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

[Michael Peck <mpeck1@ix.netcom.com>]

When I wrote

>To an excellent approximation for a nearly diffraction limited wavefront
>
>Strehl = exp(-(2*pi*RMS)^2)     [Mahajan's approximation]

I was just parroting the standard references. With nothing better to 
do this muggy late summer weekend I decided to kill a few trillion 
CPU cycles investigating just how accurate Mahajan's approximation 
is. Here's what I did: I created 1000 random wavefronts where both 
the aberration content and the total RMS wavefront error varied 
randomly. I then calculated PSFs for each wavefront under the usual 
approximation of Fraunhofer diffraction theory using Discrete Fourier 
transforms. The Strehl ratio was estimated as the ratio of the 
squared modulus of the DC component to that for a perfect wavefront. 
Here's the results shown as plots of calculated Strehl ratios vs. RMS 
error, in waves:

1) 
<http://home.earthlink.net/~mlpeck54/astro/temppix/strehlvsrms.png> 
shows the results for the entire simulation for which RMS was allowed 
to vary between 0 and 0.5 waves. The solid black line is Mahajan's 
approximation. The solid blue one is called Marechal's approximation (IIRC),

S = 1 - (2*pi*RMS)^2

2) 
<http://home.earthlink.net/~mlpeck54/astro/temppix/strehlvsrms_dl.png> 
is a close-up view of the diffraction limited regime (defined 
somewhat loosely).

3) <http://home.earthlink.net/~mlpeck54/astro/temppix/dsvsrms.png> 
shows the difference between the calculated Strehl ratios and 
Mahajan's approximation.

Conclusions: Mahajan's approximation is accurate for the usual 
definition of "diffraction limited": it's good to about 0.1% accuracy 
for RMS <= 1/14 wave. For larger wavefront errors the scatter 
increases and Mahajan's approximation errs on the pessimistic side, on average.

By the way this has nothing to do with "polychromatic Strehl ratios" 
or achromatic refractor design.

Mike Peck

_________________

Michael Peck
email mpeck1@ix.netcom.com
Wildlife photoblog! http://wildlife-pix.com
Amateur telescope making http://home.earthlink.net/~mlpeck54/astro/astro.html 

_______________________________________________
ATM mailing list http://www.atmlist.net/