Chapter 7DiffractionVersion 0407.1.K, 10 Nov 2004Please send comments, suggestions, and errata via email to [email protected] and [email protected], or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 911257.1 OverviewThe previous chapter was devoted to the classical mechanics of wave propagation. We showedhow a classical wave equation can be solved in the short wavelength approximation to yieldHamilton’s dynamical equations. We showed that, when the medium is time-independentthe 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 coincide withpaths along which the action or the integral of the phase is stationary. Our physical interpre-tation of this result was that the waves do indeed travel along every path, from some sourceto a point of observation, where they are added together but they only give a significantnet contribution when they can add coherently in phase, i.e. along the true rays. This is,essentially, Huygens’ model of wave propagation, or, in modern language, a path integral.Huygens’ principle asserts that every point on a wave front acts as a source of secondarywaves that combine so that their envelope constitutes the advancing wave front. This prin-ciple must be supplemented by two ancillary conditions, that the secondary waves are onlyformed in the direction of wave propagation and that a π/2 phase shift be introduced intothe secondary wave. The reason for the former condition is obvious, that for the latter, lessso. We shall discuss both together with the formal justification of Huygens’ constructionbelow.We begin our exploration of the “wave mechanics” of optics in this chapter, and weshall continue it in Chapters 8 and 9. Wave mechanics differs increasingly from geometricoptics as the wavelength increases relative to the scale length of the phase fronts and themedium’s inhomogeneities. The number of paths that can combine constructively increasesand the rays that connect two points become blurred. In quantum mechanics, we recognizethis phenomenon as the uncertainty principle and it is just as applicable to photons as toelectrons.Solving the wave equation exactly is very hard except in very simple circumstances.12Geometric optics is one approximate method of solving it — a method that works well in theshort wavelength limit. In this chapter and the following ones, we shall develop approximatetechniques that work when the wavelength becomes longer and geometric optics fails.We begin by making a somewhat artificial distinction between phenomena that arise whenan effectively infinite number of paths are involved, which we call diffraction and which wedescribe in this chapter, and those when a few paths, or, more correctly, a few tight bundlesof rays are combined, which we term interference, and whose discussion we defer to the nextchapter.In Sec. 7.2, we shall present the Fresnel-Helmholtz-Kirchhoff theory that underlies mostelementary discussions of diffraction, and we shall then distinguish between Fraunhoferdiffraction (the limiting case when spreading of the wavefront mandated by the uncertaintyprinciple is very important), and Fresnel diffraction (which arises when wavefront spreadingis a modest effect and geometric optics is beginning to work, at least roughly). In Sec. 7.3, weshall illustrate Fraunhofer diffraction by computing the expected angular resolution of theHubble Space Telescope, and in Sec. 7.4, we shall analyze Fresnel diffraction and illustrateit using lunar occultation of radio waves and zone plates.Many contemporary optical devices can be regarded as linear systems that take an inputwave signal and transform it into a linearly related output. Their operation, particularlyas image processing devices can be considerably enhanced by processing the signal in theFourier domain, a procedure known as spatial filtering. In Sec. 7.5 we shall introduce a toolfor analyzing such devices: paraxial Fourier optics — a close analog of the paraxial geometricoptics of Chapter 6. We shall use paraxial Fourier optics in Sec. 7.5 to analyze the phasecontrast microscope and develop the theory of Gaussian beams — the kind of light beamproduced by lasers when their optically resonating cavities have spherical mirrors. Finally,in Sec. 7.6 we shall analyze the effects of diffraction near a caustic of a wave’s phase field, alocation where geometric optics predicts a divergent magnification of the wave (Sec. 6.6 ofthe preceeding chapter). As we shall see, diffraction makes the magnification finite.7.2 Helmholtz-Kirchhoff IntegralIn this section, we shall derive a formalism for describing diffraction. We shall restrict ourattention to the simplest (and, fortunately, the most widely useful) case: a scalar wave withfield variable ψ of frequency ω = ck that satisfies the Helmholtz equation∇2ψ + k2ψ = 0 (7.1)except at boundaries. Generally ψ will represent a real valued physical quantity (although itmay, for mathematical convenience, be given a complex representation). This is in contrastto a quantum mechanical wave function satisfying the Schr¨odinger equation which is anintrinsically complex function. The wave is monochromatic and non-dispersive and themedium is isotropic and homogeneous so that k can be treated as constant. Each of theseassumptions can be relaxed with some technical penalty.The scalar formalism that we shall develop based on Eq. (7.1) is fully valid for weaksound waves in a fluid, e.g. air (Chap. 15). It is also fairly accurate, but not precisely so, for3rdS’Fig. 7.1: Surface S for Helmholtz-Kirchhoff Integral. The surface S0surrounds the observationpoint P and V is the volume bounded by S and S0. The aperture Q, the incoming wave to theleft of it, and the point P0are irrelevant to the formulation of the Helmholtz-Kirchoff integral, butappear in subsequent applications.the most widely used application of diffraction theory: the propagation of electromagneticwaves in vacuo or in a medium with homogeneous dielectric constant. In this case ψ can beregarded as one of the Cartesian components of the electric field vector, e.g. Ex(or equallywell a Cartesian component of the vector potential or the magnetic field vector). In vacuo orin a homogeneous dielectric medium, Maxwell’s equations imply that this ψ = Exsatisfiesthe scalar wave equation and thence, for fixed frequency, the Helmholtz equation
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