| Progressive addition lenses are used to treat “presbyopia”, a common visual problem for people of age over forty. A progressive addition lens comprises a large distance view zone with low power on the upper part of the lens and a small near view zone with higher power on the lower part; between these two zones, the power increases progressively and smoothly. In progressive addition lenses, the progression of the power is the result of a local variation in the curvatures of the surface. An ideal progressive surface is one with the prescribed smooth progressive power and with zero astigmatism everywhere. But in order for the astigmatism to be zero everywhere, the surface has to be either a sphere or a plane, which will not provide progressive power. In this work, we consider a variational approach to the progressive addition lens design problem, in which, a cost function which attempts to balance the power distribution with the unavoidable cylinder is created. The goal is to minimize the cost function.; By variational principal, we derive the Euler-Lagrange equation which is a nonlinear fourth-order elliptic partial differential equation. We consider two linearizations of the equation and for each linearization, we analyze three kinds of boundary conditions: (1) Clamped boundary condition; (2) Partially clamped boundary condition; (3) Natural boundary condition. For all the three boundary conditions, we prove the existence and uniqueness as well as the regularity of the solutions with reasonable choices of weight functions and domain. For the third boundary condition, a solution that is unique up to a linear ambiguity exists. To numerically construct the progressive lens surface, a finite element method using a special type of splines, chosen for its smoothness properties, is devised to solve the resulting partial differential equations. The method is shown to be both convergent and efficient. Finally we give numerical examples where we solve the lens design problem under all three boundary conditions for the equation obtained from the linearization about spherical surfaces. In summary, this thesis develops the theory and effective numerical methods for variational approach to progressive addition lens design. |
