# Rayleigh Resolution Criteria for PSF

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One of my colleagues, Michael Humphreys, wrote an interesting article on the Rayleigh Criterion, so I thought I would post it here :)

In perfect geometric optics, a point source is imaged to a perfect point. In order to achieve this perfect imaging, the Exit Pupil has to be infinite in extent. As soon as the Exit Pupil has a finite diameter, diffraction from the edges 'blur' the perfect point into a Point Spread Function.

If the finite extent pupil function in the Exit Pupil is circular and perfectly flat (no aberrations) then you will see the traditional Airy disk (mathematically represented by the Bessel function of the first kind of order one).

When two point sources are close to each other, there is an angular limit to how close they can be before the individual PSFs blur together and they become unresolvable. This limit is called the Rayleigh Resolution Criteria and occurs when the peak of one PSF falls on the first minima, also known as the Airy Disk, of the second PSF (this also corresponds to a central dip of about 20%); if two points are closer together so that their PSF peaks are closer than 1 Airy Disk, then they are said to be 'unresolvable'.

To increase the angular resolution of an optical system, you can either increase the aperture (which increases the Exit Pupil size and decreases the PSF width) or you can decrease the wavelength:

In order to model this in OpticStudio, you can use the Huygens PSF or Huygens PSF Cross Section and multiple configurations. Each configuration should represent a different point source separated by a given angle:

With the Huygens calculations, you can force the analysis to coherently sum all the PSFs across all the configurations by selecting All from the configuration drop down option:

The theory above for the Rayleigh Resolution Criteria is valid only for point sources with equal intensity. The great thing about this approach in OpticStudio is that, ignoring stray light, we can investigate the angular resolution of two sources with different intensities. As an example, in Config 2 we can add a Slide Surface to Surface 2 with a constant RGB value of (127, 127, 127); this surface will be ignored in Config 1 and considered in Config 2. This will mean that each ray in Config 2 will have 50% the intensity of Configuration 1.

The example file can be found at the following URL:

https://feito.s3-us-west-2.amazonaws.com/RayleighResolutionCriteria.zar

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