xvi Preface flashlight beam in a dusty room gives the impression that light travels in straight lines. At mirrors the lines reflect with the usual law of equal angles of incidence and reflection. Passing from air to water the lines are bent. These phenomena are described by the three fundamental principles of a physical theory called geometric optics. They are rectilinear propagation and the laws of reflection and refraction. All three phenomena are explained by Fermat’s principle of least time. The rays are locally paths of least time. Refraction at an interface is ex- plained by positing that light travels at different speeds in the two media. This description is purely geometrical involving only broken rays and times of transit. The appearance of a minimum principle had important philosoph- ical impact, since it was consistent with a world view holding that nature acts in a best possible way. Fermat’s principle was enunciated twenty years before R¨ omer demonstrated the finiteness of the speed of light based on observations of the moons of Jupiter. Today light is understood as an electromagnetic phenomenon. It is de- scribed by the time evolution of electromagnetic fields, which are solutions of a system of partial differential equations. When quantum effects are im- portant, this theory must be quantized. A mathematically solid foundation for the quantization of the electromagnetic field in 1 + 3 dimensional space time has not yet been found. The reason that a field theory involving partial differential equations can be replaced by a geometric theory involving rays is that visible light has very short wavelength compared to the size of human sensory organs and com- mon physical objects. Thus, much observational data involving light occurs in an asymptotic regime of very short wavelength. The short wavelength asymptotic study of systems of partial differential equations often involves significant simplifications. In particular there are often good descriptions involving rays. We will use the phrase geometric optics to be synonymous with short wavelength asymptotic analysis of solutions of systems of partial differential equations. In optical phenomena, not only is the wavelength short but the wave trains are long. The study of structures which have short wavelength and are in addition very short, say a short pulse, also yields a geometric theory. Long wave trains have a longer time to allow nonlinear interactions which makes nonlinear effects more important. Long propagation distances also increase the importance of nonlinear effects. An extreme example is the propagation of light across the ocean in optical fibers. The nonlinear effects are very weak, but over 5000 kilometers, the cumulative effects can be large. To control signal degradation in such fibers, the signal is treated about every 30 kilometers. Still, there is free propagation for 30 kilometers which

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