Progressive Addition Lens Anatomy

With digital lens surfacing (a.k.a freeform) one or both surfaces of a lens can be completely specified by the Lens Design System (LDS). For example the LDS can provide a height map, giving points on a grid of some chosen spacing. This surface will be produced with very high accuracy by the generator.

The optical performance of a freeform lens when used in spectacles, is difficult to predict. Either the lens must be manufactured and analyzed with a complex apparatus that can measure optical properties for a range of realistic view directions, or the lens must be simulated computationally using sophisticated ray-tracing algorithms.

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Zernike Polynomials

Truncated expansions such as Zernike polynomials provide a powerful approach for describing wavefront data. However, many simple calculations with data in this form can require significant computational effort. Important examples include recentering, renormalizing, and translating the wavefront data. This paper describes a technique whereby these operations and many others can be performed with a simple matrix approach using monomials. The technique may be applied to other expansions by reordering the data and applying transformations. The key is the use of the vectorization operator to convert data between vector and matrix descriptions. With this conversion, one-dimensional polynomial techniques may be employed to perform separable operations. Examples are also given for differentiation and integration of wavefronts.

Keith Dillon, “Bilinear wavefront transformation,” J. Opt. Soc. Am. A 26, 1839-1846 (2009) (pdf)