To: w6cwj{at}earthlink.net, atm{at}shore.net
From: mdholm{at}telerama.com
Reply-To: mdholm{at}telerama.com


Certainly it is foolish to become mesmerized by primary mirror quality and
miror cell induced deformations without also considering the quality of the
secondary mirror.

The argument that the tolerances on the secondary are much tighter because
there is a higher photon flux at it's surface doesn't pass muster with
known optical behavior.  The secondary's error budget can be analysed in
essentially the same fashion as the primary.  The 45 degree angle makes the
geometry a bit tougher (I still get hung up on it some times.) but the
basics are the same. What matters are path length differences (and dark
areas such as spider vanes).  In a mirror system, path length differences
come directly from surface errors.  Path length difference to a good
approximation is twice the surface height error.

The easiest scheme that has a good chance of being near correct for
assessing the overall performance of the system is to calculate the RMS of
the RMS values for each element. i.e.

   RMS total = Sqrt( (RMS mirror)^2 + (RMS cell)^2 + (RMS secondary) ^2 +
               (RMS tubecurrents)^2 + (RMS other)^2)

This assumes that each of the error sources is not significantly correlated
with any of the others.  It is possible, though a little unlikely, for
correlated errors in different components to add or subtract.  That would
make the system worse or better than calculated using RMS.

A consequence of the RMS calculation is that, if one component has 1/2 as
much RMS error as another, the sum of the two will be
 Sqrt(1^2 + 0.5 ^2) = Sqrt(1.25) = 1.118

If the two components have equal RMS error the result is
 Sqrt(1^2 + 1^2) = 1.414

So, if you want your secondary to contribute no more than 12% of the RMS
error attributable to the primary, it only has to be twice as good as the
primary.

If the two have equal (but of course uncorrelated) RMS error, the resulting
wavefront RMS error will only be 41% higher than the primary alone.

If it is easier to make a good primary than a good secondary, and you use
that advantage to produce a primary twice as good as the secondary, you
will lower the summed error from 1.414 to 1.118, about a 15% improvement. 
15% improvement in RMS might be worth working for.  The next increment from
another factor of two improvement in the primary probably isn't.

I have said several times that hunting for cell induced deformation less
than about 1/250 wave RMS is wasted effort.

Mark Holm
mdholm{at}telerama.com

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