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1 Introduction to Geometrical Optics

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3 Introduction to Geometrical Optics Milton Katz state university of New York US* World Scientific «New Jersey London Sim Singapore Hong Kong

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: Suite 202, 1060 Main Street, River Edge, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. First published in 1994 by Penumbra Publishing Co. Illustrations: Russel Hayes and George Zikos INTRODUCTION TO GEOMETRICAL OPTICS Copyright 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN X ISBN (pbk) Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore

5 TABLE OF CONTENTS PREFACE ACKNOWLEDGMENTS xiii xiv CHAPTER 1: LIGHT 1.1 Introduction The Electromagnetic Spectrum Wave Motion Wavelength and Frequency Range Measurement of the Velocity of Light Light Sources Self-Luminous Incandescent Sources Electric Arcs and Discharges Fluorescent Lamps Lasers Light Reflecting Sources Optical Media Transparent, Colored and Translucent Materials Reflecting and Opaque Materials Point and Extended Sources Rectilinear Propagation of Light Pinhole Camera Propagation of Light in Nonhomogeneous Media The Corpuscular and Wave Theory of Light Umbra, Penumbra and Eclipses Real and Virtual Objects and Images Stops Field Stop Aperture Stop Baffles Optical System Vergence and Diopters Pencils and Beams of Light Point Source Extended Source Visual Angle 12 CHAPTER 2: REFLECTION AT PLANE MIRRORS 2.1 Reflection The Law of Reflection Mirror Rotation The Galvanometer Measurement of Mirror Rotation Angles 16 v

6 vi Contents Hadley's Sextant Image of a Point Formed by a Plane Mirror Image of an Extended Object Inversion and Reversion of Reflected Images Constant Deviation by Two Inclined Mirrors Applications of Constant Angles of Deviation Multiple Images Produced by Two Inclined Mirrors Construction of Images Image Series J Image Series K Number of J and K Images The Kaleidoscope Ray Paths to the Eye Field of View of a Plane Mirror Reflection According to Newton's Corpuscular Theory Reflection According to Huygens' Wave Theory 29 CHAPTER 3: REFRACTION OF LIGHT 3.1 The Law of Refraction The Reversibility of Light Refraction and Particle Motion Vector Diagram - Perpendicular Component Increased by Refraction Vector Diagram - Tangential Component Decreased by Refraction Huygens'Construction for Refraction Absolute Index of Refraction and Snell's Law Construction of Refracted Ray Deviation of the Refracted Ray Dispersion Total Internal Reflection Optical Path Length Fermat's Principle 40 CHAPTER 4: REFRACTION BY PLANES, PLATES AND PRISMS 4.1 Exact Ray Tracing Through a Plane Refracting Surface Exact Ray Tracing Through a Parallel Plate Displacement of Rays Obliquely Incident on a Tilted Plate Refraction Through Prisms Prism Geometry Limits to Prism Transmission: Grazing Incidence and Emergence Minimum Deviation by a Prism Calculation of the Path of a Ray Through a Prism Calculation of the Total Deviation by a Prism 50 CHAPTER 5: PARAXIAL REFRACTION AT PLANES, PLATES AND PRISMS 5.1 Paraxial Refraction at a Plane Surface Paraxial Image Formation by Parallel Plates Reduced Thickness 56

7 Contents vii 5.4 Thin Prisms Ophthalmic Prisms, Centrads and the Prism-Diopter Obliquely Combined Prisms Risley Prisms 61 CHAPTER 6: REFRACTION AND REFLECTION AT SPHERICAL SURFACES 6.1 Sign Convention Refraction at a Single Spherical Refracting Surface The Paraxial Image Equation Focal Points and Focal Lengths Refracting Power Construction of On-Axis Image Points Construction of the Focal Points Construction of Off-Axis Image Points Image Positions as an Object Approaches the Surface Possible Positions of the Image Lateral Magnification The Image Equation for Plane Surfaces Newtonian Equations The Focal Planes and Focal Images of a Spherical Refracting Surface The Smith-Helmholtz Formula or Lagrange Invariant Reflection by Spherical Mirrors The Paraxial Image Equation The Focal Points of a Spherical Mirror Construction of Axial Image Points for Spherical Mirrors Construction of Focal Points of a Spherical Mirror Construction of Off-Axis Image Points Formed by a Mirror Image Positions as an Object Approaches a Spherical Mirror Possible Positions of the Image Formed by a Spherical Mirror Lateral Magnification by Spherical Mirrors Newtonian Equations for a Spherical Mirror The Field of View of a Spherical Mirror Angular Magnification Vergence, Power and Curvature Vergence and Power Curvature and Vergence Curvature and Sagitta The Correction of Lens Measure Readings 87 CHAPTER 7: THIN LENSES 7.1 Convex and Concave Lenses Lens Nomenclature The Optical Center of a Lens The Thin Lens The Thin Lens Equation Focal Lengths Construction of the Off-Axis Image Point 97

8 viii Contents 7.8 Lateral Magnification Vergence Equations and Power of a Thin Lens Thin Lens Power in Non-Uniform Medium Object and Image Formation by a Thin Lens Positive Lenses Negative Lenses Applications of Lenses with Various Conjugates The Minimum Separation Between a Real Object and Its Real Image Newtonian Equations The Focal Planes and Focal Images of a Thin Lens Prismatic Power of a Thin Lens Stops and Pupils of Lens Systems Procedure for Finding the Stops, Pupils and Field of View of a Thin Lens 106 CHAPTER 8: ROTATIONALLY SYMMETRICAL SYSTEMS 8.1 Introduction Paraxial Construction of the Image Paraxial Calculation of Images Through an Optical System The Vergence Form of the Refraction and Transfer Equations Iterative Use of the Vergence Equations 115 CHAPTER 9: ASTIGMATIC LENSES 9.1 Astigmatic Images Curvature and Power in Principal and Normal Sections Power in Oblique Meridians of a Cylinder The Oblique Power Equation Thin Astigmatic Lenses Forms of Astigmatic Lenses Transposition of Flat Prescriptions The Circle of Least Confusion and the Spherical Equivalent Lens The Circle of Least Confusion for Any Object Distance The Lengths of the Focal Lines and the Diameter of the CLC Obliquely Combined Cylinders 130 CHAPTER 10: THICK LENS SYSTEMS: PART I 10.1 Introduction The Cardinal Points The Principal Planes and Points The Nodal Points Construction of the Nodal Points Other "Cardinal" Points The Newtonian Equations for a Thick System The Refraction Equations for a Thick System The Lateral Magnification of a Thick Lens Summary of Vergence Equations for a Thick Lens System Lateral, Axial and Angular Magnification Lateral Magnification: Y 145

9 Contents ix Axial Magnification: X Angular Magnification: Ma 146 CHAPTER 11: THICK LENS SYSTEMS: PART II 11.1 The Gullstrand Equations Equivalent Power The Position of the Second Focal Point F' The Position of the Second Principal Point H' The Positions of the First Focal and Principal Points F and H Distances Referred to Principal Planes Summary of the Gullstrand Equations IThe Positions of the Nodal Points N and N' The Effect of Lens Thickness on the Positions of the Principal Planes The Effect of Lens Shape on the Position of the Principal Planes Telephoto and Wide-Angle Lenses Vertex Power Vertex Power and Ophthalmic Lenses Effective Power The Relationship of Effective Power to Vertex Power Combination of Two Thick Lens The Gullstrand Schematic Eye Procedure for Calculating Cardinal Points of the Schematic Eye Purkinje Images Calculation of Purkinje Image I Calculation of Purkinje Image II Calculation of Purkinje Image III 166 CHAPTER 12: STOPS, PUPILS AND PORTS 12.1 Types of Stops Pupils The Field Stop and the Entrance and Exit Ports Field of View and Vignetting Procedure for Finding the Stops, Pupils, Ports and Field of View 176 CHAPTER 13: NUMERICAL APERTURE, f-number, AND RESOLUTION 13.1 Numerical Aperture f-number Angular Resolution Linear Resolution Resolution Charts Snellen Charts Modulation Transfer Function The Numerical Aperture of a Fiber Optic 191 CHAPTER 14: MAGNIFIERS AND MICROSCOPES 14.1 Introduction Visual Angle and Angular Magnification 195

10 x Contents 14.3 The Magnifier or Simple Microscope Nominal Magnification Maximum Magnification A General Equation for Magnification The Accommodation Required to View the Image The Field of View of Magnifiers Types of Magnifiers The Compound Microscope Magnification by a Microscope Maximum Magnification by a Microscope The Compound Microscope as a Simple Magnifier The Stops of a Microscope Numerical Aperture and Resolution of Microscopes Maximum Usable Magnification of a Microscope Resolution, Wavelength and the Electron Microscope Depth of Focus of Microscope 205 CHAPTER 15: TELESCOPES 15.1 Introduction Refracting Telescopes The Astronomical or Keplerian Telescope Angular Magnification The Field of View of the Astronomical Telescope Field Lenses The Terrestrial Telescope Relay Lenses The Galilean Telescope Field and Opera Glasses The Prism Binocular Stereoscopic Range Through Binoculars Reflecting Telescopes Newtonian Telescope Gregorian Telescope Cassagrainian Telescope Catadioptric Telescopes The Maximum Useful Magnification of a Telescope 219 CHAPTER 16: CAMERAS AND PROJECTORS 16.1 Pinhole Camera and Camera Obscura Types of Cameras Camera Lenses Field of View of Cameras Depth of Field and Focus Hyperfocal Distance The Paraxial Design of a Zoom Lens Paraxial Design of a Telephoto Lens 230' 16.6 Optical Projection Systems 231

11 Contents xi CHAPTER 17: OPHTHALMIC INSTRUMENTS 17.1 Introduction Ophthalmometer Optical Principle Fixed Mire-Variable Image Size Ophthalmometers Variable Mire-Fixed Image Ophthalmometers Radius of Curvature Refractive Power of the Cornea Doubling Principle Fixed Mire Size Ophthalmometer Adjustable Mire Size Ophthalmometer Badal Optometer Badal Target Position and Ametropia Alternate Arrangement of Badal Optometer Telecentric Systems The Lensometer Lensometer Target and Reticle Slit Lamp or Biomicroscope 247 CHAPTER 18: DISPERSION AND CHROMATIC ABERRATION 18.1 Introduction Wavelength and Index of Refraction Index of Refraction Dispersion Optical Glass Crown and Flint Glasses Chromatic Aberration of Thin Prisms Achromatic Prisms The Longitudinal Chromatic Aberration of Lenses Longitudinal Chromatic Aberration in Diopters Linear Longitudinal Chromatic Aberration Achromatic Lenses in Contact Transverse Chromatic Aberration Linear Transverse Chromatic Aberration Transverse Chromatic Aberration in Prism Diopters Achromatic Combination of Two Air-Spaced Lenses The Huygens Eyepiece The Ramsden Eyepiece The Kellner and Other Eyepieces 267 CHAPTER 19: TRIGONOMETRIC RAY TRACING 19.1 Introduction Ray Tracing Sign Conventions and Nomenclature Ray Tracing Equations Transfer Equations Ray Tracing Through a Series of Surfaces 276

12 xii Contents 19.5 Longitudinal and Transverse Spherical Aberration Ray Trace Calculations for a Pencil of Rays A Format for Tracing Rays Through a Lens Image Evaluation Spherical Aberration Curves Ray Intercept Curves Spot Diagrams Radial Energy Diagrams Knife Edge Distributions Graphical Ray Trace 282 CHAPTER 20: MONOCHROMATIC ABERRATIONS 20.1 Introduction Third Order Theory Third Order Aberration Polynomial The Spherical Aberration Terms of the Polynomial Spherical Aberration of a Thin Lens Spherical Aberration and Lens Bending Correction of Spherical Aberration The Coma Terms of the Aberration Polynomial Coma of a Thin Lens The Sine Condition Astigmatism Astigmatism by a Spherical Refracting Surface Astigmatism by a Thin Lens Petzval Curvature Petzval Curvature of a Refracting Surface Petzval Curvature of a Series of Surfaces Petzval Curvature of a Thin Lens in Air The Petzval Condition for Flattening the Field Astigmatism and Petzval Curvature Polynomial Terms Distortion The Third Order Distortion Polynomial Term The Tangent Condition 301 BIBLIOGRAPHY 305 INDEX 309

13 PREFACE This work was motivated by the need to provide an intensive course in paraxial optics for students of optometry and vision science. Its organization reflects my experiences in teaching Geometrical Optics in a changing optometry curriculum. Where once optics was central to the training of optometrists, it is now one of many areas of knowledge to be mastered. As a result, course hours in optics have contracted, although the subject material has expanded with the development of sophisticated optical instrumentation, and increasing diversity in ophthalmic materials and lenses. The challenge, then, was to determine the most effective and efficient way to teach the essentials of geometrical optics, within these constraints, to students who, most commonly, are not physics or mathematics majors. The goal was for the students to retain a lifelong basic understanding of image formation by lenses and mirrors. To work toward this goal, I organized this work so that the single spherical refracting surface is the basic optical element. Spherical mirrors are treated as special cases of refraction, with the same applicable equations. Thin lens equations follow as combinations of spherical refracting surfaces. The cardinal points of the thick lens make it equivalent to a thin lens. Ultimately, one set of vergence equations is applicable to all these elements. As recently as 40 years ago we thought that everything important about optics had been discovered. We relegated optics to classical physics. Physicists primarily were interested in atomic and nuclear physics. Today optics is in the forefront of scientific and technological study. This metamorphosis is the result of the invention of the laser, and the development of holography, optical fibers for telecommunications, thin film technology and optical memory for computers. We divided classical optics into geometrical and physical branches, but today the subdivisions are many times more numerous, as can be gleaned from the titles of books: Optical Holography, Optical Physics, Optical System Design, Optical Engineering, Applied Optics, Modern Optics, and the list goes on. This book covers only a tiny fraction of geometrical optics, mainly paraxial optics. The laws of geometrical optics go back to ancient times for reflection of light, and the 17th century for the refraction of light. In fact, just one law, Snell's Law, [n sin a = n' sin a'] encapsulates most of the geometrical optics in this course. Furthermore, we mainly will consider a simplification of this law for the paraxial case: [n a = n'a]. Although the ideas are not very complicated, the reader's first challenge will be to reacquire some basic knowledge and skills in using geometry, trigonometry and algebra. Then it is essential to use only the sign convention and nomenclature consistently to solve problems. Failure to solve problems is frequently caused by confusion whether a distance is positive or negative. The second most important habit to develop is to draw clear and fully labeled diagrams. To test your understanding of the material you must do the homework problems. They should be done regularly. Cramming at exam time has been the undoing of many students. Furthermore, regularly doing homework will enable you to raise timely questions in class. Milton Katz Xlll

14 Acknowledgments I am indebted to J. P. C. Southall's long out of print book, "Mirrors, Prisms and Lenses" for introducing me to Geometrical Optics. His emphasis, in all drawings, on clearly showing the direction of measurement of a distance according to the sign convention, is one that I have adopted, along widi his nomenclature. Other influential optics books were The Fundamentals of Optics by Jenkins and White, and Modern Optical Engineering by W. J. Smith. A list of references is appended. I am grateful for the generous help of Mr. Richard Feinbloom, President of Designs For Vision, Inc., who sponsored the preparation of the illustrations and, consequently, made this book possible. Mr. Russel Hayes and my graduate student, George Zikos, produced the illustrations. I also thank the many students who identified errata, pointed out topics that needed clarification and expressed appreciation for this work. xiv

Introduction. Geometrical Optics. Milton Katz State University of New York. VfeWorld Scientific New Jersey London Sine Singapore Hong Kong

Introduction. Geometrical Optics. Milton Katz State University of New York. VfeWorld Scientific New Jersey London Sine Singapore Hong Kong Introduction to Geometrical Optics Milton Katz State University of New York VfeWorld Scientific «New Jersey London Sine Singapore Hong Kong TABLE OF CONTENTS PREFACE ACKNOWLEDGMENTS xiii xiv CHAPTER 1: