The mathematical theory of multifocal lenses
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This paper presents the fundamental optical concepts of designing multifocal ophthalmic lenses and the mathematical methods associated with them. In particular, it is shown that the design methodology is heavily based on differential geometric ideas such as Willmore surfaces. A key role is played by Hamilton’s eikonal functions. It is shown that these functions capture all the information on the local blur and distortion created by the lenses. Along the way, formulas for computing the eikonal functions are derived. Finally, the author lists a few intriguing mathematical problems and novel concepts in optics as future projects.
KeywordsOptical design Differential geometry Eikonal functions Multifocal lenses
2000 MR Subject Classification35J30 35J50 78A05
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This paper is dedicated to Professor Haim Brezis for his 70th birthday. The author thanks his collaborators GershonWolansky, Dan Katzman and Sergio Barbero. He is pleased that Haim himself is using multifocal lenses that were designed using the principles explained above.
- Alvarez, L., Two-element variable power spherical lens, USPatent, 3305294, 1967.Google Scholar
- Barbero, S. and Rubinstein, J., Wide field-of-view lenses based on the Alvarez principle, Proc. SPIE 9626, Optical Systems Design; Optics and Engineering VI, 2015, 962614.Google Scholar
- Hamilton, W. R., Systems of rays, Trans. Roy. Irish Acad. 15, 1828, 69–178.Google Scholar
- Kanolt, C. K., Multifocal ophthalmic lenses, USPatent, 2878721, 1959.Google Scholar
- Katzman, D. and Rubinstein, J., Method for the design of multifocal optical elements, USPatent, 6302540, 2001.Google Scholar
- Kealy, L. and Friedman, D. S., Correcting refractive error in low income countries, British Medical J., 343, 2011, 1–2.Google Scholar
- Maitenaz, B. F., Ophthalmic lenses with a progressively varying focal power, USPatent, 3687528, 1972.Google Scholar
- Rubinstein, J. and Wolansky, G., Wavefront method for designing optical elements, USPatent, 6655803, 2003.Google Scholar
- Rubinstein, J. and Wolansky, G., Method for designing optical elements, USPatent, 6824268, 2004.Google Scholar
- Rubinstein, J. and Wolansky, G., A mathematical theory of classical optics, in preparation.Google Scholar