** PhD thesis (August 1999)**

**
Analysis and Optimal Design of Diffractive Optical Elements**

ABSTRACT
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.

CONTENTS
Introduction

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

Conclusions

Bibliography

Last modified: August 11, 1999.

*
Send questions to Svetlana Rudnaya:
rudnaya@math.umn.edu
*