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Ali Jabini's Project ProposalElectromagnetic Wave Propagation and Scattering in Composite MediaThe behavior of electromagnetic waves in inhomogeneous and composite materials is determined mathematically by the Helmholtz equation. Methods for estimating and bounding the effective properties of the medium have been developed in the quasistatic limit where the wavelength is much longer than the microstructural scale. However, these methods have not been extended to the scattering regime where the wavelength is on the same scale as the inhomogeneities. I would like to work with Professor Golden on extending the bounding methods to the scattering regime. For propagation in a two component medium, Professor Golden has found a form for the Helmholtz equation which makes the analysis quite close mathematically to the quasistatic situation. I will begin by analyzing this problem in one dimension, where standard methods of calculus and differential and integral equations can be applied to investigate the behavior. In addition to bounding the effective properties of such media, we expect that the methods will apply to a very important class of materials, namely photonic crystals, which exhibit behavior similar to semiconductors, but for optical rather than electronic transport. These materials may one day form the key components of computers and digital networks. In this connection, Professor David Dobson has indicated his interest in helping us explore these interesting materials. Moreover, this research will fit well with my studies in Computer and Electrical Engineering. Ali Jabini's Project SummaryElectromagnetic Waves in Homogeneous MediaAt the beginning of the summer, I realized that there were a variety of research topics available and I only had time to engage in one of them. My previous interests and studies in engineering and physics, nonetheless, had a great impact on my decision making. In addition, Dr. Golden, my math professor for whom I was planning to do research, suggested that, due to my background in engineering and math, it would be more valuable and advantageous to conduct our research activities in the area of electromagnetic waves. Initially, as I had just finished my freshman year in the Department of Engineering, I realized that it would be better if I strengthened my knowledge in the area I was planning to conduct research. As a result, I decided to take several math classes in the course of summer and increase my background knowledge by engaging in a lot of reading acrivities related to the area of research. Thus, I started reading books suggested by Dr. Golden on electromagnetic waves, optics, and Maxwell's equations. One of the areas of emphasis in all the readings was Green's equation. Of course, I never used it but it was still useful to learn and I am sure it will be helpful in future research. One of the most valuable strategies used during the research process was to hold weekly meeting with Professor Golden and the rest of the research team to discuss our findings and also our learnings on the topic. A topic that kept coming up in all the meetings was the subject of random walks and electrical networks, which were very closely related to the topic of our research. After reading the books and several papers by a variety of people, I felt as though I was ready to solve some problems. I started off by examining a book on electromagnetic theory by Rietz and Milford and attempted to solve the one-dimensional case for an electric wave propagating through a layer. After probing this case closely and with the assistance of my research professor, I was able to solve this problem. Following this case, I continued my investigations by looking at an electric wave propagating through a slab or two layers. This problem was much more complex and it required more effort and time to get to the bottom of it. I was satisfied that I could finally unravel it. Due to the limited time available for the research, there was not much time left to work on the rest of the problems and I am looking forward to continuing the rest of the research during the year 2002/2003. |
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VIGRE Steering Committee Department of Mathematics University of Utah 155 South 1400 East; Room 233 Salt Lake City, UT 84112 email: viscom@math.utah.edu |
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