DIFFRACTION FROM A SINGLE SLIT

DIFFRACTION FROM A SINGLE SLIT

In this section we’ll discuss the diffraction pattern formed by plane-wave (parallel-ray) monochromatic light when it emerges from a long, narrow slit, as shown in Fig. 38-4. We call the narrow dimension the width, even though in this figure it is a vertical dimension. According to geometric optics, the transmitted beam should have the same cross section as the slit, as in Fig. 38-4a. What is actually observed is the pattern shown in Fig. 38-4b. The beam spreads out vertically after passing through the slit. The diffraction pattern consists of a centra! bright band, which may be ~uch ~roader than.the.widlh:of the slit, bordered by alternating dark and bright bands Withrapidly decreasing intensity, About 85% of the power in the transmitted beam is in the centra! bright band, whose width is found to be inversely proportional to the width of

the slit. In general, the smaller the width of the slit, the broader the entire diffraction pattern. (The horizontal spreading of the beam in Fig. 38-4b is negligible because the horizontal dimension of the slit is relatively large.) You can easily observe a similar diffraction pattern by looking at a point source, such as a distant street light, through a narrow slit formed between two fingers
in front of your eye; the retina of your eye corresponds to the screen.

Figure 38-5 shows a side view of the same setup; the long sides of the slit are perpendicular to the figure, and plane waves are incident on the slit from the left. According to Huygens’ principle, each element of area of the slit opening can be considered as a source of secondary waves. In particular, imagine dividing the slit into several narrow strips of equal width, parallel to the long edges and perpendicular to the page in Fig. 38-5a. Cylindrical secondary wavelets, shown in cross section, spread out from each strip.

In Fig. 38-5b a screen is placed to the right of the slit. We can calculate the resultant intensity at a point P on the screen by adding the contributions from the individual wavelets, taking proper account of their various phases and amplitudes. It’s easiest to do this calculation if we assume that the screen is far enough away that all the rays from various parts of the slit to a particular point P on the screen are parallel, as in Fig. 38-5c. An equivalent situation is Fig. 38-5d, in which the rays to the lens are parallel and the
lens forms a reduced image of the same pattern that would be formed on an infinitely distant screen without the lens. We might expect that the various light paths through the lens would introduce additional phase shifts, but in fact it can be shown that all the paths have equal phase shifts, so this is not a problem. The situation of Fig. 38-5b is Fresnel diffraction; those in Figs. 38-5c and 38-5d, where the outgoing rays are considered parallel, are Fraudster diffraction. We can derive quite simply the most important characteristics of the Fraudster diffraction pattern from a single slit. First consider two narrow strips, one just below the top edge of the drawing of the slit and one at its center, shown in end view in Fig. 38-6. The difference in path length to point P is (aI2) sin 8, where a is the slit width and 8 is the angle between the perpendicular to the slit and a line from the center of the slit to P. Suppose this path difference happens to be equal to ).12; then light from these two strips arrives at point P with a half-cycle phase difference, and cancellation occurs.

Similarly Iight from two strips immediately below the two in the figure also arrives P a half-cycle out of phase. In fact, the light from every strip in the top half of the slit els out the light from a corresponding strip in the bottom half. The result is complete cancellation at P for the combined light from the entire slit, giving a dark fringe in the interference pattern. That is, a dark fringe occurs whenever . The plus-or-minus (±) sign in Eq. (38-1) says that there are symmetrical dark fringes above and below point a in Fig. 38-6a. The upper fringe (8) 0) occurs at a point P where light from the bottom half of the slit travels ).J2 farther to P than does light from the top half; the lower fringe (9 < 0) occurs where light from the top half travels ).J2 farther than light from the bottom half. We may also divide the screen into quarters, sixths, and so on, and use the above argument to show that a dark fringe occurs whenever sin 8 ‘” ±2A/a, BA/a, and so on. Thus the condition for a dark fringe is For example, if the slit width is equal to ten wavelengths (a = lOA), dark fringes occur at sin 9 Between the dark fringes are bright fringes. We also note that sin 9 = 0 corresponds to a bright band; in this case, light from the entire slit arrives at P in phase. Thus it would be wrong to put m = 0 in Eq. (38-2). The central bright fringe is wider than the other bright fringes, as Fig. 38-4 shows. In the small-angle approximation that we will use below, it is exactly twice as wide. With light, the wavelength A is of the order of 500 nm = 5 x 10-7 rn. This is often much smaller than the slit width a; a typical slit width is 10-2 em = 10-4 rn. Therefore the values of 9 in Eq, (38-2) are often so small that the approximation sin 9′” 9 (where 9 is in radiant) is a very good one. In that case we can rewrite this equation as.  This equation has the same form- as the equation for the two-slit pattern, Eq. (37-6), except that in Eq. (38-3) we use x rather than R for the distance to the screen. But Eq. (38-3) gives the positions of the dark fringes in a single-slit pattern rather than the bright fringes in a double-slit pattern. Also, m = 0 is not a dark fringe. Be careful!