RE: [ATM] Accuracy of Mahajan's approximation for Strehl ratio

[Jim Burrows <burrjaw@earthlink.net>]

At 2005-08-15 13:29 -0500, Steve Koehler wrote:

>Thanks for the comparison of Marechal vs. Mahajan.  I hadn't been paying
>attention, and have used Marechal in (I believe) all of my stuff.  It 
>looks like
>Mahajan does better for more than twice the RMS error than does Marechal.

Oooh, Marechal is BAD! - it goes negative for larger RMS (not very large - 
it goes to zero at surface RMS 43.8 nm, for which Mahajan says .367 
Strehl), while Mahajan satisfies the same limits as the real Strehl <hey! 
that's a rhyme>.

>Thanks, also, for mentioning calculating Strehl directly.  It's obvious, but I
>had never tried that, before.

In the early days of developing Sixtests (in good ol' DOS), knowing about 
Marechal's problems (and not having heard of Mahajan's approximation), I 
didn't dare use Marechal (can you imagine the hoots when the Strehl comes 
up negative) and having the surface profile deviation d(y) every mm of 
radial y, I numerically integrated the Strehl definition - stand back! an 
ASCII equation:

         SR = (2/r²)²[ (int(0,r){sin d(y)y dy})² + (int(0,r){cos d(y)y dy})²]

No problem with < 0, but, be careful, check for > 1...  Then I heard about 
Mahajan, and for the expected RMS accuracies of most amateur tests (maybe 
not for the interferometer enthusiasts), it is so much easier to calculate 
the RMS and plunk it into his equation.  It's cute, in the paper (V. N. 
Mahajan, "Shrehl ratio for primary aberrations...", J. Opt. Soc. Am. 72(9), 
Sep. 1982, p. 1258-1266) that Mahajan announced his approximation he says, 
"...[it] should approximate the Strehl ratio better [than Marechal's]."  It 
turned out to be true - a very good example of a super guess by someone who 
knows what he's doing.

At 2005-08-15 17:28 -0400, Richard F.L.R. Snashall wrote:

>Michael Peck wrote:
>>
>>Maybe. I guess the way to find out is to use a larger FFT. I think I have 
>>enough memory to try it, and CPU cycles are cheap.
>
>FYI I've found that ZEMAX doesn't approach OSLO's result for Strehl until 
>at least
>256x256; at that point it is off by about 0.02 at 0.8 Strehl.  You may 
>have to go up
>to at least 1Kx1K.

On my 2.8 GHz Toshiba laptop w/512 M RAM, 2056x2056 is done in a couple of 
minutes with no disk access.

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

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