Re: ATM P-V vs. RMS

[Aplanatic@aol.com]

Scott wrote:

> Nils,
>  
>  I don't like to use RMS because I have no way to measure it.  David 
>  Rowe's program, among others, will calculate an RMS figure, but the 
>  number is not much more than a guess.  The normal Foucault analysis 
>  that one puts into the FIGURE program only involves the measurement 
>  of 5-8 real data points, and these are all  values for the slope 
>  averaged over a few square cm.  I understand that David Rowe's 
>  program, (and the corresponding Burrow's program) then fit these 
>  points to a hypothetical, spherically symmetric, smooth surface that 
>  is further analyzed.  However, when you get down to it we are only 
>  measuring a handful of points.

FIGURE is a Foucault analysis program, and is thus subject to the failings of 
the test method itself.  As you point out, astigmatism and high spatial 
frequency surface structure can give erroneous results because the test 
itself does not sample enough points on the mirror's surface to give an 
accurate and complete profile.  This problem can be mitigated, to a large 
degree, by observation and intelligence.  One should not take Foucault 
readings from a zoney mirror and trust the interpretation of these 
measurements.  In addition, one should use other methods to decide whether on 
not a mirror has astigmatism.  This is where thought, observation and 
intelligence makes a big difference.

On the other hand, I would not call the output of a Foucault analysis program 
"not much more than a guess."  If the mirror is relatively free from high 
spatial frequency structure and does not suffer from measurable astigmatism, 
then the output from these programs is a very real and reasonable estimate of 
the surface profile of the mirror.  This statement has been corroborated by 
the many users of these programs over many years.

Now, whether you like P-V, RMS, Strehl or TA is your own business entirely.  
Personally, I like RMS, Strehl, and the encircled energy ratio (EER) for the 
final evaluation of the mirror because these best measure the actual 
performance of the mirror in light of the wave optics.  But I also use P-V 
and TA when talking to other mirror makers who like and understand these 
measures.  In any case, I recommend to others that figuring should proceed 
until several conditions are met.  These are:

1) The mirror is smooth.  This should be checked with a Ronchi grating and 
with the knife edge.  We can talk about a measurable smoothness criteria, but 
I don't wish to detail it here.

2) The mirror should be tested for astigmatism.  This is a very interesting 
subject in its own right.  The ultimate test is a star test, but there are 
other methods that can be employed on the bench.

3) The mirror should have a minimum Strehl ratio of 0.85, 0.9 being better.

4) The mirror should not show more than a couple of millimeters of turned or 
rolled edge, or should be masked off.

>  Two mirrors could differ wildly in 
>  the real RMS but might both give the same Foucault numbers along an 
>  axis.   Then it becomes a judgement call, based on non-quantifiable 
>  impressions like smoothness and lack of narrow zones, minor 
>  astigmatism, etc. as to which is the better mirror.  The Foucault 
>  data, whether represented as P-V, RMS, or Strehl, is only a guide to 
>  the optical quality and are not, by itself, a complete and accurate 
>  measure of optical quality.

Quite true, but we can do something about these problems, see above.  
Obviously, the ultimate bench test is an interferogram.  However, most of us 
don't have access to such.  So, we must make the best use of what we do have, 
i.e., Ronchi, Foucault, and star test.

>  I would be glad to use Strehl or RMS numbers if I had a reasonable 
>  way to measure them.  The new programs to analyze Foucault data will 
>  produce a logical guess at these quantities, but one should not 
>  confuse these results with real data.  In fact, one of the reasons I 
>  would rather cite archaic representations like P-V and Transverse 
>  aberration is that these quantities are less likely to be 
>  misinterpreted as the results of a more advanced analytical technique 
>  like interferometry.

This is not good reasoning, I think.  Quoting a Strehl ratio does not imply 
that it was measured using an interferometer.  P-V and TA are much *less* 
accurate ways of estimating the performance of a mirror than RMS or Strehl, 
given the wave nature of light.

Dave Rowe.