Diffraction from Straight Wire

Figure 10.19: Far/near-field diffraction pattern of a straight wire. The top-left, top-right, middle-left, middle-right, bottom-left, and bottom-right panels correspond to ${\mit \Delta }u=0.25$, $0.5$, $1.0$, $2.0$, $4.0$, and $8.0$, respectively. The thick black line indicates the physical extent of the wire.

Consider the diffraction pattern of a straight wire that runs parallel to the $v$-axis and extends from $u=-{\mit\Delta }u/2$ to $u={\mit\Delta} u/2$. Because the wire is the complementary aperture of the rectangular slit discussed in the previous section, we can use Babinet's principle (see Section 10.11) to deduce that

$$f_c(u') = -\frac{1}{2}\left[C({\mit\Delta u}/2-u') + C(u'+{\mit\Delta u}/... ...ight] +\frac{1}{2}\left[S({\mit\Delta u}/2-u') + S(u'+{\mit\Delta u}/2)\right],$$ (10.136)

$$f_s(u') = 1-\frac{1}{2}\left[C({\mit\Delta u}/2-u') + C(u'+{\mit\Delta u}... ...ight] -\frac{1}{2}\left[S({\mit\Delta u}/2-u') + S(u'+{\mit\Delta u}/2)\right].$$ (10.137)

Figure 10.19 shows the diffraction pattern of a straight wire. In the far-field limit, ${\mit\Delta}u\ll 1$, the diffraction pattern consists of a bright spot, centered on the wire, surrounded by a set of interference fringes. The fringes can be thought of as the interference pattern of the light passing immediately to either side of the wire. In fact, if we think of the wire as two slits whose spacing is the diameter of the wire then straightforward interference theory (see Section 10.2) suggests that the fringe width should be

$$\delta u = \frac{2}{{\mit\Delta} u}.$$ (10.138)

It is clear from the figure that this is indeed the case. Hence, it is possible to determine the diameter of a thin wire from its diffraction pattern. In the extreme near-field limit, ${\mit\Delta u}\gg 1$, the diffraction pattern of the wire is essentially a geometric shadow, bounded on either side by a straight-edge diffraction pattern. (See Section 10.13.)