| Historical introduction |
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xxv | |
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I Basic properties of the electromagnetic field |
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1 | (74) |
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1.1 The electromagnetic field |
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1 | (10) |
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1.1.1 Maxwell's equations |
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1 | (1) |
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2 | (2) |
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1.1.3 Boundary conditions at a surface of discontinuity |
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4 | (3) |
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1.1.4 The energy law of the electromagnetic field |
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7 | (4) |
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1.2 The wave equation and the velocity of light |
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11 | (3) |
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14 | (10) |
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15 | (1) |
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16 | (1) |
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1.3.3 Harmonic waves. The phase velocity |
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16 | (3) |
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1.3.4 Wave packets. The group velocity |
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19 | (5) |
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24 | (14) |
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1.4.1 The general electromagnetic plane wave |
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24 | (1) |
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1.4.2 The harmonic electromagnetic plane wave |
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25 | (1) |
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(a) Elliptic polarization |
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25 | (4) |
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(b) Linear and circular polarization |
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29 | (2) |
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(c) Characterization of the state of polarization by Stokes parameters |
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31 | (2) |
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1.4.3 Harmonic vector waves of arbitrary form |
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33 | (5) |
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1.5 Reflection and refraction of a plane wave |
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38 | (16) |
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1.5.1 The laws of reflection and refraction |
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38 | (2) |
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40 | (3) |
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1.5.3 The reflectivity and transmissivity; polarization on reflection and refraction |
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43 | (6) |
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49 | (5) |
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1.6 Wave propagation in a stratified medium. Theory of dielectric films |
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54 | (21) |
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1.6.1 The basic differential equations |
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55 | (3) |
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1.6.2 The characteristic matrix of a stratified medium |
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58 | (3) |
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(a) A homogeneous dielectric film |
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61 | (1) |
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(b) A stratified medium as a pile of thin homogeneous films |
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62 | (1) |
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1.6.3 The reflection and transmission coefficients |
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63 | (1) |
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1.6.4 A homogeneous dielectric film |
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64 | (6) |
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1.6.5 Periodically stratified media |
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70 | (5) |
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II Electromagnetic potentials and polarization |
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75 | (41) |
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2.1 The electrodynamic potentials in the vacuum |
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76 | (4) |
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2.1.1 The vector and scalar potentials |
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76 | (2) |
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2.1.2 Retarded potentials |
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78 | (2) |
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2.2 Polarization and magnetization |
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80 | (9) |
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2.2.1 The potentials in terms of polarization and magnetization |
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80 | (4) |
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84 | (1) |
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2.2.3 The field of a linear electric dipole |
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85 | (4) |
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2.3 The Lorentz-Lorenz formula and elementary dispersion theory |
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89 | (14) |
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2.3.1 The dielectric and magnetic susceptibilities |
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89 | (1) |
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2.3.2 The effective field |
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90 | (2) |
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2.3.3 The mean polarizability: the Lorentz-Lorenz formula |
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92 | (3) |
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2.3.4 Elementary theory of dispersion |
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95 | (8) |
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2.4 Propagation of electromagnetic waves treated by integral equations |
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103 | (13) |
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2.4.1 The basic integral equation |
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104 | (1) |
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2.4.2 The Ewald-Oseen extinction theorem and a rigorous derivation of the Lorentz-Lorenz formula |
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105 | (5) |
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2.4.3 Refraction and reflection of a plane wave, treated with the help of the Ewald-Oseen extinction theorem |
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110 | (6) |
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III Foundations of geometrical optics |
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116 | (26) |
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3.1 Approximation for very short wavelengths |
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116 | (13) |
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3.1.1 Derivation of the eikonal equation |
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117 | (3) |
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3.1.2 The light rays and the intensity law of geometrical optics |
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120 | (5) |
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3.1.3 Propagation of the amplitude vectors |
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125 | (2) |
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3.1.4 Generalizations and the limits of validity of geometrical optics |
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127 | (2) |
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3.2 General properties of rays |
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129 | (6) |
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3.2.1 The differential equation of light rays |
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129 | (3) |
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3.2.2 The laws of refraction and reflection |
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132 | (2) |
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3.2.3 Ray congruences and their focal properties |
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134 | (1) |
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3.3 Other basic theorems of geometrical optics |
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135 | (7) |
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3.3.1 Lagrange's integral invariant |
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135 | (1) |
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3.3.2 The principle of Fermat |
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136 | (3) |
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3.3.3 The theorem of Malus and Dupin and some related theorems |
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139 | (3) |
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IV Geometrical theory of optical imaging |
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142 | (86) |
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4.1 The characteristic functions of Hamilton |
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142 | (10) |
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4.1.1 The point characteristic |
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142 | (2) |
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4.1.2 The mixed characteristic |
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144 | (2) |
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4.1.3 The angle characteristic |
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146 | (1) |
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4.1.4 Approximate form of the angle characteristic of a refracting surface of revolution |
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147 | (4) |
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4.1.5 Approximate form of the angle characteristic of a reflecting surface of revolution |
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151 | (1) |
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152 | (8) |
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153 | (4) |
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4.2.2 Maxwell's `fish-eye' |
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157 | (2) |
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4.2.3 Stigmatic imaging of surfaces |
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159 | (1) |
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4.3 Projective transformation (collineation) with axial symmetry |
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160 | (7) |
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161 | (3) |
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4.3.2 The telescopic case |
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164 | (1) |
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4.3.3 Classification of projective transformations |
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165 | (1) |
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4.3.4 Combination of projective transformations |
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166 | (1) |
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167 | (11) |
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4.4.1 Refracting surface of revolution |
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167 | (3) |
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4.4.2 Reflecting surface of revolution |
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170 | (1) |
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171 | (3) |
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174 | (1) |
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4.4.5 The general centred system |
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175 | (3) |
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4.5 Stigmatic imaging with wide-angle pencils |
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178 | (3) |
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179 | (1) |
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4.5.2 The Herschel condition |
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180 | (1) |
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4.6 Astigmatic pencils of rays |
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181 | (5) |
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4.6.1 Focal properties of a thin pencil |
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181 | (1) |
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4.6.2 Refraction of a thin pencil |
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182 | (4) |
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4.7 Chromatic aberration. Dispersion by a prism |
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186 | (7) |
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4.7.1 Chromatic aberration |
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186 | (4) |
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4.7.2 Dispersion by a prism |
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190 | (3) |
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4.8 Radiometry and apertures |
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193 | (11) |
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4.8.1 Basic concepts of radiometry |
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194 | (5) |
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199 | (2) |
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4.8.3 Brightness and illumination of images |
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201 | (3) |
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204 | (7) |
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4.9.1 Oblique meridional rays |
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204 | (3) |
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207 | (1) |
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208 | (3) |
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4.10 Design of aspheric surfaces |
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211 | (6) |
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4.10.1 Attainment of axial stigmatism |
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211 | (3) |
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4.10.2 Attainment of aplanatism |
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214 | (3) |
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4.11 Image-reconstruction from projections (computerized tomography) |
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217 | (11) |
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217 | (1) |
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4.11.2 Beam propagation in an absorbing medium |
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218 | (1) |
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4.11.3 Ray integrals and projections |
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219 | (2) |
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4.11.4 The N-dimensional Radon transform |
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221 | (2) |
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4.11.5 Reconstruction of cross-sections and the projection-slice theorem of computerized tomography |
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223 | (5) |
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V Geometrical theory of aberrations |
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228 | (33) |
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5.1 Wave and ray aberrations; the aberration function |
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229 | (4) |
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5.2 The perturbation eikonal of Schwarzschild |
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233 | (3) |
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5.3 The primary (Seidel) aberrations |
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236 | (8) |
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(a) Spherical aberration (B is not equal to 0) |
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238 | (1) |
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(b) Coma (F is not equal to 0) |
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238 | (2) |
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(c) Astigmatism (C is not equal to 0) and curvature of field (D is not equal to 0) |
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240 | (3) |
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(d) Distortion (E is not equal to 0) |
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243 | (1) |
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5.4 Addition theorem for the primary aberrations |
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244 | (2) |
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5.5 The primary aberration coefficients of a general centred lens system |
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246 | (8) |
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5.5.1 The Seidel formulae in terms of two paraxial rays |
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246 | (5) |
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5.5.2 The Seidel formulae in terms of one paraxial ray |
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251 | (2) |
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253 | (1) |
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5.6 Example: The primary aberrations of a thin lens |
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254 | (3) |
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5.7 The chromatic aberration of a general centred lens system |
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257 | (4) |
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VI Image-forming instruments |
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261 | (25) |
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261 | (2) |
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263 | (4) |
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6.3 The refracting telescope |
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267 | (7) |
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6.4 The reflecting telescope |
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274 | (5) |
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6.5 Instruments of illumination |
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279 | (2) |
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281 | (5) |
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VII Elements of the theory of interference and interferometers |
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286 | (126) |
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286 | (1) |
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7.2 Interference of two monochromatic waves |
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287 | (3) |
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7.3 Two-beam interference: division of wave-front |
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290 | (18) |
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290 | (2) |
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7.3.2 Fresnel's mirrors and similar arrangements |
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292 | (3) |
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7.3.3 Fringes with quasi-monochromatic and white light |
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295 | (1) |
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7.3.4 Use of slit sources; visibility of fringes |
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296 | (3) |
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7.3.5 Application to the measurement of optical path difference: the Rayleigh interferometer |
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299 | (3) |
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7.3.6 Application to the measurement of angular dimensions of sources: the Michelson stellar interferometer |
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302 | (6) |
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308 | (5) |
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7.5 Two-beam interference: division of amplitude |
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313 | (46) |
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7.5.1 Fringes with a plane-parallel plate |
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313 | (5) |
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7.5.2 Fringes with thin films; the Fizeau interferometer |
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318 | (7) |
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7.5.3 Localization of fringes |
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325 | (9) |
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7.5.4 The Michelson interferometer |
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334 | (2) |
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7.5.5 The Twyman-Green and related interferometers |
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336 | (5) |
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7.5.6 Fringes with two identical plates: the Jamin interferometer and interference microscopes |
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341 | (7) |
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7.5.7 The Mach-Zehnder interferometer; the Bates wave-front shearing interferometer |
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348 | (4) |
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7.5.8 The coherence length; the application of two-beam interference to the study of the fine structure of spectral lines |
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352 | (7) |
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7.6 Multiple-beam interference |
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359 | (50) |
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7.6.1 Multiple-beam fringes with a plane-parallel plate |
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360 | (6) |
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7.6.2 The Fabry-Perot interferometer |
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366 | (4) |
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7.6.3 The application of the Fabry-Perot interferometer to the study of the fine structure of spectral lines |
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370 | (7) |
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7.6.4 The application of the Fabry-Perot interferometer to the comparison of wavelengths |
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377 | (3) |
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7.6.5 The Lummer-Gehrcke interferometer |
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380 | (6) |
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7.6.6 Interference filters |
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386 | (5) |
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7.6.7 Multiple-beam fringes with thin films |
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391 | (10) |
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7.6.8 Multiple-beam fringes with two plane-parallel plates |
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401 | (1) |
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(a) Fringes with monochromatic and quasi-monochromatic light |
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401 | (4) |
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(b) Fringes of superposition |
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405 | (4) |
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7.7 The comparison of wavelengths with the standard metre |
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409 | (3) |
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VIII Elements of the theory of diffraction |
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412 | (105) |
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412 | (1) |
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8.2 The Huygens-Fresnel principle |
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413 | (4) |
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8.3 Kirchhoff's diffraction theory |
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417 | (13) |
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8.3.1 The integral theorem of Kirchhoff |
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417 | (4) |
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8.3.2 Kirchhoff's diffraction theory |
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421 | (4) |
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8.3.3 Fraunhofer and Fresnel diffraction |
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425 | (5) |
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8.4 Transition to a scalar theory |
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430 | (6) |
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8.4.1 The image field due to a monochromatic oscillator |
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431 | (3) |
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8.4.2 The total image field |
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434 | (2) |
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8.5 Fraunhofer diffraction at apertures of various forms |
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436 | (10) |
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8.5.1 The rectangular aperture and the slit |
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436 | (3) |
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8.5.2 The circular aperture |
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439 | (4) |
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8.5.3 Other forms of aperture |
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443 | (3) |
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8.6 Fraunhofer diffraction in optical instruments |
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446 | (30) |
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8.6.1 Diffraction gratings |
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446 | (1) |
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(a) The principle of the diffraction grating |
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446 | (7) |
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453 | (5) |
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(c) Grating spectrographs |
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458 | (3) |
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8.6.2 Resolving power of image-forming systems |
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461 | (4) |
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8.6.3 Image formation in the microscope |
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465 | (1) |
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(a) Incoherent illumination |
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465 | (2) |
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(b) Coherent illumination - Abbe's theory |
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467 | (5) |
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(c) Coherent illumination - Zernike's phase contrast method of observation |
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472 | (4) |
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8.7 Fresnel diffraction at a straight edge |
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476 | (8) |
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8.7.1 The diffraction integral |
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476 | (2) |
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8.7.2 Fresnel's integrals |
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478 | (3) |
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8.7.3 Fresnel diffraction at a straight edge |
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481 | (3) |
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8.8 The three-dimensional light distribution near focus |
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484 | (15) |
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8.8.1 Evaluation of the diffraction integral in terms of Lommel functions |
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484 | (5) |
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8.8.2 The distribution of intensity |
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489 | (1) |
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(a) Intensity in the geometrical focal plane |
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490 | (1) |
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(b) Intensity along the axis |
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491 | (1) |
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(c) Intensity along the boundary of the geometrical shadow |
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491 | (1) |
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8.8.3 The integrated intensity |
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492 | (2) |
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8.8.4 The phase behaviour |
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494 | (5) |
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8.9 The boundary diffraction wave |
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499 | (5) |
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8.10 Gabor's method of imaging by reconstructed wave-fronts (holography) |
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504 | (8) |
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8.10.1 Producing the positive hologram |
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504 | (2) |
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8.10.2 The reconstruction |
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506 | (6) |
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8.11 The Rayleigh-Sommerfeld diffraction integrals |
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512 | (5) |
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8.11.1 The Rayleigh diffraction integrals |
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512 | (2) |
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8.11.2 The Rayleigh-Sommerfeld diffraction integrals |
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514 | (3) |
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IX The diffraction theory of aberrations |
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517 | (37) |
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9.1 The diffraction integral in the presence of aberrations |
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518 | (5) |
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9.1.1 The diffraction integral |
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518 | (2) |
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9.1.2 The displacement theorem. Change of reference sphere |
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520 | (2) |
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9.1.3 A relation between the intensity and the average deformation of wave-fronts |
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522 | (1) |
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9.2 Expansion of the aberration function |
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523 | (4) |
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9.2.1 The circle polynomials of Zernike |
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523 | (2) |
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9.2.2 Expansion of the aberration function |
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525 | (2) |
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9.3 Tolerance conditions for primary aberrations |
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527 | (5) |
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9.4 The diffraction pattern associated with a single aberration |
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532 | (11) |
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9.4.1 Primary spherical aberration |
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536 | (2) |
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538 | (1) |
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9.4.3 Primary astigmatism |
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539 | (4) |
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9.5 Imaging of extended objects |
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543 | (11) |
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9.5.1 Coherent illumination |
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543 | (4) |
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9.5.2 Incoherent illumination |
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547 | (7) |
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X Interference and diffraction with partially coherent light |
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554 | (79) |
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554 | (3) |
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10.2 A complex representation of real polychromatic fields |
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557 | (5) |
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10.3 The correlation functions of light beams |
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562 | (7) |
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10.3.1 Interference of two partially coherent beams. The mutual coherence function and the complex degree of coherence |
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562 | (4) |
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10.3.2 Spectral representation of mutual coherence |
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566 | (3) |
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10.4 Interference and diffraction with quasi-monochromatic light |
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569 | (16) |
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10.4.1 Interference with quasi-monochromatic light. The mutual intensity |
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569 | (3) |
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10.4.2 Calculation of mutual intensity and degree of coherence for light from an extended incoherent quasi-monochromatic source |
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572 | (1) |
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(a) The van Cittert-Zernike theorem |
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572 | (5) |
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577 | (1) |
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578 | (2) |
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10.4.4 Propagation of mutual intensity |
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580 | (5) |
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10.5 Interference with broad-band light and the spectral degree of coherence. Correlation-induced spectral changes |
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585 | (5) |
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590 | (16) |
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10.6.1 The degree of coherence in the image of an extended incoherent quasi-monochromatic source |
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590 | (5) |
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10.6.2 The influence of the condenser on resolution in a microscope |
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595 | (1) |
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(a) Critical illumination |
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595 | (3) |
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(b) Kohler's illumination |
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598 | (1) |
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10.6.3 Imaging with partially coherent quasi-monochromatic illumination |
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599 | (1) |
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(a) Transmission of mutual intensity through an optical system |
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599 | (3) |
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(b) Images of transilluminated objects |
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602 | (4) |
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10.7 Some theorems relating to mutual coherence |
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606 | (4) |
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10.7.1 Calculation of mutual coherence for light from an incoherent source |
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606 | (3) |
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10.7.2 Propagation of mutual coherence |
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609 | (1) |
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10.8 Rigorous theory of partial coherence |
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610 | (9) |
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10.8.1 Wave equations for mutual coherence |
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610 | (2) |
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10.8.2 Rigorous formulation of the propagation law for mutual coherence |
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612 | (3) |
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10.8.3 The coherence time and the effective spectral width |
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615 | (4) |
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10.9 Polarization properties of quasi-monochromatic light |
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619 | (14) |
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10.9.1 The coherency matrix of a quasi-monochromatic plane wave |
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619 | (5) |
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(a) Completely unpolarized light (natural light) |
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624 | (1) |
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(b) Complete polarized light |
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624 | (2) |
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10.9.2 Some equivalent representations. The degree of polarization of a light wave |
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626 | (4) |
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10.9.3 The Stokes parameters of a quasi-monochromatic plane wave |
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630 | (3) |
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XI Rigorous diffraction theory |
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633 | (41) |
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633 | (2) |
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11.2 Boundary conditions and surface currents |
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635 | (1) |
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11.3 Diffraction by a plane screen: electromagnetic form of Babinet's principle |
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636 | (2) |
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11.4 Two-dimensional diffraction by a plane screen |
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638 | (5) |
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11.4.1 The scalar nature of two-dimensional electromagnetic fields |
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638 | (1) |
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11.4.2 An angular spectrum of plane waves |
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639 | (3) |
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11.4.3 Formulation in terms of dual integral equations |
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642 | (1) |
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11.5 Two-dimensional diffraction of a plane wave by a half-plane |
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643 | (14) |
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11.5.1 Solution of the dual integral equations for E-polarization |
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643 | (2) |
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11.5.2 Expression of the solution in terms of Fresnel integrals |
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645 | (3) |
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11.5.3 The nature of the solution |
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648 | (4) |
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11.5.4 The solution for H-polarization |
|
|
652 | (1) |
|
11.5.5 Some numerical calculations |
|
|
653 | (3) |
|
11.5.6 Comparison with approximate theory and with experimental results |
|
|
656 | (1) |
|
11.6 Three-dimensional diffraction of a plane wave by a half-plane |
|
|
657 | (2) |
|
11.7 Diffraction of a field due to a localized source by a half-plane |
|
|
659 | (8) |
|
11.7.1 A line-current parallel to the diffracting edge |
|
|
659 | (5) |
|
|
|
664 | (3) |
|
|
|
667 | (5) |
|
11.8.1 Two parallel half-planes |
|
|
667 | (2) |
|
11.8.2 An infinite stack of parallel, staggered half-planes |
|
|
669 | (1) |
|
|
|
670 | (1) |
|
|
|
671 | (1) |
|
11.9 Uniqueness of solution |
|
|
672 | (2) |
|
XII Diffraction of light by ultrasonic waves |
|
|
674 | (21) |
|
12.1 Qualitative description of the phenomenon and summary of theories based on Maxwell's differential equations |
|
|
674 | (6) |
|
12.1.1 Qualitative description of the phenomenon |
|
|
674 | (3) |
|
12.1.2 Summary of theories based on Maxwell's equations |
|
|
677 | (3) |
|
12.2 Diffraction of light by ultrasonic waves as treated by the integral equation method |
|
|
680 | (15) |
|
12.2.1 Integral equation for E-polarization |
|
|
682 | (1) |
|
12.2.2 The trial solution of the integral equation |
|
|
682 | (4) |
|
12.2.3 Expressions for the amplitudes of the light waves in the diffracted and reflected spectra |
|
|
686 | (1) |
|
12.2.4 Solution of the equations by a method of successive approximations |
|
|
686 | (3) |
|
12.2.5 Expressions for the intensities of the first and second order lines for some special cases |
|
|
689 | (2) |
|
12.2.6 Some qualitative results |
|
|
691 | (2) |
|
12.2.7 The Raman-Nath approximation |
|
|
693 | (2) |
|
XIII Scattering from inhomogeneous media |
|
|
695 | (40) |
|
13.1 Elements of the scalar theory of scattering |
|
|
695 | (15) |
|
13.1.1 Derivation of the basic integral equation |
|
|
695 | (4) |
|
13.1.2 The first-order Born approximation |
|
|
699 | (4) |
|
13.1.3 Scattering from periodic potentials |
|
|
703 | (5) |
|
13.1.4 Multiple scattering |
|
|
708 | (2) |
|
13.2 Principles of diffraction tomography for reconstruction of the scattering potential |
|
|
710 | (6) |
|
13.2.1 Angular spectrum representation of the scattered field |
|
|
711 | (2) |
|
13.2.2 The basic theorem of diffraction tomography |
|
|
713 | (3) |
|
13.3 The optical cross-section theorem |
|
|
716 | (8) |
|
13.4 A reciprocity relation |
|
|
724 | (2) |
|
|
|
726 | (3) |
|
13.6 Scattering of electromagnetic waves |
|
|
729 | (6) |
|
13.6.1 The integro-differential equations of electromagnetic scattering theory |
|
|
729 | (1) |
|
|
|
730 | (2) |
|
13.6.3 The optical cross-section theorem for scattering of electromagnetic waves |
|
|
732 | (3) |
|
|
|
735 | (55) |
|
14.1 Wave propagation in a conductor |
|
|
735 | (4) |
|
14.2 Refraction and reflection at a metal surface |
|
|
739 | (10) |
|
14.3 Elementary electron theory of the optical constants of metals |
|
|
749 | (3) |
|
14.4 Wave propagation in a stratified conducting medium. Theory of metallic films |
|
|
752 | (7) |
|
14.4.1 An absorbing film on a transparent substrate |
|
|
752 | (6) |
|
14.4.2 A transparent film on an absorbing substrate |
|
|
758 | (1) |
|
14.5 Diffraction by a conducting sphere; theory of Mie |
|
|
759 | (31) |
|
14.5.1 Mathematical solution of the problem |
|
|
760 | (1) |
|
(a) Representation of the field in terms of Debye's potentials |
|
|
760 | (5) |
|
(b) Series expansions for the field components |
|
|
765 | (7) |
|
(c) Summary of formulae relating to the associated Legendre functions and to the cylindrical functions |
|
|
772 | (2) |
|
14.5.2 Some consequences of Mie's formulae |
|
|
774 | (1) |
|
|
|
774 | (1) |
|
|
|
775 | (5) |
|
(c) Intensity and polarization of the scattered light |
|
|
780 | (4) |
|
14.5.3 Total scattering and extinction |
|
|
784 | (1) |
|
(a) Some general considerations |
|
|
784 | (1) |
|
(b) Computational results |
|
|
785 | (5) |
|
|
|
790 | (63) |
|
15.1 The dielectric tensor of an anisotropic medium |
|
|
790 | (2) |
|
15.2 The structure of a monochromatic plane wave in an anisotropic medium |
|
|
792 | (13) |
|
15.2.1 The phase velocity and the ray velocity |
|
|
792 | (3) |
|
15.2.2 Fresnel's formulae for the propagation of light in crystals |
|
|
795 | (4) |
|
15.2.3 Geometrical constructions for determining the velocities of propagation and the directions of vibration |
|
|
799 | (1) |
|
(a) The ellipsoid of wave normals |
|
|
799 | (3) |
|
|
|
802 | (1) |
|
(c) The normal surface and the ray surface |
|
|
803 | (2) |
|
15.3 Optical properties of uniaxial and biaxial crystals |
|
|
805 | (13) |
|
15.3.1 The optical classification of crystals |
|
|
805 | (1) |
|
15.3.2 Light propagation in uniaxial crystals |
|
|
806 | (2) |
|
15.3.3 Light propagation in biaxial crystals |
|
|
808 | (3) |
|
15.3.4 Refraction in crystals |
|
|
811 | (1) |
|
|
|
811 | (2) |
|
|
|
813 | (5) |
|
15.4 Measurements in crystal optics |
|
|
818 | (16) |
|
|
|
818 | (2) |
|
|
|
820 | (1) |
|
(a) The quarter-wave plate |
|
|
820 | (1) |
|
(b) Babinet's compensator |
|
|
821 | (2) |
|
|
|
823 | (1) |
|
|
|
823 | (1) |
|
15.4.3 Interference with crystal plates |
|
|
823 | (6) |
|
15.4.4 Interference figures from uniaxial crystal plates |
|
|
829 | (2) |
|
15.4.5 Interference figures from biaxial crystal plates |
|
|
831 | (2) |
|
15.4.6 Location of optic axes and determination of the principal refractive indices of a crystalline medium |
|
|
833 | (1) |
|
15.5 Stress birefringence and form birefringence |
|
|
834 | (6) |
|
15.5.1 Stress birefringence |
|
|
834 | (3) |
|
15.5.2 Form birefringence |
|
|
837 | (3) |
|
|
|
840 | (13) |
|
15.6.1 Light propagation in an absorbing anisotropic medium |
|
|
840 | (6) |
|
15.6.2 Interference figures from absorbing crystal plates |
|
|
846 | (1) |
|
|
|
847 | (1) |
|
|
|
848 | (1) |
|
15.6.3 Dichroic polarizers |
|
|
849 | (4) |
|
|
|
853 | (72) |
|
I The Calculus of variations |
|
|
853 | (20) |
|
1 Euler's equations as necessary conditions for an extremum |
|
|
853 | (2) |
|
2 Hilbert's independence integral and the Hamilton-Jacobi equation |
|
|
855 | (1) |
|
|
|
856 | (2) |
|
4 Determination of all extremals from the solution of the Hamilton-Jacobi equation |
|
|
858 | (2) |
|
5 Hamilton's canonical equations |
|
|
860 | (1) |
|
6 The special case when the independent variable does not appear explicitly in the integrand |
|
|
861 | (1) |
|
|
|
862 | (2) |
|
8 Weierstrass' and Legendre's conditions (sufficiency conditions for an extremum) |
|
|
864 | (2) |
|
9 Minimum of the variational integral when one end point is constrained to a surface |
|
|
866 | (1) |
|
10 Jacobi's criterion for a minimum |
|
|
867 | (1) |
|
|
|
868 | (2) |
|
12 Example II: Mechanics of material points |
|
|
870 | (3) |
|
II Light optics, electron optics and wave mechanics |
|
|
873 | (10) |
|
1 The Hamiltonian analogy in elementary form |
|
|
873 | (3) |
|
2 The Hamiltonian analogy in variational form |
|
|
876 | (3) |
|
3 Wave mechanics of free electrons |
|
|
879 | (2) |
|
4 The application of optical principles to electron optics |
|
|
881 | (2) |
|
III Asymptotic approximations to integrals |
|
|
883 | (9) |
|
1 The method of steepest descent |
|
|
883 | (5) |
|
2 The method of stationary phase |
|
|
888 | (2) |
|
|
|
890 | (2) |
|
IV The Dirac delta function |
|
|
892 | (6) |
|
V A mathematical lemma used in the rigorous derivation of the Lorentz-Lorenz formula (XXX2.4.2) |
|
|
898 | (3) |
|
VI Propagation of discontinuties in an electromagnetic field (XXX3.1.1) |
|
|
901 | (4) |
|
1 Relations connecting discontinuous changes in field vectors |
|
|
901 | (2) |
|
2 The field on a moving discontinuity surface |
|
|
903 | (2) |
|
VII The circle polynomials of Zernike (XXX9.2.1) |
|
|
905 | (6) |
|
1 Some general considerations |
|
|
905 | (2) |
|
2 Explicit expressions for the radial polynomials R^(+(-m))(m)(Rho) |
|
|
907 | (4) |
|
VIII Proof of the inequality |(Mu)(12)(Nu)| less than 1 for the spectral degree of coherence (XXX10.5) |
|
|
911 | (1) |
|
IX Proof of a reciprocity inequality (XXX10.8.3) |
|
|
912 | (2) |
|
X Evaluation of two integrals (XXX12.2.2) |
|
|
914 | (4) |
|
XI Energy conservation in scalar wavefields (XXX13.3) |
|
|
918 | (3) |
|
XII Proof of Jones' lemma (XXX13.3) |
|
|
921 | (4) |
| Author index |
|
925 | (11) |
| Subject index |
|
936 | |
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