Suppose we were to replace the double slit with an opaque screen that has $N$ closely spaced slits. When illuminated from one side, each of these slits becomes the source of a light wave that diffracts, or spreads out, behind the slit. Such a multi-slit device is called a diffraction grating. The light intensity pattern on a screen behind a diffraction grating is due to the interference of $N$ overlapped waves.

FIGURE 33.7 shows a diffraction grating in which $N$ slits are equally spaced a distance $d$ apart. This is a top view of the grating, as we look down on the experiment, and the slits extend above and below the page. Only 10 slits are shown here, but a practical grating will have hundreds or even thousands of slits. Suppose a plane wave of wavelength $\text{λ}$ approaches from the left. The crest of a plane wave arrives simultaneously at each of the slits, causing the wave emerging from each slit to be in phase with the wave emerging from every other slit. Each of these emerging waves spreads out, just like the light wave in Figure 33.2, and after a short distance they all overlap with each other and interfere.

FIGURE 33.7 Top view of a diffraction grating with $N=10$ slits.

We want to know how the interference pattern will appear on a screen behind the grating. The light wave at the screen is the superposition of $N$ waves, from $N$ slits, as they spread and overlap. As we did with the double slit, we’ll assume that the distance $L$ to the screen is very large in comparison with the slit spacing $d$; hence the path followed by the light from one slit to a point on the screen is very nearly parallel to the path followed by the light from neighboring slits. The paths cannot be perfectly parallel, of course, or they would never meet to interfere, but the slight deviation from perfect parallelism is too small to notice. You can see in Figure 33.7 that the wave from one slit travels distance $\text{Δ}r=d\text{sin}\theta$ more than the wave from the slit above it and $\text{Δ}r=d\text{sin}\theta$ less than the wave below it. This is the same reasoning we used in Figure 33.5 to analyze the double-slit experiment.

Figure 33.7 is a magnified view of the slits. FIGURE 33.8 steps back to where we can see the viewing screen. If the angle $\theta$ is such that $\text{Δ}r=d\text{sin}\theta =m\text{λ}$, where $m$ is an integer, then the light wave arriving at the screen from one slit will be exactly in phase with the light waves arriving from the two slits next to it. But each of those waves is in phase with waves from the slits next to them, and so on until we reach the end of the grating. In other words, $\mathit{N}$ light waves, from $\mathit{N}$ different slits, will all be in phase with each other and interfere constructively when they arrive at a point on the screen at angle ${\mathit{\theta }}_{\mathit{m}}$ such that

The screen will have bright constructive-interference fringes at the values of ${\theta }_m$ given by Equation 33.15. We say that the light is “diffracted at angle ${\theta }_m.\text{"}$

FIGURE 33.8 Angles of constructive interference.

Because it’s usually easier to measure distances rather than angles, the position $y_m$ of the $m\text{th}$ maximum is

The integer $m$ is called the order of the diffraction. For example, light diffracted at ${\theta }_2$ would be the second-order diffraction. Practical gratings, with very small values for $d$, display only a few orders. Because $d$ is usually very small, it is customary to characterize a grating by the number of lines per millimeter. Here “line” is synonymous with “slit,” so the number of lines per millimeter is simply the inverse of the slit spacing $d$ in millimeters.

The wave amplitude at the points of constructive interference is $Ne$ because $N$ waves of amplitude $e$ combine in phase. Because the intensity depends on the square of the amplitude, the intensities of the bright fringes of a diffraction grating are

where, as before, $I_1$ is the intensity of the wave from a single slit. You can see that the fringe intensities increase rapidly as the number of slits increases.

Not only do the fringes get brighter as $N$ increases, they also get narrower. This is again a matter of conservation of energy. If the light waves did not interfere, the intensity from $N$ slits would be $N{I}_1$. Interference increases the intensity of the bright fringes by an extra factor of $N$, so to conserve energy the width of the bright fringes must be proportional to $1/N$. For a realistic diffraction grating, with $N>100$, the interference pattern consists of a small number of very bright and very narrow fringes while most of the screen remains dark. FIGURE 33.9a shows the interference pattern behind a diffraction grating both graphically and with a simulation of the viewing screen. A comparison with Figure 33.6b shows that the bright fringes of a diffraction grating are much sharper and more distinct than the fringes of a double slit.

FIGURE 33.9 The interference pattern behind a diffraction grating.

FIGURE 33.9 Full Alternative Text

Because the bright fringes are so distinct, diffraction gratings are used for measuring the wavelengths of light. Suppose the incident light consists of two slightly different wavelengths. Each wavelength will be diffracted at a slightly different angle and, if $N$ is sufficiently large, we’ll see two distinct fringes on the screen. FIGURE 33.9b illustrates this idea. By contrast, the bright fringes in a double-slit experiment are too broad to distinguish the fringes of one wavelength from those of the other.

The science of measuring the wavelengths of atomic and molecular emissions is called spectroscopy. The two sodium wavelengths in this example are called the sodium doublet, a name given to two closely spaced wavelengths emitted by the atoms of one element. This doublet is an identifying characteristic of sodium. Because no other element emits these two wavelengths, the doublet can be used to identify the presence of sodium in a sample of unknown composition, even if sodium is only a very minor constituent. This procedure is called spectral analysis.