Knife-Edge Magnification
The actual separation between the surfaces of a sphere and a paraboloid of equal focal length, at any zone, when placed with their central zones and optical axes coincident, is equal to r4/8Rs, where r is the zonal radius of the mirror, and R is the radius of curvature of the central zone.
For the 6-inch f/8, this is 0.0000114" at the edge (PS, Fig. 37), equal to half a wave length of light. But under the knife-edge test, this departure is seen apparently highly magnified. A group of my students gave estimates of the apparent depressions of the edge zone of a paraboloidal mirror as seen with the knife-edge placed at the center of curvature of that zone; their values ranged from 1" to l¼". If their average, 1 1/8", is accepted as a reasonable value, it is found that the Foucault device yields an apparent magnification, of course in depth
only, of about 100,000.!
Now, at a distance of eight feet (the mirror's radius of curvature), a bump or dent of 1/32" on a surface is easily seen without optical aid, and its height or depth can be fairly accurately
Fig. 37. Illustrating the relationship between a paraboloid, P, and the reference
sphere, S, used in testing, when the knife-edge is placed at the center of curvature,
C, of the mirror's central zone. On the
scale of the drawing, the 6-inch f/8 is
shown by the short heavy line
estimated.
Assuredly, the presence of a zonal or other surface defect on the mirror, of apparently similar magnitude (the faint marks of the polishing tool that are occasionally seen may be much less), can be detected with the knife-edge and likewise estimated with reasonable accuracy, indicating that errors of the order of three ten-millionths of an inch can be seen.
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