A merit function construction for which the global minimum may be found

Steve C. Johnston

doi:10.1117/12.949191

14 February 1986 A merit function construction for which the global minimum may be found

Steve C. Johnston

Proceedings Volume 0554, 1985 International Lens Design Conference; (1986) https://doi.org/10.1117/12.949191
Event: 1985 International Lens Design Conference, 1985, Cherry Hill, United States

Abstract

Merit functions used in optical design are almost always extremely nonlinear in their behavior with respect to optical system parameters. This prevents the existing optimization algorithms from necessarily converging on the global minimum. They converge to the nearest local minimum which may, or may not, be the global mini- mum. It is possible that the global minimum can be achieved by successively minimizing a sequence of increasingly nonlinear merit "subfunctions" that converge to the complete merit function. The complete merit function is constructed in the usual way: Ð¤ = X23 + X25 + X27+ . . . . . (1) However, each aberration Xi is composed of a special linear combination of aberration coefficients. The sub-script i refers to the lowest order of aberration coefficient present in the aberration Xi. The complete merit function can be either mean-square spot size or wavefront variance by a proper choice of the linear combinations of aberration coefficients. Starting with the highest order aberration coefficients considered significant to the design problem (say, order nine), the sequence of merit subfunctions is constructed as Ð¤ = Ð¤3 = X23 + X25 + X27 + X29 Ð¤5 = X25 + X27 + X29 Ð¤7 = X27 + X29 Ð¤9 = X29. (2) Note that Ð¤9 behaves much more simply with respect to changes in the parameters of the optical system than does Ð¤. Also, Ð¤9 must be small at the global minimum of Ð¤ so that minimizing (09 may result in the optical system being in the region of the global minimum of Ð¤. By minimizing sequentially Ð¤9, Ð¤7, etc., the optical system may be sequentially moved to the global minimum of the complete merit function Ð¤.

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Steve C. Johnston "A merit function construction for which the global minimum may be found", Proc. SPIE 0554, 1985 International Lens Design Conference, (14 February 1986); https://doi.org/10.1117/12.949191

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KEYWORDS

Lens design

Optical design

Wavefronts

Optimization (mathematics)

Chromatic aberrations

Colorimetry

Wavefront aberrations

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