0207.1 - Chapter 7 Diraction Version 0207.1 13 Nov 02...

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Chapter 7 Diffraction Version 0207.1, 13 Nov 02 Please send comments, suggestions, and errata via email to [email protected] and to [email protected], or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125 7.1 Overview The previous chapter was devoted to the classical mechanics of wave propagation. We showed how a classical wave equation can be solved in the short wavelength approximation to yield Hamilton’s dynamical equations. We then specialized to stationary media, as we shall continue to do in this chapter. Under these conditions, the frequency of a wave packet is constant. We imported a result from classical mechanics, the principle of stationary action, to show that the true geometric-optics rays were those paths along which the action or the integral of the phase was stationary. Our physical interpretation of this result was that the waves did indeed travel along every path, from some source to a point of observation, where they were added together but they only gave a significant net contribution when they could add in phase, along the true rays. This is, essentially, Huygens’ model of wave propagation, or, in modern language, a path integral . Huygens’ principle asserted that every point on a wave front acted as a source of sec- ondary waves that combine so that their envelope constitutes the advancing wave front. This principle must be supplemented by two ancillary conditions, that the secondary waves are only formed in the direction of wave propagation and that a 90 phase shift be introduced into the secondary wave. The reason for the former condition is obvious, that for the latter, less so. We shall discuss both together with the formal justification of Huygens’ construction below. We begin our exploration of the “wave mechanics” of optics in this chapter, and we shall continue it in Chapters 8 and 9. Wave mechanics differs increasingly from geometric optics as the wavelength increases. The number of paths that can combine constructively increases and the rays that connect two points become blurred. In quantum mechanics, we recognize this phenomenon as the uncertainty principle and it is just as applicable to photons as to electrons. Solving the wave equation exactly is very hard except in very simple circumstances. 1
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2 Geometric optics is one approximate method of solving it — a method that works well in the short wavelength limit. In this chapter and the following ones, we shall develop approximate techniques that work when the wavelength becomes longer and geometric optics fails. We begin by making a somewhat artificial distinction between phenomena that arise when an effectively infinite number of paths are involved, which we call diffraction and which we describe in this chapter, and those when a few paths, or, more correctly, a few tight bundles of rays are combined, which we term interference , and whose discussion we defer to the next chapter.
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