# wavop2 - Physics 54 Wave Optics 2 It's no exaggeration to...

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Physics 54 Wave Optics 2 It's no exaggeration to say that the undecideds could go either way. — Pres. George H. W. Bush Overview When part of a wavefront is obstructed, the progress of the wave energy into the region beyond the obstruction is determined (according to Huygens's principle) by the waves emitted from points on the unobstructed part of the wavefront. (The rest of the energy is either reflected or absorbed by the obstruction.) The waves from these unobstructed points spread out spherically, including into the geometric shadow region. As a result, some energy deviates from the directions of the original rays, “bending” into the shadow. This is diffraction . It is a general property of waves. The fraction of the original unobstructed energy that diffracts into the shadow region is greater for longer wavelengths and for smaller obstacles or apertures. If the size of obstacles or apertures encountered by the wave is very large compared to the wavelength, nearly all of the energy goes in straight lines along the original rays. This is why we see fairly sharp shadows of everyday objects illuminated by visible light. But if one looks carefully enough it will be found that the shadows do not really have sharp boundaries. And in situations where the aperture or obstacle size is comparable to the wavelength, diffraction becomes a major influence in distributing the energy. The simplest situation mathematically (which we will analyze in detail here) is one where the incoming waves can be approximated by plane waves, and where the intensity pattern is detected far enough from the obstacle or aperture that the waves can again be treated as plane waves. This case is Fraunhofer diffraction . The more complicated case, where the spherical nature of the wavefronts must be taken into account, is Fresnel diffraction . We will look at some examples, but not carry out the difficult mathematical analysis for that case. Fraunhofer Diffraction by rectangular slits To illustrate the phenomenon and the method of analysis, we consider an opaque screen in which has been cut a set of identical parallel long narrow rectangular slits. Light is incident on the screen from the left, at normal incidence. We assume the light to be PHY 54 1 Wave Optics 2

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plane waves, so the rays are parallel. After passing through the slits, the light is diffracted. We seek the intensity of the light moving off in the direction indicated by angle θ . To calculate the intensity at a distant detector we must find the E-field at that point. This field is the superposition of the fields from all the point sources (according to Huygens's principle) on the part of the wavefront that passes through the slits. These sources emit waves that begin in phase in the slits, but they travel different distances to the detector, so they arrive with different phases and interfere to produce the detected intensity.
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