Classical interference of light-waves

Let us now consider the classical interference of light-waves. Figure 2 shows a standard double-slit interference experiment in which monochromatic plane light-waves are normally incident on two narrow parallel slits which are a distance $d$ apart. The light from the two slits is projected onto a screen a distance $D$ behind them, where $D\gg d$.

Figure 2: Classical double-slit interference of light.

Consider some point on the screen which is located a distance $y$ from the centre-line, as shown in the figure. Light from the first slit travels a distance $x_1$ to get to this point, whereas light from the second slit travels a slightly different distance $x_2$. It is easily demonstrated that

$${\mit\Delta} x = x_2-x_1 \simeq \frac{d}{D}\,y,$$ (46)

provided $d\ll D$. It follows from Eq. (27), and the well-known fact that light-waves are superposible, that the wave-function at the point in question can be written

$$\psi(y,t) \propto \psi_1(t)\,{\rm e}^{\,{\rm i}\,k\,x_1} + \psi_2(t)\,{\rm e}^{\,{\rm i}\,k\,x_2},$$ (47)

where $\psi_1$ and $\psi_2$ are the wave-functions at the first and second slits, respectively. However,

$$\psi_1=\psi_2,$$ (48)

since the two slits are assumed to be illuminated by in-phase light-waves of equal amplitude. (Note that we are ignoring the difference in amplitude of the waves from the two slits at the screen, due to the slight difference between $x_1$ and $x_2$, compared to the difference in their phases. This is reasonable provided $D\gg \lambda$.) Now, the intensity (i.e., the energy-flux) of the light at some point on the projection screen is approximately equal to the energy density of the light at this point times the velocity of light (provided that $y\ll D$). Hence, it follows from Eq. (39) that the light intensity on the screen a distance $y$ from the center-line is

$$I(y) \propto \vert\psi(y,t)\vert^{\,2}.$$ (49)

Using Eqs. (46)-(49), we obtain

$$I(y) \propto \cos^2\left(\frac{k\,{\mit\Delta} x}{2}\right) \simeq \cos^2\left( \frac{k\,d}{2\,D}\,y\right).$$ (50)

Figure 3 shows the characteristic interference pattern corresponding to the above expression. This pattern consists of equally spaced light and dark bands of characteristic width

$${\mit\Delta}y = \frac{D\,\lambda}{d}.$$ (51)

Figure 3: Classical double-slit interference pattern.