Engineering tolerance is the limit of permissible variation in a product's dimension or properties. Determination of tolerances is very important in the manufacture of optical systems. Yield and manufacturing cost are determined by the tolerance set. For the target yield, the cost must be minimized. Under the allowed cost, the yield must be maximized. In general, this optimization problem is large-scale because the tolerance set has tens or hundreds of variables. Monte Carlo simulation (using random numbers) of the estimated yield takes considerable time because evaluating system performance requires many repetitions. The optimization method suitable for this problem was previously unknown, but it was expected that the necessary number of Monte Carlo simulations would be on the order of the square of the parameter number. An ordinary lens optimization with 100 independent variables takes on the order of a minute to an hour (see Figure 1). Consequently, applying a similar method to tolerance optimization is estimated to require a day to week, with some researchers suggesting that it would most likely take a few days.

**Figure 1. **Section drawing of a sample facsimile lens. This type of lens typically shows good performance but has high sensitivity to manufacturing errors.

We propose a new optimization method that requires only a few or a few tens of Monte Carlo simulations, regardless of the number of independent variables.^{1} With this method, determining the optimal tolerance set would become the work of less than a few hours. The method is based on the linear relation between the variances or averages of performance criteria and the variances of tolerance parameters.

Figure 2 illustrates how the method works. Performance criteria can be characterized as a first-order model or a second-order model, depending on their nature. Tolerance parameters need to be treated as statistical variables. Variances or averages of performance criteria can be expressed as linear combinations of tolerance parameter variances, as long as the performance criteria are treated as first-order or second-order models. If the variance of a performance criterion becomes larger, the probability for the criterion to be outside of the allowed interval will increase. If the average of a performance criterion becomes lower, the probability for the criterion to be lower than the threshold will increase. In both cases the manufacturing yield would decrease. To realize the target yield, variances and average loss of the performance criteria need to be kept small.

**Figure 2. **Schematic illustration of tolerance optimization method.

The manufacturing cost, which also depends on the tolerances, can be expressed as a function of the variances of the tolerance parameters. Instead of controlling the yield of the Monte Carlo simulation, constraints are placed on variances or averages of performance criteria, and the manufacturing cost is minimized under these constraints. To see whether the given constraints are appropriate, we use a Monte Carlo simulation with the optimized tolerance set. This means that the yield and the manufacturing cost are determined as functions of the constraints on performance criteria averages or variances. Finding the ideal combination of yield and cost then becomes an optimization problem with constraints on variances or averages of performance criteria as independent variables. This optimization is simple because the number of critical performance criteria is very small, and there are no constraints on the values of independent variables.

We have applied this method to a variety of lens types, such as facsimile lenses, CCTV lenses, and a night scope objective lens. In each case, it took about five Monte Carlo simulation trials to find the optimal tolerance set.

This rapid optimization method could enable manufacturing cost reductions and reliable yield improvements. However, for the optimization result to be truly optimal, the yield estimation and the cost model need to reflect reality. Both of these elements involve challenges and are not yet satisfactory. For example, for axially asymmetric lens manufacturing errors such as center displacement, tilt, or surface irregularity, modeling the perturbation distribution from tolerance values remains a complicated problem. The relation between the tolerance and the manufacturing cost needs to be investigated through large-scale statistical processing. Now that breakthroughs in optimization are ongoing, the importance of these tasks is greater than ever. As a next step, we are planning a project to improve the modeling of perturbed optical systems.

**Akira Yabe**

Landsberg am Lech, Germany.

Akira Yabe received a BSc in physics from Tokyo University (1978). He worked for Fuji Photo Optical from 1978 to 2003. Since 2004 he has worked as an independent lens design consultant.

References:

1. A. Yabe, Rapid optimization of cost-based tolerancing, *Appl. Opt.* 51(7), p. 855-860, 2012.