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

（a）Elliptic polarization

（b）Linear and circular polarization

（c）Characterization of the state of polarization by Stokes parameters

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 formulae

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 difierential equations

1.6.2 The characteristic matrix of a stratified medium

（a）A homogeneous dielectric film

（b）A stratified medium as a pile of thin homogeneous films

1.6.3 The reflection and transmission coefficients

1.6.4 A homogeneous dielectric film

1.6.5 Periodicallv 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 efiective 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 formulae

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 centred 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 ofa thin pencil

4.7 Chromatic aberration.Dispersion by a prism

4.7.1 Chromatic aberration

4.7.2 Dispersion by a prism

4.8 Radiometry and apertures

4.8.1 Basic concepts of radiometry

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 Attainmcnt of aplanatism

4.11 Image-reconstruction from projections（computerized tomography）

4.11.1 Introduction

4.11.2 Beam propagation in an absorbing medium

4.11.3 Ray integrals and projections

4.11.4 The N-dimensional Radon transform

4.11.5 Reconstruction of cross-sections and the Projection-slice theorem of computerized tomography

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

（a）Spherical aberration（B≠0）

（b）Coma（F≠0）

（c）Astigmatism（C≠0）and curvature of field（D≠0）

（d） Distortion（E≠0）

5.4 Addition theorem for the primary aberrations

5.5 The primary aberration coefficients of a general centred lens system

5.5.1 The Seidel formulae in terms of two paraxial rays

5.5.2 The Seidel formulae 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 centred 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 interferenee 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 inter-ferometer

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

（a）Fringes with monochromatic and quasi-monochromatic light

（b）Fringes of superposition

7.7 The comparison of wavelengths with the standard metre

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 totalimage 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

（a）The principle of the diffraction grating

（b）Types of grating

（c）Grating spectrographs

8.6.2 Resolving power of image-forming systems

8.6.3 Image formation in the microscope

（a）Incoherent illumination

（b）Coherent illumination-Abbe's theory

（c）Coherent illumination-Zemike's phase contrast method of observation

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

（a）Intensity in the geometrical focal plane

（b）Intensity along the axis

（c）Intensity along the boundary of the geometrical shadow

8.8.3 The integrated intensity

8.8.4 The phase behaviour

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

8.11 The Rayleigh-Sommerfeld diffraction integrals

8.11.1 The Rayleigh diffraction integrals

8.11.2 The Rayleigh-Sommerfeld diffraction integrals

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 Zemike

9.2.2 Expansion of the aberration function

9.3 Tolerance conditions for primary aberrations

9.4 The diffraction pattem 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

（a）The van Cittert-Zernike theorem

（b）Hopkins'formula

10.4.3 An example

10.4.4 Propagation of mutual intensity

10.5 Interference with broad-band light and the spectral degree of coherence.Correlation-induced spectral changes

10.6 Some applications

10.6.1 The degree of coherence in the image of an extended incoherent quasi-monochromatic source

10.6.2 The influence of the condenser on resolution in a microscope

（a） Critical illumination

（b）K？hler's illumination

10.6.3 Imaging with partially coherent quasi-monochromatic illumination

（a）Transmission of mutual intensity through an optical system

（b）Images of transilluminated objects

10.7 Some theorems relating to mutual coherence

10.7.1 Calculation of mutual coherence for light from an incoherent source

10.7.2 Propagation of mutual coherence

10.8 Rigorous theory of partial coherence

10.8.1 Wave equations for mutual coherence

10.8.2 Rigorous formulation of the propagation law for mutual coherence

10.8.3 The coherence time and the effective spectral width

10.9 Polarization properties of quasi-monochromatic light

10.9.1 The coherency matrix of a quasi-monochromatic plane wave

（a）Completely unpolarized light（natural light）

（b）Complete polarized light

10.9.2 Some equivalent representations.The degree of polarization of a light wave

10.9.3 The Stokes parameters of a quasi-monochromatic plane wave

Ⅺ Rigorous difiraction 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-dimensioual diffiaction 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 uature 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 field due to 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 Difiraction of light by ultrasonie waves

12.1 Qualitative description of the phenomenon and summary of theories based on Maxwell's difierential 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 Scattering from inhomogeneous media

13.1 Elements of the scalar theory of scattering

13.1.1 Derivation of the basic integral equation

13.1.2 The first-order Born approximation

13.1.3 Scattering from periodic potentials

13.1.4 Multiple scattering

13.2 Principles of diffraction tomography for reconstruction of the scattering potential

13.2.1 Angular spectrum representation of the scattered field

13.2.2 The basic theorem of diffraction tomography

13.3 The optical cross-section theorem

13.4 A reciprocity relation

13.5 The Rytov series

13.6 Scattering of electromagnetic waves

13.6.1 The integro-differential equations of electromagnetic scattering theory

13.6.2 The far field

13.6.3 The optical cross-section theorem for scattering o felectromagnetic waves

ⅩⅣ Optics of metals

14.1 Wave propagation in a conductor

14.2 Refraction and reflection at a metal surface

14.3 Elementary electron theory of the optical constants of metals

14.4 Wave propagation in a stratified conducting medium.Theory of metallic films

14.4.1 An absorbing film on a transparent substrate

14.4.2 A transparent film on an absorbing substrate

14.5 Diffraction by a conducting sphere;theory of Mie

14.5.1 Mathematical solution of the problem

（a）Representation of the field in terms of Debye's potentials

（b）Series expansions for the field components

（c）Summary of formulae relating to the associated Legendre func-tions and to the cylindrical functions

14.5.2 Some consequences of Mie's formulae

（a）The partial waves

（b） Limiting cases

（c）Intensity and polarization of the scattered light

14.5.3 Total scattering and extinction

（a） Some general considerations

（b）Computational results

ⅩⅤ Opries of crystals

15.1 The dielectric tensor of an anisotropic medium

15.2 The structure of a monochromatic plane wave in an anisotropic medium

15.2.1 The phase velocity and the ray velocity

15.2.2 Fresnel's formulae for the propagation of light in crystals

15.2.3 Geometrical constructions for determining the velocities of propagation and the directions of vibration

（a）The ellipsoid of wave normals

（b）The ray ellipsoid

（c）The normal surface and the ray surface

15.3 Optical properties of uniaxial and biaxial crystals

15.3.1 The optical classification of crystals

15.3.2 Light propagation in uniaxial crystals

15.3.3 Light propagation in biaxial crystals

15.3.4 Refraction in crystals

（a）Double refraction

（b）Conical refraction

15.4 Measurements in crystal optics

15.4.1 The Nicol prism

15.4.2 Compensators

（a）The quarter-wave plate

（b）Babinet's compensator

（c）Soleil's compensator

（d）Berek's compensator

15.4.3 Interference with crystal plates

15.4.4 Interference figures from uniaxial crystal plates

15.4.5 Interference figures from biaxial crystal plates

15.4.6 Location of optic axes and determination of the principal refractive indices of a crystalline medium

15.5 Stress birefringence and form birefringence

15.5.1 Stress birefringence

15.5.2 Form birefringence

15.6 Absorbing crystals

15.6.1 Light propagation in an absorbing anisotropic medium

15.6.2 Interference figures from absorbing crystal plates

（a）Uniaxial crystals

（b）Biaxial crystals

15.6.3 Dichroic polarizers

Appendices

Ⅰ The Calculus of variations

1 Euler's equations as necessary conditions for an extremurn

2 Hilbert'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 Hamiltoniananalogy 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 formula（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 R±m m

Ⅷ Proof of the inequality｜μ12(v)｜≤1 for the spectral degree of coherence（10.5）

Ⅸ Proof of a reciprocity inequality（10.8.3）

Ⅹ Evaluation of two integrals（12.2.2）

Ⅺ Energy conservation in scalar wavefields（13.3）

Ⅻ Proof of Jones'lemma（13.3）

Author index

Subject index