Tolerancing is an important part of optical design.  Throughout tolerancing, various tolerances are relaxed or tightened, depending upon their contribution to the change in criteria, and/or whether the tolerance is within budget.  This process in repeated again and again until we are satisfied that the performance is within specification, and that the tolerances are realistic and do not unnecessarily increase the cost of manufacture. 

In this repetitive process of changing tolerances and re-evaluating the performance of our optical system, it is likely that only a few tolerances need to be adjusted.  Therefore, it isn’t necessary to recalculate every tolerance operand over and over again.  Furthermore, for those tolerances which were adjusted, we can quickly compute the criteria change using a fitted formula.  The benefit: potentially hundreds of hours of computation time can be saved!  This is Polynomial Sensitivity Tolerancing.

For most tolerances, the criteria value has a smooth varying shape as a function of the tolerance perturbation.  In most cases, this change in criteria curve can be accurately fit using either a 3 or 5 term polynomial.  Only if tolerances were to loose that the design could find a different local minimum would this not be true.

Imagine a multi-element imaging system where we plot the RMS Spot Radius as a function of the decentration of the first lens element.

Within the tolerance range, the curve is well behaved, and can easily be fit to a polynomial.  In ZEMAX, we can fit the actual criteria curve with a 3 or 5 term polynomial of the following form:

where δ is the tolerance perturbation value and P is the resulting criterion.  For the 3-term fit, a total of 4 equally spaced points are used within the minimum and maximum tolerance values.  A total of 6 points are used for the 5-term fit. 

Using the same example as above, a 3 term fit would require an evaluation of the criteria at 4 equally spaced perturbations, as is shown in the figure below.

Once the criteria values are calculated, a 3 term polynomial is fit.  Note that the fit is created to minimize the amount of error between the resulting criteria at the 4 perturbation points.  Using a finite number of polynomial terms, the fit may not actually pass through each (or any) of the actual criteria values at these four points. 

In the current example, the fit would look something like:

There is virtually no difference between the fit (determined by only 4 data points) and the curve of the actual criteria as a function of the tolerance perturbation!

The beauty of this is that once we have a mathematical equation for how the criterion varies as a function of the tolerance perturbation, subsequent calculations are almost instantaneous!  For example, if we choose to tighten the tolerance on the decenter of our first lens element (as in the demonstration above), there is no need to recalculate the criteria by brute force.  Instead, we simply plug the tolerance perturbation into our fitted polynomial equation and evaluate the new criteria.