Diffraction is a physical phenomenon in which a wave (could be an acoustic wave, water wave, electromagnetic wave, etc) bends around the edges of some kind of obstacle. When we look at the passage of light through a narrow slit for example, we note how most of the light in the original beam is blocked, and the projected pattern on a back-screen resembles the shape of the aperture through which the light beam travels. However, if the aperture is smaller than the wavelength of the incoming light, an interesting phenomenon arises: the light interferes with itself, forming a diffraction pattern. In this article, we will look at the physics and mathematics of diffraction through a narrow slit.
Single-slit diffraction in 1D
One and two-dimensional diffraction patterns can be formed by blocking a source of light with a screen possessing a small aperture of arbitrary shape, and allowing a small portion of that light to be projected on a screen, as shown in Figure 1. In most practical cases, spherical wavefronts from a source can be considered as planar as long as the distance between the projecting screen and the source is considerably larger than the largest feature on the aperture or obstacle. In order to predict how these diffraction patterns will look, we can analyse aperture interference patterns based on Huygen’s principle, and applying suitable approximations depending on the scale of the problem.
The simplest possible case we can look at is the single-slit with width d and length L, as shown in Figure 2. We will consider the case in which each point on the slit acts as its own source of spherical waves. Since in practical situations the distance between the aperture and the screen is much larger than the aperture dimensions (D >> d), we can assume planar wave fronts will hit the screen. Furthermore, if we take two consecutive sources separated by a distance y as shown in Figure 3, then we can define the phase difference as
This is of course assuming that D>>d, which means that the wave-vectors will be almost parallel to each other. Now, assuming a planar source wave propagating along the z-axis
is incident on the aperture, every virtual source at the aperture will be described by
Now, the resulting combination of waves at any point P on the screen will be the sum of all the virtual sources we have at our aperture, namely the integral
In this case, we express dS = L dy, and hence the double-integral reduces to a single-integral
At this point, we can exploit the property
by way of the substitution
Now, since the source is located at a fixed distance D along the z-axis, we make the substitution
which allows us to write
A plot of Eq(3) as a function of β is shown in Figure 4. We can see how the center peak is much larger than every consecutive peak, and this corresponds to the region of maximum constructive interference between all the wave-fronts propagating from the slit.
Single-slit diffraction in 2D
We now extend our analysis to a two-dimensional rectangular aperture. Consider a rectangular aperture of width w and height h centred at the origin, and oriented along the x and y axes (and the light source travelling along the z-direction). This will yield a diffraction pattern similar to that of an infinitely long slit in 1D. To construct the pattern, we will consider the sum of all point sources inside the aperture
In this case, we will express the field distribution ψ in terms of the wave-vector components kx and ky in order to simplify calculations. Since the domain of integration in this case is constant along both axes, and the integrand is separable, we can use the following property of double integrals (known as Fubini’s theorem)
Hence we write
Computing each integral yields
and simplifying the exponential expressions yields
with the substitutions
Finally, setting a constant C = w h ψ0(D)/4 yields
which is plotted in Figure 5 for a rectangular aperture of size 2x2 units.