Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 647–660

# The mathematical theory of multifocal lenses

Article

## Abstract

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.

### Keywords

Optical design Differential geometry Eikonal functions Multifocal lenses

### 2000 MR Subject Classification

35J30 35J50 78A05

## Notes

### Acknowledgments

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.

### References

1. [1]
Alvarez, L., Two-element variable power spherical lens, USPatent, 3305294, 1967.Google Scholar
2. [2]
Barbero, S. and Portilla, L., Geometrical interpretation of dioptric blurring and magnification in ophthalmic lenses, Optic Express, 23, 2015, 13185–13199.
3. [3]
Barbero, S. and Rubinstein, J., Adjustable-focus lenses based on the Alvarez principle, J. Optics, 13, 2011, 125705.
4. [4]
Barbero, S. and Rubinstein, J., Power-adjustable sphero-cylindrical refractor comprising two lenses, Optical Eng., 52, 2013, 063002.
5. [5]
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
6. [6]
Bourdoncle, B., Chauveau, J. P. and Mercier, J. L., Traps in displaying optical performance of a progressive addition lens, Applied Optics, 31, 1992, 3586–3593.
7. [7]
Campbell, C., The refractive group, Optometry and Vision Science, 74, 1997, 381–387.
8. [8]
Hamilton, W. R., Systems of rays, Trans. Roy. Irish Acad. 15, 1828, 69–178.Google Scholar
9. [9]
Kanolt, C. K., Multifocal ophthalmic lenses, USPatent, 2878721, 1959.Google Scholar
10. [10]
Katzman, D. and Rubinstein, J., Method for the design of multifocal optical elements, USPatent, 6302540, 2001.Google Scholar
11. [11]
Kealy, L. and Friedman, D. S., Correcting refractive error in low income countries, British Medical J., 343, 2011, 1–2.Google Scholar
12. [12]
Keller, J. B. and Lewis, R. M., Asymptotic methods for partial differential equations: The reduced wave equation and Maxwell’s equations, Surveys in Applied Mathematics, 1, 1993, 1–82.
13. [13]
Landau, L. D. and Lifshitz, E. M., Theory of Elasticity, Pergamon Press, New York, 1986.
14. [14]
Luneburg, R. K., The Mathematical Theory of Optics, UCLA Press, California, 1964
15. [15]
Maitenaz, B. F., Ophthalmic lenses with a progressively varying focal power, USPatent, 3687528, 1972.Google Scholar
16. [16]
Minkwitz, G., Uber den Flachenastigmatismus Bei Gewissen Symmetruschen Aspharen, Opt. Acta, 10, 1963, 223–227.
17. [17]
Rubinstein, J., On the relation between power and astigmatism of a spectacle lens, J. Opt. Soc. Amer., 28, 2011, 734–737.
18. [18]
Rubinstein, J. and Wolansky, G., A class of elliptic equations related to optical design, Math. Research Letters, 9, 2002, 537–548.
19. [19]
Rubinstein, J. and Wolansky, G., Wavefront method for designing optical elements, USPatent, 6655803, 2003.Google Scholar
20. [20]
Rubinstein, J. and Wolansky, G., Method for designing optical elements, USPatent, 6824268, 2004.Google Scholar
21. [21]
Rubinstein, J. and Wolansky, G., A mathematical theory of classical optics, in preparation.Google Scholar
22. [22]
Walther, A., The Ray and Wave Theory of Lenses, Cambridge University Press, Cambridge, 1995.
23. [23]
Wang, J., Gulliver, R. and Santosa, F., Analysis of a variational approach to progressive lens design, SIAM J. Appl. Math., 64, 2003, 277–296.