Zhou, Fanhuan2011-03-102011-03-102010-09https://hdl.handle.net/11299/101437University of Minnesota Ph.D. dissertation. September 2010. Major: Mathematics. Advisor:Prof. Fadil Santosa. 1 computer file (PDF); vi, 88 pages. Ill. (some col.)Progressive additional lenses is a new approach to compensate for the defects of presbyopia for the human visual system. While using two or more single-vision lenses with different power, progressive additional lenses require the specification of free surfaces. A progressive additional lens comprises a large distance zone with low power on the upper portion of the lens, a small near zone with higher power on the lower part, and a progressive corridor of increasing power connects these two zones smoothly and progressively. In this work, we proposed an optimization approach to the progressive additional lens design problem, in which, a cost function which balances the power distribution and other optical aberrations up to the second order aberrations. The goal is to minimize the cost function. In previous progressive additional lens design, the formulas for the power and astigmatism of the lens is an approximation of the refraction on the front and back lens surface. Also, the assumption of the thickness of the lens $d ll 1$ limits the design to be a thin lens design. To better evaluate the exact properties of the optical system, one should study the ray tracing though the lens and the wavefront passing through a lens. Therefore, geometric optics is employed to evaluate how a wavefront is deformed by a lens and how the curvature of a wavefront is transformed when propagating through a homogeneous medium and refracting on a lens surface. To understand the second order aberration of the optical system, a concept of Third Order Surface coefficients is also introduced. Wavefront tracing method can then be used to evaluate the exact property of the refracted wavefront of the lens system. We derived a series of formulas describing First Fundamental Form coefficients, Second Fundamental Form coefficients and Third Order Surface coefficients of the wavefront on propagation and on refraction. A full process of ray tracing through the lens and wavefront deformation is also derived explicitly. We also discuss the process of front lens surface design and back lens surface design. To numerically construct the progressive lens surface, tensor product B-spline functions is used to solve the optimization problem. Numerical Methods such as Gradient method, Newton's method and Quasi-Newton method are used. One numerical example of front surface design with the fixed back spherical lens surface is shown in this work. In summary, this thesis develops a model of progressive additional lens design up to second order aberration by wavefront tracing method.en-USProgressive additional lensTensor product B-spline functionWavefront tracing methodMathematicsThesis or Dissertation