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### Principles of Optics

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Principles of Optics

Max Born & Emil Wolf

855 pages

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Preface

The idea of writing this book was a result of frequent enquiries about the possibility of publishing in the English language a book on optics written by one of us. more than twenty-five years ago. A preliminary survey of the literature showed that numerous researches on almost every aspect of optics have been carried out in the intervening years, so that the book no longer gives a comprehensive and balanced picture of the field. In consequence it was felt that a translation was hardly appropriate; instead a substantially new book was prepared, which we are now placing before the reader. In planning this book it soon became apparent that even if only the most important developments which took place since the publication of Optic were incorporated, the book would become impracticably large. It was, therefore, deemed necessary to restrict its scope to a narrower field. Optic itself did not treat the whole of optics. The optics of moving media, optics of X-rays and y rays, the theory of spectra and the Ml connection between optics and atomic physics were not discussed ; nor did the old book consider the effects of light on our visual sense organ—the eye. These subjects can be treated more appropriately in connection with other fields such as relativity, quantum mechanics, atomic and nuclear physics, and physiology. In this book not only are these subjects excluded, but also the classical molecular optics which was the subject-matter of almost half of the German book. Thus our discussion is restricted to those optical phenomena which may be treated in terms of Maxwell’s phenomenological theory. This includes all situations in which the atomistic structure of matter plays no decisive part. The connection with atomic physics, quantum mechanics, and physiology is indicated only by short references wherever necessary. The fact that, even after this limitation, file book is much larger than Optic, gives some indication about the extent of the researches that have been carried out in classical optics in recent times.

We have aimed at giving, within the framework just outlined, a reasonably complete picture of our present knowledge. 'We have attempted to present the theory in such a way that practically all the results can be traced back to the basic equations of Maxwell’s electromagnetic theory, from which our whole consideration starts. In Chapter I the main properties of the electromagnetic field are discussed and the effect of matter on the propagation of the electromagnetic disturbance is described formally, in terms of the usual material constants. A more physical approach to the question of influence of matter is developed in Chapter II; it is shown that in the presence of an external incident field, each volume element of a material medium may be assumed to give rise to a secondary (scattered) wavelet and that the combination of these wavelets leads to the observable, macroscopic field. This approach is of considerable physical significance and its power is illustrated in a later chapter (Chapter XII) in connection with the diffraction of light by ultrasonic waves, first treated in this way by A. B. Bhatia and W. J. Noble; Chapter XII was contributed by Prof. Bhatia himself.

A considerable part of Chapter III is devoted to showing how geometrical optics follows from Maxwell's wave theory as a limiting case of short wavelengths. In addition to discussing the main properties of rays and wave-fronts, the vectorial aspects of the problem (propagation of the directions of the field vectors) are also considered. A detailed discussion of the foundations of geometrical optics seemed to us desirable in view of the important developments made in recent years in the related field of microwave optics (optics of short radio waves). These developments were often stimulated by the close analogy between the two fields and have provided new experimental techniques for testing the predictions of the theory. We found it convenient to separate the mathematical apparatus of geometrical optics—the calculus of variations—from the main text ; an appendix on this subject {Appendix I) is based in the main part on unpublished lectures given by D. Hilbert at Gottingen University in the early years of this century. The following appendix (Appendix XI), contributed by Prof. X). Gabor, shows the close formal analogy that exists between geometrical optics, classical mechanics, and electron optics, when these subjects are presented in the language of the calculus of variations.

We make no apology for basing our treatment of geometrical theory of Imaging (Chapter IV) on Hamilton's classical methods of characteristic functions. Though these methods have found little favor in connection with the design of optical instruments, they represent nevertheless an essential tool for presenting in a unified manner the many diverse aspects of the subject. It is, of course, possible to derive some of the results more simply from assumptions ; but, however valuable such an approach may be for the solution of individual problems, it cannot have more than illustrative value in a book concerned with a systematic development of a theory from a few simple postulates.

The defect of optical images (the influence of aberrations) may be studied either by geometrical optics (appropriate when the aberrations are large), or by diffraction theory (when they are sufficiently small). Since one usually proceeds from quite different starting points in the two methods of treatments, a comparison of results has in the past not always been easy. We have attempted to develop a more unified treatment, based on the concept of the deformation of wave- fronts. In the geometrical analysis of aberrations (Chapter V) we have found it possible and advantageous to follow, after a slight modification of his eikonal, the old method of K. Schwarzschild. The chapter on diffraction theory of aberrations (Chapter IX) gives an account of the Nijboer-Zrrnike theory and also includes an introductory section on the imaging of extended objects, in coherent and in incoherent illumination, based on the techniques of Fourier transforms.

TOC

I BASIC PROPERTIES OF THE ELECTROMAGNETIC FIELD I

II. The Electromagnetic Field I

III Maxwell equations 1

1.1.2. Material equations 2

1.1.3. Boundary conditions at a surface of discontinuity 4

1.1.4, The energy law of the electromagnetic field 7

1.2. The Wave Equation and the Velocity of Light 10

1.. Scalar Waves 14

1.3.1 Plane waves 14

1.3.2. Spherical waves 15

1.3.3. Harmonic waves. The phase velocity 16

1.3.4. Wave packets. The group velocity 18

1.4. Vector Waves 23

1.4.1, The general electromagnetic plane wave 23

1.4.2, The harmonic electromagnetic plane wave 24

(a) Elliptic polarization 25

(b) Linear and circular polarization 28

(c) Characterization of the state of polarization by Stokes parameters 30

1.4.3, Harmonic vector waves of arbitrary form 32

1.5. Reflection and Refraction of a Plane Wave 36

1.5.1. The laws of reflection and refraction 36

1.5.2, Fresnel formulae 38

1.5.3, The reflectivity and transmissivity; polarization on reflection and refraction 41

1.5.4. Total reflection 47

1.6. Wave Propagation in a Stratified Medium. Theory of Dielectric Films 51

1.6.1. The basic differential equations 52

1.6.2. The characteristic matrix of a stratified medium 55

(a) A homogeneous dielectric film 57

(b) A stratified medium as a pile of thin homogeneous films 58

1.6.3. The reflection and transmission coefficients 59

1.6.4. A homogeneous dielectric film 61

1.6.5. Periodically stratified media 66

II, ELECTROMAGNETIC POTENTIALS AND POLARIZATION 71 2.1.

The Electrodynamic Potentials in the Vacuum 72

2.1.1, The vector and scalar potentials 72

2.1.2, Retarded potentials 74

2.2. Polarization and Magnetization 76

2.2.1. The potentials in terms of polarization and magnetization 76

2.2.2. Hertz vectors 79

2.2.3. The field of a linear electric dipole 81

2.3. The Lorentz-Lorenz Formula and Elementary Dispersion Theory 84

2.3.1. The dielectric and magnetic susceptibilities S4

2.3.2. The effective field 85

2.3.3. The mean polarizability: the Lorentz-Lorenz formula 87

2,3.4. Elementary theory of dispersion 90

2,4. Propagation of Electromagnetic Waves Treated by Integral Equations 98

2,4.1 . The basic integral equation 99

2.4.2. The Ewald-Qseen extinction theorem and a rigorous derivation of the Lorentz-Lorenz formula 100

2.4.3. Refraction and reflection of a plane wave, treated with the help of the Ewald-Oseen extinction theorem 104

III. FOUNDATIONS OF GEOMETRICAL OPTICS 109

3.1. Approximation for Very Short Wavelengths 109

3.1.1. Derivation of the eikonal equation 110

3.1.2. The light rays and the intensity law of geometrical optics 113

3.1.3. Propagation of the amplitude vectors 117

3.1.4. Generalizations and the limits of validity of geometrical optics 119

3.2. General Properties of Rays 121

3.2.1 The differential equation of light rays 121

3.2.2. The laws of refraction and reflection 124

3.2.3. Ray congruences and their focal properties 126

3.8. Other Basic Theorems of Geometrical Optics 127

3.3.1. Lagrange’s integral invariant 127

3.3.2, The principle of Fermat 12S

3.3.3. The theorem of Maius and Dupin and some related theorems 130

IV. GEOMETRICAL THEORY OF OPTICAL IMAGING 133

4.1. The Characteristic Functions of Hamilton 133

4.1.1. Tiie point characteristic 133

4.1.2. The mixed characteristic 135

4.1.3. The angle characteristic 137

4.1.4. Approximate form of the angle characteristic of a refracting surface of revolution 138

4.1.5. Approximate form of the angle characteristic of a reflecting surface of revolution 141

4.2. Perfect Imaging 143

4.2.1. General theorems 143

4,2.2, Maxwell's “fish-eye” 147

4,2.3. Stigmatie imaging of surfaces 149

4.3. Projective Transformation (Collineation) with Axial Symmetry 150

4.3.1. General formulae 151

4.3.2. The telescopic case 1 54

4.3.3. Classification of projective transformations 154

4.3.4. Combination of projective transformations 155

4.4. Gaussian Optics 157

4.4.1. Refracting surface of revolution 157

4.4.2. Reflecting surface of revolution 160

4.4.3. The thick lens 161

4.4.4. The thin lens 163

4.4.5. The general centred S3rstem 164

4.5. Stigmatic Imaging with Wide-angle Pencils 166

4.5.1. The sine condition 167

4.5.2. The Hcrschel condition 169

4.6. Astigmatic Pencils of Rays 169

4.6.1. Focal properties of a thin pencil 169

4.6.2. Refraction of a thin pencil 171

4.7. Chromatic Aberration. Dispersion by a Prism 174

4.7.1. Chromatic aberration 174

4.7.2. Dispersion by a prism 177

4.8. Photometry and Apertures 181

4.8.1. Basic concepts of photometry 181

4.8.2. Stops and pupils 186

4.8.3. Brightness and illumination of images 188

4.9. Ray Tracing 190

4.9.1. Oblique meridional rays 191

4.9.2. Paraxial rays 193

4.9.3. Skew rays 194

4.10. Design of Aspheric Surfaces 197

4.10.1, Attainment of axial stigmatism 197

4.10.2, Attainment of aplanatism 200

V. GEOMETRICAL THEORY OP ABERRATIONS 203

5.1. Wave and Ray Aberrations; the Aberration Function 203

5.2. The Perturbation Eikonal of Schvvarzschild 207

5.3. The Primary (Seidel) Aberrations 211

5.4. Addition Theorem for the Primary Aberrations 218

5.5. The Primary Aberration Coefficients of a General Centred Lens System 220

5.5.1. The Seidel formulae in terms of two paraxial rays 220

5.5.2. The Seidel formulae in terms of one paraxial ray 224

5.5.3. Petzvafs theorem 225

5.6. Example: The Primary Aberrations of a Thin Lens 226

5.7. The Chromatic Aberration of a General Centred Lens System 230

VL IMAGE-FGRMING INSTRUMENTS 233

6.1, The Eye 233

6.2, The Camera 235

6.3, The Refracting Telescope 239

6.4, The Reflecting Telescope 245

6.5. Instruments of Illumination 250

6.6. The Microscope

VII. ELEMENTS OP THE THEORY OF INTERFERENCE AND

INTERFEROMETERS 256

7.1 Introduction 257

7.2. Interference of Two Monochromatic Waves 257

7.3. Two-beam Interference: Division of Wave-front 260

7.3.1. Young’s experiment 260

7.3.2. FresneTs mirrors and similar arrangements 261

7.3.3. Fringes with quasi-monochromatic and white light 264

7.3.4. Use of slit sources; visibility of fringes 265

7.3.5. Application to the measurement of optical path difference: the Rayleigh interferometer 268

7 .3 .6. Application to the measurement ofangular dimensions ofsources the Michelson stellar interferometer 271

7.4. Standing Waves 277

7.5. Two-beam Interference : Division of Amplitude 281

7.5.1, Fringes with a plane parallel plate 281

7.5.2, Fringes with thin films; the Fizeau inteiTerbmeter 286

7.5.3, Localization of fringes 291

7.5.4, The Michelson interferometer 300

7.5.5, The Twyman-Green and related interferometers 302

7.5.6, Fringes with two identical plates: the Jamin interferometer and interference microscopes 306

7.5.7, The Mach-Zehndcr interferometer; the Bates wave-front shearing interferometer 312

7.5.8, The coherence length; the application of two-beam interference to the study of the fine structure of spectral lines 316

7.6. Multiple-beam Interference 323

7.6.1. Multiple-beam fringes with a plane parallel plate 323

7.6.2. The Fabry-Perot interferometer 329

7.6.3. The application of the Fabry-Perot interferometer to the study of the fine structure of spectral lines 333

7.0.4. The application of the Fabry-Perot interferometer to the comparison of wavelengths 338

7.6.5. The Lummer-Gehrcke interferometer 341

7.6.6. Interference filters 347

7.6.7. Multiple-beam fringes with thin films 351

7.6.8. Multiple-beam fringes with two plane parallel plates 360

(a) Fringes with monochromatic and quasi -monochromatic light 360

(b) Fringes of superposition 364

7.7. The Comparison of Wavelengths with the Standard Metre 367

VIH. ELEMENTS OF THE THEORY OF DIFFRACTION 370

8.1. Introduction 370

8.2. The Huygens-Fresnel Principle 370

8.3. Kirchhoff's Diffraction Theory 375

8.3.1 . The integral theorem of Kirchhoff 375

8.3.2. Kirchhoff’s diffraction theory 378

8.3.3. Fraunhofer and Fresnel diffraction 382

8.4. Transition to a Scalar Theory 387

8.4. The image field due to a monochromatic oscillator 387

8.4.2. The total image field 390

8.5. Fraunhofer Diffraction at Apertures of Various Forms 392

8.5.1. The rectangular aperture and the slit 393

8.5.2. The circular aperture 395

8.5.3. Other forms of aperture 398

8.6. Fraunhofer Diffraction in Optical Instruments 401

8.6.1. Diffraction gratings 401

(a) The principle of the diffraction grating 401

(b) Types of grating 407

(c) Grating spectrographs 412

8.6.2. Resolving power of image-forming systems 414

8.6.3. Image formation in the microscope 418

(a) Incoherent illumination 418

(b) Coherent illumination—Abbe's theory 419

(e) Coherent illumination-—Zernike's phase contrast method of observation 424

8.7. Fresnel Diffraction at a Straight Edge 428

8.7.. The diffraction integral 428

8.7.2. Fresnel's integrals 430

8.7.3. Fresnel diffraction at a straight edge 433

8.8. The Three-dimensional Light Distribution near Focus 435

8.8.1. Evaluation of the diffraction integral in terms of Lommel functions 435

8.8.2. The distribution of intensity 439

(a) Intensity in the geometrical focal plane 441

(b) Intensity along the axis 441

(c) Intensity along the boundary of the geometrical shadow 441

8.8.3. The integrated intensity 442

8.8.4. The phase behaviour 445

8.9. The Boundary Diffraction Wave 449

8.10. Gabor's Method of Imaging by Reconstructed Wave-fronts (Holography) 453

8.10.L Producing the positive hologram 453

8.10,2. The reconstruction 455

IX. THE DIFFRACTION THEORY OF ABERRATIONS 459

9.1. The Diffraction Integral in the Presence of Aberrations 460

9.1.1. The diffraction integral 462

9.1.2. The displacement theorem. Change of reference sphere 462

9.1.3. A relation between the intensity and the average deformation of wave-fronts 463

9.2. Expansion of the Aberration Function 464

9.2.1. The circle polynomials of Zernike 464

9.2.2. Expansion of the aberration function 466

9.3. Tolerance Conditions for Primary Aberrations 468

9.4. The Diffraction Pattern Associated with a Single Aberration 473

9.4.1 . Primary spherical aberration 475

9.4.2. Primary coma 477

9.4.3. Primary astigmatism 479

9.5. Imaging of Extended Objects 480

9.5.L Coherent illumination 481

9.5.2, Incoherent illumination 484

X, INTERFERENCE AND DIFFRACTION WITH PARTIALLY COHERENT LIGHT 49i

10.2. A Complex Representation of Real Polychromatic Fields 494

10.3. The OorrelatlonFunetions of Light Beams 499

10.3.1, Interference of two partially coherent beams. The mutual coherence function and the complex degree of coherence 499

10.3.2. Spectral representation of mutual coherence 503

10.4. Interference and Diffraction with Quasi-monochromatic Light 505

10.4.1. Interference with quasi -monochromatic light. The mutual intensity 505

10.4.2, Calculation of mutual intensity and degree of coherence for light from an extended incoherent quasi-monochromatic source 508

(a) Tiie Van Cittert-Zernike theorem 508

(b) Hopkins' formula 512

10.4.3, An example 513

10.4.4. Propagation of mutual intensity 516

10.5. Some Applications 51

10. 5. The degree of coherence in the image of an extended incoherent

quasi-monochromatic source 51S

10.5.2. The influence of the condenser on resolution in a microscope 522

(a) Critical illumination 522

(b) Kohler's illumination 524

10.5.3. Imaging with partially coherent quasi-monochromatic illumination 525

(a) Transmission of mutual intensity through an optical system 526

(b) Images of transiHumiliated objects 52S

10.6. Some Theorems Relating to Mutual Coherence 532

10.6.1. Calculation of mutual coherence for light from an incoherent source 532

10.6.2. Propagation of mutual coherence 534

10,7. Rigorous Theory of Partial Coherence 535

10.7,L Wave equations for mutual coherence 535

10.7,2, Rigorous formulation of the propagation law for mutual coherence 537

10,7.3. The coherence time and the effective spectral width 540

10.8. Polarization Properties of Quasi-monochromatic Light 544

10.8.1. The coherency matrix of a quasi-monochromatic plane wave 544

(a) Completely unpolarized light (Natural light) 548

(b) Completely polarized light 549

10.8.2. Some equivalent representations. The degree of polarization of a light wave 550

1 0.8.3. The Stokes parameters of a quasi-monochromatic plane wave 554

XI. RIGOROUS DIFFRACTION THEORY 556

11.1. Introduction 556

11.2. Boundary Conditions and Surface Currents 557

11.3. Diffraction by a Plane Screen: Electromagnetic Form of Rabinet's Principle 559

11.4. Two-dimensional Diffraction by a Plano Screen 560

11.4.1. The scalar nature of two-dimensional electromagnetic fields 560

11.4.2. An angular spectrum of plane waves 561

11.4.3. Formulation in terms of dual integral equations 564

11.5, Two-dimensional Diffraction of a Plane Wave by a Half-plane 565

11.5.1. Solution of the dual integral equations for E-polarization 565

11.5.2. Expression of the solution in terms of Fresnel integrals 567

11.5.3. The nature of the solution 570

11.5.5. Some numerical calculations 575

11 .5,6. Comparison with approximate theory and with experimental results 577

11.6. Three-dimensional Diffraction of a Plane Wave by a Half-plane 578

11.7, Diffraction of a Localized Source by a Half-plane 580

11.7.1. A line-current parallel to the diffracting edge 580

11.8. Other Problems 587

11.3.1. Two parallel half-planes 587

11.8.2. An infinite stack of parallel, staggered half-planes 589

11.8.3. A strip 589

11.8.4. Further problems 591

11,9. Uniqueness of Solution 591

XII. DIFFRACTION OF LIGHT BY ULTRASONIC WAVES 593

12.1. Qualitative Description of the Phenomenon and Summary of Theories Based on Maxwell’s Differential Equations 593

12.1.1. Qualitative description of the phenomenon 593

12.1.2. Summary of theories based on Maxwell's equations 596

12.2. Diffraction of Light by Ultrasonic Waves as Treated by the Integral Equation Method 599

12.2.1. Integral equation for #- polarization 600

12.2.2. The trial solution of the integral equation 601

12.2.3. Expressions for the amplitudes of the light waves in the diffracted and reflected spectra 603

12.2.4. Solution of the equations by a method of successive approximations 604

12.2.5. Expressions for the intensities of the first and second order lines for some special cases 607

12.2.6. Some qualitative results 608

12.2.7. The Raman-Nath approximation 609

XIII. OPTICS OF METALS 61,1

13.1. Wave Propagation in a Conductor 611

13.2. Refraction and Reflection at a Metal Surface 615

13.3. Elementary Electron Theory of the Optical Constants of Metals 624

13.4. Wave Propagation in a Stratified Conducting Medium. Theory of Metallic Films 627

18.4.1. An absorbing film on a transparent substrate 627

13.4.2. A transparent him on an absorbing substrate, 632

13.5. Diffraction by a Conducting Sphere ; Theory of Mie 633

13.5.1, Mathematical solution of the problem 634

(a) Representation of the field in terms of Debye’s potentials 634

(b) Series expansions for the field components 639

(c) Summary of formulae relating to the associated Legendre functions and to the cylindrical functions 645

13.5.2. Some consequences of Mia’s formulae 647

(a) The partial waves 647

(b) Limiting cases 649

(c) Intensity and polarization of the scattered light 652

13.5.3. Total scattering and extinction 656

(a) Some general considerations 656

(b) Computational results 661

XIV. OPTICS OF CRYSTALS 665

14. L The Dielectric Tensor of an Anisotropic Medium 665

14.2. The Structure of a Monochromatic Plane Wave in an Anisotropic Medium 667

L4.2.L The phase velocity and the ray velocitjr 667

14.2.2. Fresnel’s formulae for the propagation of light in crystals 670

14.2.3. Geometrical constructions for determining the velocities of propagation and the directions of vibration 673

(a) The ellipsoid of wave normals 673

(b) The ray ellipsoid 676

(c) The normal surface and the ray surface 676

14.3. Optical Properties of Uniaxial and Biaxial Crystals 678

14.3.1. The optical classification of crystals 678

14.3.2. Light propagation in uniaxial crystals 679

14.3.3. Light propagation in biaxial crystals 681

14.3.4. Refraction in crystals 684

(a) Double refraction 684

(b) Conical refraction 686

14.4. Measurements in Crystal Optics 690

14.4.1. The Ntcol prism 690

14.4.2. Compensators 691

(a) The quarter-wave plate 691

(b) Babinet’s compensator 692

(c) Soleil’s compensator 694

(d) Berek’s compensator 694

14.4.3. Interference with crystal plates 694

14.4.4. Interference figures from uniaxial crystal plates 698

14.4.5. Interference figures from biaxial crystal plates 701

14.4.6. Location of optic axes and determination of the principal refractive indices of a crystalline medium 702

14.5. Stress Birefringence and Form Birefringence 703

14.5.1. vStress birefringence ?03

14.5.2, Form birefringence 705

14.6. Absorbing Crystals 708

14.6.1. Light propagation in an absorbing anisotropic medium 708

14.6.2. Interference figures from absorbing crystal plates 713

(a) Uniaxial crystals 714

(b) Biaxial crystals 715

14.6.3. Dichroic polarizers 716

APPENDICES 719

I, The Calculus of Variations 719

1, Euler’s equations as necessary conditions for an extremum 719

2, Hilbert.s independence integral and the Hamilton-Jacobi equation 720

3, The field of extremals 722

4, Determination of all extremals from the solution of the Hamilton- Jacobi equation 724

5, Hamilton’s canonical equations 725

6, The special case when the independent variable does not appear explicitly in the integrand 726

7, Discontinuities 7^7

S. Weierstrass. and Legendre’s conditions sufficiency conditions for an extremum) 7^9

9. Minimum of the variational integral when one end point is constrained to a surface 7^1

10. Jacobi’s criterion for a minimum 732

11. Example I: Optics 7^2

12. Example II: Mechanics of material points 734

II. Light Optics, Electron Optics and Wave Mechanics 733

L The Hamiltonian analogy in elementary form 738

2. The Hamiltonian analogy in variational form 740

3. Wave mechanics of free electrons 743

4. The application of optical principles to electron optics 745

III. Asymptotic Approximations to Integrals 747

1. The method of steepest descent 747

2. The method of stationary phase 752

3. Double integrals 753

IV. The Dirac Della Function 755

V. A Mathematical Lemma used in the Rigorous Derivation of the Lorentz-

Lormz Law (§2.4.2) 700

VI, Propagation of Discontinuities in an Electromagnetic Field (§3.1 .1) 763

L Relations connecting discontinuous changes in field vectors 763

2, The field on a moving discontinuity surface 765

VII. The Circle Polynomials of Zernike (§9,2,1) 767

2. Explicit expressions for the radial polynomials R£m (p) 769

VIII. Proof of an Inequality (§ 10.7,3} 773

IX. Evaluation of Two Integrals (§ 12,2.2) 775