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Diffraction through a circular aperture spreads the wave over a larger bright circular area surrounded by alternating dark and bright rings. The circular aperture may be just a circular hole in a opaque object. The circular aperture may also be a circular lens through which light could pass such as in the viewing tube of a telescope or microscope. Because of the spreading of light, diffraction limits images that can be resolved in instruments that use lens.

Consider a circular aperture of diameter \(D\) and we wish to study the diffraction pattern observed on a screen a distance \(L\) away as shown in Figure 51.6.1. We find that the radius \(R\) of the central bright spot on the screen is not necessarily equal to the diameter of the aperture but depends on the wavelength \(\lambda\) and the distance \(L\) also.

Sometimes it is more convenient to write this equation for the central bright spot in terms of the angle \(\theta\) subtended at the slit. As the distance to the screen is far greater than the radius of the bright spot, we can use small angle approximation and write

Light from a point object passing through a circular aperture will spread out in accordance with Eq. (51.6.1) producing a bright circle on the screen instead of a point along with larger circular rings around the circle. As a result, central bright cicles corresponding to two close by sources, such as two stars, may overlap making it difficult, or even impossible, to distinguish them.

As in other instances, we use Raleigh criterion to set resolvability of two images. Specifically, two images on a screen will be considered just resolvable if the center of one circle is at the edge of the other circle as illustrated in Figure 51.6.2.

In terms of angular separation of the centers of the two images, the Raleigh criterion states that the angle of separation of the centers of the images must greater than a minimum angle given by the first dark ring of the diffraction pattern of one of the objects.

Figure 51.6.3 shows two stars whose directions in the sky have angular separation \(\theta\text{.}\) Suppose we project their images by a lens of diameter \(D\text{.}\) Each will form diffraction pattern at the focal plane of the lens. If stars are too close, central disks of the images will overlap. Therefore, unless, their separation angle \(\theta\) is larger than the angle of airy disk, \(\theta_R\text{,}\) the images will blur into each other.

Since loss of resolution due to diffraction cannot be eliminated by grinding a better lens or adding additional optical elements we say that the resolution is diffraction-limited. An image that is diffraction-limited can be improved by changing the aperture or observing in a different part of the electromagnetic spectrum, e.g., by using a smaller \(\lambda\) will make \(\theta_R\) smaller.