Figure 33.4 showed qualitatively how interference is produced behind a double slit by the overlap of the light waves spreading out behind each slit. Now let’s analyze the experiment more carefully. FIGURE 33.5 shows a double-slit experiment in which the spacing between the two slits is $d$ and the distance to the viewing screen is $L$. We will assume that $L$ is very much larger than $d$. Consequently, we don’t see the individual slits in the upper part of Figure 33.5.

FIGURE 33.5 Geometry of the double-slit experiment. The experiment is being viewed from above.

Let P be a point on the screen at angle $\theta$. To determine whether the interference at P is constructive, destructive, or in between, we need to examine $\text{Δ}r$, the path-length difference of light coming from the two slits. The insert to Figure 33.5 shows the individual slits and the paths from these slits to point P. Because P is so far away on this scale, the two paths are virtually parallel, both at angle $\theta$. Both slits are illuminated by the same wave front from the laser; hence the slits act as sources of identical, in-phase waves $(\text{Δ}{\varphi }_0=0)$. Thus the interference at point P is constructive, producing a bright fringe, if $\text{Δ}r=m\text{λ}$ at that point.

The midpoint on the viewing screen at $y=0$ is equally distant from both slits $(\text{Δ}r=0)$ and thus is a point of constructive interference. This is the bright fringe identified as the central maximum in Figure 33.4b. The path-length difference increases as you move away from the center of the screen, and the $m=1$ fringes occur at the points where $\text{Δ}r=1\text{λ}$—that is, where one wave has traveled exactly one wavelength farther than the other. In general, the $\mathit{m}\text{th}$ bright fringe occurs where the wave from one slit travels $\mathit{m}$ wavelengths farther than the wave from the other slit and thus $\text{Δ}\mathit{r}=\mathit{m}\text{λ}$.

You can see from the magnified portion of Figure 33.5 that the wave from the lower slit travels an extra distance

If we use this result for the path-length difference in Equation 33.1, we find that bright fringes (constructive interference) occur at angles ${\theta }_m$ such that

We added the subscript $m$ to denote that ${\theta }_m$ is the angle of the $m\text{th}$ bright fringe, starting with $m=0$ at the center.

In practice, the angle $\theta$ in a double-slit experiment is very small $(<1\text{°})$. We can use the small-angle approximation $\text{sin}\theta \approx \theta$, where $\theta$ must be in radians, to write Equation 33.3 as

This gives the angular positions in radians of the bright fringes in the interference pattern.