Ulf Leonhardt and Thomas Philbin

Geometry and LightContents Chapter 1 Prologue 1 Chapter 2 Fermat's Principle 7 1. Letters from Pierre de Fermat 7 2. Variational calculus 15 3. Newtonian analogy 20 4. Hamilton's equations 27 5. Optical conformal mapping 33 6. Transmutation 44 7. Spherical symmetry 48 8. Tomography 58 9. From invisible spheres to perfect lenses 66 Chapter 3 Differential geometry 83 10. Coordinate transformations 83 11. The metric tensor 87 12. Vectors and bases 92 13. One-forms and general tensors 99 14. Vector products and the Levi-Civita tensor 103 15. The covariant derivative of a vector 107 16. Covariant derivatives of tensors and of the metric 115 17. Divergence, curl and Laplacian 119 18. Curvature 126 19. Geodesics 129 20. Parallel transport and covariant derivatives 133 21. Geodesic deviation and the Riemann tensor 136 22. Parallel transport around a closed loop 144 23. Conformally flat spaces 151 24. The hypersphere 155 25. Space-time geometry 161 Chapter 4 Maxwell's equations 165 26. Spatial geometrics and media 166 27. Planar media 170 28. Transformation media 173 29. Electromagnetic waves 175 30. Geometrical optics 186 31. Space-time geometries and media 199 Chapter 5 Geometries and media 205 32. Spatial transformation media 206 33. Perfect invisibility devices 210 34. Negative refraction and perfect lenses 216 35. Cloaking at a distance 221 36. Perfect imaging with positive refraction 227 37. Moving media 239 38. Optical Aharonov-Bohm effect 246 39. Analogue of the event horizon 251 Appendix 261 Bibliography 265

The science of invisibility combines two of physics' greatest concepts: Einstein's general relativity and Maxwell's principles of electromagnetism. Recent years have witnessed major breakthroughs in the area, and the authors of this volume — Ulf Leonhardt and Thomas Philbin of Scotland's University of St. Andrews — have been active in the transformation of invisibility from fiction into science. Their work on designing invisibility devices is based on modern metamaterials, inspired by Fermat's principle, analogies between mechanics and optics, and the geometry of curved space.
Suitable for graduate students and advanced undergraduates of engineering, physics, or mathematics, and scientific researchers of all types, this is the first authoritative textbook on invisibility and the science behind it. The book is two books in one: it introduces the mathematical foundations — differential geometry — for physicists and engineers, and it shows how concepts from general relativity become practically useful in electrical and optical engineering, not only for invisibility but also for perfect imaging and other fascinating topics. More than one hundred full-color illustrations and exercises with solutions complement the text.