Figure 33.4 showed double-slit interference occurring between two overlapping waves as they “spread out behind the two slits.” The waves are spreading because light has passed through two narrow slits, and each slit is causing single-slit diffraction. What we see in double-slit interference is actually interference between two overlapping single-slit diffraction patterns. Interference produces the fringes, but the diffraction pattern—the amount of light reaching the screen—determines how bright the fringes are.

We earlier calculated the ideal double-slit intensity, $I_{\text{double}}=4I_1{\text{cos}}^2(\pi dy/\text{λ}L)$, for two slits separated by distance $d$. But each slit has width $a$, so the double-slit pattern is modulated by the single-slit diffraction intensity for a slit of width $a$. Thus a realistic double-slit intensity, for small angles, is

The cosine term produces the fringe oscillations, but now the overall intensity is determined by the diffraction of the individual slits. If the slits are extremely narrow $(a\ll d)$, which we tacitly assumed before, then the central maximum of the single-slit pattern is very broad and we see many fringes with only a slow decline in the fringe intensity. This was the case in Figure 33.6b.

But many double slits have a width $a$ that is only slightly smaller than the slit spacing $d$, and this leads to a complex interplay between diffraction and interference. FIGURE 33.20 shows a double-slit interference pattern for two 0.055-mm-wide slits separated by 0.35 mm and, for comparison, the diffraction pattern of a single 0.055-mm-wide slit. Diffraction of the individual slits determines the overall brightness on the screen—it is the spreading of the light behind the slit—and within this we see the interference between light waves from the two slits. It seems there could be no better proof that light is a wave!

FIGURE 33.20 The overall intensity of a double-slit interference pattern is governed by the single-slit diffraction through each slit.

Full Alternative Text

Full Alternative Text

Notice that the $m=7$ interference fringe is missing (and $m=8$ is so weak as to be almost invisible). If an interference maximum falls exactly on a minimum (a zero) in the single-slit diffraction pattern, then we have what is called a missing order. Interference maxima occur at $y_m=m\text{λ}L\text{/}d$ and diffraction minima are at $y_p=p\text{λ}L\text{/}a$, where $m$ and $p$ are integers. Equating these, to find where they overlap, we see that order $m$ is missing if

$m$ has to be an integer, so the order is truly missing only for certain slit-spacing-to-width ratios $d\text{/}a$. In the case of Figure 33.20, $d\text{/}a=7$ and so the $m=7$ order is missing (it falls on the $p=1$ diffraction minimum), as is $m=14$. In practice, one or more interference maxima may be too weak to be seen even if they don’t satisfy Equation 33.29 exactly.