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According to the physical reasoning of the preceding paragraph, we obtain the diffraction pattern of a single slit by substituting d=a/N into Eq. 17 (so that we always have the correct total width a) and then taking the limit tex2html_wrap_inline1282 (so that we have in the end a single, continuous slit). Recalling that tex2html_wrap_inline1214 , the factor in numerator then becomes tex2html_wrap_inline1286 . As a similar term appears in the denominator, it is convenient to define the path difference between the top and bottom of the slit (Figure 8)


With this, the phase factor in the numerator becomes tex2html_wrap_inline1288 . Finally, noting that the argument of the sine function in the denominator is the same but for a missing factor of N, the denominator is just tex2html_wrap_inline1292 .

This leaves for us to take the tex2html_wrap_inline1282 limit of


As tex2html_wrap_inline1282 , the argument of the sine function in the denominator becomes quite small, so that we may use the small angle formula, tex2html_wrap_inline1300 . With this, we have


From the discussion in Section 3.3.2, we know the intensity at the central maximum to be tex2html_wrap_inline1304 . Defining this intensity as tex2html_wrap_inline1306 , we have our final result for the single slit,


where, recall, tex2html_wrap_inline1308 .

Tomas Arias
Thu Sep 13 15:26:14 EDT 2001