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### Rayleigh resolution limit

#### Short tutorial

When doing optical imaging, there is always a certain 'smearing' of the resulting image compared to what is being imaged. This degradation of the image can be characterized by explaining what the imaging system does to a single point, that is, how it smears that point out. A function characterizing this is known as a point spread function (psf).

In the applet above, we image two points (or at least two point-like sources). As we can see in the lower half of the applet, the point sources are smeared out (having a psf of a certain width). The wider the psf, the further apart two points must be for us to be able to separate them visually in the image. The upper part of the applet shows a 1D function of the center-slice of the image.

In a classical imaging system, no matter how well we focus the image or tune the lens, diffraction will cause a certain width and shape of the psf. When we address this diffraction effect alone, we can get an analytic expression for the minimum distance between point-sources that can be resolved.

A typical criterion for resolvability is when the first dip of the psf of one source corresponds to the maximum of the other. This is known as the Rayleigh criterion. Based on the aperture diameter (lens diameter), D, and the wavelength, λ, the criterion reads:

$\sin(\theta) = 1.22 \frac{\lambda}{D}$
where θ is the minimum angle between two resolvable point sources.

Note the simple relation between the resolution limit and the wavelength and aperture size. Halving the wavelength, or doubling the aperture size, will lead to twice as good resolution capabilities.

#### Applet instructions

The lower part of the applet shows how two point sources are imaged, and the upper part shows a 1D function of the center-slice of this image. The left scrollbar guides how close the point sources are apart. The middle scrollbar lets you alter the wavelength used for imaging, effectively setting the imaging resolution in our setting. Clicking the button on the right sets the points exactly to the Rayleigh limit for the chosen wavelength.