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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?

CoffeeBiscuit
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All/most of our coherent sources are also monochrome as well (i.e. lasers), but in practice incoherent sources interfere just fine .... the mix of wavelengths just creates a an overlap of patterns. For your spot equation just use the longest (i.e. red) wavelength for spot size.

Interference/Huygens principle is 1700s/1800s but is still taught today ... it works mathematically but is misleading physically. Photons never cancel each other (violates energy conservation), in the DSE dark areas have no photons, bright areas all. Photons take paths per Feynman path integral, photons are only observed when they are absorbed. The path integral says photons travel paths that have lengths that are integer times the wavelength ... i.e. it appears the EM field is resonant ... that math works out the same as Huygens.

PhysicsDave
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  • Coherent (spatial) source is monochrome doesn't mean monochrome source are spatially coherent. If I place a narrow band filter in front of a light bulb the light is not spatially coherent right? That said, I don't believe we can consider incoherent source as superposition of many different wavelengths while each of them having spatial coherent and the result is just a summation. Is this wrong? – CoffeeBiscuit Nov 08 '23 at 01:36
  • Temporarily coherent = all photons same phase. Spatially coherent = all photons same direction. In the DSE or in your Abbe experiment you are effectively constraining the sources geometrically/spatially ... with slits (DSE) or tight apertures for the source (Abbe) and the disk edge. Typically lasers are both spatially and temporarily coherent. – PhysicsDave Nov 08 '23 at 01:53
  • "Spatially coherent = all photons same direction" Why is that? Think of spherical wave. The k vectors of all points are converging/diverging. Spatial coherence need not be only for plane wave. – CoffeeBiscuit Nov 08 '23 at 07:01
  • Yes that's true ..... spatial coherence is probably always a relative term ... it's more or less spatially coherent than some other situation. A small source is more sp coherent than a big one ... but it could depend on how far the observer is from the source. When we use slits/apertures we tend to constrain the light .... making it more sp coherent. – PhysicsDave Nov 08 '23 at 13:55
  • Yes and my problem is why do people define the point spread function as such without considering for these effects? "...but it could depend on how far the observer is from the source" I believe this is temporal coherence of the individual emitters. Which further adds to the spatial incoherence and hence my confusion – CoffeeBiscuit Nov 09 '23 at 01:21