Chapter 4, Problem 120CP

(a) By differentiating Equation 4.4, show that the higher-order maxima of the single-slit diffraction pattern occur at values of $\beta$ that satisfy $\tan \beta =\beta$ . (b) Plot $y=\tan \beta$ and $y=\beta$ versus $\beta$ and find the intersections of these two curves. What information do they give you about the locations of the maxima? (c) Convince yourself that these points do not appear exactly $\beta =\left(n+\frac{1}{2}\right)\pi$ , where $n=0,1,2,\dots ,$ but are quite close to these values.

   $I=I_0{\left(\frac{\sin \beta }{\beta }\right)}^2$   (4.4)