39.1.1 A Wave Analysis of Interference

The interference of light can be analyzed from either a wave perspective or a photon perspective. Let’s start with a wave analysis. Figure 39.1 shows light waves passing through a double slit with slit separation $d$ . You should recall that the lines in a wave-front diagram represent wave crests, spaced one wavelength apart. The bright fringes of constructive interference occur where two crests or two troughs overlap. The graphs and the picture of the detection screen (notice that they’re aligned vertically) show the outcome of the experiment.

The figure consists of four parts that illustrate the double-slit experiment with light.

FIGURE 39.1 Full Alternative Text

You studied interference and the double-slit experiment in 《 CHAPTERS 17 and 33. The two waves traveling from the slits to the viewing screen are traveling waves with electric fields

\begin{array}{l} E_{1} & = & a sin (k r_{1} - ω t) \\ E_{2} & = & a sin (k r_{2} - ω t) \end{array}

where $a$ is the amplitude of each wave, $k = 2 π / λ$ is the wave number, and $r_{1}$ and $r_{2}$ are the distances from the two slits.

According to the principle of superposition, these two waves add together where they meet at a point on the screen to give a wave with net electric field $E = E_{1} + E_{2}$ . Previously we found that the amplitude of the superposition of two sinusoidal waves is

A (x) = 2 a cos (\frac{π d x}{λ L})

(39.1)

where $x$ is the horizontal coordinate on the screen, measured from $x = 0$ in the center.

The function $A (x)$ , the top graph in Figure 39.1, is called the amplitude function. It describes the amplitude $A$ of the light wave as a function of the position $x$ on the viewing screen. The amplitude function has maxima where two crests from individual waves overlap and add constructively to make a larger wave with amplitude 2a. $A (x)$ is zero at points where the two individual waves are out of phase and interfere destructively.

If you carry out a double-slit experiment in the lab, what you observe on the screen is the light’s intensity, not its amplitude. A wave’s intensity $I$ is proportional to the square of the amplitude. That is, $I \propto A^{2}$ , where $\propto$ is the “is proportional to” symbol. Using Equation 39.1 for the amplitude at each point, we find the intensity $I (x)$ as a function of position $x$ on the screen is

I (x) = C {cos}^{2} (\frac{π d x}{λ L})

(39.2)

where $C$ is a proportionality constant.

The lower graph in Figure 39.1 shows the intensity as a function of position along the screen. This graph shows the alternating bright and dark interference fringes that you see in the laboratory. In other words, the intensity of the wave is the experimental reality that you observe and measure.