<title>A Statistical Approach To Lens Tolerancing</title>

A common procedure in modern lens tolerancing is to use Monte Carlo, or direct random simulation models of manufacturing errors to determine the probability of achieving certain performance levels. The results of such calculations are usually the probability of achieving specific performance levels as a minimum, and provide an excellent overview of the impact of a set of manufacturing tolerances. Unfortunately, the use of such methods generally requires a high speed computer, and the results are not easily modified for tolerance changes without performing additional simulations. As an alternative approach to direct simulation, a second moment statistical method is described that provides a simple algebraic computational scheme to estimate the same probabilities. The methods allow direct computation on a desk calculator once parameter sensitivities are known and are also useful in substantially reducing the amount of time required when using a high speed computer. Applications to linearly related phenomena (such as focal length, distortion, back focal distance, etc.) as well as non-linear phenomena (e.g., RMS wavefront error, MTF at some frequency, etc.) are discussed. Commonly used techniques such as finding the root of the sum of the squares of perturbations (RSS) are reviewed and shown to be a subset of this more accurate and general approach. Some discussion of accuracy compared to direct simulation is included.