waves03

waves03 - 5 04 Interference and Diffra c tion Sec. 9.7 505...

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504 Interference and Diffraction Sec. 9.7 505 Concave spheri cal mirror. A sphere is said to be "nestled" at the apex of a paraboloid if it is tangent to th e paraboloid there and has the same radi us as the radius of curvatu re of the para boloid there. It is not difficult to show tha t the radius of suc h a nestled sphe re is 2[. Sec Fig. 9.24. far to the right, the ellipsoid degenerates into a paraboloid. Rays emitted from F then form a parallel bea m (because they still focus at F' , infinitely far away). This is shown in Fig. 9.23. If the parabolic mir ror ape rture has a dia met er D, th en a point source at F doe s not form a perfectly parall el beam. Th e angular wid th of the interfere nce maxim um is liB ::: 'A/ D. D is "infinite:' we get a perfect plane wav e from the point source. Conversely, an incident plan e wave (perfectly well defined in angle) focuses to an image at F tha t is not a poin t unless D is infinite. The image has a width liz z. J 1I8 z. JX/ D. Fig. 9.24 Concave spherical mirro r ("in contact" with an imagined nestled para- bolic mirTor). The sphere's center is at C; its radius is 2f Ray a reflected from the sphere is not parallel to. the am; ray a' reflected f rom the paraboloid u. Thil illuetrates $pherical aberration: Fig. 9.23 Concave parabolic mi'rror. Ellip soidal mirror. In Fig. 9.22 we see a hollow ellipsoid of revolution with a spec uJarly refiecung inn er surface and with a point source of light locat ed at F, one of th e two prin cipal foci. Fr om th e de finition of an ellipse , the distances from F to the other focus F ' are the same for all pa ths (except for the direct path not involving a reflection]. Therefore the focus F ' is a region of complete constructive interference for radia tion emitted by elect rons in the surface that are dri ven by radiation from F. We say that the sour ce at F is imaged at the poin t F ' . The image at F ' is not a poin t; the phase of the resultant field at a point near F ' is within ab out ± 11 of the phase at F ' provided the point lies within a sphere with radius about >"/ 4 ce ntered at F '. Therefore that is roughly the size of the image at F '. Concave parabolic mirror. Imagine th at the focal point F an d the focal length J of the ellipsoid in Fig. 9.22 are held fixed, but tha t the focal point F ' is moved to the righ t; the ellipse is "str etched.' F ' is moved infinitely I I I . , I I I "-1 f..- f-+l I · - , Fig. 9.22 Ellipsoida l mirror.
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The deviation a is a co nstant, independent of the angle of inciden ce, as long as we stay near normal incidence. Equation (89) is easily derived as follows (See Fig. 9.25 ): At the base of the prism, the wavefront transverses the distan ce I at velocity er n. At the apex, the velocity is n times larger (since the prism thickness is zero there), and the same wavefront therefore travels a distance nl in the same time. Thus the wa vefront is ahead by a distance (11 - 1)1at the top. Thi s distan ce divided by the width IV of the prism is (for small angles) the angle of deviation 8 = (11 - 1)(l/ IV) = (11 - 1)" , which is Eq. (89).
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waves03 - 5 04 Interference and Diffra c tion Sec. 9.7 505...

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