Abstract

When designing an optical system it is generally accepted that it is too costly to generate the entire Hessian for every iteration of an optimizer. However, the Hessian also is useful in tolerance analysis for which it needs to be calculated only once. We propose using the Hessian as part of a cost-based tolerancing procedure. Considerations for the general implementation of the proposed ideas are discussed, and the utility of this approach is demonstrated by way of an example. In the example optimal manufacturing tolerances are determined for a doublet. As expected, the optimal tolerances change as quantities such as the requisite image quality for finished systems, manufacturing yields, and relative expenses of meeting given tolerances are varied.

© 2000 Optical Society of America

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  1. 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.
  2. 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]
  3. 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.
  4. D. S. Grey, “Tolerance sensitivity and optimization,” Appl. Opt. 9, 523–526 (1970); D. S. Grey, “The inclusion of tolerance sensitivities in the merit function for lens optimization,” in Computer-Aided Optical Design, R. E. Fischer, ed., Proc. SPIE147, 63–65 (1978).
    [CrossRef] [PubMed]
  5. 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]
  6. A. P. Grammatin, A. B. Degen, “Evaluating the ease of manufacturing of optical systems,” Sov. J. Opt. Technol. 40, 464–465 (1973).
  7. 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]
  8. 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]
  9. 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]
  10. 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).
  11. 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).
  12. 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]
  13. 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.
  14. 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.
  15. 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.
  16. V. M. Chirkov, S. V. Pozdnov, Y. I. Sverchkov, “Calculation of tolerances on optical system parameters,” Sov. J. Opt. Technol. 53, 19–21 (1986).
  17. A. P. Grammatin, N. E. Kundeleva, “Calculation of tolerances on the construction elements of optical systems taking technological limits into account,” Sov. J. Opt. Technol. 48, 376–377 (1981).
  18. 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]
  19. 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.
  20. 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.
  21. 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.
  22. 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]
  23. 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.
  24. 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]
  25. 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]
  26. 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]
  27. This procedure is customarily referred to as a Monte Carlo analysis. The use of Monte Carlo analyses in assigning manufacturing tolerances is discussed, for example, by D. Heshmaty-Manesh, G. Y. Haig, “Lens tolerancing by desk-top computer,” Appl. Opt. 25, 1268–1270 (1986).
    [CrossRef] [PubMed]
  28. H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
    [CrossRef]
  29. M. P. Rimmer, “Analysis of perturbed lens systems,” Appl. Opt. 9, 533–537 (1970). Linear corrections to the optical path difference are used in the above-referenced paper. The second-order path-length correction are not included. These second-order corrections are essential, however, for the accurate modeling of figures of merit such as the mean-squared spot size, the mean-squared wave-front error, and the modulation transfer function.
    [CrossRef] [PubMed]
  30. 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.
  31. 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.
  32. 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.
  33. 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.
  34. 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.
  35. A local optimization is performed to determine the optimal focal-shift compensation for the perturbed systems in the exact case.
  36. 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.
  37. oslo is a registered trademark of Sinclair Optics, Inc., 6780 Palmyra Road, Fairport, N.Y. 14450.

1986 (2)

1981 (1)

A. P. Grammatin, N. E. Kundeleva, “Calculation of tolerances on the construction elements of optical systems taking technological limits into account,” Sov. J. Opt. Technol. 48, 376–377 (1981).

1973 (1)

A. P. Grammatin, A. B. Degen, “Evaluating the ease of manufacturing of optical systems,” Sov. J. Opt. Technol. 40, 464–465 (1973).

1970 (2)

1966 (1)

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Adams, G.

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]

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]

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]

Chirkov, V. M.

V. M. Chirkov, S. V. Pozdnov, Y. I. Sverchkov, “Calculation of tolerances on optical system parameters,” Sov. J. Opt. Technol. 53, 19–21 (1986).

Crawford, C.

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.

Degen, A. B.

A. P. Grammatin, A. B. Degen, “Evaluating the ease of manufacturing of optical systems,” Sov. J. Opt. Technol. 40, 464–465 (1973).

Drake, P.

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]

Durham, M. E.

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.

Flannery, B. P.

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.

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.

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.

George, R.

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]

Grammatin, A. P.

A. P. Grammatin, N. E. Kundeleva, “Calculation of tolerances on the construction elements of optical systems taking technological limits into account,” Sov. J. Opt. Technol. 48, 376–377 (1981).

A. P. Grammatin, A. B. Degen, “Evaluating the ease of manufacturing of optical systems,” Sov. J. Opt. Technol. 40, 464–465 (1973).

Grey, D. S.

Haig, G. Y.

Heshmaty-Manesh, D.

Hopkins, H. H.

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Hoyle, L.

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.

Jiang, H.

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]

Koch, D. G.

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]

Kundeleva, N. E.

A. P. Grammatin, N. E. Kundeleva, “Calculation of tolerances on the construction elements of optical systems taking technological limits into account,” Sov. J. Opt. Technol. 48, 376–377 (1981).

Li, G.

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]

Maxwell, J.

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]

Nelson, W.

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]

Odell, J.

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]

Papoulis, A.

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.

Parks, R. E.

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.

Plummer, J. L.

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]

Pozdnov, S. V.

V. M. Chirkov, S. V. Pozdnov, Y. I. Sverchkov, “Calculation of tolerances on optical system parameters,” Sov. J. Opt. Technol. 53, 19–21 (1986).

Press, W. H.

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.

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.

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.

Rimmer, M. P.

Slyusarev, G. G.

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.

Sverchkov, Y. I.

V. M. Chirkov, S. V. Pozdnov, Y. I. Sverchkov, “Calculation of tolerances on optical system parameters,” Sov. J. Opt. Technol. 53, 19–21 (1986).

Teukolsky, S. A.

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.

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.

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.

Thompson, K. P.

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.

Tiziani, H. J.

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Todd, C. D.

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]

Vetterling, W. T.

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.

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.

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.

Wang, Y.

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]

Welford, W. T.

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.

Willey, R. R.

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).

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.

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.

Appl. Opt. (3)

Br. J. Appl. Phys. (1)

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Sov. J. Opt. Technol. (3)

A. P. Grammatin, A. B. Degen, “Evaluating the ease of manufacturing of optical systems,” Sov. J. Opt. Technol. 40, 464–465 (1973).

V. M. Chirkov, S. V. Pozdnov, Y. I. Sverchkov, “Calculation of tolerances on optical system parameters,” Sov. J. Opt. Technol. 53, 19–21 (1986).

A. P. Grammatin, N. E. Kundeleva, “Calculation of tolerances on the construction elements of optical systems taking technological limits into account,” Sov. J. Opt. Technol. 48, 376–377 (1981).

Other (30)

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]

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.

oslo is a registered trademark of Sinclair Optics, Inc., 6780 Palmyra Road, Fairport, N.Y. 14450.

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.

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]

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Figures (8)

Fig. 1
Fig. 1

Sample CPDF: Systems with a figure of merit larger than the value of M max are rejected.

Fig. 2
Fig. 2

Flowchart for the proposed cost-based tolerancing procedure.

Fig. 3
Fig. 3

Variables to be toleranced in the doublet: (a) the perturbations that maintain symmetry, i.e., the three radii of curvature and the two glass thicknesses, (b) the tilt (shown in the figure about the x axis), (c) the roll of the last element about the center of curvature of the cemented surfaces.

Fig. 4
Fig. 4

Plots of (a) the PDF and (b) the relative expense function used in the example. The PDF shows that each parameter’s probability density is centered about the nominal system value ŝ and has limits that are defined by the tolerance range Δs. The relevant quantities for the individual expenses (specified in Table 4) are illustrated in (b).

Fig. 5
Fig. 5

Two plots of the exact and the estimated figures of merit for lines running between the corners of the variable space: (a) The case in which each parameter is taken from -Δs d to Δs d . (b) The parameters vary from minimum to maximum except for the last radius of curvature and the two glass thicknesses.

Fig. 6
Fig. 6

Bar chart that represents the tolerances for case 1 divided by the tolerances for case 2 for the seven variables in the example.

Fig. 7
Fig. 7

Bar charts that represent the relative costs for the optimal tolerances in cases 2–5.

Fig. 8
Fig. 8

Exact and approximate histograms representing the CPDF’s of the figures of merit for (a) case 2 and (b) case 5 of example. Parameters for (a): a bin size of M 0/600; lowest value of M of 0.992M 0; highest value of M of 1.094M 0; M max = 1.05M 0; R max = 1%. Parameters for (b): a bin size of M 0/100; lowest value of M of 0.981M 0; highest value of M of 11.15M 0; M max = 1.20M 0; R max = 5%.

Tables (6)

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Table 1 Lens Prescriptiona

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Table 2 Gradient of the Uncompensated Example System: (∂M/∂s i ) is Given for the s i That Appear in the First Row

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Table 3 Hessian of the Uncompensated Example System with the Elements (∂2M/∂s i s j ) a

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Table 4 Parameters That Specify the Individual Expense Functionsa

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Table 5 Parameters for the Five Different Cases of the Examplea

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Table 6 Resultant Optimal Tolerances for the Given Variables after Optimization of the Five Different Casesa

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

C=wR-Rmax2+i=1NciΔsiR>Rmaxi=1NciΔsiotherwise,
k=1N k=12N+1N
M=M0+s-sˆMs+12s-sˆ2Ms2s-sˆT+O3,
Ms=Ms1, Ms2, , MsN,
2Ms2=2Ms122Ms1s22Ms1sN2Ms2s12Ms222Ms2sN2MsNs12MsNs22MsN2.
MsiMsi+δsi-Msi-δsi2δsi,
2Msi2Msi+δsi+Msi-δsi-2Msiδsi2,
2MsisjMsi+δsi, sj+δsj-Msi-δsi, sj+δsj-Msi+δsi, sj-δsj+Msi-δsi, sj-δsj4δsiδsj,
M=M0+sv-sˆvMsv+sc-sˆcMsc+12sv-sˆv2Msv2sv-sˆvT+sc-sˆc2Msc2×sc-sˆcT+2sc-sˆc2Mscsvsv-sˆvT,
2Mscsv=2Msc,1sv,12Msc,1sv,22Msc,1sv,P2Msc,2sv,12Msc,2sv,22Msc,2sv,P2Msc,Qsv,12Msc,Qsv,22Msc,Qsv,P.
sc-sˆc=-2Msc2-1Msc+2Mscsvsv-sˆvT.
M=M0-12M2sc2Msc2-1Msc+sv-sˆvMsv-2Msvsc2Msc2-1Msc+12sv-sˆv2Msv2-2Msvsc×2Msc2-12Mscsvsv-sˆvT.

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