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The Rayleigh criterion for optical resolution suggests that the closest two resolvable objects are separated by a distance $r \sim \frac{\lambda}{2NA}$, where $\lambda$ is the wavelength of my light source and $NA$ is the numerical aperture of the lens. This condition for optical resolution is based on the Fraunhofer diffraction pattern formed at the focus of our lens.

Since the Fraunhofer diffraction scheme assumes that the source is temporally coherent (one wavelength) and spatially coherent (well-defined phase front), does the Rayleigh criterion generalize to incoherent light? Should my theoretical resolution be better or worse if I use a coherent source?

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    $\begingroup$ I believe this book chapter has explained all you need: spie.org/samples/FG17.pdf. $\endgroup$
    – WDC
    Commented Jul 23, 2019 at 10:57

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An imaging system's ability can be limited by either aberration or diffraction.

Which one dominates, is interpreted as the Point Spread Function.

If the PSF is dominated by diffraction (the narrower the aperture of the lens), the angular resolution can be determined by the Rayleigh criterion.

two point sources are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other.1[2] If the distance is greater, the two points are well resolved and if it is smaller, they are regarded as not resolved. Rayleigh defended this criteria on sources of equal strength.

https://en.wikipedia.org/wiki/Angular_resolution

The answer to your question is that the Rayleigh criterion can be used with coherent and incoherent light sources too.

In a coherent imaging system, the amplitudes of signals from different parts of the image add, and the result is then squared to form the intensity image, while in an incoherent system the intensities add directly. Consequently, the use of the simple definition of resolution based on the half-power width is not always adequate. It is, therefore, common to employ the Rayleigh criterion, which states that two closely space illuminated points are distinguishable from each other if the maximum response to point A is located at the zero of the response to the point B.

enter image description here

You can see the image intensity for two mutually coherent point sources, and for two incoherent point sources.

https://www.sciencedirect.com/topics/engineering/rayleigh-criterion

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The NA in your formula is the system NA, which depends on all sorts of things, but a decent approximation is NA = (NAo + NAi) where NAo is the NA of the objective and NAi is the objective of the illumination (often, NAcond, the NA of the condenser).

The NA of coherent illumination (e.g. a laser) is generally very small.

With incoherent illumination, it's fairly easy to make the NA whatever you want. In conventional brightfield you match the condenser NA to the objective NA to balance resolution and contrast. With dark field, your NAi is greater than that of the objective.

So in general, the resolution of incoherent imaging is going to be greater than coherent, often around 2x.

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