Principles of Optics: 60th Anniversary Edition

Contents
Foreword by Sir Peter Knight
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
			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 differential 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 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 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 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 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 Attainment 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 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
			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 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
			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 - Zernike'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 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
			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) Kohler'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 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 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 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 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 of electromagnetic 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 functions 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 Optics 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 extremum
		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 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 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
	VIII Proof of the inequality 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