Progressive addition lenses (PALs) not only solve the problem that the elderly need different focal powers for distance and near visions but also overcome the image jump of bifocal lenses owing to their continuously varying focal power along the meridian. Therefore, PAL design methods have a broad development prospect. PALs are mainly designed by a direct method and an indirect method. Although the indirect method offers convenient design, its prism is large when the focal power changes. The design of PALs by the direct method has the disadvantage that the maximum peripheral astigmatism exceeds two times the addition power (ADD) according to the Minkwitz theorem. Nevertheless, the direct method has relatively favorable advantages for the distance and near areas. The optimization of the direct method in China and abroad mainly focused on the changes in the focal power along the meridian, the optimization of the local average focal power, and the calculation equations for different surface heights. In contrast, how to obtain reasonable focal power profile distributions on the lenses has rarely been reported. In addition, a conic parametric equation can be changed into an equation satisfying the conditions of the focal power profile distribution along the meridian. Therefore, this paper proposes a method based on a conic parametric equation to reasonably distribute the focal power profile on the entire lens, achieve the design of PALs by the direct method, and ultimately reduce peripheral astigmatism.

On the basis of the principles of geometry and the direct method, the law of curvature change along the meridian of the lens is determined to satisfy the trigonometric function in this study, and the surface height equation is calculated as a spherical equation. Specifically, the function of the focal power profile distribution is solved by the circular parametric equation and also by the hyperbolic, parabolic, and elliptic equations. Then, the four sets of surface height data are used to simulate the focal power and astigmatism of the lenses in the simulation software. The designed lenses are processed and tested by the free-form surface machining machine, and the experimental results are verified. Finally, the influences of different conic parametric equations, i.e., hyperbolic, parabolic, elliptic, and circular equations, are analyzed for the optical properties, such as the focal power and astigmatism, of the lenses.

Theoretical analysis, actual processing, detection, and comparison reveal that the method of solving the focal power profiles of lenses by conic parametric equations is feasible (Fig. 7). The actual focal power in the distance area and ADD of the four groups of lenses meet the national standard (GB 10810.1—2005). The conic parametric equations mainly include the hyperbolic, parabolic, elliptic, and circular equations, and different equations can be used to design the focal power profile on the whole lens. Moreover, the width of the distance and near areas can be set as required, and the distribution of peripheral astigmatism can be adjusted. With the same parameter (Table 1), the maximum peripheral astigmatism of the lens obtained by solving the hyperbolic equation is 1.36 times the ADD, and the visual effect is relatively poor (Table 2); the distortion of the lens on the periphery of its area of the fixed focal power calculated by the elliptic equation is the smallest. The maximum peripheral astigmatism of the lens obtained by solving the parabolic equation is the smallest, and the corresponding ratio of the maximum peripheral astigmatism of the lens to the ADD is also the smallest. The area with peripheral astigmatism larger than 1.75 D on the lens obtained by solving the elliptic equation is relatively small (Fig. 8), and the width of the actual visible area at the fixed focal power point in the distance area is the largest. Therefore, the elliptic equation can be used as the basis for further optimization design in the future.

In this paper, the focal power profile distributions obtained by four different conic equations are proposed to design PALs, and the four groups of lenses are simulated, evaluated, and processed. The results show that the focal power profile distributions obtained by different conic equations have an impact on the design of PALs. With the design parameters, the peripheral astigmatism of the lens obtained by solving the hyperbolic equation is large and thus highly likely to cause severe vertigo when people wear such lenses to look around. In contrast, the peripheral astigmatism of the lenses calculated by the elliptic, circular, and parabolic equations is all smaller than that calculated by the hyperbolic equation. On this basis, variables are added to control the size of the distance and near areas so that the visible area at the fixed focal power point can be adjusted. However, a larger area of the fixed focal power corresponds to larger distortion and dispersion of the spherical lens. Consequently, the actual width at the fixed focal power point will be smaller than the theoretical value. When the area of the fixed focal power is small, the distortion of the PAL designed on the basis of the circular equation on the periphery of its area of the fixed focal power is the smallest. In future research, the inner surface of PALs can be transformed from a spherical design to an aspherical design on the basis of this design method to further study the optimal design of PALs, reduce the tangential error, and obtain more accurate actual results.