G. Adams, “Selection of tolerances,” in Simulation and Modeling of Optical Systems, R. E. Fischer, D. C. O’Shea, eds., Proc. SPIE892, 173–185 (1988).

[CrossRef]

R. R. Willey, M. E. Durham, “Maximizing production yield and performance in optical instruments through effective design and tolerancing,” in Optomechanical Design, P. R. Yoder, ed., Vol. CR43 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1992), pp. 76–108.

R. R. Willey, R. E. Parks, “Dimensional tolerances and budgets,” in Handbook of Optomechanical Engineering, A. Ahmad, ed. (CRC Press, Boca Raton, Fla., 1997), pp. 14–38.

In this paper, we discuss and give an example incorporating only individual expenses for the variables to be toleranced. In general, any individual expense associated with the inclusion of compensators should be taken into account in a manner analogous to the method prescribed here.

A recent paper recommends that the system’s mechanical performance be fed into the optical design. For more details on this procedure, see P. Drake, “Auto-tolerancing on optical systems (six sigma approach to designing optical systems),” in Lens Design, Illumination, and Optomechanical Modeling, R. E. Fischer, ed., Proc. SPIE3130, 136–147 (1997).

[CrossRef]

For an introduction to PDF’s, see, for example, A. Papoulis, in Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991), Chaps. 4 and 5.

Examples of a variety of individual PDF’s are given, for example, in D. G. Koch, “A statistical approach to lens tolerancing,” in Computer-Aided Optical Design, R. E. Fischer, ed., Proc. SPIE147, 71–82 (1978).

[CrossRef]

D. G. Koch, “A statistical approach to lens tolerancing,” in Computer-Aided Optical Design, R. E. Fischer, ed., Proc. SPIE147, 71–82 (1978). That the central-limit theorem is not valid is mentioned in the errata sheet published in G. Wiese, ed., Selected Papers on Optical Tolerancing, Vol. MS36 of the SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1992), p. 383.

[CrossRef]

G. Adams, “Some statistical aspects of tolerancing,” in Optical System Design, Analysis, and Production for Advanced Technology Systems, R. E. Fischer, P. J. Rogers, eds., Proc. SPIE655, 67–79 (1986).

[CrossRef]

J. L. Plummer, “Tolerancing for economies in mass production of optics,” in Contemporary Optical Systems and Components Specifications, R. E. Fischer, ed., Proc. SPIE181, 90–92 (1979).

[CrossRef]

R. R. Willey, R. George, J. Odell, W. Nelson, “Minimized cost through optimized tolerance distribution in optical assemblies,” in Optical Systems Engineering III, W. H. Taylor, ed., Proc. SPIE389, 12–17 (1983).

[CrossRef]

R. R. Willey, “Economics in optical design, analysis, and production,” in Optical System Design, Analysis, and Production, P. J. Rogers, R. E. Fischer, eds., Proc. SPIE399, 371–377 (1983).

[CrossRef]

R. R. Willey, “The impact of tight tolerances and other factors on the cost of optical components,” in Optical Systems Engineering III, P. R. Yoder, ed., Proc. SPIE518, 106–111 (1984).

R. R. Willey, “Optical design for manufacture,” in Recent Trends in Optical Systems Design II, R. E. Fischer, R. C. Juergens, eds., Proc. SPIE1049, 96–102 (1989).

H. Jiang, G. Li, Y. Wang, “Making optical tolerances with regard to cost and performance,” in Current Developments in Optical Engineering III, R. E. Fischer, W. J. Smith, eds., Proc. SPIE965, 174–177 (1988).

[CrossRef]

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, Bristol, UK, 1984), Chap. 8; W. J. Smith, Modern Optical Engineering, 2nd ed. (McGraw-Hill, New York, 1990), Sec. 14.2. Both volumes mention the practical considerations in tolerancing, such as fixing manufacturing parameters by use of test-plate fittings as well as other optical shop legerdemain to eliminate complexity.

K. P. Thompson, L. Hoyle, “Key to cost-effective optical systems: maximize the number of qualified fabricators,” in Proceedings of the International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 208–213.

C. Crawford, “Extensions of the linear model for tolerance computation,” in Proceedings of the International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 162–166.

Inclusion of aberrational weight adjustments by use of tolerances is investigated in C. D. Todd, J. Maxwell, “The maximum transverse ray aberrations permissible in an optical system,” in Design and Engineering of Optical Systems, J. J. M. Braat, ed., Proc. SPIE2774, 28–45 (1996); C. D. Todd, J. Maxwell, “Aberrational weight adjustment by tolerance-based weighting in damped least-squares optimization,” in Design and Engineering of Optical Systems, J. J. M. Braat, ed., Proc. SPIE2774, 89–105 (1996).

[CrossRef]

Myriad figures of merit are used to quantify the image quality of optical systems. The best choice for any given system depends on the particular application and constraints. It is assumed in this study that lower values for the figure of merit correspond to better performance. The mean-squared wave-front error is taken to be the figure of merit in the example discussed in Section 5. The mean-squared wave-front error is defined by use of the wave-aberration function that can be found, for example, in W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1991), Chap. 7, pp. 92–105.

The effects of manufacturing errors on the mean-squared wave-front error are generally nonlinear. For a discussion of these types of figures of merit in a statistical framework, see G. Adams, “Some statistical aspects of tolerancing,” in Optical System Design, Analysis, and Production for Advanced Technology Systems, R. E. Fischer, P. J. Rogers, eds., Proc. SPIE655, 67–79 (1986).

[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art and Science of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1995), Sec. 10.5.

During manufacture some system parameters can be adjusted to compensate for measured errors in other parameters. Such compensators are an integral part of the tolerancing process, and any tolerancing method must take them into account.

This estimate is obtained by the noting of the Hessian as a symmetric matrix. Note that this is a worst-case situation, as some derivatives do not require evaluation in practice. For instance, a mixed derivative with respect to a variable that breaks symmetry and another variable that maintains symmetry is zero. This point is discussed further in Subsection 3.C.

Numerical differentiation is discussed by W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art and Science of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1995), Sec. 5.7.

Recall that the Hessian is a symmetric matrix. This fact is explicitly used here because the transpose of a symmetric matrix equals the matrix itself.

The curvatures of the cemented surfaces are taken to be equal in this example. Thus when the second element rolls with respect to the first they remain perfectly in contact.

A local optimization is performed to determine the optimal focal-shift compensation for the perturbed systems in the exact case.

The simplex algorithm of Nelder and Mead is described, for example, by W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art and Science of Scientific Computing, 2nd ed. (Cambridge U. Press, New Your, 1995), Sec. 10.4.

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