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Re: [ATM] How much cell induced deformation is too much?

At 2004-07-08 22:15 -0400, Mark wrote:

>I am aware of the rough correspondence between the Raliegh criterion and 1/14
>wavlength RMS (is that the Marechal criterion?).  (Can't put the Raliegh
>criterion in absolute terms, because it is defined in reference to 
>wavelength.)

The point to start from is the Strehl ratio.  "The general implication of a 
good-quality image is that the Strehl ratio is 0.8." (V. N. Mahajan, 
"Strehl ratio for primary aberrations...", J. Opt. Soc. Am., (72)9, Sept. 
1982, p.1258-1266).  All optical papers agree with this - including Lord 
Rayleigh, who said performance deteriorates beyond 1/4 wave of primary 
spherical aberration (which has Strehl ratio 0.8003 (same wavelength 
assumption)).

There's nothing in the literature that I've found relating to how much the 
image improves for SR (Strehl ratio) > .8, or how the image varies with SR 
= .8, but different aberrations.  So it's up to the mirror maker how far he 
carries this obsession.

Now to the correspondence between on-the-glass RMS and Strehl 
ratio.  Mahajan, in the quoted reference (and in Suiter, p. 238, Eq. 
(13.2)), says the relationship is SR = 1/exp((4ps/L)˛) for on-the-glass RMS 
s, and wavelength L (same units). It's cute - Mahajan didn't derive this 
approximation - he just says it "... should approximate the Strehl ratio 
better [than the Maréchal formula]."  It's good enough to avoid having to 
integrate the Strehl ratio definition numerically for empirical mirror 
surface profiles.

Turning the Mahajan formula around, for the standard visual wavelength L = 
550 nm and SR = .8, the requirement is

         surface RMS = (L/4p)*sqrt(-ln(SR)) = 20.7 nm

To include diagonal and cell aberrations, Mark says,

>SQRT(Mirror RMS ^2 + cell induced deformation RMS ^2 + Secondary RMS ^2) 
><= 9nM

Except for the final "9 nm" (unless you want to work harder than maybe 
necessary), I agree with this, assuming the aberrations are random and 
uncorrelated (it's a statistical theorem that the variances add along with 
the sum of uncorrelated random variables).

>Anybody
>know if they reference HeNe laser (633nM) or something else?

Boy, I sure hope they don't!  Another cheat like quoting surface 
aberrations without saying it, while everybody is assuming wavefront.  It's 
certainly OK to test with lasers, but the final answer should be, if given 
as RMS, PV, or Strehl, referenced to the standard 550 nm visual wavelength.

>I'd like to see reasoned thoughts on the subject.

An excuse for this long-winded post,

         -- Jim Burrows
         -- mailto://burrjaw@earthlink.net
         -- http://home.earthlink.net/~burrjaw
         -- Seattle N47.4723 W122.3662 (WGS84)  
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