# Abbe diffraction limit and Fourier

I am trying to understand the diffraction limited spot calculation. If my understanding is correct, the calculation uses the idea of Huygens wavelet where there are multiple point emitters on the lens. Summing the contributions from these spot we get an expression that can be approximated as the Fourier transform of a plane wave. The result of this is a spot with radius of $$0.61\lambda$$/NA where $$\lambda$$ is wavelength of the plane wave and NA the numerical aperture of the lens.

So my first question is does this equation work as a good approximation to cases where the light source is incoherent? I always see people calculate microscope resolution using this formula while using LED light source.

Second question is why do we still see the Gaussian-like distribution regardless of the coherence of the light source? Based on the explanation using Huygen wavelet, the distribution is a result of interference much like double slit experiment. If we were to use this idea of Huygen wavelet, doesn't the input have to have good spatial coherence such that these wavelets are in phase to interfere?