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: LIGHT 1.1 Introduction 1 1.2 The Electromagnetic Spectrum 1 1.2.1 Wave Motion 1 1.2.2 Wavelength and Frequency Range 2 1.2.3 Measurement of the Velocity of Light 3 1.3 Light Sources 4 1.3.1 Self-Luminous 4 1.3.1.1 Incandescent Sources 4 1.3.1.2 Electric Arcs and Discharges 4 1.3.1.3 Fluorescent Lamps 4 1.3.1.4 Lasers 4 1.3.2 Light Reflecting Sources 4 1.4 Optical Media 4 1.4.1 Transparent, Colored and Translucent Materials 5 1.4.2 Reflecting and Opaque Materials 5 1.5 Point and Extended Sources 5 1.6 Rectilinear Propagation of Light 6 1.6.1 Pinhole Camera 6 1.6.2 Propagation of Light in Nonhomogeneous Media 6 1.7 The Corpuscular and Wave Theory of Light 7 1.8 Umbra, Penumbra and Eclipses 8 1.9 Real and Virtual Objects and Images 10 1.10 Stops 10 1.10.1 Field Stop 10 1.10.2 Aperture Stop 10 1.10.3 Baffles 11 1.11 Optical System 11 1.11.1 Vergence and Diopters 11 1.12 Pencils and Beams of Light 12 1.12.1 Point Source 12 1.12.2 Extended Source 12 1.13 Visual Angle 12 CHAPTER 2: REFLECTION AT PLANE MIRRORS 2.1 Reflection 15 2.2 The Law of Reflection 15 2.2.1 Mirror Rotation 16 2.2.1.1 The Galvanometer 16 2.2.1.2 Measurement of Mirror Rotation Angles 16
vi Contents 2.2.1.3 Hadley's Sextant 17 2.3 Image of a Point Formed by a Plane Mirror 17 2.3.1 Image of an Extended Object 18 2.3.2 Inversion and Reversion of Reflected Images 18 2.4 Constant Deviation by Two Inclined Mirrors 19 2.4.1 Applications of Constant Angles of Deviation 20 2.5. Multiple Images Produced by Two Inclined Mirrors 21 2.5.1 Construction of Images 21 2.5.1.1 Image Series J 21 2.5.1.2 Image Series К 23 2.5.2 Number of J and К Images 23 2.5.3 The Kaleidoscope 25 2.6. Ray Paths to the Eye 26 2.7. Field of View of a Plane Mirror 26 2.8 Reflection According to Newton's Corpuscular Theory 28 2.9. Reflection According to Huygens' Wave Theory 29 CHAPTER 3: REFRACTION OF LIGHT 3.1 The Law of Refraction 33 3.2 The Reversibility of Light 33 3.3 Refraction and Particle Motion 33 3.3.1 Vector Diagram - Perpendicular Component Increased by Refraction 34 3.3.2 Vector Diagram - Tangential Component Decreased by Refraction 34 3.3.3 Huygens'Construction for Refraction 35 3.4 Absolute Index of Refraction and Snell's Law 36 3.5 Construction of Refracted Ray 37 3.6 Deviation of the Refracted Ray 38 3.7 Dispersion 38 3.8 Total Internal Reflection 38 3.9 Optical Path Length 39 3.10 Fermat's Principle 40 CHAPTER 4: REFRACTION BY PLANES, PLATES AND PRISMS 4.1 Exact Ray Tracing Through a Plane Refracting Surface 43 4.2 Exact Ray Tracing Through a Parallel Plate 44 4.3 Displacement of Rays Obliquely Incident on a Tilted Plate 46 4.4 Refraction Through Prisms 47 4.5 Prism Geometry 48 4.6 Limits to Prism Transmission: Grazing Incidence and Emergence 48 4.7 Minimum Deviation by a Prism 49 4.8 Calculation of the Path of a Ray Through a Prism 50 4.9 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 53 5.2 Paraxial Image Formation by Parallel Plates 54 5.3 Reduced Thickness 56
Contents vii 5.4 Thin Prisms 56 5.5 Ophthalmie Prisms, Centrads and the Prism-Diopter 57 5.7 Obliquely Combined Prisms 59 5.8 Risley Prisms 61 CHAPTER 6: REFRACTION AND REFLECTION AT SPHERICAL SURFACES 6.1 Sign Convention 65 6.2 Refraction at a Single Spherical Refracting Surface 66 6.2.1 The Paraxial Image Equation 66 6.2.2 Focal Points and Focal Lengths 67 6.2.3 Refracting Power 69 6.2.4 Construction of On-Axis Image Points 69 6.2.5 Construction of the Focal Points 70 6.2.6 Construction of Off-Axis Image Points 71 6.2.7 Image Positions as an Object Approaches the Surface 72 6.2.8 Possible Positions of the Image 73 6.2.9 Lateral Magnification 73 6.3 The Image Equation for Plane Surfaces 74 6.4 Newtonian Equations 74 6.5 The Focal Planes and Focal Images of a Spherical Refracting Surface 75 6.6 The Smith-Helmholtz Formula or Lagrange Invariant 76 6.7 Reflection by Spherical Mirrors 76 6.7.1 The Paraxial Image Equation 76 6.7.2 The Focal Points of a Spherical Mirror 78 6.7.3 Construction of Axial Image Points for Spherical Mirrors 79 6.7.4 Construction of Focal Points of a Spherical Mirror 79 6.7.5 Construction of Off-Axis Image Points Formed by a Mirror 79 6.7.6 Image Positions as an Object Approaches a Spherical Mirror 80 6.7.7 Possible Positions of the Image Formed by a Spherical Mirror 81 6.7.8 Lateral Magnification by Spherical Mirrors 81 6.7.9 Newtonian Equations for a Spherical Mirror 81 6.7.10 The Field of View of a Spherical Mirror 82 6.8 Angular Magnification 83 6.9 Vergence, Power and Curvature 84 6.9.1 Vergence and Power 84 6.9.2 Curvature and Vergence 85 6.9.3 Curvature and Sagitta 86 6.9.4 The Correction of Lens Measure Readings 87 CHAPTER 7: THIN LENSES 7.1 Convex and Concave Lenses 93 7.2 Lens Nomenclature 94 7.3 The Optical Center of a Lens 94 7.4 The Thin Lens 95 7.5 The Thin Lens Equation 95 7.6 Focal Lengths 97 7.7 Construction of the Off-Axis Image Point 97
viii Contents 7.8 Lateral Magnification 98 7.9 Vergence Equations and Power of a Thin Lens 99 7.9.1 Thin Lens Power in Non-Uniform Medium 99 7.10 Object and Image Formation by a Thin Lens 100 7.10.1 Positive Lenses 100 7.10.2 Negative Lenses 100 7.10.3 Applications of Lenses with Various Conjugates 102 7.10.4 The Minimum Separation Between a Real Object and Its Real Image 102 7.11 Newtonian Equations 103 7.12 The Focal Planes and Focal Images of a Thin Lens 103 7.13 Prismatic Power of a Thin Lens 104 7.14 Stops and Pupils of Lens Systems 105 7.14.1 Procedure for Finding the Stops, Pupils and Field of View of a Thin Lens 106 CHAPTER 8: ROTATIONALLY SYMMETRICAL SYSTEMS 8.1 Introduction 111 8.2 Paraxial Construction of the Image 111 8.3 Paraxial Calculation of Images Through an Optical System 112 8.4 The Vergence Form of the Refraction and Transfer Equations 115 8.4.1 Iterative Use of the Vergence Equations 115 CHAPTER 9: ASTIGMATIC LENSES 9.1 Astigmatic Images 119 9.2 Curvature and Power in Principal and Normal Sections 121 9.3 Power in Oblique Meridians of a Cylinder 123 9.3.1 The Oblique Power Equation 123 9.4 Thin Astigmatic Lenses 124 9.4.1 Forms of Astigmatic Lenses 125 9.5 Transposition of Flat Prescriptions 127 9.6 The Circle of Least Confusion and the Spherical Equivalent Lens 128 9.7 The Circle of Least Confusion for Any Object Distance 129 9.8 The Lengths of the Focal Lines and the Diameter of the CLC 130 9.9 Obliquely Combined Cylinders 130 CHAPTER 10: THICK LENS SYSTEMS: PART I 10.1 Introduction 139 10.2 The Cardinal Points 139 10.2.1 The Principal Planes and Points 139 10.2.2 The Nodal Points 140 10.2.3 Construction of the Nodal Points 141 10.2.4 Other "Cardinal" Points 143 10.3 The Newtonian Equations for a Thick System 143 10.4 The Refraction Equations for a Thick System 144 10.5 The Lateral Magnification of a Thick Lens 144 10.6 Summary of Vergence Equations for a Thick Lens System 145 10.7 Lateral, Axial and Angular Magnification 145 10.7.1 Lateral Magnification: Y 145
Contents ix 10.7.2 Axial Magnification: X 145 10.7.3 Angular Magnification: Ma 146 CHAPTER 11: THICK LENS SYSTEMS: PART II 11.1 The Gullstrand Equations 149 11.1.1 Equivalent Power 149 11.1.2 The Position of the Second Focal Point F' 151 11.1.3 The Position of the Second Principal Point H' 151 11.1.4 The Positions of the First Focal and Principal Points F and H 151 11.2 Distances Referred to Principal Planes 152 11.2.1 Summary of the Gullstrand Equations 152 11.2.1. IThe Positions of the Nodal Points N and N' 153 11.2.2 The Effect of Lens Thickness on the Positions of the Principal Planes 154 11.2.3 The Effect of Lens Shape on the Position of the Principal Planes 155 11.2.4 Telephoto and Wide-Angle Lenses 155 11.3 Vertex Power 157 11.4 Vertex Power and Ophthalmic Lenses 157 11.5 Effective Power 158 11.5.1 The Relationship of Effective Power to Vertex Power 159 11.6 Combination of Two Thick Lens 159 11.7 The Gullstrand Schematic Eye 161 11.7.1 Procedure for Calculating Cardinal Points of the Schematic Eye 161 11.8 Purkinje Images 164 11.8.1 Calculation of Purkinje Image I 164 11.8.2 Calculation of Purkinje Image II 165 11.8.3 Calculation of Purkinje Image III 166 CHAPTER 12: STOPS, PUPILS AND PORTS 12.1 Types of Stops 173 12.2 Pupils 173 12.3 The Field Stop and the Entrance and Exit Ports 174 12.4 Field of View and Vignetting 175 12.5 Procedure for Finding the Stops, Pupils, Ports and Field of View 176 CHAPTER 13: NUMERICAL APERTURE, f-number, AND RESOLUTION 13.1 Numerical Aperture 183 13.2 f-number 183 13.3 Angular Resolution 185 13.4 Linear Resolution 187 13.5 Resolution Charts 187 13.6 Snellen Charts 188 13.7 Modulation Transfer Function 189 13.8 The Numerical Aperture of a Fiber Optic 191 CHAPTER 14: MAGNIFIERS AND MICROSCOPES 14.1 Introduction 195 14.2 Visual Angle and Angular Magnification 195
x Contents 14.3 The Magnifier or Simple Microscope 195 14.3.1 Nominal Magnification 196 14.3.2 Maximum Magnification 197 14.3.3 A General Equation for Magnification 198 14.3.4 The Accommodation Required to View the Image 199 14.3.5 The Field of View of Magnifiers 199 14.3.6 Types of Magnifiers 200 14.4 The Compound Microscope 200 14.4.1 Magnification by a Microscope 201 14.4.2 Maximum Magnification by a Microscope 202 14.4.3 The Compound Microscope as a Simple Magnifier 202 14.4.4 The Stops of a Microscope 202 14.5 Numerical Aperture and Resolution of Microscopes 203 14.6.1 Maximum Usable Magnification of a Microscope 204 14.6.2 Resolution, Wavelength and the Electron Microscope 205 14.6.3 Depth of Focus of Microscope 205 CHAPTER 15: TELESCOPES 15.1 Introduction 209 15.2 Refracting Telescopes 210 15.2.1 The Astronomical or Keplerian Telescope 210 15.2.1.1 Angular Magnification 210 15.2.1.2 The Field of View of the Astronomical Telescope 212 15.2.1.3 Field Lenses 213 15.2.2 The Terrestrial Telescope 214 15.2.2.1 Relay Lenses 214 15.2.3 The Galilean Telescope 214 15.2.3.1 Field and Opera Glasses 215 15.2.4 The Prism Binocular 216 15.2.4.1 Stereoscopic Range Through Binoculars 217 15.3 Reflecting Telescopes 218 15.3.1 Newtonian Telescope 218 15.3.2 Gregorian Telescope 219 15.3.3 Cassagrainian Telescope 219 15.4 Catadioptric Telescopes 219 15.5 The Maximum Useful Magnification of a Telescope 219 CHAPTER 16: CAMERAS AND PROJECTORS 16.1 Pinhole Camera and Camera Obscura 223 16.2 Types of Cameras 223 16.3 Camera Lenses 225 16.3.1 Field of View of Cameras 225 16.3.2 Depth of Field and Focus 225 16.3.3 Hyperfocal Distance 227 16.4 The Paraxial Design of a Zoom Lens 229 16.5 Paraxial Design of a Telephoto Lens 230 16.6 Optical Projection Systems 231
Contents xi CHAPTER 17: OPHTHALMIC INSTRUMENTS 17.1 Introduction 235 17.2 Ophthalmometer 235 17.2.1 Optical Principle 235 17.2.1.1 Fixed Mire-Variable Image Size Ophthalmometers 236 17.2.1.2 Variable Mire-Fixed Image Ophthalmometers 236 17.2.2 Radius of Curvature 236 17.2.3 Refractive Power of the Cornea 237 17.3 Doubling Principle 238 17.3.1 Fixed Mire Size Ophthalmometer 239 17.3.2 Adjustable Mire Size Ophthalmometer 241 17.4 Badal Optometer 242 17.4.1 Badal Target Position and Ametropia 243 17.4.2 Alternate Arrangement of Badal Optometer 244 17.4.3 Telecentric Systems 244 17.5 The Lensometer 245 17.5.1 Lensometer Target and Reticle 247 17.6 Slit Lamp or Biomicroscope 247 CHAPTER 18: DISPERSION AND CHROMATIC ABERRATION 18.1 Introduction 251 18.2 Wavelength and Index of Refraction 251 18.2.1 Index of Refraction 252 18.2.2 Dispersion 253 18.3 Optical Glass 254 18.3.1 Crown and Flint Glasses 254 18.4 Chromatic Aberration of Thin Prisms 255 18.5 Achromatic Prisms 256 18.6 The Longitudinal Chromatic Aberration of Lenses 257 18.6.1 Longitudinal Chromatic Aberration in Diopters 258 18.6.2 Linear Longitudinal Chromatic Aberration 258 18.7 Achromatic Lenses in Contact 260 18.8 Transverse Chromatic Aberration 261 18.8.1 Linear Transverse Chromatic Aberration 261 18.8.2 Transverse Chromatic Aberration in Prism Diopters 263 18.9 Achromatic Combination of Two Air-Spaced Lenses 263 18.9.1 The Huygens Eyepiece 264 18.9.2 The Ramsden Eyepiece 266 18.9.3 The Kellner and Other Eyepieces 267 CHAPTER 19: TRIGONOMETRIC RAY TRACING 19.1 Introduction 271 19.2 Ray Tracing Sign Conventions and Nomenclature 271 19.3 Ray Tracing Equations 272 19.3.1 Transfer Equations 275 19.4 Ray Tracing Through a Series of Surfaces 276
xii Contents 19.5 Longitudinal and Transverse Spherical Aberration 277 19.6 Ray Trace Calculations for a Pencil of Rays 278 19.7 A Format for Tracing Rays Through a Lens 278 19.8 Image Evaluation 278 19.8.1 Spherical Aberration Curves 278 19.8.2 Ray Intercept Curves 280 19.8.3 Spot Diagrams 280 19.8.4 Radial Energy Diagrams 281 19.8.5 Knife Edge Distributions 282 19.9 Graphical Ray Trace 282 CHAPTER 20: MONOCHROMATIC ABERRATIONS 20.1 Introduction 285 20.2 Third Order Theory 285 20.3 Third Order Aberration Polynomial 286 20.3.1 The Spherical Aberration Terms of the Polynomial 286 20.3.2 Spherical Aberration of a Thin Lens 287 20.3.3 Spherical Aberration and Lens Bending 288 20.3.4 Correction of Spherical Aberration 290 20.4 The Coma Terms of the Aberration Polynomial 291 20.4.1 Coma of a Thin Lens 293 20.4.2 The Sine Condition 293 20.5 Astigmatism 295 20.5.1 Astigmatism by a Spherical Refracting Surface 295 20.5.2 Astigmatism by a Thin Lens 297 20.6 Petzval Curvature 297 20.6.1 Petzval Curvature of a Refracting Surface 297 20.6.2 Petzval Curvature of a Series of Surfaces 298 20.6.3 Petzval Curvature of a Thin Lens in Air 299 20.6.4 The Petzval Condition for Flattening the Field 299 20.7 Astigmatism and Petzval Curvature Polynomial Terms 299 20.8 Distortion 300 20.8.1 The Third Order Distortion Polynomial Term 300 20.8.2 The Tangent Condition 301 BIBLIOGRAPHY 305 INDEX 309