Freeform surface of progressive addition lens represented by Zernike polynomials

We used the explicit expression of Zernike polynomials in Cartesian coordinates to fit and describe the freeform surface of progressive addition lens (PAL). The derivatives of Zernike polynomials can easily be calculated from the explicit expression and used to calculate the principal curvatures of freeform surface based on differential geometry. The surface spherical power and surface astigmatism of the freeform surface were successfully derived from the principal curvatures. By comparing with the traditional analytical method, Zernike polynomials with order of 20 is sufficient to represent the freeform surface with nanometer accuracy if dense sampling of the original surface is achieved. Therefore, the data files which contain the massive sampling points of the freeform surface for the generation of the trajectory of diamond tool tip required by diamond machine for PAL manufacture can be simplified by using a few Zernike coefficients.