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Optical imaging starts with geometrical optics, and ray tracing lies at its forefront. This book starts with Fermat’s principle and derives the three laws of geometrical optics from it. After discussing imaging by refracting and reflecting systems, paraxial ray tracing is used to determine the size of imaging elements and obscuration in mirror systems. Stops, pupils, radiometry, and optical instruments are also discussed. The chromatic and monochromatic aberrations are addressed in detail, followed by spot sizes and spot diagrams of aberrated images of point objects. Each chapter ends with a summary and a set of problems. The book ends with an epilogue that summarizes the imaging process and outlines the next steps within and beyond geometrical optics.

### Table of Contents

**Preface****Acknowledgments****Symbols and Notation****CHAPTER 1: FOUNDATIONS OF GEOMETRICAL OPTICS**- 1.1 Introduction
- 1.2 Sign Convention
- 1.3 Fermat's Principle
- 1.4 Rays and Wavefronts
- 1.5 Laws of Geometrical Optics
- 1.5.1 Rectilinear Propagation
- 1.5.2 Refraction in 2D
- 1.5.3 Reflection in 2D
- 1.5.4 Refraction in 3D
- 1.5.5 Reflection in 3D
- 1.6 Exact Ray Tracing
- 1.6.1 Ray Incident on a Spherical Surface
- 1.6.2 Rectilinear Propagation from the Object Plane to the First Refracting Surface
- 1.6.3 Refraction of a Ray by a Spherical Refracting Surface
- 1.6.4 Rectilinear Propagation from the First Refracting Surface to the Second
- 1.6.5 Reflection of a Ray by a Spherical Reflecting Surface
- 1.6.6 Conic Surface and Surface Normal
- 1.6.7 Refraction of a Ray by a Conic Refracting Surface
- 1.6.8 Reflection of a Ray by a Conic Reflecting Surface
- 1.6.9 Tracing a Tangential Ray
- 1.6.10 Determining Wave and Ray Aberrations
- 1.7 Paraxial Ray Tracing
- 1.7.1 Snell's Law
- 1.7.2 Point on a Spherical Surface
- 1.7.3 Distance between Two Points
- 1.7.4 Unit Vector along a Surface Normal
- 1.7.5 Unit Vector along a Ray
- 1.7.6 Transfer of a Ray
- 1.7.7 Refraction of a Ray
- 1.7.8 Reflection of a Ray
- 1.8 Gaussian Approximation and Imaging
- 1.8.1 Gaussian Approximation
- 1.8.2 Gaussian Imaging by a Refracting Surface
- 1.8.3 Gaussian Imaging by a Reflecting Surface
- 1.8.4 Gaussian Imaging by a Multisurface System
- 1.9 Imaging beyond Gaussian Approximation
- 1.10 Summary of Results
- 1.10.1 Sign Convention
- 1.10.2 Fermat's Principle
- 1.10.3 Laws of Geometrical Optics
- 1.10.4 Exact Ray Tracing
- 1.10.4.1 Transfer Operation
- 1.10.4.2 Refraction Operation
- 1.10.4.3 Reflection Operation
- 1.10.4.4 Ray Tracing a Conic Surface
- 1.10.4.5 Tracing a Tangential Ray
- 1.10.5 Paraxial Ray Tracing
- 1.10.6 Gaussian Optics
- 1.10.6.1 Gaussian Imaging by a Refracting Surface
- 1.10.6.2 Gaussian Imaging by a Reflecting Surface
- References
- Problems
**CHAPTER 2: REFRACTING SYSTEMS**- 2.1 Introduction
- 2.2 Spherical Refracting Surface
- 2.2.1 Gaussian Imaging Equation
- 2.2.2 Object and Image Spaces
- 2.2.3 Focal Lengths and Refracting Power
- 2.2.4 Magnifications and Lagrange Invariant
- 2.2.5 Graphical Imaging
- 2.2.6 Newtonian Imaging Equation
- 2.3 Thin Lens
- 2.3.1 Gaussian Imaging Equation
- 2.3.2 Focal Lengths and Refracting Power
- 2.3.3 Magnifications and Lagrange Invariant
- 2.3.4 Graphical Imaging
- 2.3.5 Newtonian Imaging Equation
- 2.3.6 Image Throw
- 2.3.7 Thin Lens Not in Air
- 2.3.8 Thin Lenses in Contact
- 2.4 General System
- 2.4.1 Introduction
- 2.4.2 Cardinal Points and Planes
- 2.4.3 Gaussian Imaging, Focal Lengths, and Magnifications
- 2.4.4 Nodal Points and Planes
- 2.4.5 Newtonian Imaging Equation
- 2.4.6 Graphical Imaging
- 2.4.7 Reference to Other Conjugate Planes
- 2.4.8 Comparison of Imaging by a General System and a Refracting Surface or a Thin Lens
- 2.4.9 Determination of Cardinal Points
- 2.5 Afocal Systems
- 2.5.1 Introduction
- 2.5.2 Lagrange Invariant for an Infinite Conjugate
- 2.5.3 Imaging by an Afocal System
- 2.6 Plane-Parallel Plate
- 2.6.1 Introduction
- 2.6.2 Imaging Relations
- 2.7 Petzval Image
- 2.7.1 Spherical Refracting Surface
- 2.7.2 General System
- 2.7.3 Thin Lens
- 2.8 Misaligned Surface
- 2.8.1 Decentered Surface
- 2.8.2 Tilted Surface
- 2.8.3 Despaced Surface
- 2.9 Misaligned Thin Lens
- 2.9.1 Decentered Lens
- 2.9.2 Tilted Lens
- 2.9.3 Despaced Lens
- 2.10 Anamorphic Imaging Systems
- 2.11 Summary of Results
- 2.11.1 Imaging Equations
- 2.11.1.1 General System
- 2.11.1.2 Refracting Surface
- 2.11.1.3 Thin Lens
- 2.11.1.4 Afocal System
- 2.11.1.5 Plane-Parallel Plate
- 2.11.2 Petzval Image
- 2.11.3 Misalignments
- 2.11.3.1 Misaligned Surface
- 2.11.3.2 Misaligned Thin Lens
- 2.11.4 Anamorphic Imaging Systems
- Problems
**CHAPTER 3: REFLECTING SYSTEMS**- 3.1 Introduction
- 3.2 Spherical Reflecting Surface (Spherical Mirror)
- 3.2.1 Gaussian Imaging Equation
- 3.2.2 Focal Length and Reflecting Power
- 3.2.3 Magnifications and the Lagrange Invariant
- 3.2.4 Graphical Imaging
- 3.2.5 Newtonian Imaging Equation
- 3.3 Two-Mirror Telescopes
- 3.4 Beam Expander
- 3.5 Petzval Image
- 3.5.1 Single Mirror
- 3.5.2 Two-Mirror System
- 3.5.3 System of
*k*Mirrors - 3.6 Misaligned Mirror
- 3.6.1 Decentered Mirror
- 3.6.2 Tilted Mirror
- 3.6.3 Despaced Mirror
- 3.7 Misaligned Two-Mirror Telescope
- 3.7.1 Decentered Secondary Mirror
- 3.7.2 Tilted Secondary Mirror
- 3.7.3 Despaced Secondary Mirror
- 3.8 Summary of Results
- 3.8.1 Imaging by a Mirror
- 3.8.2 Imaging by a Two-Mirror Telescope
- Problems
**CHAPTER 4: PARAXIAL RAY TRACING**- 4.1 Introduction
- 4.2 Refracting Surface
- 4.3 General System
- 4.3.1 Determination of Cardinal Points
- 4.3.2 Combination of Two Systems
- 4.4 Thin Lens
- 4.5 Thick Lens
- 4.6 Two-Lens System
- 4.7 Reflecting Surface (Mirror)
- 4.8 Two-Mirror System
- 4.8.1 Focal Length
- 4.8.2 Obscuration
- 4.9 Catadioptric System: Thin-Lens - Mirror Combination
- 4.10 Two-Ray Lagrange Invariant
- 4.11 Summary of Results
- 4.11.1 Ray-Tracing Equations
- 4.11.2 Thick Lens
- 4.11.3 Two-Lens System
- 4.11.4 Two-Mirror System
- 4.11.5 Two-Ray Lagrange Invariant
- Problems
**CHAPTER 5: STOPS, PUPILS, AND RADIOMETRY**- 5.1 Introduction
- 5.2 Stops, Pupils, and Vignetting
- 5.2.1 Introduction
- 5.2.2 Aperture Stop, and Entrance and Exit Pupils
- 5.2.3 Chief and Marginal Rays
- 5.2.4 Vignetting
- 5.2.5 Size of an Imaging Element
- 5.2.6 Telecentric Aperture Stop
- 5.2.7 Field Stop, and Entrance and Exit Windows
- 5.3 Radiometry of Point Object Imaging
- 5.3.1 Flux Received by an Aperture
- 5.3.2 Inverse-Square Law of Irradiance
- 5.3.3 Image Intensity
- 5.4 Radiometry of Extended Object Imaging
- 5.4.1 Introduction
- 5.4.2 Lambertian Surface
- 5.4.3 Illumination by a Lambertian Disc
- 5.4.4 Flux Received by an Aperture
- 5.4.5 Image Radiance
- 5.4.6 Image Irradiance: Aperture Stop in front of the System
- 5.4.7 Image Irradiance: Aperture Stop in back of the System
- 5.4.8 Telecentric Systems
- 5.4.9 Throughput
- 5.4.10 Interrelations among Invariants in Imaging
- 5.4.11 Concentric Systems
- 5.5 Photometry
- 5.5.1 Photometric Quantities and Spectral Response of the Human Eye
- 5.5.2 Imaging by the Human Eye
- 5.5.3 Brightness of a Lambertian Surface
- 5.6 Summary of Results
- 5.6.1 Stops, Pupils, Windows, and Field of View
- 5.6.2 Radiometry of Point Object Imaging
- 5.6.3 Radiometry of Extended Object Imaging
- 5.6.3.1 Illumination by a Lambertian Disc
- 5.6.3.2 Image Radiance
- 5.6.3.3 Image Irradiance
- 5.6.4 Visual Observations
- References
- Problems
**CHAPTER 6: OPTICAL INSTRUMENTS**- 6.1 Introduction
- 6.2 Eye
- 6.2.1 Anatomy and Structure
- 6.2.2 Paraxial Models
- 6.2.3 Accommodation
- 6.2.4 Visual Acuity
- 6.2.5 Spectacles (or Eyeglasses)
- 6.3 Magnifier
- 6.4 Microscope
- 6.5 Telescope
- 6.6 Ocular
- 6.7 Telephoto Lens and Wide-Angle Camera
- 6.8 Resolution
- 6.8.1 Introduction
- 6.8.2 Airy Pattern
- 6.8.3 Rayleigh Criterion of Resolution
- 6.8.4 Resolution of an Imaging System
- 6.8.5 Resolution of the Eye
- 6.8.6 Resolution of a Microscope
- 6.8.7 Resolution of a Telescope
- 6.9 Pinhole Camera
- 6.10 Summary of Results
- 6.10.1 Eye
- 6.10.2 Magnifier
- 6.10.3 Microscope
- 6.10.4 Telescope
- 6.10.5 Resolution
- 6.10.6 Pinhole Camera
- References
- Problems
**CHAPTER 7: CHROMATIC ABERRATIONS**- 7.1 Introduction
- 7.2 Refracting Surface
- 7.3 Thin Lens
- 7.4 Plane-Parallel Plate
- 7.5 General System
- 7.6 Doublet
- 7.6.1 Lenses of Different Materials
- 7.6.2 Lenses of the Same Material
- 7.6.3 Doublet with Two Separated Components
- 7.6.4 Thin-Lens Doublet
- 7.7 Summary of Results
- 7.7.1 General System
- 7.7.2 Thin Lens
- 7.7.3 Plane-Parallel Plate
- 7.7.4 Doublet
- References
- Problems
**CHAPTER 8: MONOCHROMATIC ABERRATIONS**- 8.1 Introduction
- 8.2 Wave and Ray Aberrations
- 8.2.1 Definitions
- 8.2.2 Relationship between Wave and Ray Aberrations
- 8.3 Wavefront Defocus Aberration
- 8.4 Wavefront Tilt Aberration
- 8.5 Aberrations of a Rotationally Symmetric System
- 8.5.1 Explicit Dependence on Object Coordinates
- 8.5.2 No Explicit Dependence on Object Coordinates
- 8.6 Additivity of Primary Aberrations
- 8.6.1 Introduction
- 8.6.2 Primary Wave Aberrations
- 8.6.3 Transverse Ray Aberrations
- 8.6.4 Off-Axis Point Object
- 8.6.5 Higher-Order Aberrations
- 8.7 Strehl Ratio and Aberration Balancing
- 8.7.1 Strehl Ratio
- 8.7.2 Aberration Balancing
- 8.8 Zernike Circle Polynomials
- 8.8.1 Introduction
- 8.8.2 Polynomials in Optical Design
- 8.8.3 Polynomials in Optical Testing
- 8.8.4 Characteristics of Polynomial Aberrations
- 8.8.4.1 Isometric Characteristics
- 8.8.4.2 Interferometric Characteristics
- 8.9 Relationship between Zernike Polynomials and Classical Aberrations
- 8.9.1 Introduction
- 8.9.2 Wavefront Tilt Aberration
- 8.9.3 Wavefront Defocus Aberration
- 8.9.4 Astigmatism
- 8.9.5 Coma
- 8.9.6 Spherical Aberration
- 8.9.7 Seidel Coefficients from Zernike Coefficients
- 8.10 Aberrations of an Anamorphic System
- 8.10.1 Introduction
- 8.10.2 Classical Aberrations
- 8.10.3 Polynomial Aberrations Orthonormal over a Rectangular Pupil
- 8.10.4 Expansion of a Rectangular Aberration Function in Terms of Orthonormal Rectangular Polynomials
- 8.11 Observation of Aberrations
- 8.11.1 Primary Aberrations
- 8.11.2 Interferograms
- 8.11.3 Random Aberrations
- 8.12 Summary of Results
- 8.12.1 Wave and Ray Aberrations
- 8.12.2 Wavefront Defocus Aberration
- 8.12.3 Wavefront Tilt Aberration
- 8.12.4 Primary Aberrations
- 8.12.5 Strehl Ratio and Aberration Balancing
- 8.12.6 Zernike Circle Polynomials
- 8.12.6.1 Use of Zernike Polynomials in Wavefront Analysis
- 8.12.6.2 Polynomials in Optical Design
- 8.12.6.3 Zernike Primary Aberrations
- 8.12.6.4 Polynomials in Optical Testing
- 8.12.6.5 Isometric and Interferometric Characteristics
- 8.12.7 Relationship between Zernike and Seidel Coefficients
- 8.12.8 Aberrations of an Anamorphic System
- Appendix: Combination of Two Zernike Polynomial Aberrations with the Same
*n*Value and Varying as cos*m*θ and sin*m*θ - References
- Problems
**CHAPTER 9: SPOT SIZES AND DIAGRAMS**- 9.1 Introduction
- 9.2 Theory
- 9.3 Application to Primary Aberrations
- 9.3.1 Spherical Aberration
- 9.3.2 Coma
- 9.3.3 Astigmatism and Field Curvature
- 9.3.4 Field Curvature and Depth of Focus
- 9.3.5 Distortion
- 9.4 Balanced Aberrations for the Minimum Spot Sigma
- 9.5 Spot Diagrams
- 9.6 Aberration Tolerance and a Golden Rule of Optical Design
- 9.7 Summary of Results
- 9.7.1 Spherical Aberration
- 9.7.2 Coma
- 9.7.3 Astigmatism and Field Curvature
- 9.7.4 Field Curvature and Defocus
- 9.7.5 Distortion
- 9.7.6 Aberration Tolerance
- 9.7.7 A Golden Rule of Optical Design
- References
- Problems
**EPILOGUE**- E1 Introduction
- E2 Principles of Geometrical Optics and Imaging
- E3 Ray Tracing: Exact and Paraxial
- E4 Gaussian Optics
- E4.1 Tangent Plane or Paraxial Surface
- E4.2 Sign Convention
- E4.3 Cardinal Points
- E4.4 Graphical Imaging
- E4.5 Lagrange Invariant
- E4.6 Matrix Approach to Gaussian Imaging
- E4.7 Petzval Image
- E4.8 Field of View
- E4.9 Chromatic Aberrations
- E5 Image Brightness
- E6 Image Quality
- E6.1 Wave and Ray Aberrations
- E6.2 Primary Aberrations
- E6.3 Spot Size and Aberration Balancing
- E6.4 Strehl Ratio and Aberration Balancing
- E7 Reflecting Systems
- E8 Anamorphic Imaging Systems
- E9 Aberration Tolerance and a Golden Rule of Optical Design
- E10 General Comments
- References
**Index**

### Preface

Portions of this book have their origin in the author's lectures given as an adjunct professor in the electrical engineering/electrophysics department of the University of Southern California from about 1984 to 1998. It is a precursor to the author's "*Optical Imaging and Aberrations* books (Part I: *Ray Geometrical Optics*; Part II: *Wave Diffraction Optics*; and Part III: *Wavefront Analysis*)," all published by SPIE Press. It is an expanded yet simplified version of some of the material from Part I, and contains some new material. The focus is on Gaussian imaging, ray tracing, radiometry, basic optical instruments, optical aberrations, and spot diagrams. The primary aberrations of simple systems, such as a thin lens or a two-mirror telescope, that are derived in Part I are not discussed here. The book can be used as a textbook for a senior undergraduate or a first-year graduate class.

Geometrical optics is fundamental to optical imaging. Chapter 1 lays out its foundations. It starts with the sign convention of Cartesian geometry, states the Fermat's principle, and derives the three laws of geometrical optics from it. These laws are used to obtain the equations for exact ray tracing, and those for paraxial ray tracing are obtained from them as an approximation. The latter equations are used to obtain the basic equations of Gaussian optics. In Chapter 2, the Gaussian and Newtonian imaging equations are derived for a refracting surface using the small-angle approximation of Snell's law. The equations thus obtained are applied to derive the imaging equations for a thin lens, and for a general imaging system. Afocal systems, as applied to astronomical telescopes, and telephoto and wide-angle camera lenses are discussed. The Petzval image describing the defocus error of the Gaussian image of an off-axis point object is considered. Also discussed is how the Gaussian image is displaced due to a misalignment of a surface or a thin lens. Imaging by an anamorphic system is briefly considered. Imaging by reflecting systems is discussed in Chapter 3, including Gaussian imaging by two-mirror telescopes.

The imaging equations obtained in Chapters 2 and 3 are rederived in Chapter 4 by using the paraxial ray-tracing equations. These ray-tracing equations are also used to determine the size of the imaging elements, vignetting of rays by them for off-axis point objects, and obscurations in mirror systems. Stops, pupils, and radiometry are discussed in Chapter 5. How to determine the aperture stop of a system and its images in the object and image spaces, i.e., the entrance and exit pupils, is described. The intensity of the image of a point object, invariance of the radiance of a ray bundle as it is refracted or reflected, and the irradiance distribution of the image of an extended object in terms of its radiance distribution are discussed. A brief discussion of photometry is also given. Some of the familiar optical instruments such as the eye, magnifier, microscope, telescope, and pinhole camera are addressed in Chapter 6. The most common and interesting among them is the eye, which is discussed in detail. The resolution of such common optical instruments is discussed based on Rayleigh's criterion of resolution, thus necessitating a brief discussion of the aberration-free diffraction image of a point object, i.e., the Airy pattern. The chromatic aberrations of a system are discussed in Chapter 7. A refracting surface, a thin lens, a plane-parallel plate, and a doublet are considered as simple examples of systems.

The monochromatic aberrations of a system with an emphasis on primary aberrations are considered in Chapter 8. The wave and ray aberrations are introduced, and a simple derivation of the relationship between them is given. The Strehl ratio of an image as a measure of its quality is introduced, and the balancing of wave aberrations to minimize their variance is discussed. The aberrations are also discussed in terms of the Zernike circle polynomials because of their widespread use in optical design and testing. The aberrations of an anamorphic imaging system are also discussed. The spot sizes and diagrams for primary aberrations are addressed in Chapter 9. Aberration balancing for minimum standard deviation of the ray distribution of an image spot is discussed. The aberration tolerances for primary aberrations based on their spot radius are derived, and the golden rule of optical design is described.

The content of each chapter is summarized in its last section. This section is written to be comprehensive enough that it can be read on its own without reading the whole chapter. Each chapter ends with a set of problems, which are an integral part of the book. They help develop and test how to apply the results obtained in a chapter to practical situations.

The book ends with an epilogue, which gives a summary of the imaging process, and outlines the next steps within and beyond geometrical optics.

**Virendra N. Mahajan**

El Segundo, California

April 2014

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