# Dissertation/Thesis Abstract

Mathematical studies on the human eye
by Nam, Jayoung, Ph.D., Indiana University, 2007, 101; 3274988
Abstract (Summary)

We study several mathematical problems in visual optics. We develop a method to quantify the visual quality of a human eye and attempt to objectively determine refractive imperfection of the optical system of the eye. We devise the Zernike-Gauss functions Z[special characters omitted](ρ,&phis;,γ) which are mutually orthonormal on the unit disc. We adjust the parameter γ to best approximate the subjective description of the error of the optics of the eye. Among 36 eyes measured in 575 nm light, we found that the optimal γ = 5.79 resulted in an error of less than 0:25 diopter. This γ determines an effective radius of vision, Re = 1:89 mm. Although, γ tends to be universal, some individuals have shown different optimal values for γ.

In an effort to describe the individual differences of the optimal value of γ, we consider a general eye model. As a first step for developing multi-surface eye models, we provide an algorithm for a single surface eye model and verify that the algorithm gives the exact solution. The algorithm may depend on how many surfaces we assume, but we make no prior assumptions on the details of the surfaces. One application of this algorithm is in converting a wavefront measured in one wavelength to another wavelength. We find that, in general, the conversion between the Zernike representations of wavefronts of two different wavelengths does not follow the fixed formula that is widely used.

In all the above, the phase is determined by solving a first order partial differential equation of the form ∇u = F( x, y, u), allowing uncertainties on F(x, y, u). The same equation occurs in the reγector design. We consider several mathematical formulations for the differential equation and employ numerical algorithms to find a solution. Among which are the Integration by Many Orbits method, the Finite Difference Method, and the Least Square Finite Difference Method.

Indexing (document details)
 Advisor: Rubinstein, Jacob Commitee: Glassey, Robert, Purdom, Paul, Sternberg, Peter, Thibos, Larry N. School: Indiana University Department: Mathematics School Location: United States -- Indiana Source: DAI-B 68/07, Dissertation Abstracts International Source Type: DISSERTATION Subjects: Mathematics, Optics Keywords: Eye, Refractive imperfections, Visual optics, Zernike-Gauss functions Publication Number: 3274988 ISBN: 978-0-549-15848-6