Geometrical and Visual Optics Schw_FM_i-xii.indd i 10/05/13 8:19 AM Notice Medicine is an ever-changing science. As new research and clinical experience broaden our knowledge, changes in treatment and drug therapy are required. The author and the publisher of this work have checked with sources believed to be reliable in their efforts to provide information that is complete and generally in accord with the standards accepted at the time of publication. However, in view of the possibility of human error or changes in medical sciences, neither the author nor the publisher nor any other party who has been involved in the preparation or publication of this work warrants that the information contained herein is in every respect accurate or complete, and they disclaim all responsibility for any errors or omissions or for the results obtained from use of the information contained in this work. Readers are encouraged to confirm the information contained herein with other sources. For example and in particular, readers are advised to check the product information sheet included in the package of each drug they plan to administer to be certain that the information contained in this work is accurate and that changes have not been made in the recommended dose or in the contraindi- cations for administration. This recommendation is of particular importance in connection with new or infrequently used drugs. Schw_FM_i-xii.indd ii 10/05/13 8:19 AM Geometrical and Visual Optics A Clinical Introduction SECOND EDITION Steven H. Schwartz, OD, PhD Professor Department of Biological and Vision Sciences College of Optometry State University of New York New York, New York New York Chicago San Francisco Athens London Madrid Mexico City Milan New Delhi Singapore Sydney Toronto Schw_FM_i-xii.indd iii 10/05/13 8:19 AM Copyright © 2013 by McGraw-Hill Education. All rights reserved. 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 database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-179083-3 MHID: 0-07-179083-7 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-179082-6, MHID: 0-07-179082-9. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefi t of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill Education books are available at special quantity discounts to use as premiums and sales promotions or for use in corporate training programs. To contact a representative, please visit the Contact Us pages at www.mhprofessional.com. Previous edition copyright © 2002 by The McGraw-Hill Companies, Inc. TERMS OF USE This is a copyrighted work and McGraw-Hill Education, LLC. and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill Education’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL EDUCATION AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMA- TION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill Education and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill Education nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill Education has no responsibility for the content of any information accessed through the work. Under no circumstanc- es shall McGraw-Hill Education and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. Contents Preface .......................................................................................................................xi 1. Basic Terms and Concepts ..................................................................................1 Sources, Light Rays, and Pencils .......................................................................3 Vergence .............................................................................................................4 Refraction and Snell’s Law .................................................................................7 Summary ..........................................................................................................10 Self-Assessment Problems ...............................................................................13 2. Refraction at Spherical Surfaces ....................................................................15 Converging and Diverging Spherical Surfaces ...............................................15 A Word on Sign Conventions ..........................................................................19 Primary and Secondary Focal Points ...............................................................19 A Very Handy Formula ....................................................................................21 Image Formation by Spherical Surfaces ..........................................................24 Real Images ..................................................................................................24 Virtual Images ..............................................................................................26 Summary ..........................................................................................................27 Self-Assessment Problems ...............................................................................28 3. The Vergence Relationship ..............................................................................29 More on Vergence ............................................................................................29 Linear Sign Convention...................................................................................33 Vergence Relationship .....................................................................................34 Sample Problems ..............................................................................................35 Sample Problem 1: Converging Surface ......................................................36 Sample Problem 2: Diverging Surface ........................................................38 Sample Problem 3: Locating the Object .....................................................40 Sample Problem 4: A Flat (Plane) Refracting Surface ................................41 Summary ..........................................................................................................44 Self-Assessment Problems ...............................................................................45 v Schw_FM_i-xii.indd v 10/05/13 8:19 AM vi Contents 4. Thin Lenses ......................................................................................................47 Ray Tracing ......................................................................................................49 Vergence Relationship .....................................................................................54 Newton’s Relation ............................................................................................59 Summary ..........................................................................................................60 Self-Assessment Problems ...............................................................................60 5. Optical Systems with Multiple Surfaces .........................................................63 Multiple Thin Lens Systems............................................................................63 Virtual Objects .................................................................................................68 Thick Lenses ....................................................................................................70 Summary ..........................................................................................................75 Self-Assessment Problems ...............................................................................75 6. Thick Lenses .....................................................................................................77 Definitions ........................................................................................................77 Back and Front Vertex Power ..........................................................................79 Equivalent Lenses ............................................................................................81 Equivalent Power .........................................................................................81 Locating the Principal Planes ......................................................................81 Sample Problem ...........................................................................................83 Nodal Points .................................................................................................85 Summary ..........................................................................................................86 Self-Assessment Problems ...............................................................................87 7. Ametropia ........................................................................................................89 Myopia ..............................................................................................................91 Hyperopia .........................................................................................................95 Lens Effectivity ................................................................................................98 Correction of Ametropia with Laser and Surgical Procedures .....................103 Summary ........................................................................................................105 Self-Assessment Problems .............................................................................105 8. Accommodation ............................................................................................107 Vergence Relationship for Accommodation ..................................................109 Accommodation in Ametropia .......................................................................110 Near Point of Accommodation ......................................................................114 Accommodation when Ametropia is Corrected with Spectacles ..................117 Correction of Presbyopia ...............................................................................122 Summary ........................................................................................................124 Self-Assessment Problems .............................................................................124 9. Cylindrical Lenses and the Correction of Astigmatism ................................127 Lens Crosses ...................................................................................................127 Lens Formulae/Prescriptions ........................................................................130 Image Formation: Point Sources ...................................................................133 Image Formation: Extended Sources ............................................................135 Schw_FM_i-xii.indd vi 10/05/13 8:19 AM Contents vii Power in an Oblique Meridian of a Cylindrical Lens ...................................136 Astigmatism: Definitions and Classifications ................................................138 Jackson Crossed-Cylinder Test ......................................................................141 Spherical Equivalency ....................................................................................144 What Does a Person with Astigmatism See? ................................................145 Summary ........................................................................................................146 References ......................................................................................................147 Self-Assessment Problems .............................................................................147 10. Prisms ............................................................................................................149 Thick and Thin Prisms ..................................................................................149 Prism Diopters ...............................................................................................151 Prismatic Effects of Lenses ............................................................................151 Clinical Applications ......................................................................................158 Summary ........................................................................................................162 Reference ........................................................................................................162 Self-Assessment Problems .............................................................................162 11. Depth of Field ................................................................................................165 Blur Circles, Visual Acuity, and Pinholes ......................................................165 Diffraction Caused by Apertures ...................................................................168 Depth of Field and Depth of Focus ...............................................................171 Hyperfocal Distance ......................................................................................176 Summary ........................................................................................................178 Self-Assessment Problems .............................................................................178 12. Magnification and Low Vision Devices ........................................................179 Angular Magnification Produced by Plus Lenses .........................................180 Use of Plus Lenses for Low Vision ................................................................183 Magnifying Lens and Bifocal Add in Combination ..........................................................................................186 Fixed-Focus Stand Magnifiers ...................................................................187 Effective Magnification ..............................................................................191 Electronic Magnifiers for Near .....................................................................193 Telescopes .......................................................................................................194 Galilean Telescopes ....................................................................................194 Keplerian Telescopes ..................................................................................196 An Alternative Formula to Determine a Telescope’s Angular Magnification ...............................................................................197 A Convenient Clinical Method to Determine a Telescope’s Angular Magnification ...............................................................................197 Telescope Use in Ametropia ......................................................................199 Telemicroscopes .........................................................................................201 Summary ........................................................................................................203 Further Reading .............................................................................................204 Self-Assessment Problems .............................................................................204 Schw_FM_i-xii.indd vii 10/05/13 8:19 AM viii Contents 13. Retinal Image Size ........................................................................................207 Spectacle Magnification .................................................................................207 Retinal Image Size in Uncorrected Ametropia .............................................210 Retinal Image Size in Corrected Ametropia .................................................211 Summary ........................................................................................................215 Self-Assessment Problems .............................................................................215 14. Reflection .......................................................................................................217 Ray Tracing: Concave, Convex, and Plane Mirrors ......................................217 Concave Mirrors ........................................................................................217 Convex Mirrors ..........................................................................................220 Plane Mirrors .............................................................................................220 Power of Mirrors ............................................................................................221 The Vergence Relationship ...........................................................................225 Reflections and Antireflection Coatings ........................................................231 Purkinje Images ..............................................................................................233 Corneal Topography ......................................................................................237 Summary ........................................................................................................239 Self-Assessment Problems .............................................................................240 15. Aberrations ....................................................................................................241 The Paraxial Assumption and Seidel Aberrations .........................................241 Longitudinal Spherical Aberration ............................................................242 Coma ..........................................................................................................245 Oblique Astigmatism ..................................................................................246 Curvature of Field ......................................................................................249 Distortion ...................................................................................................250 Wavefront Sensing and Adaptive Optics .......................................................250 Measurement of the Eye’s Monochromatic Aberrations ..........................250 Supernormal Vision ....................................................................................253 Imaging the Fundus ...................................................................................256 Chromatic Aberration ....................................................................................257 Dispersive Power and Constringence ........................................................257 Lateral (Transverse) Chromatic Aberration ..............................................258 Longitudinal Chromatic Aberration ..........................................................260 Chromatic Aberration in the Human Eye .................................................261 The Red–Green Refraction Technique .....................................................263 Summary ........................................................................................................264 References ......................................................................................................265 Self-Assessment Problems .............................................................................266 Answers to Self-Assessment Problems ................................................................267 Chapter 1: Basic Terms and Concepts .........................................................267 Chapter 2: Refraction at Spherical Surfaces ................................................268 Chapter 3: The Vergence Relationship .......................................................270 Chapter 4: Thin Lenses ...............................................................................275 Schw_FM_i-xii.indd viii 10/05/13 8:19 AM Contents ix Chapter 5: Optical Systems with Multiple Surfaces ...................................279 Chapter 6: Thick Lenses ..............................................................................284 Chapter 7: Ametropia ..................................................................................290 Chapter 8: Accommodation .........................................................................292 Chapter 9: Cylindrical Lenses and the Correction of Astigmatism ...........298 Chapter 10: Prisms .........................................................................................302 Chapter 11: Depth of Field ............................................................................304 Chapter 12: Magnification and Low Vision Devices ....................................305 Chapter 13: Retinal Image Size .....................................................................308 Chapter 14: Reflection ...................................................................................309 Chapter 15: Aberrations .................................................................................314 Appendix A: Entrance and Exit Pupils of Telescopes ..........................................317 Appendix B: Location of Purkinje Image III ........................................................323 Appendix C: Fluid Lenses .....................................................................................329 Appendix D: Javal’s Rule .....................................................................................335 Appendix E: Derivation of the Paraxial Relationship .........................................337 Appendix F: Correction of Chromatic Aberration ...............................................339 Longitudinal Chromatic Aberration .............................................................339 Lateral (Transverse) Chromatic Aberration ..................................................343 Practice Examinations ..........................................................................................345 Practice Examination 1 ..................................................................................345 Practice Examination 2 ..................................................................................349 Answers to Practice Examinations .......................................................................355 Practice Examination 1 ..................................................................................355 Practice Examination 2 ..................................................................................357 Index .......................................................................................................................361 Schw_FM_i-xii.indd ix 10/05/13 8:19 AM