ContentsII OPTICS ii6 Geometric Optics 16.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Waves in a Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . 26.2.1 Monochromatic, Plane Waves . . . . . . . . . . . . . . . . . . . . . . 26.2.2 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.3 Waves in an Inhomogeneous, Time-Varying Medium: The Eikonal Approxi-mation and Geometric Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 76.3.1 Geometric Optics for Light and Sound Waves . . . . . . . . . . . . . 76.3.2 Connection of Geometric Optics to Quantum Theory . . . . . . . . . 116.3.3 Geometric Optics for a General Wave . . . . . . . . . . . . . . . . . . 126.3.4 Examples of Propagation Laws . . . . . . . . . . . . . . . . . . . . . 146.3.5 Relation to Wave Packets; Breakdown of the Eikonal Approximationand Geometric Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3.6 Fermat’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4 Paraxial Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4.1 Axisymmetric, Paraxial Systems . . . . . . . . . . . . . . . . . . . . . 236.4.2 Converging Magnetic Lens . . . . . . . . . . . . . . . . . . . . . . . . 256.5T2 Caustics and Catastrophes—Gravitational Lenses . . . . . . . . . . . . 296.5.1T2 Formation of Multiple Images . . . . . . . . . . . . . . . . . . . 296.5.2T2 Catastrophe Optics — Formation of Caustics . . . . . . . . . . 336.6 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.6.1 Polarization Vector and its Geometric-Optics Propagation Law . . . . 386.6.2 T2 Geometric Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 39iPart IIOPTICSiiOpticsVersion 0806.2.K.pdf, 06 Nov 2 008Prior to the opening up of the electromagnetic spectrum and the development of quantummechanics, the study of optics wa s only concerned with visible light. Reflection and refractionwere first described by the Greeks and further studied by the medieval schola stics like RogerBacon, who explained the rainbow, a nd used refraction in the design of crude magnifyinglenses and spectacles. However, it was not until the seventeenth century that there arosea strong commercial interest in developing the telescope and the compound microscope.Naturally, the discovery of Snell’s law in 1621 and the observation of diffractive phenomenastimulated serious speculation about the physical nature of light. The corpuscular and wavetheories were propounded by Newton and Huygens, respectively. The corpuscular theoryinitially held sway, but the studies of interference by Young and the derivat io n of a waveequation for electromagnetic disturbances by Maxwell seemed to settle the matter in favorof the undulatory theory, only fo r the debate to be resurrected with the discovery of thephotoelectric effect. After quantum mechanics was developed in the 1920’s, the disputewas abandoned, the wave a nd particle descriptions of light became “complementary”, andHamilton’s optics-inspired formulation o f classical mechanics was modified to produce theSchr¨odinger equation.Physics students are all too familiar with this potted history and may consequently re-gard optics as an ancient precursor to modern physics that has been completely subsumedby quantum mechanics. However, this is not the case. Optics has developed dramaticallyand independently from quantum mechanics in recent decades, and is now a major branchof classical physics. It is no longer concerned primarily with light . The principles of opticsare routinely applied to all types of wave pro pagation: from all parts of the electromagneticspectrum, to quantum mechanical waves, e.g. of electrons and neutrinos, to waves in elas-tic solids (Part III of this book), fluids (Part IV), plasmas (Part V) and the geometry ofspacetime (Part VI). There is a commonality, for instance, to seismology, oceanography andradio physics that allows ideas to be freely tra nsported between these different disciplines.Even in the study of visible light, there have been major developments: the invention of thelaser has led to the modern theory of coherence and has begotten the new field of nonlinearoptics.An even greater revolution has occured in optical technology. From the credit card andwhite light hologram to the laser scanner at a supermarket checkout, from laser printers toCD’s and DVD’s, from radio telescopes capable of nanoradian angular resolution to Fabry-Perot systems that detect displacements smaller than the size of an elementary particle, weiiiivare surrounded by sophisticated optical devices in our everyday and scientific lives. Manyof these devices turn o ut to be clever and direct applications of the fundamental principlesthat we shall discuss.The treatment of optics in this text differs from that found in traditional texts in thatwe shall assume familiarity with basic classical and quantum mechanics and, consequently,fluency in the language of Fourier transforms. This inversion of the historical developmentreflects contempo r ary priorities and allows us to emphasize those aspects of the subject thatinvolve fresh concepts and modern applications.In Chapter 6, we shall discuss optical (wave-propagation) phenomena in the geometricoptics approximation. This approximation is accurate whenever the wavelength and thewave period are short compared with the lengthscales and timescales on which the waveamplitude and the waves’ environment vary. We shall show how a wave equation can besolved approximately in such a way that optical rays become the classical trajectories ofparticles, e.g. photo ns, and how, in general, ray systems develop singularities or causticswhere the geometric optics approximation breaks down and we must revert to the wavedescription.In Chapter 7 we will develop the theory of diffraction that arises when the geometricoptics approximation fails and the waves’ energy spreads in a non-particle-like way. Weshall analyze diffraction in two limiting regimes, called Fresne l and Fraunhofer, after thephysicists who discovered them, in which the …
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