ATM What Rayleigh said (was: aargh! maybe)

["James W. Burrows" <burrjaw@earthlink.net>]

At 19:10 2002-06-17 -0400, Mark Holm wrote:

>>The 1/4 wave value comes from a statement made in the 19th century by an 
>>English
>>physicist, Lord Rayleigh.  He said that if the path lengths of the waves 
>>arriving
>>at the final focus differ by less than 1/4 wave, that one will not be able to
>>tell the difference between that image and a perfect one.

Lord Rayleigh (Philos. Mag. 8, p. 403 (1879)) said that a quarter wave of 
primary spherical aberration "reduces the irradiance at the Gaussian focus 
by 20%", i.e., the Strehl ratio for this aberration is 0.8.  And, as Nils 
Olof Carlin quoted, more aberration causes marked image 
deterioration.  Amateurs since have focused on the wrong part of that 
statement - it's not the 1/4 wave, it's the 20% that's important.  For 
different aberrations, the connection between the Strehl ratio and the P-V 
is very loose.

For a talk at the Bellingham workshop, I played on the computer to find two 
theoretical surfaces:  1) what's the biggest RMS that a 1/4 wave P-V can 
have, and, on the other hand, 2) what's the biggest P-V that a mirror can 
have that has a passing surface RMS of 20 nm?  The answer to 1) was 
something like 22 nm - so you'd have to be awfully unlucky to have a 
1/4-wave mirror that doesn't pass.  The answer to 2) was quite 
surprising:  something like 1 wave!  Most of the P-V was a peak in the 
center of the mirror (unobstructed, of course).

So the moral is - 1/4-wave is almost always good enough, but bigger P-V 
than that could also be good enough, depending on where the error is.

         -- Jim Burrows
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         --              http://home.earthlink.net/~burrjaw
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