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PhD thesis (August 1999)
Analysis and Optimal Design of Diffractive Optical Elements
The problem we study arose in an industrial application. For an optical system, Diffractive Optical Elements (DOE) are used to produce a certain light intensity pattern in the near field. Of our particular interest is an Inverse problem: given a target image, determine the DOE configuration, e. g. thickness, that would produce this image. The problem can be complicated by specific constraints such as finite number of thickness levels that the DOE can have.
Diffraction theory and Green's function approach are applied to construct a mathematical model for the light propagating through the DOE. Asymptotic methods of stationary phase and multiple-scale analysis are used to derive analytic solutions for periodic and quasi-periodic cases. These analytical expressions do not involve integration, save computational resources, and allow us to solve the Inverse problem analytically. Numerical results for particular applications are presented.
The Inverse problem can be posed a large optimization problem with finite discrete variables, which can not be solved by traditional methods. We propose Genetic Algorithms based on analogies to natural evolution and representing a combination of random and directed search. A modification of the method that suits better to our problem, the Micro-Genetic Algorithm (MGA), is proposed. The MGA operates on a small set of potential solutions and restarts, using an adaptive mutation scheme, each time the local convergence is achieved. We prove convergence for the MGA using the Markov chain analysis. Numerical results of the MGA optimization are provided.
1. Mathematical model
1.1. Preliminaries
1.2. Derivation of the Rayleigh-Sommerfeld solution
. . . 1.2.1. 3D case
. . . 1.2.2. 2D case
1.3. Forward problem
1.4. Inverse problem
2. Asymptotic Analysis
2.1. Introduction
2.2. Method of stationary phase
2.3. Multiple-scale analysis
2.4. Generalization
2.5. Applications
. . . 2.5.1. First approximation (independent of z)
. . . 2.5.2. Second approximation (z-dependent)
. . . 2.5.3. Inverse problem
2.6. Numerical results
2.7. 3D case
. . . 2.7.1. Asymptotic formulas
. . . 2.7.2. Applications
3. Genetic Algorithm Optimization
3.1. Overview of Genetic Algorithms
3.2. Micro-Genetic Algorithm
3.3. Convergence Analysis
. . . 3.3.1. Stochastic matrices properties
. . . 3.3.2. Markov chain analysis
. . . 3.3.3. Convergence of the Micro-Genetic Algorithm
3.4. Numerical results
3.5. Continuous optimization and rounding
Last modified: August 11, 1999.
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