FRESNEL AND FRAUNHOFER DIFFRACTION

FRESNEL AND FRAUNHOFER DIFFRACTION

According to geometric optics, when an opaque object is placed between a point light source and a screen, as in Fig. 38-1 (page 1166), the shadow of the object forms a perfectly sharp line. No light at all strikes the screen at points within the shadow, and the area outside the shadow is illuminated nearly uniformly. But as we saw in Chapter 37, the wave nature of light causes effects that can’t be understood with the simple model of geometric optics. An important class of such effects occurs when light strikes a barrier that has an aperture or an edge. The interference patterns formed in such a situation are grouped under the heading diffraction. An example of diffraction is shown in Fig. 38-2. The photograph in Fig. 38-2a (on the right) was made by placing a razor blade halfway between a pinhole, illuminated by monochromatic light, and a photographic film. The film recorded the shadow cast by the blade. Figure

FRESNEL AND FRAUNHOFER DIFFRACTION
FRESNEL AND FRAUNHOFER DIFFRACTION

38-2b (on the left) is an enlargement of a region near the shadow of the left edge of the blade. The position of the geometrical shadow line is indicated by arrows. The area outside the geometric shadow is bordered by alternating bright and dark bands. There is some light in the shadow region, although this is not very visible in the photograph. The first bright band in Fig. 38-2b, just to the left of the geometrical shadow, is actually brighter than in the region of uniform illumination to the extreme left. This simple experiment gives us some idea of the richness and complexity of what might seem to be a simple idea, the casting of a shadow by an opaque object. We don’t often observe diffraction patterns. such as Fig. 38-2 in everyday life because most ordinary light sources are not monochromatic and are not point sources. If we use a white frosted light bulb instead of a point source in Fig. 38-1, each wavelength of the light from every point of the bulb forms its own diffraction pattern, but the patterns overlap to such an extent that we can’t see any individual pattern. Figure 38-3 shows a diffraction pattern formed by a steel ball about 3 rom in diameter.

Note the rings in the pattern, both outside and inside the geometrical shadow area, and the bright spot at the very center of the shadow. The existence of this spot was predicted in 1818, on the basis of a wave theory of light, by the French mathematician Sime on-Denis Poisson during an extended debate in the French Academy of Sciences concerning the nature of light. Ironically, Poisson was not a believer in the wave theory of light, and he published this apparently absurd prediction as a death blow to the wave theory. But the prize committee of the Academy arranged for an experimental test, and soon the bright spot was actually observed. (It had in fact been seen as early as 1723, but those earlier experiments had gone unnoticed.) Diffraction patterns can be analyzed by use of Huygens’ principle (Section 34-8). Let’s review that principle briefly. Every point of a wave front can be considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the wave. The position of the wave front at any later time is the envelope of the secondary waves at that time. To find the resultant displacement at any point, we combine all the individual displacements produced by these secondary waves, using the superposition principle and taking into account their amplitudes and relative phases. In Fig. 38-1, both the point source and the screen are relatively close to the obstacle forming the diffraction pattern. This situation is described as near-field diffraction or Fresnel diffraction, pronounced “Freh-nell” (after the French scientist Augustin Jean Fresnel, 1788-1827). If the source, obstacle, and screen are far enough away that all lines from the source to the obstacle can be considered.

Finally, we emphasize that there is no fundamental distinction between interference and diffraction. In Chapter 37 we used the term interference for effects involving waves from a small number of sources, usually two. Diffraction usually involves a continuous distribution of Huygens’ wavelets across the area of an aperture, or a very large number of sources or apertures. But both categories of phenomena are governed by the same basic physics of superposition and Huygens’ principle.

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