Re: [ATM] Re: FigureXP and Foucault test Analysis

[Jim Burrows <burrjaw@earthlink.net>]

At 2005-04-10 19:30 +0200, Andreas Reifke wrote:

>First step: Calculating the mirror slope for the measured zones.
>Second step: Interpolating the slope values to get a function for the 
>mirror's slope
>-------------------------------
>As a result from step one you get single slope values for each mirror 
>zone. The problem is, that you can't describe the mirror's slope with 
>single data poins. So you have to interpolate the slope-values to get a 
>function for the mirror's slope. And here, all programs uses different 
>methods for interpolation:
>
>- Sixtest uses zernike coefficients
>Third Step: Integrating the slope-function to get a function which 
>describes the mirror-profile
>Fourth step: finding the equation for the reference-parabola

Actually, Sixtests only uses Zernike coefficients in the Fourth step - 
finding the best-fit conic.  The original Sixtests used only the first 2 
coefficients, had problems with fast spheres, so I upped the number of 
coefficients to 5, and haven't heard any gripes since.

First, we have to decide what Foucault is telling us about the mirror 
profile, i.e., model the test.  Geometric optics says the profile comes 
from solving a differential equation,

         x' = dx/dy = y/(f - x)

(x = mirror surface height at zone radius y, f = moving-source Foucault 
reading), not just integrating slopes.  If you want to solve that 
differential equation using standard numerical techniques (4th order 
Runge-Kutta rules the roost), you have to have the reading f for every half 
integration step in y, every .5 mm, say.  So the choice comes down to 
interpolating the Foucault readings themselves, and it turns out not to 
make too much difference.  Sixtests uses a linear spline interpolation in 
y˛, saying that the mirror is conic between readings (thank you, Mike Peck, 
for making that recommendation).

This gives you the mirror profile, and you are good to go - find best-fit 
conics, RMS, Strehl, deviation plots, etc.  BTW, the second formula for the 
Strehl ratio is Mahajan's and it is much better than the first (Maréchal's).

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

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