Principles of Optics

Historical Introduction

I. Basic Properties of the Electromagnetic Field


1.1. The Electromagnetic Field


1.1.1. Maxwell's Equations


1.1.2. Material Equations


1.1.3. Boundary Conditions at a Surface of Discontinuity


1.1.4. The Energy Law of the Electromagnetic Field


1.2. The Wave Equation and the Velocity of Light


1.3. Scalar Waves


1.3.1. Plane Waves


1.3.2. Spherical Waves


1.3.3. Harmonic Waves. The Phase Velocity


1.3.4. Wave Packets. The Group Velocity


1.4. Vector Waves


1.4.1. The General Electromagnetic Plane Wave


1.4.2. The Harmonic Electromagnetic Plane Wave


1.4.3. Harmonic Vector Waves of Arbitrary Form


1.5. Reflection and Refraction of a Plane Wave


1.5.1. The Laws of Reflection and Refraction


1.5.2. Fresnel Formula


1.5.3. The Reflectivity and Transmissivity; Polarization on Reflection and Refraction


1.5.4. Total Reflection


1.6. Wave Propagation in a Stratified Medium. Theory of Dielectric Films


1.6.1. The Basic Differential Equations


1.6.2. The Characteristic Matrix of a Stratified Medium


1.6.3. The Reflection and Transmission Coefficients


1.6.4. A Homogeneous Dielectric Film


1.6.5. Periodically Stratified Media

II. Electromagnetic Potentials and Polarization


2.1. The Electrodynamic Potentials in the Vacuum


2.1.1. The Vector and Scalar Potentials


2.1.2. Retarded Potentials


2.2. Polarization and Magnetization


2.2.1. The Potentials in Terms of Polarization and Magnetization


2.2.2. Hertz Vectors


2.2.3. The Field of a Linear Electric Dipole


2.3. The Lorentz-Lorenz Formula and Elementary Dispersion Theory


2.3.1. The Dielectric and Magnetic Susceptibilities


2.3.2. The Effective Field


2.3.3. The Mean Polarizability: the Lorentz-Lorenz Formula


2.3.4. Elementary Theory of Dispersion


2.4. Propagation of Electromagnetic Waves Treated by Integral Equations


2.4.1. The Basic integral Equation


2.4.2. The Ewald-Oseen Extinction Theorem and a Rigorous Derivation of the Lorentz-Lorenz Formula


2.4.3. Refraction and Reflection of a Plane Wave, Treated with the Help of the Ewald-Oseen Extinction Theorem

III. Foundations of Geometrical Optics


3.1. Approximation for Very Short Wavelengths


3.1.1. Derivation of the Eikonal Equation


3.1.2. The Light Rays and the Intensity Law of Geometrical Optics


3.1.3. Propagation of the Amplitude Vectors


3.1.4. Generalizations and the Limits of Validity of Geometrical Optics


3.2. General Properties of Rays


3.2.1. The Differential Equation of Light Rays


3.2.2. The Laws of Refraction and Reflection


3.2.3. Ray Congruences and Their Focal Properties


3.3. Other Basic Theorems of Geometrical Optics


3.3.1. Lagrange's Integral Invariant


3.3.2. The Principle of Fermat


3.3.3. The Theorem of Malus and Dupin and Some Related Theorems

IV. Geometrical Theory of Optical Imaging


4.1. The Characteristic Functions of Hamilton


4.1.1. The Point Characteristic


4.1.2. The Mixed Characteristic


4.1.3. The Angle Characteristic


4.1.4. Approximate Form of the Angle Characteristic of a Refracting Surface of Revolution


4.1.5. Approximate Form of the Angle Characteristic of a Reflecting Surface of Revolution


4.2. Perfect Imaging


4.2.1. General Theorems


4.2.2. Maxwell's "Fish-Eye"


4.2.3. Stigmatic Imaging of Surfaces


4.3. Projective Transformation (Collineation) with Axial Symmetry


4.3.1. General Formula


4.3.2. The Telescopic Case


4.3.3. Classification of Projective Transformations


4.3.4. Combination of Projective Transformations


4.4. Gaussian Optics


4.4.1. Refracting Surface of Revolution


4.4.2. Reflecting Surface of Revolution


4.4.3. The Thick Lens


4.4.4. The Thin Lens


4.4.5. The General Centered System


4.5. Stigmatic Imaging with Wide-angle Pencils


4.5.1. The Sine Condition


4.5.2. The Herschel Condition


4.6. Astigmatic Pencils of Rays


4.6.1. Focal Properties of a Thin Pencil


4.6.2. Refraction of a Thin Pencil


4.7. Chromatic Aberration. Dispersion by a Prism


4.7.1. Chromatic Aberration


4.7.2. Dispersion by a Prism


4.8. Photometry and Apertures


4.8.1. Basic Concepts of Photometry


4.8.2. Stops and Pupils


4.8.3. Brightness and Illumination of Images


4.9. Ray Tracing


4.9.1. Oblique Meridional Rays


4.9.2. Paraxial Rays


4.9.3. Skew Rays


4.10. Design of Aspheric Surfaces


4.10.1. Attainment of Axial Stigmatism


4.10.2. Attainment of Aplanatism

V. Geometrical Theory of Aberrations


5.1. Wave and Ray Aberrations; the Aberration Function


5.2. The Perturbation Eikonal of Schwarzschild


5.3. The Primary (Seidel) Aberrations


5.4. Addition Theorem for the Primary Aberrations


5.5. The Primary Aberration Coefficients of a General Centered Lens System


5.5.1. The Seidel Formula in Terms of Two Paraxial Rays


5.5.2. The Seidel Formula in Terms of one Paraxial Ray


5.5.3. Petzval's Theorem


5.6. Example: The Primary Aberrations of a Thin Lens


5.7. The Chromatic Aberration of a General Centered Lens System

VI. Image-Forming Instruments


6.1. The Eye


6.2. The Camera


6.3. The Refracting Telescope


6.4. The Reflecting Telescope


6.5. Instruments of Illumination


6.6. The Microscope

VII. Elements of the Theory of Interference and Interferometers


7.1. Introduction


7.2. Interference of Two Monochromatic Waves


7.3. Two-Beam Interference: Division of Wave-Front


7.3.1. Young's Experiment


7.3.2. Fresnel's Mirrors and Similar Arrangements


7.3.3. Fringes with Quasi-Monochromatic and White Light


7.3.4. Use of Slit Sources; Visibility of Fringes


7.3.5. Application to the Measurement of optical Path Difference: the Rayleigh Interferometer


7.3.6. Application to the Measurement of Angular Dimensions of Sources: the Michelson Stellar Interferometer


7.4. Standing Waves


7.5. Two-Beam Interference: Division of Amplitude


7.5.1. Fringes with a Plane Parallel Plate


7.5.2. Fringes with Thin Films; the Fizeau Interferometer


7.5.3. Localization of Fringes


7.5.4. The Michelson Interferometer


7.5.5. The Twyman-Green and Related Interferometers


7.5.6. Fringes with Two Identical Plates: the Jamin Interferometer and Interference Microscopes


7.5.7. The Mach-Zehnder Interferometer; the Bates Wave-Front Shearing Interferometer


7.5.8. The Coherence Length; the Application of Two-Beam Interference to the Study of the Fine Structure of Spectral Lines


7.6. Multiple-Beam Interference


7.6.1. Multiple-Beam Fringes with a Plane Parallel Plate


7.6.2. The Fabry-Perot Interferometer


7.6.3. The Application of the Fabry-Perot Interferometer to the Study of the Fine Structure of Spectral Lines


7.6.4. The Application of the Fabry-Perot Interferometer to the Comparison of Wavelengths


7.6.5. The Lummer-Gehrcke Interferometer


7.6.6. Interference Filters


7.6.7. Multiple-Beam Fringes with Thin Films


7.6.8. Multiple-Beam Fringes with Two Plane Parallel Plates


7.7. The Comparison of Wavelengths with the Standard Meter

VIII. Elements of the Theory of Diffraction


8.1. Introduction


8.2. The Huygens-Fresnel Principle


8.3. Kirchhoff's Diffraction Theory


8.3.1. The Integral Theorem of Kirchhoff


8.3.2. Kirchhoff's Diffraction Theory


8.3.3. Fraunhofer and Fresnel Diffraction


8.4. Transition to a Scalar Theory


8.4.1. The Image Field Due to a Monochromatic oscillator


8.4.2. The Total Image Field


8.5. Fraunhofer Diffraction at Apertures of Various Forms


8.5.1. The Rectangular Aperture and the Slit


8.5.2. The Circular Aperture


8.5.3. Other Forms of Aperture


8.6. Fraunhofer Diffraction in Optical Instruments


8.6.1. Diffraction Gratings


8.6.2. Resolving Power of Image-forming Systems


8.6.3. Image Formation in the Microscope


8.7. Fresnel Diffraction at a Straight Edge


8.7.1. The Diffraction Integral


8.7.2. Fresnel's Integrals


8.7.3. Fresnel Diffraction at a Straight Edge


8.8. The Three-Dimensional Light Distribution near Focus


8.8.1. Evaluation of the Diffraction Integral in Terms of Lommel Functions


8.8.2. The Distribution of Intensity


8.8.3. The Integrated Intensity


8.8.4. The Phase Behavior


8.9. The Boundary Diffraction Wave


8.10. Gabor's Method of Imaging by Reconstructed Wave-Fronts (Holography)


8.10.1. Producing the Positive Hologram


8.10.2. The Reconstruction

IX. The Diffraction Theory of Aberrations


9.1. The Diffraction Integral in the Presence of Aberrations


9.1.1. The Diffraction Integral


9.1.2. The Displacement Theorem. Change of Reference Sphere


9.1.3. A Relation between the Intensity and the Average Deformation of Wave-Fronts


9.2. Expansion of the Aberration Function


9.2.1. The Circle Polynomials of Zernike


9.2.2. Expansion of the Aberration Function


9.3. Tolerance Conditions for Primary Aberrations


9.4 The Diffraction Pattern Associated with a Single Aberration


9.4.1. Primary Spherical Aberration


9.4.2. Primary Coma


9.4.3. Primary Astigmatism


9.5. Imaging of Extended Objects


9.5.1. Coherent Illumination


9.5.2. Incoherent Illumination

X. Interference and Diffraction with Partially Coherent Light


10.1. Introduction


10.2. A Complex Representation of Real Polychromatic Fields


10.3. The Correlation Functions of Light Beams


10.3.1. Interference of Two Partially Coherent Beams. The Mutual Coherence Function and the Complex Degree of Coherence


10.3.2. Spectral Representation of Mutual Coherence


10.4. Interference and Diffraction with Quasi-monochromatic Light


10.4.1. Interference with Quasi-monochromatic Light. The Mutual Intensity


10.4.2. Calculation of Mutual Intensity and Degree of Coherence for Light from an Extended Incoherent Quasi-Monochromatic Source


10.4.3. An Example


10.4.4. Propagation of Mutual Intensity


10.5. Some Applications


10.5.1. The Degree of Coherence in the Image of an Extended Incoherent Quasi-Monochromatic Source


10.5.2. The Influence of the Condenser on Resolution in a Microscope


10.5.3. Imaging with Partially Coherent Quasi-monochromatic Illumination


10.6. Some Theorems Relating to Mutual Coherence


10.6.1. Calculation of Mutual Coherence for Light from an Incoherent Source


10.6.2. Propagation of Mutual Coherence


10.7. Rigorous Theory of Partial Coherence


10.7.1. Wave Equations for Mutual Coherence


10.7.2. Rigorous Formulation of the Propagation Law for Mutual Coherence


10.7.3. The Coherence Time and the Effective Spectral Width


10.8. Polarization Properties of Quasi-Monochromatic Light


10.8.1. The Coherency Matrix of a Quasi-Monochromatic Plane Wave


10.8.2. Some Equivalent Representations. The Degree of Polarization of a Light Wave


10.8.3. The Stokes Parameters of a Quasi-Monochromatic Plane Wave

XI. Rigorous Diffraction Theory


11.1. Introduction


11.2. Boundary Conditions and Surface Currents


11.3. Diffraction by a Plane Screen: Electromagnetic Form of Babinet's Principle


11.4. Two-Dimensional Diffraction by a Plane Screen


11.4.1. The Scalar nature of Two-dimensional Electromagnetic Fields


11.4.2. An Angular Spectrum of Plane Waves


11.4.3. Formulation in Terms of Dual Integral Equations


11.5. Two-Dimensional Diffraction of a Plane Wave by a Half-Plane


11.5.1. Solution of the Dual Integral Equations for E-Polarization


11.5.2. Expression of the Solution in Terms of Fresnel Integrals


11.5.3. The nature of the Solution


11.5.4. The Solution for H-Polarization


11.5.5. Some numerical Calculations


11.5.6. Comparison with Approximate Theory and with Experimental Results


11.6. Three-Dimensional Diffraction of a Plane Wave by a Half-Plane


11.7. Diffraction of a Localized Source by a Half-Plane


11.7.1. A Line-Current Parallel to the Diffracting Edge


11.7.2. A Dipole


11.8. Other Problems


11.8.1. Two Parallel Half-Planes


11.8.2. An Infinite Stack of Parallel, Staggered Half-Planes


11.8.3. A Strip


11.8.4. Further Problems


11.9. Uniqueness of Solution

XII. Diffraction of Light by Ultrasonic Waves


12.1. Qualitative Description of the Phenomenon and Summary of Theories Based on Maxwell's Differential Equations


12.1.1. Qualitative Description of the Phenomenon


12.1.2. Summary of Theories Based on Maxwell's Equations


12.2. Diffraction of Light by Ultrasonic Waves as Treated by the Integral Equation Method


12.2.1. Integral Equation for E-Polarization


12.2.2. The Trial Solution of the Integral Equation


12.2.3. Expressions for the Amplitudes of the Light Waves in the Diffracted and Reflected Spectra


12.2.4. Solution of the Equations by a Method of Successive Approximations


12.2.5. Expressions for the Intensities of the First and Second Order Lines for some Special Cases


12.2.6. Some Qualitative Results


12.2.7. The Raman-Nath Approximation

XIII. Optics of Metals


13.1. Wave Propagation in a Conductor


13.2. Refraction and Reflection at a Metal Surface


13.3. Elementary Electron Theory of the Optical Constants of Metals


13.4. Wave Propagation in a Stratified Conducting Medium. Theory of Metallic Films


13.4.1. An Absorbing Film on a Transparent Substrate


13.4.2. A Transparent Film on an Absorbing Substrate


13.5. Diffraction by a Conducting Sphere; Theory of Mie


13.5.1. Mathematical Solution of the Problem


13.5.2. Some Consequences of Mie's Formula


13.5.3. Total Scattering and Extinction

XIV. Optics of Crystals


14.1. The Dielectric Tensor of an Anisotropic Medium


14.2. The Structure of a Monochromatic Plane Wave in an Anisotropic Medium


14.2.1. The Phase Velocity and the Ray Velocity


14.2.2. Fresnel's Formula for the Propagation of Light in Crystals


14.2.3. Geometrical Constructions for Determining the Velocities of Propagation and the Directions of Vibration


14.3. Optical Properties of Uniaxial and Biaxial Crystals


14.3.1. The Optical Classification of Crystals


14.3.2. Light Propagation in Uniaxial Crystals


14.3.3. Light Propagation in Biaxial Crystals


14.3.4. Refraction in Crystals


14.4. Measurements in Crystal Optics


14.4.1. The Nicol Prism


14.4.2. Compensators


14.4.3. Interference with Crystal Plates


14.4.4. Interference Figures from Uniaxial Crystal Plates


14.4.5. Interference Figures from Biaxial Crystal Plates


14.4.6. Location of Optic Axes and Determination of the Principal Refractive Indices of a Crystalline Medium


14.5. Stress Birefringence and Form Birefringence


14.5.1. Stress Birefringence


14.5.2. Form Birefringence


14.6. Absorbing Crystals


14.6.1. Light Propagation in an Absorbing Anisotropic Medium


14.6.2. Interference Figures from Absorbing Crystal Plates


14.6.3. Dichroic Polarizers

Appendices

I. The Calculus of Variations


1. Euler's Equations as necessary Conditions for an Extremum


2. Hubert's Independence Integral and the Hamilton-Jacobi Equation


3. The Field of Extremals


4. Determination of All Extremals from the Solution of the Hamilton-Jacobi Equation


5. Hamilton's Canonical Equations


6. The Special Case When the Independent Variable Does not Appear Explicitly in the Integrand


7. Discontinuities


8. Weierstrass' and Legendre's Conditions (Sufficiency Conditions for an Extremum)


9. Minimum of the Variational Integral When one End Point is Constrained to a Surface


10. Jacobi's Criterion for a Minimum


11. Example I: Optics


12. Example II: Mechanics of Material Points

II. Light Optics, Electron Optics and Wave Mechanics


1. The Hamiltonian Analogy in Elementary Form


2. The Hamiltonian Analogy in Variational Form


3. Wave Mechanics of Free Electrons


4. The Application of Optical Principles to Electron Optics

III. Asymptotic Approximations to Integrals


1. The Method of Steepest Descent


2. The Method of Stationary Phase


3. Double Integrals

IV. The Dirac Delta Function

V. A Mathematical Lemma Used in the Rigorous Derivation of the Lorentz-Lorenz Law (§2.4.2)

VI. Propagation of Discontinuities in an Electromagnetic Field (§3.1.1)


1. Relations Connecting Discontinuous Changes in Field Vectors


2. The Field on a Moving Discontinuity Surface

VII. The Circle Polynomials of Zernike (§9.2.1)


1. Some General Considerations


2. Explicit Expressions for the Radial Polynomials Rn±m(ρ)

VIII. Proof of an Inequality (§10.7.3)

IX. Evaluation of Two Integrals (§12.2.2)

Author Index

Subject Index