The uncertainty equations presented here have been derived using standard
principles of error propagation (see for example D.C. Baird, "Experimentation:
*An Introduction to measurement theory and experiment design*", 3^{rd}
edition, Prentice Hall, Inc., New Jersey, 1995). I have also tested all of
the (more complex) uncertainty equations using a numerical minimum-maximum
searching routine (explained below) to ensure that the error estimates produced
are reasonable.

To find the radius of curvature R using a spherometer with a spherometer constant r and a sagitta s, we use the equation (spherometer equation):

The spherometer constant r is determined by taking the average distance between the central contact point and the three outer contact points with the spherometer zeroed. The uncertainty in r determined in this way is the uncertainty for a single measurement multiplied by the square root of 3.

The uncertainty R in the radius of curvature R is given by,

where s is the uncertainty in the sagitta and r is the uncertainty in the spherometer constant. Now, if one wants to know the sagitta that will give a particular radius of curvature then one uses,

The tolerance T_{s} required in the measurement of the sagitta,
given the tolerance T_{R} in the radius of curvature is given by,

For example, if a tolerance of 0.02% is required in the radius of curvature
R=10" using a spherometer with arms of length 3.5", the required accuracy
in reading the sagitta must exceed 0.00015" (which is close to the limit of
mechanical length measurement).

Now here's the tricky part; determining the uncertainty in the difference
(R_{2} - R_{1}), where R_{1} and R_{2} are
the radii curvature of the front and back of the lens respectively. This
isn't as simple as calculating the difference (R_{2} - R_{1})
and taking the uncertainty as being the square root of the sum of the squared
uncertainties of R_{1} and R_{2}. The uncertainty would be
overestimated if one were to use this approach. Instead we compute the expression
for the difference in terms of the sagittae s_{1} and s_{2}
on both sides of the corrector directly.

This statement may appear confusing, even self-contradictory and you might
think: "Weren't the radii calculated from the sagittas anyway, so what does
it matter if you compute (R_{2} - R_{1}) directly from the
radii instead of using the sagittas?" Well, in fact you're right; it doesn't
matter how you compute (R_{2} - R_{1}), of course you'll
come out with the same answer either way; but when it comes to computing the
uncertainty in the difference, (R_{2} - R_{1}) it matters
a lot. This is because the radius of curvature is not linearly related to
the sagitta and therefore the uncertainties propagate non-linearly as well.

Here's how the uncertainty is determined. We begin with the spherometer equation [1] to find the difference between the radii of curvature; explicitly,

where s_{1} and s_{2} are the sagittas of the front and
back of the lens respectively. A bit of rearrangement of [5] gives,

You can try this equation to determine (R_{2} - R_{1})
and find that it gives the same result as computing (R_{2} - R_{1})
directly. The advantage now is that we can propagate the errors properly because
we have an expression for (R_{2} - R_{1}) written in terms
of primitive variables only (i.e. the actual variables you measure). The
uncertainty in (R_{2} - R_{1}) is then given by the somewhat
ghastly expression,

I tried this equation using some typical data and it agreed very well with
an estimate of the uncertainty based on finding the absolute maxima and minima
of (R_{2} - R_{1}) in the parameter space defined by the
span of the variables s_{1}, s_{2} and r (this is the intuitive
way most of us try to work out uncertainties when the algebra gets too messy).
This is a long way of saying: it works.

Just to give you an idea of the magnitude of the accuracy typically available
in determining (R_{2} - R_{1}), lets choose:

We then find that the difference in the radii of curvature (R_{2}
- R_{1}) is 0.4313 ± 0.0025", which amounts to an uncertainty
of ~0.02% of R_{1}. Had we simply added the uncertainties for R_{1}
and R_{2 }in quadrature, specifically,

Then we would have found (R_{2} - R_{1}) = 0.017", which
is almost 7 times greater than the uncertainty computed using equation [7].

When using my wedge control device,
I found it convenient to also use it to determine the central thickness of
the lens. This was achieved by measuring the thickness t_{r} of the
lens (as measured in a direction parallel to the optical axis) at a distance
r_{m} from the centre of the lens.

The central thickness t was then calculated using t_{r} and the
radii of curvature R_{1} and R_{2} of the front and back
of the lens respectively. For a meniscus lens where R_{1} is shorter
than R_{2} we have,

When computing the uncertainty here one must be careful because the absolute
uncertainty in (R_{2} - R_{1}) is much less than that in
R_{2} and R_{1} taken individually, as explained earlier
(the relative uncertainties however are roughly the same in both cases).
One way to proceed is to expand [8] in terms of (R_{2} - R_{1});
this is convenient because we've already worked out the uncertainty (R_{2}
- R_{1}). Only keeping the lowest order terms in the expansion we
get,

This relation only gives a rough estimate of t, and it is only valid for
a meniscus lens where (R_{2} - R_{1}) is much smaller than
either R_{2} or R_{1}, however it contains the principal variations
of t with respect to R (i.e.: let R be either R_{2} or R_{1},
whichever has the greater uncertainty) and the difference (R_{2}
- R_{1}). It is then a simple (though somewhat tedious) matter
to compute the uncertainty in t,

Where we have defined R as (R_{2} - R_{1}), with it's associated
uncertainty R. This isn't a completely rigorous expression (because it is
derived from [9] which is only an approximate expression), however it agrees
well with the uncertainty determined by numerically finding the minimum of
t within the space defined by the span of the variables R, R, r_{m}
and t_{r}.

The accuracy of the thickness determined in this way isn't great but it
suffices for working a Maksutov corrector lens. For example, if t_{r}
= 0.800 ± 0.001, R_{1} = 10 ± 0.01, R_{2} =
10.44 ± 0.01 and r = 3.00 ± 0.003, then the centre thickness
will be; 0.780 ± 0.0017. The relative error in the thickness here is
0.2%, which is well within the 2% limit specified for the thickness in a
Maksutov corrector lens.

Sometimes one needs to propagate uncertainties through a complex expression
(as in some of the ones discussed above) and there won't be a clear way to
proceed analytically. The best (i.e. most robust) technique to use in such
a case is to make an exhaustive search for the maximum and minimum of the
expression by trying all possible values of the variables (within the span
of their uncertainties) in the expression. This is a perfectly valid
way of propagating uncertainties; in fact, all of the uncertainty relations
discussed previously were tested by this method. If several variables
are involved, and a fine search grid is used this might take a long time
to compute, but you'll * always* get the right answer. Note
that you shouldn't expect the uncertainties computed in this way to exactly
match those calculated using algebraic error propagation (e.g. the previously
defined equations), but the two should agree within a factor of 1.5 or so
...and that's good enough for us. This approach is also useful to make
sure that you haven't made mistakes when coding the above equations into
your analysis program.