ATM Foucault and Diffraction

[Aplanatic@aol.com]

Dear List:

I just wrote a simple C program that performs the diffraction integral for the masked Foucault test in order to see for myself if there are systematic errors when diffraction (physical optics) are taken into full consideration.  The short answer is a qualified no, as long as the zone centers are correctly choosen.  The correct zone centers are well approximated by Nils Olof's formula:

Xc = ( (X1+X2)/2 + sqrt((X1^2 + X2^2)/2) )/2

where X1 is the inner edge radius and X2 is the outer edge radius.  This is, in fact, the zone center that I use in Figure, having been previously convinced by Nils Olof of its virtue.  For all cases that I ran, the true zone center was within 0.5% of the above.

In simulation I have explored mirrors as fast as f/2.5 and see no systematic effects from diffraction with the above choice.  Of course, visually balancing two mask openings near the edge of an f/2.5 mirror is rather difficult as I have found even for spherical mirrors, and much more difficult for paraboloidal ones.

I also explored the errors induced when there is a second opening close to the one being nulled.  Depending on the exact circumstances, a second opening can change the longitudinal KE position at null by as much as 1%, not a huge effect but worth noting.

Of further interest is the fact that the total integrated intensity is equal (nulls) at the same KE position that nulls the brightness at the above zone center.  This may not be a surprise to you but I didn't expect it.  This suggests to me that the best method for comparing the brightness of the two mask openings is to integrate the total brightness across the opening instead of trying to compare the brightnesses at the zone center.  Keep this in mind if you are using a camera to detect brightness.

All of the above is based on a transverse KE position that cuts the light intensity by a factor of two.  This happens when the edge of the knife is on the optical axis (as is the slit).  I have yet to explore other transverse KE positions.  

Dave Rowe