I started out looking at this while puzzling over the problem of how to turn my polished f/3.9 mirror into a paraboloid, which by a naive calculation was going to require something like 6 waves of total correction (at the surface) to make. Taking off from comments of Jim Burrows I ended up working on the problem of solving for the paraboloid having minimum area weighted rms deviations from a given curve. As a preliminary comment, I'm working at the surface of the glass here. The same analysis works for the wavefront evaluated at the center of curvature or focus, modulo factors of 2 or so (which in my universe don't count anyway).

**Notation**

I more or less follow the notational conventions of Rutten & van Venrooij (1988; R&vV) here: the optical surface is taken as tangent to the x-y plane, with the z axis corresponding to the optical axis. The origin is at the optical center. Radial distance in the x-y plane is denoted by h.

The mirror's paraxial radius of curvature is r, and it's physical radius is R.

**Model and solution**

The simple way to approach this is to represent our existin>Model and solution

The simple way to approach this is to represent our existing curve with a generic power series in h,

Our target paraboloid has equation z_{t} = h^{2}/(2r_{t}).
I'm imagining working actual glass here, so I allow a change of origin
c between the existing curve and the target paraboloid. The area weighted
integrated square difference between the existing and target curve is (ignoring
a factor 2pi):

and the weighted mean square error is

We want to minimize Q (or equivalently WMSE) with respect to the free
parameters r_{t} and c. I'll briefly outline how to do this --
it's pretty straightforward, but tedious. The first order necessary conditions
(which I think are also sufficient for this case) for a minimum are:

The rest is just paperwork. Here's the solutions:

The rest is just paperwork. Here's the solutions:

and

Substituting equations 6 and 7 into 2 and 3 we get, with more tedious
algebra:

I don't see a correspondingly simple way to get P-V. The equation of
the surface deviation is of course a power series in h, which in general
could have many local extrema. Except in special cases one would have to
evaluate all of them to calculate P-V -- not a simple task. One special
case that is easy though is the 2^{nd} order approximation (ie
terms in h^2 and h^4).

**2 ^{nd} order approximation**

Solutions:

and

Substituting the solution for r_{t} back in the expression for
surface deviation we get (we don't need c)

&_{t} back in the expression for
surface deviation we get (we don't need c)

Now this has local extrema at h=0, R/sqrt(2), and R, and therefore

and

independent of the shape of the curve, implying among other things that at this level of approximation statements about "P-V" errors are equivalent to statements about WRMS errors (apart from a constant factor). This confirms analytically what I speculated based on plugging numbers into a spreadsheet. The ratio I get differs by a few percent from the value given by Born & Wolf of 2 sqrt(3). Perhaps they based their analysis on unweighted RMS errors. Of course a mistake on my part can't be ruled out.

For a conic section, a binomial expansion gives A_{2} = (1+b)/(8r^{3}).
Substituting D=2R and f=r/2 we can find a criterion for acceptable defective
surfaces:

The value of k corresponding to Rayleigh's criterion (1/8 wave at the surface) is 12 sqrt(5), so rearranging we get

so rearranging we get

or at 550 nm,

which is within a fraction of a percent of the value given in Texereau (I think he adopts a default wavelength of 560nm, which would account for the small difference.

**Comments**

All of this analysis assumes the target is a paraboloid of freely adjustable focal length. For surfaces of interest to the ATM the difference in r between a given curve and it's best fitting target paraboloid will usually be measured in fractions of a millimeter -- for most ATM projects this is of no consequence. The same assumption is implicit in Texereau's foucault data reduction procedures. Of course if we're truly aiming at a fixed target the analysis will be different.

I haven't looked at the case where the target is a surface other than paraboloid. My guess is that the analysis would be much messier, but the mathematics here isn't particularly deep.

I haven't looked in any detail at cases where higher order terms in
the power series for the existing curve are significant. I anticipate though
that the constant relation between P-V and WRMS breaks down for realistic
curves. For example the broad defects that I'm getting right now on my
f/4 mirror result in estimated P-V/WRMS ~ 2 (I'm usingdefects that I'm getting right now on my
f/4 mirror result in estimated P-V/WRMS ~ 2 (I'm using a 3^{rd}
order estimate of the existing curve from foucault test data). Conversely
a few large mirrors for professional observatories that I've happened across
on the web have reported P-V/RMS ~ 10 or more. In the latter case I suspect
what's happening is that the use of local polishers is resulting in lots
of low amplitude, small scale errors, with a few much larger departures
thrown in.

Michael Peck

10 Sep 97

mailto:mpeck1@ix.netcom.com

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