| Optical phase space, Hamiltonian systems and Lie algebras | |
| Introduction | p. 3 |
| Two fundamental postulates | p. 5 |
| Geometric postulate | p. 5 |
| Dynamic postulate | p. 6 |
| Conservation laws at discontinuities | p. 7 |
| Descartes' sphere and the Ibn Sahl law of refraction | p. 8 |
| Geometric optics and classical mechanics | p. 11 |
| Optical phase space | p. 15 |
| Ray coordinates and their manifold | p. 15 |
| Hamilton equations on the screen | p. 18 |
| Guides and their index profile | p. 21 |
| Paraxial optics and mechanics | p. 23 |
| Canonical transformations | p. 25 |
| Beams and the conservation of light | p. 25 |
| Conservation of the Hamiltonian structure | p. 28 |
| Hamiltonian evolution with Poisson brackets | p. 33 |
| One-parameter Lie groups | p. 34 |
| Hamiltonian flow of phase space | p. 37 |
| Some aberrations and their Fourier conjugates | p. 38 |
| Spherical aberration and pocus | p. 39 |
| Distorsion and coma | p. 40 |
| Multiparameter Lie algebras and groups | p. 42 |
| The roots of refraction and reflection | p. 47 |
| Refraction equations in screen coordinates | p. 48 |
| Factorization of refraction | p. 49 |
| Canonicity of the root transformation | p. 51 |
| Aberration series expansion for the root transformation | p. 53 |
| Refraction between inhomogeneous media | p. 54 |
| Factorization of reflection | p. 55 |
| Some historical comments | p. 57 |
| Antiquity | p. 57 |
| Age of Reason | p. 58 |
| Fermat's principle and the Lagrangian | p. 59 |
| Geometric optics in the nineteenth century | p. 60 |
| Hamiltonian formulations | p. 61 |
| The evolution of Sophus Lie | p. 61 |
| Symmetries and dynamics in the twentieth century | p. 63 |
| Symmetry and dynamics of optical systems | |
| Introduction | p. 67 |
| Euclidean and Lorentzian maps | p. 69 |
| Presentations and realizations of symmetry | p. 69 |
| Translations in 3-space | p. 72 |
| Rotations of 3-space | p. 74 |
| Rotations of the screen | p. 77 |
| Euclidean and semidirect product groups | p. 79 |
| Lorentz boost of light-like vectors | p. 81 |
| Relativistic aberration of images | p. 84 |
| The Lorentz Lie algebra and group | p. 87 |
| Other global optical transformations | p. 88 |
| Conformai optics - Maxwell fish-eyes | p. 91 |
| On the eyes of fish and point rotors | p. 91 |
| Phase space and rotations | p. 94 |
| Restriction to conics | p. 96 |
| Stereographic map of phase space | p. 97 |
| Hidden symmetry and the Hamiltonian | p. 103 |
| Dynamical Lie algebra of the fish-eye | p. 106 |
| Conformal Lie algebra | p. 108 |
| The Kepler system and its hidden rotor | p. 110 |
| Axial symmetry reduction | p. 113 |
| Symmetry-adapted coordinates of phase space | p. 113 |
| Hamiltonian knife cuts hyperbolic onion | p. 116 |
| Stability of trajectories and critical rays | p. 117 |
| The reduced phase space of axis-symmetric systems | p. 120 |
| Hamiltonian structure on reduced phase space | p. 122 |
| Reconstruction of the ignored coordinate | p. 123 |
| Anisotropic optical media | p. 127 |
| Direction and momentum of rays | p. 127 |
| Hamilton equations for anisotropic media | p. 128 |
| Angular dependences of the refractive index | p. 129 |
| Comparison with Maxwellian anisotropy | p. 131 |
| Euclidean optical models | p. 135 |
| Manifolds of rays, planes and frames | p. 135 |
| Coset spaces for geometric and wave models | p. 138 |
| Conservation of volume and structure | p. 141 |
| Signal and Helmholtz models | p. 144 |
| Hilbert space of Helmholtz wavefields | p. 145 |
| Euclidean algebra in Helmholtz optics | p. 148 |
| The recipe for wavization | p. 149 |
| The paraxial régime | |
| Introduction | p. 155 |
| Optical elements of the symplectic group | p. 157 |
| Free spaces, thin lenses, and action on phase space | p. 157 |
| Linear canonical maps and symplectic matrices | p. 162 |
| Orthogonal and unitary matrices | p. 165 |
| Bargmann parameters and group covers | p. 168 |
| The Iwasawa and other decompositions | p. 173 |
| Construction of optical systems | p. 179 |
| Plane optical systems | p. 179 |
| Astigmatic lenses and magnifiers | p. 184 |
| Lenses | p. 184 |
| Magnifiers | p. 185 |
| Reflectors and rotators | p. 187 |
| U(2) fractional Fourier transformers | p. 188 |
| Central Fourier transforms | p. 189 |
| Separable Fourier transforms | p. 189 |
| SU(2)-Fourier transforms | p. 190 |
| Systems cum reflection | p. 191 |
| Minimal lens arrangements | p. 196 |
| The abc-parameters | p. 196 |
| One-lens DLD configurations | p. 197 |
| Two-lens configurations | p. 199 |
| Three-lens configurations | p. 202 |
| Classical Lie algebras | p. 205 |
| Lie algebras of the linear groups | p. 205 |
| The classical Cartan algebras | p. 211 |
| The Weyl trick for symplectic algebras | p. 215 |
| Phase space functions, operators and matrices | p. 221 |
| Roots and multiplets of the symplectic algebras | p. 224 |
| Roots and multiplets of the unitary algebras | p. 232 |
| Roots of the orthogonal algebras | p. 237 |
| Hamiltonian orbits | p. 249 |
| Orbits in sp(2, R) for plane systems | p. 249 |
| Trajectories in Sp(2, R) | p. 252 |
| sp(4, R) Hamiltonians and eigenvalues | p. 254 |
| Hamiltonians in the so(3, 2) basis | p. 258 |
| Equivalence under Fourier transformers and magnifiers | p. 262 |
| Separable, Lorentzian and Euclidean Hamiltonians | p. 264 |
| Evolution along sp(4, R)-guides | p. 267 |
| Inhomogeneous Hamiltonians | p. 270 |
| Canonical Fourier optics | p. 273 |
| The Royal Road to Fourier optics | p. 273 |
| Linear canonical transforms | p. 277 |
| Hyperdifferential forms | p. 282 |
| D-dim and radial canonical transforms | p. 286 |
| Hamilton-Lie aberrations | |
| Introduction | p. 291 |
| Polynomials and aberrations in one dimension | p. 293 |
| Monomials in multiplets | p. 293 |
| Rank-K aberration algebras | p. 296 |
| Rank-K aberration groups | p. 298 |
| Aberrations of phase space | p. 302 |
| Axis-symmetric aberrations | p. 309 |
| Axis-symmetric aberrations in the Cartesian basis | p. 309 |
| The harmonic basis of aberrations | p. 315 |
| Harmonic aberration family features | p. 321 |
| Concatenation of aberrating systems | p. 329 |
| Aberration coefficients for optical elements | p. 333 |
| Parametric correction of fractional Fourier transformers | p. 341 |
| The lens arrangement | p. 342 |
| The warped-face guide arrangement | p. 347 |
| The cat's eye arrangement | p. 349 |
| Afterword | p. 353 |
| References | p. 355 |
| Index | p. 365 |
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