Diffraction from Straight Edge

Consider the diffraction pattern of a semi-infinite opaque plane bounded by a sharp straight edge. Suppose that plane occupies the region $u<0$, which implies that the edge corresponds to $u=0$. In other words, suppose that $u_1=0$ and $u_2=\infty$. It is easily demonstrated that

$$C(u',0,\infty) = \frac{1}{2}+C(u'),$$ (10.128)

$$S(u',0,\infty) = \frac{1}{2}+S(u').$$ (10.129)

Thus, Equations (10.125) and (10.126) yield

$$f_c(u') = \frac{1}{2}\left[ C(u')-S(u')\right],$$ (10.130)

$$f_s(u') =\frac{1}{2}\left[ 1+ C(u')+S(u')\right],$$ (10.131)

whereas Equation (10.127) gives

$$\frac{{\cal I}(u')}{{\cal I}_0} = \frac{1}{2}\left[C(u')+\frac{1}{2}\right]^{\,2} + \frac{1}{2}\left[S(u')+\frac{1}{2}\right]^{\,2}.$$ (10.132)

Figure 10.17: Near-field diffraction pattern of a semi-infinite opaque plane occupying the region $u<0$.

The intensity of the diffraction pattern of a semi-infinite opaque plane is shown in Figure 10.17. Note that this is an intrinsically near-field diffraction pattern [because the aperture is of infinite extent; see Equation (10.96)]. In the directly illuminated region, $u'>0$, the intensity oscillates with diminishing amplitude, as the distance from the edge increases, and asymptotically approaches the value ${\cal I}_0$, as would be expected on the basis of geometric optics. In the shadow region, $u'<0$, the intensity decreases monotonically towards zero as the distance from the edge increases. Note that the maximum value of the intensity is not at the edge of the geometric shadow (i.e., $u'=0$), but some distance away from it, in the directly illuminated region. Finally, at the edge of the shadow, ${\cal I}= {\cal I}_0/4$. This is to be expected because half the wavefront is obstructed, the amplitude of the wave at the projection screen is thus halved, and the intensity consequently drops to one quarter of the unobstructed intensity.