Field Mathematics for Electromagnetics, Photonics, and Materials Science

Excerpt

The overriding objective of this book is to offer a review of vector calculus needed for the physical sciences and engineering. This review includes necessary excursions into tensor analysis intended as the reader's first exposure to tensors, making aspects of them understandable at the undergraduate level. A secondary objective of this book is to prepare the reader for more advanced studies in these areas.

As the world embarks on new horizons in photonics and materials science, honing one's skills in vector calculus and learning the essential role that tensors play are paramount. New inroads in engineering are driving the need for a revamp of engineering mathematics in these areas. Profound new paradigms in optical engineering and new advances in composites are necessitating these changes. The author has found that there is an ever-increasing need for vector calculus concepts to be extended to tensors and that his undergraduates can indeed grasp tensorial concepts if taught following the lines of thinking presented here.

Whereas the classical approach to teaching electromagnetics at the junior level has been to avoid any mention of tensors, the high-tech world entering the third millennium warrants a rethinking of this practice. This is especially true as nonlinear optical effects become more common in the design of optical systems. Advanced materials, especially composites and nanodesigned materials, provide further evidence supporting the teaching of tensor fundamentals to upper-division students. Even for isotropic materials, the fundamental relationship between stress, strain, and elastic modulus—which are rank-two and rank-four tensors—requires a fundamental understanding of tensor analysis. For anisotropic materials such as composites, piezoelectric materials, and magnetostrictive materials, tensorial relationships are unavoidable even in the linear regime.

Furthermore, the development of new photonics devices in optoelectronics, acousto-optics, magneto-optics, and fiber optics is playing an ever-increasing role in contemporary communications system design. Pollock states The drive for faster systems has led to… [an] electronic speed bottleneck…This has motivated the study of integrated optics, where light, which has a much higher implicit frequency limit, is used to control light… Without a doubt the biggest research task…will be the development of optical switches and devices, and better communication architectures. These devices include laser sources, optical switches, rare-earth-doped fiber amplifiers, nonlinear-effect fiber amplifiers, nonlinear-effect fiber soliton waves, optical detectors, and new dispersion-managed optical fibers.

Uses of this Guide

This is a guide, and was not planned as a text book. As such, it is intended for multiple uses, including its use as a

1. reference to salient differential and integral forms for problem solving,

2. supplement to an engineering or science course, used in conjunction with and as a counterpart to it,

3. study guide before entering such courses,

4. reference manual in an R&D laboratory or design group,

5. complement to required or elective math courses, or just as a

6. refresher and reference source to vector calculus and an introduction to tensor analysis, or a

7. text, provided the instructor devises problem sets to provide the usual practical experience with numerical examples.