This project concerns analytical studies and the development of computational approaches to two problems arising in optics. The first problem arises in the design of progressive lenses which must have desired corrective powers and minimal aberrations. The optical phenomenon in this case is well described by geometrical optics. The second problem arises in the design of optical devices that exploit photonic bandgap phenomena. In this problem, the wavelength of light is about the same size as the structure, and calls for modeling of the light's propagation using Maxwell's equation. In both cases, the design problem may be posed as an inverse problem where the desired performance can be viewed as data and the design parameters as unknowns. The aim of this research is to use analytical tools to guide in the development of effective computational methods to solve these inverse problems.
Many people over 45 years old suffer a condition called presbyopia, which is the loss of the ability of the eye to focus on near objects. Progressive lenses, which correct for seeing both far and near objects, are often prescribed to presbyopia patients. The design of such lenses, taking into account the patient's need and the comfort, is a mathematical problem that involves optics, differential geometry, and optimization. The dream of creating optical computers that have the capability of high processing speeds requires the development of optical devices that are counterparts of electronic devices. The design of optical devices requires modeling of light propagation in complex nanostructures, which can be described by partial differential equations. In order to create devices that perform certain functions, it is necessary to solve a mathematical problem of optimal design. The unifying theme in the design of these two optical devices is the mathematics of inverse problems. The research to be carried out will help create lenses of unprecedented performance and optical computing subsystems for future optical computers.
The research work carried out under this grant dealt with problems in design of optical devices. The list of outcomes are: 1. Development of a method for design of progressive addition lenses. Presbyopia, the loss of the eyes' ability to focus, is an impairment that affects over a billion people. The most common remedy for presbyopia is to prescribe progressive lenses. These lenses have corrective powers that depend on the eye's gaze direction. Therefore, progressive lenses can correct for far-distance vision as well as near-distance vision in a single lens. We develop new mathematical formulas that calculates the optical properties of a progressive lens in a given gaze direction. The formulas are based on geometrical optics which is the most accurate theory for modeling human vision. We also developed a lens design strategy using an optimization approach. This work has the potential to create progressive lenses that are comfortable because of the low distortion. 2. Development of computational methods for photonic devices. Photonic devices are nano-structures designed to manipulate light and are expected to play an important role in optical transmission and processing of information. One of the challenges in designing photonic devices is the "curse of dimensionality" -- the computational problem to simulate light propagation is prohibitively complex. We devised an approximation method for optical resonators, a basic element in a photonic device, that is both accurate and mathematically rigorous. This breakthrough potentially removes "the curse" and will allow optical engineers an efficient way to explore the design space for a specific optical device. 3. Development of methods for design of masks in photolithography. Photolithography is a step in the process by which computer chips are made. In this step, a chemically-treated silicon layer is exposed to UV light which has been made to go through a photo-mask. Portions of the surface that receive light intensity greater than some threshold can be etched away so that other material can be deposited in their place. The problem that needs to be solved is "Given the geometry of the surface that must be etched, find a mask that produces such a geometry". Diffractive optics models the propagation of light through the mask and onto the silicon surface. The design problem, which is to find the mask, can be posed as an optimization problem. We develop mathematical theory and computational approaches for this problem. Our work has the potential to further improve photolithographic process which is key to better performing computer chips. 4. Development of human resources. Three graduate students directly benefited from the project: Michael Aschenbeck (PhD 2009 currently at GeoEye Inc), Fanhuan Zhou (PhD 2010 currently at Citi Group), and Jose Orozco (PhD 2011 currently at Vision Ease Lens). Two undergraduate students, Benjamin Weitz and Tom VandenBoom, were supported by this grant. Weitz graduated from Caltech and is now graduate student in Computer Science at University of California Berkeley. VandenBoom is a senior at the University of Minnesota.
