Diffraction from Rectangular Slit

Figure 10.18: Far/near-field diffraction pattern of a rectangular slit. The top-left, top-right, middle-left, middle-right, bottom-left, and bottom-right panels correspond to ${\mit \Delta }u=0.1$, $1.0$, $2.0$, $4.0$, $6.0$, and $10.0$, respectively. The thick black line indicates the physical extent of the slit.

Consider the diffraction pattern of a rectangular slit that runs parallel to the $v$-axis and extends from $u=-{\mit\Delta }u/2$ to $u={\mit\Delta} u/2$. In other words, let $u_1=-{\mit\Delta }u/2$ and $u_2={\mit\Delta}u/2$. It is easily demonstrated, from Equations (10.121), (10.122), (10.125), and (10.126), that

$$f_c(u') = \frac{1}{2}\left[C({\mit\Delta u}/2-u') + C(u'+{\mit\Delta u}/2)\right] -\frac{1}{2}\left[S({\mit\Delta u}/2-u') + S(u'+{\mit\Delta u}/2)\right],$$ (10.133)

$$f_s(u') = \frac{1}{2}\left[C({\mit\Delta u}/2-u') + C(u'+{\mit\Delta u}/2)\right] +\frac{1}{2}\left[S({\mit\Delta u}/2-u') + S(u'+{\mit\Delta u}/2)\right].$$ (10.134)

It follows from Equation (10.127) that

$$\frac{{\cal I}(u')}{{\cal I}_0 }= \frac{1}{2}\left[C({\mit\Delta ... ... \frac{1}{2}\left[S({\mit\Delta u}/2-u') + S(u'+{\mit\Delta u}/2)\right]^{\,2}.$$ (10.135)

Note that the far-field limit corresponds to ${\mit\Delta}u\ll 1$ [see Equations (10.96) and (10.98)], whereas the near-field limit corresponds to ${\mit\Delta}u\gtrsim 1$ [see Equation (10.97)].

Figure 10.18 shows the diffraction pattern of a rectangular slit. It can be seen that in the far-field limit, ${\mit\Delta}u\ll 1$, the diffraction pattern is the same as that calculated in Section 10.7. However, in the near-field limit, ${\mit\Delta}u>1$, the diffraction pattern is significantly modified. For instance, in the far-field limit, the diffraction pattern is much wider than the slit, whereas, in the near-field limit, the diffraction pattern is similar in size to the slit. In fact, in the extreme near-field limit, ${\mit\Delta}u\gg 1$, the diffraction pattern is fairly similar in form to the geometric image of the slit, apart from the presence of fringes within the image.