Resolving Power of Optical Instruments and coherent light

Question

Generally, optical instruments are formed by circular openings. When light passes through these openings, diffraction occurs and we have the formation of rings of light, interspersed with dark rings. This leads to the Rayleigh criterion, which establishes a minimum angle for distinguishing two objects: $\theta_{min}\approx 1.22 \lambda/D$, where $\lambda$ is the wavelenght and $D$ the diameter of the opening. However, I was thinking: to have interference effects, and consequently diffraction, the light that passes through the aperture must be coherent light. However, most light sources in nature are not coherent. Stars are not, so in a telescope, for example, I shouldn't observe these diffraction patterns. In this case, doesn't the Rayleigh criterion lose its meaning?

The books I read, when they talked about this subject, simply ignored the need for light to be coherent, what sounds a bit strange to me and made me question whether this condition is really necessary.

Answer 1

In fact, the $\theta = 1.22\lambda/D$ result in a sense relies on the sources being incoherent.

We know that due to diffraction, an aperture will form an Airy disk with characteristic angular size $\theta = 1.22\lambda/D$ where $\lambda$ is the wavelength and $D$ is the size of the limiting aperture (assumed to be circular). This result does require coherent light, specifically spatial or transverse coherence. For example, a star is so far away and looks so small on the sky that it is effectively a point source, and light from it is in phase across all points of the telescope aperture. This is different from temporal or longitudinal coherence, which tells us how pure the wavelength of the light is, and you are correct that in general light from a star will not be temporally coherent. For our purposes however, we will consider it to be quasimonochromatic, meaning it can be approximately described by a single wavelength, which is necessary for the $\theta = 1.22\lambda/D$ formula to make sense anyway.

Now, the Rayleigh criterion comes in when we want to resolve another point source, which is incoherent with respect to the first source. For example, if we try to image two stars close to each other on the sky, the light from one star will be statistically completed unrelated to that of the other, as the two stars are independent giant balls of gas. For incoherent light, we add the intensities, so each star forms its own independent Airy disk on the camera of the telescope, and these can be resolved (according to the Rayleigh criterion) if the angular separation of the stars is less than $\theta = 1.22\lambda/D$. If, hypothetically, the two stars were coherent with respect to each other, then their amplitudes would add, and we would get a more weird pattern formed by adding their amplitudes, and then squaring to find the intensity.