Rayleigh criterion for diffraction limited resolution through a circular aperture: Rayleigh equation
In this video, we motivate the Rayleigh criterion for diffraction limited resolution through a circular aperture by starting with a familiar picture of 1-slit diffraction.
Diffraction through a circular aperture is the circularly symmetric version of 1-slit diffraction, and it leads to an intensity pattern called an "Airy pattern". The Airy pattern shows us precisely how light spreads out when we pass it through a circular aperture.
Because light diffracts through a circular aperture, two point sources of light with a small angular separation can have some overlap in the diffraction patterns. If the angular separation between point sources is rather large, the overlap isn't too bad, and we can still distinguish the two sources. However, if the angular separation is too small, the overlap between the Airy patterns of the two sources can make the point sources indistinguishable, and we perceive a single source.
Where is the cutoff for the minimum resolvable angle through a circular aperture? This is obtained from the Rayleigh criterion. We require the angular separation to be at least large enough for the central maximum of one Airy pattern to land on the first minimum of the other, and we can reasonably distinguish the point sources. This leads to the Rayleigh criterion formula theta=1.22*lambda/D, where theta is measured in radians, lambda is the wavelength of the light, and D is the diameter of the circular aperture.
We comment that the minimum resolvable angle is larger for a larger wavelength of light, because diffraction patterns spread out farther when the wavelength is large. On the other hand, the minimum resolvable angle is smaller when the aperture diameter is larger. This explains why radio telescopes must be so large to obtain decent resolution: the wavelength of light is larger by five or six orders of magnitude compared to optical wavelengths, and we compensate for this by making radio telescopes of enormous diameter, such as the Arecibo telescope, which has a diameter of 1000 ft. or 305m.
Finally, we work two examples with the Rayleigh criterion. First, we estimate the diameter of the pupil by using Rayleigh equation by using the distance at which letters on a page become resolvable.
Second, we calculate the angular resolution of the Arecibo telescope while observing at a wavelength of 10cm.
