Spherometry Equations

In the following I will assume that the reader has not had exposure to error analysis (...or promptly forgot it after their first college year), so that just about anyone with a calculator (preferably a programmable one, or better yet a computer) can get this right. It's worth fussing over this because the tolerances involved in making a Maksutov corrector are rather severe. It is impossible to proceed without having accurate estimates of the uncertainties involved in the radius of curvature measurements.

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", 3rd 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):

, [1]

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,

, [2]

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,

, [3]

The tolerance Ts required in the measurement of the sagitta, given the tolerance TR in the radius of curvature is given by,

, [4]

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 (R2 - R1), where R1 and R2 are the radii curvature of the front and back of the lens respectively. This isn't as simple as calculating the difference (R2 - R1) and taking the uncertainty as being the square root of the sum of the squared uncertainties of R1 and R2. 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 s1 and s2 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 (R2 - R1) directly from the radii instead of using the sagittas?" Well, in fact you're right; it doesn't matter how you compute (R2 - R1), of course you'll come out with the same answer either way; but when it comes to computing the uncertainty in the difference, (R2 - R1) 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,

, [5]

where s1 and s2 are the sagittas of the front and back of the lens respectively. A bit of rearrangement of [5] gives,

, [6]

You can try this equation to determine (R2 - R1) and find that it gives the same result as computing (R2 - R1) directly. The advantage now is that we can propagate the errors properly because we have an expression for (R2 - R1) written in terms of primitive variables only (i.e. the actual variables you measure). The uncertainty in (R2 - R1) is then given by the somewhat ghastly expression,

, [7]

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 (R2 - R1) in the parameter space defined by the span of the variables s1, s2 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 (R2 - R1), lets choose:

r = 3.5", r = 0.002", s1 = 0.6000", s2 = 0.5975", s1 = s2 = 0.0001"

We then find that the difference in the radii of curvature (R2 - R1) is 0.4313 ± 0.0025", which amounts to an uncertainty of ~0.02% of R1. Had we simply added the uncertainties for R1 and R2 in quadrature, specifically,


Then we would have found (R2 - R1) = 0.017", which is almost 7 times greater than the uncertainty computed using equation [7].

Thickness equations:

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 tr of the lens (as measured in a direction parallel to the optical axis) at a distance rm from the centre of the lens.

The central thickness t was then calculated using tr and the radii of curvature R1 and R2 of the front and back of the lens respectively. For a meniscus lens where R1 is shorter than R2 we have,

, [8]

When computing the uncertainty here one must be careful because the absolute uncertainty in (R2 - R1) is much less than that in R2 and R1 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 (R2 - R1); this is convenient because we've already worked out the uncertainty (R2 - R1). Only keeping the lowest order terms in the expansion we get,

,   or  , [9]

This relation only gives a rough estimate of t, and it is only valid for a meniscus lens where (R2 - R1) is much smaller than either R2 or R1, however it contains the principal variations of t with respect to R (i.e.: let R be either R2 or R1, whichever has the greater uncertainty) and the difference (R2 - R1).  It is then a simple (though somewhat tedious) matter to compute the uncertainty in t,

, [10]

Where we have defined R as (R2 - R1), 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, rm and tr.

The accuracy of the thickness determined in this way isn't great but it suffices for working a Maksutov corrector lens. For example, if tr = 0.800 ± 0.001, R1 = 10 ± 0.01, R2 = 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.

What to do when intuition fails

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.

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