VIGRE2 Vertical Intergration of Research and Education Department of Mathematics, University of Utah

MATH 4800

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Fall 2009 course

Topic: Metamaterials and invisibility, a confluence of ideas
Instructor: Daniel Onofrei
Catalog Number: MATH 4800-1
Credits: 3
Time: MWF, 2:00-2:50pm

Students interested in the course should give a current 'Unofficial Transcript' (obtained from the Campus Information System, after log-in under the 'Grades' section) to the instructor. Registering requires obtaining the class number from the instructor.

Tuition Benefit:
The NSF VIGRE program provides a $500 tuition benefit for US students (US citizens, nationals, and permanent residents).

Description:

Metamaterials are artificially constructed composites with unnatural or "exotic properties" not encountered in real life materials.

One of the most unusual properties of metamaterials is related to the index of refraction. The refractive index (or index of refraction) of a medium is a measure of how much the speed of light (or other waves such as sound waves) is reduced inside the medium. Mathematically, the refractive index of a medium, n, is defined as n = c/vp, where c is usually the speed of light in vacuum and vp is the phase velocity of the light wave in the respective medium.

The refractive index is usually greater than one except in metamaterials, where it may have a negative value. This, for example, implies, that if you introduce a pencil in a metamaterial fluid, say from the north-east direction (somewhere in the first quadrant), the pencil will appear bent towards the south-east direction (somewhere in the fourth quadrant), when normally, if you use water for example instead of a metafluid, the pencil will appear bent towards the south-west direction (third quadrant)!!! On the same note, it is worth mentioning that by using metamaterials one can achieve for example, extremely high or almost zero conductivities, which would be impossible to find in nature.

Because of these exotic properties, metamaterials are used in the implementation of certain cloaking schemes, i.e., the so called transformation based cloaking schemes. But what cloaking means, in general?

First, let us observe that almost all the detection mechanisms of today's world are based on the following principle:
  1. Generation of an outgoing wave

  2. Analysis of the reflected wave to obtain information about the target, such as its location or speed.
The idea behind cloaking is to make objects invisible to external detection mechanisms, radar devices, sonar devices, or the human eye, etc. Thus, heuristically, we say that material Mc distributed in an annuli around a certain region D, cloaks the region D, if the entire structure, region D and the cloak material annuli around it, will be invisible to a detection mechanism. Or equivalently, any incident wave propagating towards the cloak will not reflect of it, but instead bent around it, exactly like water waves around a big stone. We will try to understand how one could achieve this with the use of metamaterials.

The first part of the class will introduce the basic ideas about acoustic and electromagnetic wave propagation. The mathematical models corresponding to these phenomena will be introduced and discussed. We will also try to understand the principles behind various detection mechanisms, such as, sonar and radar.

The second part of the class will be devoted to the study of metamaterials trying to understand a few of their properties and current applications.

The last part of the class will be focused on the understanding of the mathematical and physical arguments behind the transformation based cloaking schemes. We will present the current progress in this very new field of research, describing and analyzing the main ideas.

Past Courses:
Fall 2006, Math Finance
Spring 2007, Fractals
Fall 2007, Metric Spaces, The Contraction Mapping Principle, Fractals & Other Applications
Spring 2008, Knot Theory
Fall 2008, Random Walk: Modeling, Theory, and Applications
Spring 2009, Graph Theory