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# Multi-Slit Diffraction

Suppose that the opaque screen in our interference/diffraction apparatus contains identical, equally spaced, parallel slits of finite width. Let the slit spacing be , and the slit width , where . It follows that the aperture function for the screen is written (1051)

where (1052)

and (1053)

We recognize as the aperture function for a single slit, of finite width , that is centered on . [See Equation (1046).]

Assuming normal incidence (i.e., ), the interference/diffraction function, which is the Fourier transform of the aperture function, takes the form [see Equation (1044)] (1054)

Hence,     (1055)

where . Here, we have made use of the result , for any , which follows because is even in , whereas is odd. We have also employed the trigonometric identity . (See Appendix B.) The previous expression reduces to (1056)

Here [cf., Equation (1030)],   (1057)

is the interference/diffraction function for identical parallel slits of negligible width that are equally spaced a distance apart, and (1058)

is the corresponding aperture function. Furthermore [cf., Equation (1047)], (1059)

is the interference/diffraction function for a single slit of width . We conclude, from the preceding analysis, that the interference/diffraction function for identical, equally spaced, parallel slits of finite width is the product of the interference/diffraction function for identical, equally spaced, parallel slits of negligible width, , and the interference/diffraction function for a single slit of finite width, . We have already encountered both of these functions. The former function (see Figure 69, which shows ) consists of a series of sharp maxima of equal amplitude located at [see Equation (1031)] (1060)

where is an integer. The latter function (see Figure 71, which shows ) is of order unity for , and much less than unity for . It follows that the interference/diffraction pattern associated with identical, equally spaced, parallel slits of finite width, which is given by (1061)

is similar to that for identical, equally spaced, parallel slits of negligible width, , except that the intensities of the various maxima in the pattern are modulated by . Hence, those maxima lying in the angular range are of similar intensity, whereas those lying in the range are of negligible intensity. This is illustrated in Figure 73, which shows the multi-slit interference/diffraction pattern calculated for , , and . As expected, the maxima lying in the angular range have relatively large intensities, whereas those lying in the range have negligibly small intensities.   Next: Two-Dimensional Fourier Optics Up: Wave Optics Previous: Single-Slit Diffraction
Richard Fitzpatrick 2013-04-08