190508 Foucault test tolerance origin

You asked about the origin of the tolerance (x=2pR/r), and I didn't know, as it came from page 3 of Tom's 4-page Foucault test description.  The detailed description is also avoided in Harbour's paper.  Harbour says:  The amount of error that is allowed for the location of the focal plane to deviate from its ideal location for any given zone on a mirror has been worked out for us with the science of geometry. For our purposes it is not necessary to elucidate the entire method of determining the allowed error.  Thank you very much Mr Harbour!

My curiosity was piqued, so I did a Google search.  It comes from the Rayleigh criterion and the Airy disk first minima width (small angle sine approximation used, i.e. sin(x) = x).  See: https://www.quora.com/Where-does-this-%CE%B8-1-22-%CE%BB-D-come-from-Whats-its-derivation. 

The variable names used in the quora paper are different than in Tom's derivation, i.e., Tom uses D as the mirror focal length, while the quora reference uses D for the lens (mirror) diameter (I'll call it D' herein to avoid confusion).  They are related by f/, the focal ratio (your mirror is f/5.09). 

So, the quora paper has p=1.22λ/D', and Tom uses 1.22λf/D.  These formulas are equivalent and confusing because of the D's represent different measurements.  Note that if D' is the mirror focal length, and Tom's mirror diameter is D,  then D = D'/f, where f is the f/ ratio.

I know the above is a lot of information.  If you have any questions, I welcome them.

Enjoy,

Bruce