDAMENTALS OF OPTICS must be very small, because its size determines the amount of blurring in the image. A small square hole is quite satisfactory. A piece of household aluminum foil is folded twice and the corner fold cut off with a razor blade, leaving good clean edges. After several such trials, and examination with a magnifying glass, a good square hole can be selected. The photograph reproduced in Fig. IB was taken with such a pinhole camera. Note the undistorted perspective lines as well as the depth of focus in the picture. 1.2 THE SPEED OF LIGHT The ancient astronomers believed that light traveled with an infinite speed. Any major event that occurred among the distant stars was believed to be observable instantly at all other points in the universe. It is said that around 1600 Galileo tried to measure the speed of light but was not successful. He stationed himself on a hilltop with a lamp and his assistant on a distant hilltop with another lamp. The plan was for Galileo to uncover his lamp at an agreed signal, thereby sending a flash of light toward his assistant. Upon seeing the light the assistant was to uncover his lamp, sending a flash oflight back to Galileo, who observed the total elapsed time. Many repetitions of this experiment, performed at greater and greater distances between the two observers, convinced Galileo that light must travel at an infinite speed. We now know that the speed of light is finite and that it has an approximate value of v = 300,000 km/s = 186,400 mils In 1849 the French physicist Fizeau. became the first man to measure the speed of light here on earth. His apparatus is believed to have looked like Fig. 1C. His account of this experiment is quite detailed, but no diagram of his apparatus is given in his notes. An intense beam of light from a source S is first reflected from a halfsilvered mirror G and then brought to a focus at the point 0 by means of lens L1• The diverging beam from 0 is made into a parallel beam by lens Lz. After traveling a distance of 8.67 km to a distant lens L3 and mirror M, the light is reflected back toward the source. This returning beam retraces its path through Lz, 0, and Lt, half of it passing through G and entering the observer's eye at E. The function of the toothed wheel is to cut the light beam into short pulses and to measure the time required for these pulses to travel to the distant mirror and back. When the wheel is at rest, light is permitted to pass through one of the openings at O. • Armand H. L. Fizeau (18191896), French physicist, was born of a wealthy French family that enabled him to be financially independent. Instead of shunning work, however, he devoted his life to diligent scientific experiment. His most important achievement was the measurement of the speed of light in 1849, carried on in Paris between Montmartre and Suresnes. He also gave the correct explanation of the Doppler principle as applied to light coming from the stars and showed how the effect could be used to measure stellar velocities. He carried out his experiments on the velocity oflight in a moving medium in 1851 and showed that light is dragged along by a moving stream of water. PROPERTIES OF LIGHT 7 s FIGURE Ie Experimental arrangement described by the French physicist Fizeau, with which he determined the speed of light in air in 1849. In this position all lenses and the distant mirror are aligned so that an image of the light source S can be seen by the observer at E. The wheel is then set rotating with slowly increasing speed. At some point the light passing through 0 will return just in time'to be stopped by tooth a. At this same speed light passing through opening 1 will return in time to be stopped by the next tooth b. Under these circumstances the light S is completely eclipsed from the observer. At twice this speed the light will reappear and reach a maximum intensity. This condition occurs when the light pulses getting through openings I, 2, 3, 4, ... return just in time to get through openings 2, 3, 4, 5, ... , respectively. Since the wheel contained 720 teeth, Fizeau found the maximum intensity to occur when its speed was 25 rev/so The time required for each light pulse to travel over and back could be calculated by (7h)(is) = 1/18,000 s. From the measured distance over and back of 17.34 km, this gave a speed of v = ~ = 17.34 km = 312,000 km/s t 1/18,000 s In the years that followed Fizeau's first experiments on the speed of light, a number of experimenters improved on his apparatus and obtained more and more accurate values for this universal constant. About threequarters of a century passed, however, before A. A. Michelson, and others following him, applied new and improved 8 FUNDAMENTALS OF OPTICS methods to visible light, radio waves, and microwaves and obtained the speed of light accurate to approximately six significant figures. Electromagnetic waves of all wavelengths, from X rays at one end of the spectrum to the longest radio waves, are believed to travel with exactly the same speed in a vacuum. These more recent experiments will be treated in detail in Chap. 19, but we give here the most generally accepted value of this universal constant, • c = 299,792.5 km/s = 2.997925 x 108 m/s (la) For practical purposes where calculations are to be made to four significant figures, the speed of light in air or in a vacuum may be taken to be c = 3.0 X 108 m/s (lb) One is often justified in using this rounded value since it differs from the more accurate value in Eq. (la) by less than 0.1 percent. 1.3 THE SPEED OF LIGHT IN STATIONARY MATTER In 1850, the French physicist Foucault* completed and published the results of an experiment in which he had measured the speed of light in water. Foucault's experiment was of great importance for it settled a long controversy over the nature of light. Newton and his followers in England and on the Continent believed light to be made up of small particles emitted by every light source. The Dutch physicist Huygens, on the other hand, believed light to be composed of waves, similar to water or sound waves. According to Newton's corpuscular theory, light should travel faster in an optically dense medium like water than in a less dense medium like air. Huygens' wave theory required light to travel slower in the more optically dense medium. Upon sending a beam of light back and forth through a long tube containing water, Foucault found the speed of light to be less than in air. This result was considered by many to be a strong confirmation of the wave theory. Foucault's apparatus for this experiment is shown in Fig. 10. Light coming through a slit S is reflected from a plane rotating mirror R to the equidistant concave mirrors Mt and M2• When R is in the position I, the light travels to Mt, back along the same path to R, through the lens L, and by reflection to the eye at E. When R is in position 2, the light travels the lower path through an auxiliary lens E and tube • Jean Bernard Leon Foucault (18191868), French physicist. After studying medicine he became interested in experimental physics and with A. H. L. Fizeau carried out experiments on the speed of light. After working together for some time, they quarreled over the best method to use for "chopping" up a light beam, and thereafter went their respective ways. Fizeau (using a toothed wheel) and Foucault (using a rotating mirror) did admirable work, each supplementing the work of the other. With a rotating mirror Foucault in 1850 was able to measure the speed of light in a number of different media. In 1851 he demonstrated the earth's rotation by the rotation of the plane of oscillation of a long, freely suspended, heavy pendulum. For the development of this device, known today as a Foucault pendulum, and his invention of the gyroscope, he received the Copley medal of the Royal Society of London, in 1855. He also discovered the eddy currents induced in a copper disk moving in a strong magnetic field and invented the optical polarizer which bears his name. PROPERTIES OF LIGHT 9 FIGURE 10 Foucault's apparatus for determining the speed of light in water. T to M2, back to R, through L to G, and then to the eye E. If now the tube Tis filled with water and the mirror is set into rotation, there will be a displacement of the images from E to E1 and E2• Foucault observed that the light ray through the tube was more displaced than the other. This means that it takes the light longer to travel the lower path through water than it does the upper path through the air. The image observed was due to a fine wire parallel to, and stretched across, the slit. Since sharp images were desired at E1 and E2, the auxiliary lens L' was necessary to avoid bending the light rays at the ends of the tube T. Over 40 years later the American physicist Michelson (first American Nobel laureate 1907) measured the speed of light in air and water. For water he found the value of 225,000 kmfs, which is just threefourths the speed in a vacuum. In ordinary optical glass, the speed was still lower, about twothirds the speed in a vacuum. The speed of light in air at normal temperature and pressure is about 87 kmfs less than in a vacuum, or v = 299,706 kmfs. For many practical purposes this difference may be neglected and the speed of light in air taken to be the same as in a vacuum, v = 3.0 X 108 mfs. 1.4 THE REFRACTIVE INDEX The index of refraction, or refractive index, of any optical medium is defined as the ratio between the speed of light in a vacuum and the speed of light in the medium: . d speed in vacuum Ref rae t.lve In ex = (Ie) speed in medium 10 FUNDAMENTALS OF OPTICS In algebraic symbols • n= c v (ld) The letter n is customarily used to represent this ratio. Using the speeds given in Sec. 1.3, we obtain the following values for the refractive indices: For glass: n = 1.520 (Ie) For water: n = 1.333 (If) For air: n = 1.000 (lg) Accurate determination of the refractive index of air at standard temperature (O°C) and pressure (760 mmHg) give (lh) for air n = 1.000292 Different kinds of glass and plastics have different refractive indices. The most commonly used optical glasses range from 1.52 to 1.72 (see Table IA). The optical density of any transparent medium is a measure of its refractive index. A medium with a relatively high refractive index is said to have a high optical density, while one with a low index is said to have a low optical density. 1.5 OPTICAL PATH To derive one ofthe most fundamental principles in geometric optics, it is appropriate to define a quantity called the optical path. The path d of a ray of light in any medium is given by the product velocity times time: d = vt Since by definition n = clv, which gives v = cln, we can write or nd = ct A = nd The product nd is called the optical path A: The optical path represents the distance light travels in a vacuum in the same time it travels a distance d in the medium. If a light ray travels through a series of optical media of thickness d, d', d", ... and refractive indices n, n', n", ... , the total optical path is just the sum of the separate values: • A = nd + n'd' + nild" + ... (Ii) A diagram illustrating the meaning of optical path is shown in Fig. 1E. Three media of length d, d', and d", with refractive indices n, n', and n", respectively, are shown touching each other. Line AB shows the length of the actual light path through these media, while the line CD shows the distance A, the distance light would travel in a vacuum in the same amount of time t. PROPERTIES Id ell .1. d' .1. d".f Equivalent optical path in a vacuum ~ OF LIGHT 11 .( D FIGURE IE The optical path through a series of optical media. 1.6 LAWS OF REFLECTION AND REFRACTION Whenever a ray of light is incident on the boundary separating two different media, part of the ray is reflected back into the first medium and the remainder is refracted (bent in its path) as it enters the second medium (see Fig. IF). The directions taken by these rays can best be described by two wellestablished laws of nature. According to the simplest of these laws, the angle at which the incident ray strikes the interface MM' is exactly equal to the angle the reflected ray makes with the same interface. Instead of measuring the angle of incidence and the angle of reflection from the interface MM', it is customary to measure both from a common line perpendicular to this surface. This line NN' in the diagram is called the normal. As the angle of incidence 4> increases, the angle of reflection also increases by exactly the same amount, so that for all angles of incidence • angle of incidence = angle of reflection (lj) A second and equally important part of this law stipulates that the reflected ray lies in the plane of incidence and on the opposite side of the normal, the plane of incidence being defined as the plane containing the incident ray and the normal. In other words, the incident ray, the normal, and the reflected ray all lie in the same plane, which is perpendicular to the interface separating the two media. The second law is concerned with the incident and refracted rays of light, and states that the sine of the angle of incidence and the sine of the angle of refraction bear a constant ratio one to the other, for all angles of incidence: sin 4> = const sin 4>' (lk) 12 FUNDAMENTALS OF OPTICS FIGURE IF Reflection and refraction at the boundary separating two media with refractive indices nand n', respectively. Furthermore, the refracted ray also lies in the plane of incidence and on the opposite side of the normal. This relationship, experimentally established by Snell, * is known as Snell's law. In addition the constant is found to have exactly the ratio of the refractive indices of the two media nand n'. Hence we can write sin </J n' = sin </J' n (ll) n sin </J = n' sin </J' (1m) which can be written in the symmetrical form • as By Eqs. (Ie) and (Id) the refractive indices of different optical media are defined c n =v and where c is the speed of light in a vacuum (c = 2.997925 are the speeds of light in the two media. + I C n= Vi (In) 108 m/s) and v and Vi • Willebrord Snell (15911626), Dutch astronomer and mathematician, was born at Leyden. At twentyone he succeeded his father as professor of mathematics at the University of Leyden. In 1611, he determined the size of the earth from measurements of its curvature between Alkmaar and BergenopZoom. He announced what is essentially the law of refraction in an unpublished paper in 1621. His geometrical construction requires that the ratios of the cosecants of ,p and ,p' be constant. Descartes was the first to use the ratio of the sines, and the law is known as Descartes' law in France. PROPERTIES By the substitution OF LIGHT 13 of Eqs. (lc) in Eq. (11), we obtain, sin 4J = sin 4J' .£. (10) v' If one or both indices are different from unity, the ratio n' /n is often called the relative index n' and Snell's law can be written sin 4J = n' sin 4J' • (lp) If the first medium is a vacuum, n = 1.0, the relative index has just the value of the second index and Eq. (lp) is again valid. If the first index is air at normal temperature and pressure (n = 1.000292), and if threefigure accuracy is satisfactory, Eq. (Ip) is again used. Wherever practical, we shall use unprimed symbols to refer to the first medium, primed symbols for the second medium, double primed symbols for the third medium, etc. When the angles of incidence and refraction are very small, a good approximation is obtained by setting the sines of angles equal to the angles themselves, obtaining (lq) 1.7 GRAPHICAL CONSTRUCTION FOR REFRACTION A simple method for tracing a ray of light across a boundary separating two optically transparent media is shown in Fig. IG. Because the principles involved in this construction are readily extended to complicated optical systems, the method is useful in the preliminary design of many different kinds of optical instruments. After the line GH is drawn, representing the boundary separating the two media of index nand n', the angle of incidence 4J of the incident ray JA is selected and the construction proceeds as follows. At one side of the drawing, and as reasonably close as possible, a line OR is drawn parallel to JA. With a point of origin 0, two circular arcs are drawn with their radii proportional to the two indices nand n', respectively. Through the point of intersection R a line is drawn parallel to the boundary normal NN', intersecting the arc n' at P. The line OP is next drawn in; parallel to it, through A, the refracted ray AB is drawn. The angle p between the incident and refracted ray, called the angle of deviation, is given by p = 4J  4J' (Ir) To prove that this construction sines to the triangle 0 RP: follows Snell's law exactly, we apply the law of OR OP =sin 4J' sin (1t Since sin (1t  4J) = sin 4J, OR = n, and OP = n', substitution n sin which is Snell's law [Eq. (ll)]. 4J)  gives directly n' 4J' =sin 4J (Is) 14 FUNDAMENTALS OF OPTICS o H n' FIGURE IG Graphical construction for refraction at a smooth surface separating two media of index nand n'. 1.8 THE PRINCIPLE OF REVERSIBILITY The symmetry of Eqs. (1j) and (1m) with respect to the symbols used shows at once that if a reflected or refracted ray is reversed in direction, it will retrace its original path. For any given pair of media with indices nand n' anyone value of 4J is correlated with a corresponding value of n'. This will be equally true when the ray is reversed and 4J' becomes the angle of incidence in the medium of n'; the angle of refraction will then be 4J. Since reversibility holds at each reflecting and refracting surface, it holds also for even the most complicated light paths. This useful principle has more than a purely geometrical foundation, and later it will be shown to follow from the application of wave motion to a principle in mechanics. 1.9 FERMAT'S PRINCIPLE The term optical path was introduced in Sec. 1.5, where it was defined as the distance a light ray would travel in a vacuum in the same time it travels from one point to another, a specified distance, through one or more optical media. The real path of a ray of light through a prism, with media of different refractive index on either side, is shown in Fig. lH. The optical path from the point Q in medium n, through medium n', and to the point Q" in medium n" is given by • A = nd + n'd' + n"d" (1t) One can also define an optical path in a medium of continuously varying refractive index by replacing the summation by an integral. The paths of the rays are then curved, and Snell's law of refraction loses its meaning. PROPERTIES OF LIGHT IS FIGURE IH The refraction of light by a prism and the meaning of optical path A. We shall now consider Fermat's. principle, which is applicable to any type of variation of n and hence contains within it the laws of reflection and refraction as well: The path taken by a light ray in going from one point to another through any set of media is such as to render its optical path equal, in the first approximation, to other paths closely adjacent to the actual one. The other paths must be possible ones in the sense that they may undergo deviations only where there are reflecting or refracting surfaces. Fermat's principle will hold for a ray whose optical path is a minimum with respect to adjacent hypothetical paths. Fermat himself stated that the time required by the light to traverse the path is a minimum and the optical path is a measure of this time. But there are plenty of cases in which the optical path is a maximum or neither a maximum nor a minimum but merely stationary (at a point of inflection) at the position of the true ray. Consider a ray of light that must pass through a point Q and then, after reflection from a plane surface, pass through a second point Q" (see Fig. 11). To find the real path, we first drop a perpendicular to GH and extend it an equal distance on the other side to Q'. The straight line Q' Q" is drawn in, and from its intersection B the line QB • Pierre de Fermat (16011665), French mathematician, born at BeaumontdeLomagne. In his youth, with Pascal, he made discoveries about the properties of numbers, on which he later built his method of calculating probabilities. His brilliant researches in the theory of numbers rank him as the founder of modern theory. He also studied the reflection of light and enunciated his principle of least time. His justification for this principle was that nature is economical, but he was unaware of circumstances where exactly the opposite is true. Fermat was a counselor for the parliament of Toulouse, distinguished for both legal knowledge and for strict integrity of conduct. He was also an accomplished general scholar and linguist. 16 FUNDAMENTALS OF OPTICS Q I I I I , I G 1M H I I I I : FIGURE 11 Fermat's principle applied to reflection at a plane surface. t ~ /// 1,/# ..•...~ I 1 I Q'''' I I I A Be x is drawn. The real light path is therefore QBQ", and, as can be seen from the symmetry relations in the diagram, it obeys the law of reflection. Consider now adjacent paths to points like A and C on the mirror surface close to B. Since a straight line is the shortest path between two points, both the paths Q' A Q" and Q' CQ" are greater than Q' BQ". By the above construction and equivalent triangles, QA = Q'A, and QC = Q'C, so that QAQ" > QBQ" and QCQ" > QBQ". Therefore the real path QBQ" is a minimum. A graph of hypothetical paths close to the real path QBQ", as shown in the lower right of the diagram, indicates the meaning of a minimum, and the flatness of the curve between A and C illustrates that to a first approximation adjacent paths are equal to the real optical path. Consider finally the optical properties of an ellipsoidal reflector, as shown in Fig. lJ. All rays emanating from a point source Q at one focus are reflected according to the law of reflection and come together at the other focus Q'. Furthermore all paths are equal in length. It will be remembered that an ellipse can be drawn with a string of fixed length with its ends fastened at the foci. Because all optical paths are equal, this is a stationary case, as mentioned above. On the graph in Fig. lK(b) equal path lengths are represented by a straight horizontal line. Some attention will be devoted here to other reflecting surfaces like a and c shown dotted in Fig. lJ. If these surfaces are tangent to the ellipsoid at the point B, F FIGURE 1] Fermat's principle applied to an ellipti. cal reflector. PROPERTIES FIGURE lK Graphs of optical paths involving reflection illustrating conditions for (a) maximum, (b) stationary, and (c) minimum light paths. Fermat's principle. OF LIGHT (a) (b) (c) B B B 17 8 the line NB is normal to all three surfaces and QBQ' is a real path for all three. Adjacent paths from Q to points along these mirrors, however, will give a minimum condition for the real path to and from reflector c and a maximum condition for the real path to and from reflector a (see Fig. lK). It is readily shown mathematically that both the laws of reflection and refraction follow from Fermat's principle. Figure lL, which represents the refraction of a ray at a plane surface, can be used to prove the law of refraction [Eq. (1m)]. The length of the optical path between the point Q in the upper medium of index n and another point Q' in the lower medium of index n' passing through any point A on the surface is A = nd + n'd' (lu) where d and d' represent the distances QA and AQ', respectively. Now if we let h and hi represent perpendicular distances to the surface and p the total length of the x axis intercepted by these perpendiculars, we can invoke the pythagorean theorem concerning right triangles and write d2 = h2 + (p _ X)2 d'2 = h'2 + x2 When these values of d and d' are substituted in Eq. (Ii), we obtain A = n[h2 + (p  x)2r/2 + n'(h'2 + X2)1/2 (Iv) According to Fermat's principle, A must be a minimum or a maximum (or in general stationary) for the actual path. One method of finding a minimum or maximum for the optical path is to plot a graph of A against x and find at what value of x FIGURE lL Geometry of a refracted ray used in illustrating Fermat's principle. 18 FUNDAMENTALS OF OPTICS a tangent to the curve is parallel to the x axis (see Fig. IK). The mathematical means for doing the same thing is, first, to differentiate Eq. (I v) with respect to the variable x, thus obtaining an equation for the slope of the graph, and, second, to set this resultant equation equal to zero, thus finding the value of x for which the slope of the curve is zero. By differentiating Eq. (I v) with respect to x and setting the result equal to zero, we obtain dl!!. = dx which gives x n  P [hZ + (p _ x)Zr/z = n'  x Z (h'Z + X )l/Z p  x , x n=nd d' or simply By reference to Fig. IL it will be seen that the multipliers of nand n' are just the sines of the corresponding angles, so that we have now proved Eq. (1m), namely n sin <p = n' sin <p' (I w) A diagram for reflected light, similar to Fig. IL, can be drawn and the same mathematics applied to prove the law of reflection. 1.10 COLOR DISPERSION It is well known to those who have studied elementary physics that refraction causes a separation of white light into its component colors. Thus, as is shown in Fig. 1M, the incident ray of white light gives rise to refracted rays of different colors (really a continuous spectrum) each of which has a different value of <p'. By Eq. (1m) the value of n' must therefore vary with color. It is customary in the exact specification of indices of refraction to use the particular colors corresponding to certain dark lines in the spectrum of the sun. These Fraunhofer* lines, which are designated by the letters A, B, C, ... , starting at the extreme red end, are given in Table IA. The ones most commonly used are those in Fig. 1M. The angular divergence of rays F and C is a measure of the dispersion produced, and has been greatly exaggerated in the figure relative to the average deviation of the • Joseph von Fraunhofer (17871826) was the son of a Bavarian glazier. He learned glass grinding from his father and entered the field of optics from the practical side. Fraunhofer gained great skill in the manufacture of achromatic lenses and optical instruments. While measuring the refractive index of different kinds of glass and its variation with color or wavelength, he noticed and made use of the yellow D lines of the sodium spectrum. He was one of the first to produce diffraction gratings, and his rare skill with these devices enabled him to produce better spectra than his predecessors. Although the dark lines of the solar spectrum were first observed by W. H. Wollaston, they were carefully observed by Fraunhofer, under high dispersion and resolution, and the wavelengths of the most prominent lines were measured with precision. He mapped 576 of these lines, the principal ones, denoted by the letters A through K, being known by his name. 19 PROPERTIES OF UGHT FIGURE 1M Upon refraction, white light is spread out into a spectrum. This is called dispersion. spectrum, which is measured by the angle through which ray D is bent. To take a typical case of crown glass, the refractive indices as given in Table IA are nF = 1.52933 nD = 1.52300 nc = 1.52042 Now it is readily shown from Eq. (lq) that for a given small angle f/J the dispersion of the F and Crays (f/J~  f/Jc) is proportional to nF  nc = 0.00891 while the deviation of the Dray (f/J  f/J~) depends on nD  I which is equal to 0.52300. Thus it is nearly 60 times as great. The ratio of these two quantities varies greatly for different kinds of glass and is an important characteristic of any optical substance. It is called the dispersive power and is defined by the equation v = nF  nc (Ix) nD  I The reciprocal of the dispersive power is called the dispersive index v: • (Iy) For most optical glasses v lies between 20 and 60 (see Table 1B and Appendix III). Table lA FRAUNHOFER'S DESIGNATIONS, ELEMENT SOURCE, WAVELENGTH, AND REFRACTIVE INDEX FOR FOUR OPTICAL GLASSES. Designadon Chemical element Wavelength, At Spectacle crown Light flint Dense flint Extra dense flint C D F G' H Na H H 6563 5892 4861 4340 1.52042 1.52300 1.52933 1.53435 1.57208 1.57600 1.58606 '1.59441 1.66650 1.67050 1.68059 1.68882 1.71303 1.72000 1.73780 1.75324 • For other glasses and crystals see Appendices III and IV. (A) to nanometers (nm), move decimal point one place to the left (see Appendix VI). t To change wavelengths in angstroms 20 FUNDAMENTALS OF OPTICS t n FIGURE IN The variation of refractive index with color. Foe Violet Blue Green Yellow Red Figure IN illustrates schematically the type of variation of n with color that is usually encountered for optical materials. The denominator of Eq. (ly), which is a measure of the dispersion, is determined by the difference in the index at two points near the ends of the spectrum. The numerator, which measures the average deviation, represents the magnitude in excess of unity of an intermediate index of refraction. It is customary in most treatments of geometrical optics to neglect chromatic effects and assume, as we have in the next seven chapters, that the refractive index of each specific element of an optical instrument is that determined for yellow sodium D light. Table 1B DISPERSION GLASSES. INDEX FOR FOUR OPTICAL Glass Spectacle crown Light flint Dense flint Extra dense flint v 58.7 41.2 47.6 29.08 • See Table IA. PROBLEMS. 1.1 1.2 1.3 A boy makes a pinhole camera out of a cardboard box with the dimensions 10.0 cm x 10.0 cm x 16.0 cm. A pinhole is located in one end, and a film 8.0 cm x 8.0 cm is placed in the other end. How far away from a tree 25.0 m high should the boy place his camera if the image of the tree is to be 6.0 cm high on the film? Ans. 66.7 m A physics student wishes to repeat Fizeau's experiment for measuring the speed of light. If he uses a toothed wheel containing 1440 teeth and his distant mirror is located in a laboratory window across the college campus 412.60 m away, how fast must his wheel be rotated if the returning light pulses show the first maximum intensity? If the mirror R in Foucault's experiment were to rotate at 12,000 rev/min, find (a) the rotational speed of the mirror R in revolutions per second and (b) the rotational speed of the sweeping beam RM1 in radians per second. Find the time it takes the light to traverse the path (c) RM1R and (d) RMzR. What is the observed slit deflection (e) EEl> and (f) EE2? Assume the distances RM1 = RM2 = 6.0 m, RS = RE = 6.0 m, the • Before solving any problems in this text, read Appendix VI. PROPERTIES OF LIGHT 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 21 length of the water tube T = 5.0 m, the refractive index of water is 1.3330, and the speed of light in air is 3.0 x 108 m/s. If the refractive index for a piece of optical glass is 1.5250, calculate the speed of light in the glass. Ans. 1.9659 x 108 mls Calculate the difference between the speed of light in kilometers per second in a vacuum and the speed of light in air if the refractive index of air is 1.0002340. Use velocity values to seven significant figures. If the moon's distance from the earth is 3.840 x 105 km, how long will it take microwaves to travel from the earth to the moon and back again? How long does it take light from the sun to reach the earth? Assume the earth's distance from the sun to be 1.50 X 108 km. Ans. 500 s, or 8 min 20 s A beam of light passes through a block of glass 10.0 cm thick, then through water for a distance of 30.5 cm, and finally through another block of glass 5.0 cm thick. If the refractive index of both pieces of glass is 1.5250 and of water is 1.3330, find the total optical path. A water tank is 62.0 cm long inside and has glass ends which are each 2.50 cm thick. If the refractive index of water is 1.3330 and of glass is 1.6240, find the overall optical path. A beam of light passes through 285.60 cm of water of index 1.3330, then through 15.40 cm of glass of index 1.6360, and finally through 174.20 cm of oil of index 1.3870. Find to three significant figures (a) each of the separate optical paths and (b) the total optical path. Ans. (a) 380.7, 25.19, and 241.6 cm, (b) 647 cm A ray of light in air is incident on the polished surface of a block of glass at an angle of 10°. (a) Ifthe refractive index of the glass is 1.5258, find the angle of refraction to four significant figures. (b) Assuming the sines of the angles in Snell's law can be repllJced by the angles themselves, what would be the angle of refraction? (c) Find the percentage error. Find the answers to Prob. 1.11, if the angle of incidence is 45.0° and the refractive index is 1.4265. A ray of light in air is incident at an angle of 54.0° on the smooth surface of a piece of glass. (a) If the refractive index is 1.5152, find the angle of refraction to four significant figures. (b) Find the angle of refraction graphically. (See Fig. P1.13). Ans. (a) 32.272°, (b) 32.3° Air Glass n' FIGURE P1.13 Graph for part (b) of Prob. 1.13. = 1.5152 22 FUNDAMENTALS OF OPTICS 1.14 A straight hollow pipe exactly 1.250 m long, with glass plates 8.50 mm thick to close the two ends, is thoroughly evacuated. (0) If the glass plates have a refractive index of 1.5250, find the overall optical path between the two outer glass surfaces. (b) By how much is the optical path increased if the pipe is filled with water of refractive index 1.33300. Give answers to five significant figures. 1.15 ReferringtoFig.1L,thedistancex = 6.0cm,h = 12.0cm,h' = 15.0cm,n = 1.3330, and n' = 1.5250. Find ,p', ,p, d, d', p, and 11, to three significant figures. Ans. ,p' = 21.80°, ; = 25.14°, d = 13.26 em, d' = 16.16 cm, p = 11.63 cm, 11 = 42.3 cm 1.16 Solve Prob. 1.15 graphically. 1.17 In studying the refraction of light Kepler arrived at a refraction formula ,p' ,p = 1 ksec,p' where n'  1 k=n' n' being the relative index of refraction. Calculate the angle of incidence ,p for a piece 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 of glass for which n' = 1.7320 and the angle of refraction ,p' = 32.0° according to (0) Kepler's formula and (b) Snell's law. Note that sec ,p' = 1/(cos ,p'). White light is incident at an angle of 55.0° on the polished surface of a piece of glass. If the refractive indices for red C light and blue F light are nc = 1.53828, and nF = 1.54735, respectively, what is the angular dispersion between these two colors? (0) Find the two angles to five significant figures and (b) the dispersion to three significant figures. Ans. (0) ,p'c = 32.1753°, ,p'F = 31.9643°, (b) 0.2110° A piece of dense flint glass is to be made into a prism. If the refractive indices for red, yellow, and blue light are specified as nc = 1.64357, no = 1.64900, and nF = 1.66270, find (0) the dispersive power and (b) the dispersion constant for this glass. A block of spectacle crown glass is to be made into a lens. The refractive indices furnished by the glass manufacturer are specified as nc = 1.52042, no = 1.52300, and nF = 1.52933. Determine the value of (0) the dispersion constant and (b) the dispersive power. A piece of extra dense flint glass is to be made into a prism. The refractive indices furnished by the glass manufacturer are those given in Table 1A. Find the value of (0) the dispersive power and (b) the dispersion constant. Ans. (0) 0.034403, (b) 29.067 Two plane mirrors are inclined to each other at an angle ex. Applying the law of reflection show that any ray whose plane of incidence is perpendicular to the line of intersection of the two mirrors is deviated by two reflections by an angle ~ which is independent of the angle of incidence. Express this deviation in terms of ex. An ellipsoidal mirror has a major axis of 10.0 cm, a minor axis of 8.0 cm, and foci 6.0 cm apart. If there is a point source of light at one focus Q, there are only two rays of light that pass through the point C, midway between Band Q', as can be drawn in Fig. IJ. Draw such an ellipse and graphically determine whether these two paths QBC and QDC are maxima, minima, or stationary. A ray of light in air enters the center of one face of a prism at an angle making 55.0° with the normal. Traveling through the glass, the ray is again refracted into the air beyond. Assume the angle between the two prism faces to be 60.0° and the glass to have a refractive index of 1.650. Find the deviation of the ray (0) at the first surface and (b) the second surface. Find the total deviation (c) by calculation and (d) graphically. One end of a glass rod is ground and polished to the shape of a hemisphere with a diameter of 10.0 em. Five parallel rays of light 2.0 em apart and in the same plane are PROPERTIES OF UGHT 23 incident on this curved end, with one ray traversing the center of the hemisphere parallel to the rod axis. If the refractive index is 1.5360, calculate the distances from the front surface to the point where the refracted rays cross the axis. 1.26 Crystals of clear strontium titanate are made into semiprecious gems. The refractive indices for different colors of light are as follows: A,A n Red Yellow Blue Violet 6563 2.37287 5892 2.41208 4861 2.49242 4340 2.57168 Calculate the value of (a) the dispersion constant and (b) the dispersive power. Plot a graph of the wavelength A.against the refractive index n. Use the blue, yellow, and red indices. . / ! 2 PLANE SURFACES AND PRISMS The behavior of a beam of light upon reflection or refraction at a plane surface is of basic importance in geometrical optics. Its study will reveal several of the features that will later have to be considered in the more difficult case of a curved surface. Plane surfaces often occur in nature, e.g., as the cleavage surfaces of crystals or as the surfaces of liquids. Artificial plane surfaces are used in optical instruments to bring about deviations or lateral displacements of rays as well as to break light into its colors. The most important devices of this type are prisms, but before taking up this case of two surfaces inclined to each other, we must examine rather thoroughly what happens at a single plane surface. 2.1 PARALLEL BEAM In a beam or pencil of parallel light, each ray meets the surface traveling in the same direction. Therefore anyone ray may be taken as representative of all the others. The parallel beam remains parallel after reflection or refraction at a plane surface, as shown in Fig. 2A(a). Refraction causes a change in width ofthe beam which is easily seen to be in the ratio (cos 4>')/(cos4», whereas the reflected beam remains of the same PLANE SURFACES AND PRISMS (e) (b) (a) 25 N I I I I I n' "',ii",:"i,.,:;,i,:,.;:.,i,!,i:,!, Nii ..•. l,.,,!,;r,,: ;,., .•...;.'.! .•:, •..•!.. i.i~. :,'.'.:,.i.. :;:;:;:;:::;:;:;:;:;:::;:::;:::::;:; n <n' n>n' n >n' FIGURE 2A Reflection and refraction of a parallel beam: (a) external reflection; (b) internal reflection at an angle smaller than the critical angle; (c) total reflection at or greater than the critical angle. width. There is also chromatic dispersion of the refracted beam but not of the reflected one. Reflection at a surface where n increases, as in Fig. 2A(a), is called external reflection. It is also frequently termed raretodense reflection because the relative magnitudes of n correspond roughly (though not exactly) to those of the actual densities of materials. In Fig. 2A(b) is shown a case of internal reflection or densetorare reflection. In this particular case the refracted beam is narrow because tP' is close to 90°. 2.2 THE CRITICAL ANGLE AND TOTAL REFLECTION We have already seen in Fig. 2A(a) that as light passes from one medium like air into another medium like glass or water the angle of refraction is always less than the angle of incidence. While a decrease in angle occurs for all angles of incidence, there exists a range of refracted angles for which no refracted light is possible. A diagram illustrating this principle is shown in Fig. 2B, where for several angles of incidence, from 0 to 90°, the corresponding angles of refraction are shown from 0° to tPe, respectively. It will be seen that in the limiting case, where the incident rays approach an angle of 90° with the normal, the refracted rays approach a fixed angle tPe beyond which no refracted light is possible. This particular angle tPe, for which tP = 90°, is called the critical angle. A formula for calculating the critical angle is obtained by substituting tP = 90°, or sin tP = I, in Snell's law (Eq. (1m)], n x 1 • so that = n' sin . '" =n sm'l'c n' tPc (2a) 26 FUNDAMENTALS OF OPTICS 2 N 1 (a) N (b) FIGURE 2B Refraction and total reflection: (0) the critical angle is the limiting angle of refraction; (b) total reflection beyond the critical angle. a quantity which is always less than unity. For a common crown glass of index 1.520 surrounded by air sin <Pc = 0.6579, and <Pc = 41°8'. If we apply the principle of reversibility of light rays to Fig. 2B(a), all incident rays will lie within a cone subtending an angle of 2<pc, while the corresponding refracted rays will lie within a cone of 180°. For angles of incidence greater than <Pc there can be no refracted light and every ray undergoes total reflection as shown in Fig.2B(b). The critical angle for the boundary separating two optical media is defined as the smallest angle of incidence, in the medium of greater index,for which light is totally reflected. Total reflection is really total in the sense that no energy is lost upon reflection. In any device intended to utilize this property there will, however, be small losses due to absorption in the medium and to reflections at the surfaces where the light enters and leaves the medium. The commonest devices of this kind are called total reflection prisms, which are glass prisms with two angles of 45° and one of 90°. As shown in Fig. 2C(a), the light usually enters perpendicular to one of the shorter faces, is totally reflected from the hypotenuse, and leaves at right angles to the other short face. This deviates the rays through a right angle. Such a prism may also be used in two other ways which are illustrated in (b) and (c) of the figure. The Dove prism (c) interchanges the two rays, and if the prism is rotated about the direction of the light, they rotate around each other with twice the angular velocity of the prism. Many other forms of prisms which use total reflection have been devised for special purposes. Two common ones are illustrated in Fig. 2C(d) and (e). The roof prism accomplishes the same purpose as the total reflection prism (a) except that it introduces an extra inversion. The triple mirror (e) is made by cutting off the corner of a cube by a plane which makes equal angles with the three faces intersecting at that PLANE SURFACES AND PRISMS Tolol refleclion 27 Dove or inverting Porro (G) ,,"... " " " 2 (e) )~, "" 2 1 2 Amici or roof Triple mirror LummerBrodhun FIGURE 2C Reflecting prisms utilizing the principle of total reflection. corner.* It has the useful property that any ray striking it will, after being internally reflected at each of the three faces, be sent back parallel to its original direction. The LummerBrodhun "cube" shown in (f) is used in photometry to compare the illumination of two surfaces, one of which is viewed by rays (2) coming directly through the circular region where the prisms are in contact, the other by rays (I) which are totally reflected in the area around this region. Since, in the examples shown, the angles of incidence can be as small as 45°, it is essential that this exceed the critical angle in order that the reflection be total. Supposing the second medium to be air (n' = 1), this requirement sets a lower limit on the value of the index n of the prism. By Eq. (2a) we must have n' 1 . 450 =~SIn n n J2 so that n ~ = 1.414. This condition always holds for glass and is even fulfilled for optical materials having low refractive indices such as Lucite (n = 1.49) and fused quartz (n = 1.46). The principle of most accurate rejractometers (instruments for the determination of refractive index) is based on the measurement of the critical angle cPc' In both the Pulfrich and Abbe types a convergent beam strikes the surface between the unknown sample, of index n, and a prism of known index n'. Now n' is greater than n, so the • A 46cm array of 100 of these prisms is located on the moon's surface, 3.84 x 10' m from the earth. This retrodirector, placed there during the Apollo 11 moon flight, is used to return light from a laser beam from the earth to a point on the earth close to the source. Such a marker can be used to accurately determine the distance to the moon at different times. See J. E. Foller and E. J. Wampler, The Lunar Reflector, Sci. Am., March 1970, p. 38. For more details see Sec. 30.13. / 28 FUNDAMENTALS OF OPTICS FIGURE 2D Refraction by the prism in a Pulfrich refractometer. two must be interchanged in Eq. (2a). The beam is so oriented that some of its rays just graze the surface (Fig. 2D) so that one observes in the transmitted light a sharp boundary between light and dark. Measurement of the angle at which this boundary occurs allows one to compute the value of ljJc and hence of n. There are important precautions that must be observed if the results are to be at all accurate. * 2.3 PLANEPARALLEL PLATE When a single ray traverses a glass plate with plane surfaces that are parallel to each other, it emerges parallel to its original direction but with a lateral displacement d which increases with the angle of incidence ljJ. Using the notation shown in Fig. 2E, we may apply the law of refraction and some simple trigonometry to find the displacement d. Starting with the right triangle ABE, we can write d = I sin (ljJ  ljJ') (2b) which, by the trigonometric relation for the sine of the difference between two angles, can be written d = I(sin ljJ cos ljJ'  sin ljJ' cos ljJ) (2c) From the right triangle ABC we can write t 1=cos ljJ' which, substituted in Eq. (2c), gives d = t (sin ljJ cos ljJ' _ sin ljJ' cos ljJ) cos ljJ' cos ljJ' (2d) From Snell's law [Eq. (1m)] we obtain sin ljJ' = !!.. sin ljJ n' • For a valuable description of this and other methods of determining indices of refraction see A. C. Hardy and F. H. Perrin, "Principles of Optics," pp. 359364, McGrawHill Book Company, New York, 1932. PLANE SURFACES AND PRISMS 29 FIGURE 2E Refraction by a planeparallel plate. which upon substitution in Eq. (2d), gives d = t (sin d=tsIDc/J. c/J  cos c/J .! sin c/J) cos c/J' n' c/J) ( 1n cos n' cos c/J' (2e) From 0° up to appreciably large angles, d is nearly proportional to c/J, for as the ratio of the cosines becomes appreciably less than 1, causing the righthand factor to increase, the sine factor drops below the angle itself in almost the same proportion. * 2.4 REFRACTION BY A PRISM In a prism the two surfaces are inclined at some angle ex so that the deviation produced by the first surface is not annulled by the second but is further increased. The chromatic dispersion (Sec. 1.10) is also increased, and this is usually the main function of a prism. First let us consider, however, the geometrical optics of the prism for light of a single color, i.e., for monochromatic light such as is obtained from a sodium arc. • This principle is made use of in most of the home movingpicture filmeditor devices in common use today. Instead of starting and stopping intermittently, as it does in the normal film projector, the film moves smoothly and continuously through the filmeditor gate. A small eightsided prism, immediately behind the film, produces a stationary image of each picture on the viewing screen of the editor. See Prob. 2.2 at the end of this chapter. 30 FUNDAMENTALS OF OPTICS FIGURE 2F The geometry associated with refraction by a prism. The solid ray in Fig. 2F shows the path of a ray incident on the first surface at the angle 4>1' Its refraction at the second surface, as well as at the first surface, obeys Snell's law, so that in terms of the angles shown sin 4>1 = n' = sin 4>2 (2f) sin 4>~ n sin 4>2 The angle of deviation produced by the first surface is P = 4>1  4>~, and that produced by the second surface is '}' = 4>2  4>2' The total angle of deviation ~ between the incident and emergent rays is given by ~ = P + '}' Since NN' and MN' are perpendicular to the two prism faces, angle at N'. From triangle ABN' and the exterior angle 0(, we obtain (2g) 0( is also the 4>~ + cP2 (2h) cP1 + cP2  (cP~ + cP2) ~ = cP1 + cP2  0( (2i) 0( = Combining the above equations, we obtain ~= P + '}'= cP1  cP~ + cP2  cP2 or 2.5 MINIMUM = DEVIATION When the total angle of deviation ~ for any given prism is calculated by the use of the above equations, it is found to vary considerably with the angle of incidence. The angles thus calculated are in exact agreement with the experimental measurements. If during the time a ray of light is refracted by a prism the prism is rotated continuously in one direction about an axis (A in Fig. 2F) parallel to the refracting edge, the angle of deviation ~ will be observed to decrease, reach a minimum, and then increase again, as shown in Fig. 2G. The smallest deviation angle, called the angle of minimum deviation <5m,occurs at that particular angle of incidence where the refracted ray inside the prism makes equal angles with the two prism faces (see Fig. 2H). In this special case 4>2 4>~ = 4>2 P = '}' (2j) To prove these angles equal, assume 4>1 does not equal cP2 when minimum cP1 = deviation occurs. By the principle of the reversibility of light rays (see Sec. 1.8), PLANE SURFACES AND PRISMS 31 60 50 30 20 20 30 40 60 q,1 80 70 90 FIGURE 2G A graph of the deviation produced by a 60° glass prism of index n' = 1.50. At minimum deviation Om = 37.r, ~1 = 48.6°, and ~1' = 30.0°. there would be two different angles of incidence capable of giving minimum deviation. Since experimentally we find only one, there must be symmetry and the above equalities must hold. In the triangle ABC in Fig. 2H the exterior angle ~m equals the sum of the opposite interior angles p + y. Similarly, for the triangle ABN', the exterior angle a equals the sum <P~ + <P2' Consequently a Solving these three equations = 2<p~ ~m = 2p for <P~ and <PIgives <P~ = !a <PI = <P~ + p Since by Snell's law n'/n = (sin <pl)/(sin <PI.), ~ = sin !(a n FIGURE 2H The geometry of a light ray traversing a prism at minimum deviation. + sin !a ~m) (2k) 32 FUNDAMENTALS OF OPTICS The most accurate measurements of refractive index are made by placing the sample in the form of a prism on the table of a spectrometer and measuring the angles (jm and IX, the former for each color desired. When prisms are used in spectroscopes and spectrographs, they are always set as nearly as possible at minimum deviation because otherwise any slight divergence or convergence of the incident light would cause astigmatism in the image. 2.6 THIN PRISMS The equations for the prism become much simpler when the refracting angle IX becomes small enough to ensure that its sine and the sine of the angle of deviation (j may be set equal to the angles themselves. Even at an angle of 0.1 rad, or 5.7°, the difference between the angle and its sine is less than 0.2 percent. For prisms having a refracting angle of only a few degrees, we can therefore simplify Eq. (2k) by writing n' n sin t(om + IX) om + = """"'= sin !IX IX IX o = (n'  1)0( Thin prism in air • and (21) The subscript on 0 has been dropped because such prisms are always used at or near minimum deviation, and n has been dropped because it will be assumed that the surrounding medium is air, n = 1. It is customary to measure the power of a prism by the deflection of the ray in centimeters at a distance of 1 m, in which case the unit of power is called the prism diopter (D). A prism having a power of 1 prism diopter therefore displaces the ray on a screen 1 m away by 1 cm. In Fig. 21(0) the deflection on the screen is x cm and is numerically equal to the power of the prism. For small values of 0 it will be seen that the power in prism diopters is essentially the angle of deviation 0 measured in units of 0.01 rad, or 0.573°. For the dense flint glass of Table lA, n~ = 1.67050, and Eq. (21) shows that the refracting angle of a 10 prism should be 0( 2.7 COMBINATIONS = 0.57300 0.67050 = 0.85459° OF THIN PRISMS In measuring binocular accommodation, ophthalmologists make use of a combination of two thin prisms of equal power which can be rotated in opposite directions in their own plane [Fig. 2I(b)]. Such a device, known as the Risley or Herschel prism, is equivalent to a single prism of variable power. When the prisms are parallel, the power is twice that of either one; when they are opposed, the power is zero. To find how the power and direction of deviation depend on the angle between the 33 PLANE SURFACES AND PRISMS (e) FIGURE 21 Thin prisms: (a) the displacement x in centimeters at a distance of 1 m gives the power of the prism in diopters; (b) Risley prism of variable power; (c) vector addition of prism deviations. components, we use the fact that the deviations add vectorially. In Fig. 2I(e) it will be seen that the resultant deviation c; will in general be, from the law of cosines, c; = •. Jc;/ + 0/ + 20102 cos p (2m) where P is the angle between the two prisms. To find the angle y between the resultant deviation and that due to prism 1 alone (or, we may say, between the "equivalent" prism and prism 1) we have the relation 02 sin P t any=(2n) 01 + 02 cos P Since almost always 01 = 02' we may call the deviation and the equations simplify to ° = ../20/(1 + cos P) = J40/ cos2 tan y = and I by either component ~ 2 = 20, cos ~ sin P = + cos P GRAPHICAL METHOD (20) tan ~ 2 y=~ so that 2.8 2 0" 2 (2p) OF RAY TRACING It is often desirable in the process of designing optical instruments to be able to trace rays of light through the system quickly. For prism instruments the principles presented below are extremely useful. Consider first a 60° prism of index n' = 1.50 surrounded by air of index n 1.00. After the prism has been drawn to scale, as in Fig. 2J, and the angle of incidence tPl has been selected, the construction begins as in Fig. 10. Line OR is drawn parallel to lA, and, with an origin at 0, the two circular arcs are drawn with radii proportional to nand n'. Line RP is drawn parallel to NN', = 34 FUNDAMENTALS OF OPTICS n n' FIGURE 2J A graphical method for ray tracing through a prism. and OP is drawn to give the direction of the refracted ray AB. Carrying on from the point P, a line is drawn parallel to MN' to intersect the arc nat Q. The line OQ then gives the correct direction of the final refracted ray BT. In the construction diagram at the left the angle RPQ is equal to the prism angle a, and the angle ROQ is equal to the total angle of deviation b. 2.9 DIRECTVISION PRISMS As an illustration of ray tracing through several prisms, consider the design of an important optical device known as a directvision prism. The primary function of such an instrument is to produce a visible spectrum the central color of which emerges from the prism parallel to the incident light. The simplest type of such a combination usually consists of a crownglass prism of index n' and angle a' opposed to a f1intglass prism of index nil and angle a", as shown in Fig. 2K. The indices n' and nil chosen for the prisms are those for the central color of the spectrum, namely, for the sodium yellow D lines. Let us assume that the angle a" of the flint prism is selected and the construction proceeds with the light emerging perpendicular to the last surface and the angle a' of the crown prism as the unknown. The flint prism is first drawn with its second face vertical. The horizontal line OP is next drawn, and, with a center at 0, three arcs are drawn with radii proportional to n, n', and nil. Through the intersection at P a line is drawn perpendicular to AC intersecting n' at Q. The line RQ is next drawn, and normal to it the side AB of the crown prism. All directions and angles are now known. OR gives the direction of the incident ray, OQ the direction of the refracted ray inside the crown prism, OP the direction of the refracted ray inside the flint prism, and finally OP the direction of the emergent rayon the right. The angle a' of the crown prism is the supplement of angle RQP. If more accurate determinations of angles are required, the construction diagram will be found useful in keeping track of the trigonometric calculations. Ifthe dispersion of white light by the prism combination is desired, the indices n' and nil for the red and violet light can be drawn in and new ray diagrams constructed proceeding now PLANE SURFACES AND PRISMS 35 A B o FIGURE 2K Graphical ray tracing applied to the design of a directvision prism. from left to right in Fig. 2K(b). These rays, however, will not emerge perpendicular to the last prism face. The principles just outlined are readily extended to additional prism combinations like those shown in Fig. 2L. It should be noted that the upper directvision prism in Fig. 2L is in principle two prisms of the type shown in Fig. 2K placed back to back. V R FIGURE 2L Directvision prisms used for producing a spectrum with its central color in line with the incident white light. I 36 FUNDAMENTALS OF OPTICS Object "" 1 j. Q I I I I I IA FIGURE 2M The reflection of divergent rays of light from a plane surface. 2.10 REFLECTION OF DIVERGENT RAYS When a divergent pencil of light is reflected at a plane surface, it remains divergent. All rays originating from a point Q (Fig. 2M) will after reflection appear to come from another point Q' symmetrically placed behind the mirror. The proof of this proposition follows at once from the application of the law of reflection (Eq. (lj)], according to which all the angles labeled 4J in the figure must be equal. Under these conditions the distances QA and AQ' along the line QAQ' drawn perpendicular to the surface must be equal; i.e., s = s' object distance = image distance The point Q' is said to be a virtual image of Q since when the eye receives the reflected rays, they appear to come from a source at Q' but do not actually pass through Q', as would be the case if it were a real image. In order to produce a real image a surface other than a plane one is required. 2.11 REFRACTION OF DIVERGENT RAYS If an object is embedded in clear glass or plastic or is immersed in a transparent liquid such as water, the image appears closer to the surface. Fig. 2N has been drawn accurately to scale for an object Q located in water of index 1.3330 at a depth PLANE SURFACES AND PRISMS 37 ~E N' I G I I I I I H I I I I I I 1 s' 1 N FIGURE2N Image positions of an object under water as seen by an observer above; n > n'. s below the surface. Light rays diverging from this object arrive at the surface at angles <p. There they are refracted at larger angles <p', only to diverge more rapidly as shown. Extending these emergent rays backward, we locate their intersections in pairs. These are image points, or virtual images. As the observer changes his position, the virtual image moves closer to the surface and along the curve formed by the successive images. If the object is located in the less dense medium and is observed from the medium of higher index, we obtain an entirely different view (see Fig. 20). An object Q in air is observed by an underwater swimmer or fish. Rays of light diverging from any point of this object are refracted according to Snell's law. Extended backward to their intersections, their virtual images are located. Note how far away these images are for large angles of <p and <p'. / 38 FUNDAMENTALS OF OPTICS FIGURE 20 Image positions of an object in air as seen by an observer under water; n < n'. 2.12 IMAGES FORMED BY PARAXIAL RAYS Of particular interest to many observers are the object and image distances sand s' for rays making small angles 4J and 4J'. Rays for which angles are small enough to permit setting the cosines equal to unity and the sines and tangents equal to the angles are called paraxial rays. Consider the right triangles QAB and Q' AB in Fig. 2N, redrawn in Fig. 2P. Since there is a common side AB = h, we can write h = stan 4J = s' tan 4J' 4J 4J' s sin 4J cos cos 4J sin 4J' 4J' From this we find s' = s tan tan = (2q) PLANE SURFACES AND PRISMS 39 FIGURE 2P Paraxial rays for an object in water and observed from the air above. Applying Snell's law, sin cP n' n =sin cP' we obtain on substitution in Eq. (2q) , n' cos cP' s = sn cos cP For paraxial rays like the ones shown in the diagram, angles Eq. (2q) can be written s' = sf cP' or (2r) cP and cP' are very small; s' cP = (2s) _cP __ n_' cP' n (2t) s cP' and Eq. (2r) written 40 FUNDAMENTALS OF OPTICS FIGURE 2Q Light from a flashlight follows a bent transparent rod by total reflection. Together Eqs. (2s) and (2t) provide the simple relation s' n' s n Paraxial rays • (2u) The ratio of the image to object distancefor paraxial rays is just equal to the ratio of the indices of refraction. 2.13 FIBER OPTICS When light in an optically dense medium approaches the boundary of a less dense medium at an angle cP, greater than the critical angle cPe, it is totally reflected [see Fig. 2B(b)]. Using this knowledge, the British physicist John Tyndall demonstrated that light rays in a tank of water shining through a hole in the side follow the stream of water emerging from the orifice. This effect is commonly observed today in fountains illuminated by lights from under the water. The transmission of light from a flashlight through a glass or plastic rod is shown in Fig. 2Q. Bundles of tiny rods or fibers of clear glass or plastic provide the basis for the sizable industry of fiber optics. Tests on individual fibers over 50 m long show that there are essentially no losses due to reflection on the sides. All attenuation of an incident beam is attributable to reflection from the two ends and absorption by the fiber material. An ordered array or bundle of tiny transparent fibers can be used to transmit light images around corners and over long distances. A bundle of hundreds and even thousands of fibers is frequently made to follow a path with many turns and ends up at a distant or nearby point (see Fig. 2R). If the individual fibers in a bundle are not arranged in an orderly array as in the figure but are randomly interwoven, the emerging image will be scrambled and meaningless. Fibers are usually coated with a thin transparent layer of glass or other material of lower refractive index. Total reflection will still take place between the two. This separates the fibers of a bundle from one another, thereby preventing light leakage between touching fibers and at the same time protecting the firepolished reflecting surfaces. 41 PLANE SURFACES AND PRISMS Fiber bundle ... , \ \ \, , ,t I I I I " / Image A' FIGURE 2R An ordered array of fine glass fibers can be used to transmit images from one end A to the other A' along any curved path. One method for producing coated fibers is to insert a thick, highrefractiveindex glass rod in tubing of lower index. In a special furnace the two are then drawn down to nfoo in. diameter, and the thickness is controlled within narrow limits. A bundle of these fibers can then be fused together to form a solid mass and drawn down a second time so that individual fibers are about 2 Jlm in diameter. This is about two wavelengths of visible light. Such bundles can resolve approximately 250 lines per millimeter. If fibers are drawn down until their diameters are close to the wavelength of light, they cease to act like pipes and behave more like waveguides used in conducting microwaves. * Two wavelengths of light is an approximate limit for image • For an introductory treatment of microwaves and waveguides, see Harvey E. White, "Modern College Physics," 5th ed., pp. 547551, D. Van Nostrand, Princeton, N.J., 1966. For further details on fiber optics, see Narinder S. Kapany, Fiber Optics, Sci. Am., November, 1960, pp. 7180. 42 FUNDAMENTALS OF OPTICS transmission. Of the numerous practical applications of fiber optics, one of the most important is in the field of medicine. A cystoscope, or cathetertype instrument, enables the surgeon to observe and operate by remote control on tiny areas deep within the body. PROBLEMS 2.1 A ray of light is incident on a piece of glass at an angle of 45.0°. If the angle of refraction is 25.37°, find (a) the refractive index and (b) the critical angle. (c) Solve (b) graphically (See Fig. P2.1). Ans. (a) 1.6504, (b) 37.30°, (c) 1.650 and 37.3° n n' FIGURE P2.1 Graph for Prob. 2.1. 2.2 2.3 2.4 2.5 2.6 Calculate the lateral displacements of rays of light incident on a block of glass with parallel sides at the following angles: (a) 5.0°, (b) 10.0°, (c) 15.0°, (d) 20.0°, (e) 30.0°, and (f) 40.0°. (g) Plot a graph of dversus 4J. Assume the glass thickness to be 5.0 cm. A rectangular aquarium is to be filled with water. The sides are made of glass plates 8.0 mm thick. Inside, the walls are 35.0 cm apart, and the refractive index of the glass is 1.5250. If a ray of light is incident on one side at an angle of 50.0°, find the lateral displacement produced when the tank is (a) empty and (b) filled with water. A Pulfrich refractometer is used to measure the refractive index of a clear transparent oil. The glass prism has a refractive index of 1.52518 and a refracting angle (X of 80.0°. If the boundary between light and dark field makes an angle of 29.36° with the normal to the second face, find the refractive index. Ans. 1.3371 A 55.0° prism made of dense flint glass is used at an angle of incidence of 4Jl = 60.0°. Using the refractive index for D light given in Table lA, find (a) the angle of deviation P at the first surface, (b) the angle of deviation l' at the second surface, and (c) the total deviation by the prism. A 50.0° crownglass prism has a refractive index no = 1.52300 for sodium yellow light. If a ray of this yellow light is incident on one surface at an angle of 45.0°, find (a) the PLANE SURFACES AND PRISMS 2.7 2.8 2.9 2.10 2.11 2.12 angle of deviation P at the first surface, (b) the angle of deviation )I at the second surface, and (c) the total deviation by the prism. A 45.0° flint glass prism has a refractive index of 1.6705 for sodium yellow light, and it is adjusted for minimum deviation. Find (a) the angle of minimum deviation and (b) the angle of incidence. (c) Solve graphically. A 60.0° prism produces an angle of minimum deviation of 43.60° for blue light. Find (a) the refractive index, (b) the angle of refraction, and (c) the angle of incidence. Ans. (a) 1.572, (b) 30.0°, (c) 51.81° A 55.0° prism has a refractive index of 1.68059 for blue light. (a) Graphically determine the angle of deviation for each of the following angles of incidence: 40.0, 45.0, 50.0, 55.0,60.0, and 65.0°. (b) Plot a graph of 0 against ~ (see Fig. 2G). Two thin prisms have powers of 6.0 D each. At what angles should their axes be superimposed to produce powers of 2.0, 4.0, 6.0, 8.0, 10.0, and 12.0 D? Ans. 160.8, 141.1, 120.0, 96.4, 67.1, and 0° Two thin prisms of 5.0 and 7.0 D, respectively, are superimposed so their axes make an angle of 75.0° with each other. Find (a) the resultant deviation they produce in degrees, (b) the power of the resultant deviation in diopters, and (c) the angle the resultant makes with the stronger of the two prisms. A directvision prism is to be made of two elements like the one shown in Fig. 2K. The flintglass prism of index 1.720 has an angle IX = 55.0°. Find the angle IX' for the crownglass prism if its refractive index is 1.520. Solve by (a) graphical methods and (b) calculation. A coin lies on the bottom of a bathtub. If the water is 36.0 cm deep and the refractive index of water is 1.3330, find the image depth of the coin as seen from straight above. Assume the sines of angles can be replaced by the angles themselves. H 2.13 43 3 SPHERICAL SURFACES Many common optical devices contain not only mirrors and prisms having flat polished surfaces but lenses having spherical surfaces with a wide range of curvatures. Such spherical surfaces, in contrast with plane surfaces treated in the last chapter, are capable of forming real images. Crosssectional diagrams of several standard forms of lenses are shown in Fig. 3A. The three converging, or positive, lenses, which are thicker at the center than at the edges, are shown as (a) equiconvex, (b) planoconvex, and (c) positive meniscus. The three diverging, or negative, lenses, which are thinner at the center, are (d) equiconcave, (e) planoconcave, and (/) negative meniscus. Such lenses are usually made of optical glass as free as possible from inhomogeneities, but occasionally other transparent materials like quartz; fluorite, rock salt, and plastics are used. Although we shall see that the spherical form for the surfaces may not be the ideal one in a particular instance, it gives reasonably good images and is much the easiest to grind and polish. This chapter treats the behavior of refraction at a single spherical surface separating two media of different refractive indices, and the following chapters show how the treatment can be extended to two or more surfaces in succession. These combinations form the basis for the treatment of thin lenses in Chap. 4, thick lenses in Chap. 5, and spherical mirrors in Chap. 6. 45 SPHERICAL SURFACES (a) (b) (d) (e) .Can verging .or pasitive lenses ~. Diverging .or nogotive lenses FIGURE 3A Cross sections of common types of thin lenses. 3.1 FOCAL POINTS AND FOCAL LENGTHS Characteristic diagrams showing the refraction of light by convex and concave spherical surfaces are given in Fig. 3B. Each ray in being refracted obeys Snell's law as given by Eq. (1m). The principal axis in each diagram is a straight line through the center of curvature C. The point A where the axis crosses the surface is called the vertex. In diagram (a) rays are shown diverging from a point source F on the axis in the first medium and refracted into a beam everywhere parallel to the axis in the second medium. Diagram (b) shows a beam converging in the first medium toward the point F and then refracted into a parallel beam in the second medium. F in each of these two cases is called the primary focal point, and the distance f is called the primary focallength. In diagram (c) a parallel incident beam is refracted and brought to a focus at the point F',and in diagram (d) a parallel incident beam is refracted to diverge as if it came from the point P. F' in each case is called the secondary focal point, and the distance f' is called the secondary focal length. Returning to diagrams (a) and (b) for reference, we now state that the primary focal point F is an axial point having the property that any ray coming from it or proceeding toward it travels parallel to the axis after refraction. Referring to diagrams (c) and (d), we make the similar statement that the secondary focal point F' is an axial point having the property that any incident ray traveling parallel to the axis will, after refraction, proceed toward, or appear to come from, F'. A plane perpendicular to the axis and passing through either focal point is called afocal plane. The significance of a focal plane is illustrated for a convex surface in Fig. 3.C Parallel incident rays making an angle (J with the axis are brought to a focus in the focal plane at a point Q'. Note that Q' is in line with the undeviated ray through the center of curvature C and that this is the only ray that crosses the boundary at normal incidence. It is important to note in Fig. 3B that the primary focallengthffor the convex surface [diagram (a)] is not equal to the secondary focallengthj' of the same surface [diagram (c)]. It will be shown in Sec. 3.4 that the ratio of the focal lengths f'Lf is equal to the ratio n'/n of the corresponding refractive indices [see Eq. (3e)]: f' f n' =  n (3a) 46 FUNDAMENTALS OF OPTICS (a) Axis (c) FIGURE 3B The focal points F and F' and focal lengths f and f' associated with a single spherical refracting surface of radius r separating two media of index nand n'. In optical diagrams it is common practice to show incident light rays traveling from left to right. A convex surface therefore is one in which the center of curvature C lies to the right of the vertex, while a concave surface is one in which C lies to the left of the vertex. If we apply the principle of the reversibility of light rays to the diagrams in Fig. 3B, we should turn each diagram endforend. Diagram (a), for example, would then become a concave surface with converging properties, while diagram (b) would become a convex surface with diverging properties. Note that we would then have the incident rays in the denser medium, i.e., the medium of greater refractive index. 3.2 IMAGE FORMATION A diagram illustrating image formation by a single refracting surface is given in Fig. 3D. It has been drawn for the case in which the first medium is air with an index n = 1 and the second medium is glass with an index n' = 1.60. The focal lengths f and/' therefore have the ratio 1: 1.60 [see Eq. (3a)]. Experimentally it is observed that if the object is moved closer to the primary focal plane, the image will be formed farther to the right away from F' and will be larger, i.e., magnified. If the object is moved to the left, farther away from F, the image will be found closer to F' and will be smaller in size. All rays coming from the object point Q are shown brought to a focus at Q'. SPHERICAL SURFACES 47 F FIGURE 3C How parallel incident rays are brought to a focus at Q' in the secondary focal plane of a single spherical surface. Rays from any other object point like M will also be brought to a focus at a corresponding image point like M'. This ideal condition never holds exactly for any actual case. Departures from it give rise to slight defects of the image known as aberrations. The elimination of aberrations is the major problem of geometrical optics and will be treated in detail in Chap. 9. If the rays considered are restricted to paraxial rays, a good image is formed with monochromatic light. Paraxial rays are defined as those rays which make very small angles with the axis and lie close to the axis throughout the distance from object to image (see Sec. 2.12). The formulas given in this chapter are to be taken as applying to images formed only by paraxial rays. 3.3 VIRTUAL IMAGES The image M'Q' in Fig. 3D is a real image in the sense that if a flat screen is located there, a sharply defined image of the object MQ will be formed on the screen.