ContentsII OPTICS ii7GeometricOptics 17.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Waves in a Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . 27.2.1 Monochromatic, Plane Waves . . . . . . . . . . . . . . . . . . . . . . 27.2.2 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47.3 Waves in an Inhomogeneous, Time-Varying Medium: The Eikonal Approxi-mation and Geometric Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 77.3.1 Geometric Optics for Sound Waves . . . . . . . . . . . . . . . . . . . 87.3.2 Connection of Geometric Optics to Quantum Theory . . . . . . . . . 117.3.3 Geometric Optics for a General Wave . . . . . . . . . . . . . . . . . . 157.3.4 Examples of Geometric-Optics Wave Propagation . .......... 167.3.5 Relation to Wave Packets; Breakdown of the Eikonal Approximationand Geometric Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 187.3.6 Fermat’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.4 Paraxial Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.4.1 Axisymmetric, Paraxial Systems . . . . . . . . . . . . . . . . . . . . . 247.4.2 Converging Magnetic Lens . . . . . . . . . . . . . . . . . . . . . . . . 267.5T2 Caustics and Catastrophes—Gravitational Lenses . . . . . . . . . . . . 307.5.1T2 Formation of Multiple Images . . . . . . . . . . . . . . . . . . . 307.5.2T2 Catastrophe Optics — Formation of Caustics . . . . . . . . . . 347.6 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.6.1 Polarization Vector and its Geometric-Optics Propagation Law .... 397.6.2T2 Geometric Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 40iPart IIOPTICSiiOpticsVersion 1107.1.K.pdf, 06 Nov 2011Prior to the opening up of the electromagnetic spectrum and the development of quantummechanics, the study of optics was only concerned with visible light.Reflection and refraction of light we re first described by the Greeks and further studied bymedieval scholastics like Roger Bacon, who explained the rainbow and used refraction in thedesign of crude magnifying lenses and spectacles. However, it was not until the seventeenthcentury that there arose a strong commercial interest in developing the telescope and thecompound microscope.Naturally, the discovery of Snell’s law in 1621 and the observation of diffractive phe-nomena stimulated serious speculation about the physical nature of light. The corpuscularand wave theories were propounded by Newton in YYYY and Huygens in YYYY, respec-tively. The corpuscular theory initially held sway, but the studies of interference by Young inYYYY and the derivation of a wave equation for electromagnetic disturbances by Maxwellin YYYY seemed to settle the matter in favor of the undulatory theory, only for the debateto be resurrected in YYYY with the discovery of the photoelectric effect. After quantummechanics was developed in the 1920’s, the dispute was abandoned, the wave and particledescriptions of light became “complementary”, and Hamilton’s optics-inspired formulationof classical mec hanics was modified to produce the Schr¨odinger equation.Many physics students are all too familiar withthispottedhistoryandmayconsequentlyregard 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 indep endently 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 propagation: from all parts of the electromagneticspectrum, to quantum mechanical waves, e.g. of electrons and neutrinos, to waves in elasticsolids (Part IV of this book), fluids (Part V), plasmas (Part VI) and the geometry of space-time (Part VII). There is a commonality, for instance, to seismology, oceanography andradio physics that allows ideas to be freely transported 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 coherenceandhasbegottenthenewfieldofnonlinearoptics.An even greater revolution has occured in optical technology. From the credit card andwhite ligh t hologram to the laser scanner at a supermarket chec kout, from laser printersto CD’s, DVD’s and BD’s, from radio telescopes capable of nanoradian angular resolutioniiiivto Fabry-Perot systems that detect displacements smaller than the size of an elementaryparticle, we are surrounded by sophisticated optical devices in our everyday and scientificlives. Many of these devices turn out to be clever and direct applications of the fundamentaloptical principles that we shall discuss.The treatment of optics in this text differs from that found in traditional texts in thatwe shall assume familiarity with basic classicalandquantummechanicsand,consequently,fluency in the language of Fourier transforms. This inversion of the historical developmentreflects contemporary priorities and allows us to emphasize those aspects of the subject thatinvolve fresh concepts and modern applications.In Chap. 7, 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. photons, and how, in general, raysystemsdevelopsingularitiesorcausticswhere the geometric optics approximation breaks down and we must revert to the wavedescription.In Chap. 8 we will develop the theory of diffraction that arises when the geometric opticsapproximation fails and the waves’ energy spreads in a non-particle-like way. We shall analyzediffraction in two limiting regimes, called Fresnel and Fraunhofer,afterthephysicistswhodiscovered them, in which the wavefronts are approximately planar or spherical, respectively.Insofar as we are working with a linear theory of wave propagation, we shall make heavy useof Fourier methods and shall show how elementary applications of Fourier transforms canbe used to design powerful optics
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