Metamaterial-Enabled Transformation Optics
Transformation Optics is a design methodology that uses the form invariance of Maxwell's equations to distort electromagnetic fields. This distortion is imposed on a region of space by mimicking a curvilinear coordinate system with prescribed magnetoelectric material parameters. By simply specifying the correct coordinate transformation, researchers have created such exotic devices as invisibility cloaks, ``perfect'' lenses, and illusion devices.
Unfortunately, these devices typically require correspondingly exotic material parameters that do not occur in Nature. Researchers have therefore turned to complex artificial media known as metamaterials to approximate the desired responses. However, the metamaterial design process is complex, and there are limitations on the responses that they achieve.
In this dissertation, we explore both the applicability and limitations of metamaterials in Transformation Optics design. We begin in Chapter 2 by investigating the freedoms available to use in the transformation optics design process itself. We show that quasi-conformal mappings may be used to alleviate some of the complexity of material design in both two- and three-dimensional design. We then go on in Chapter 3 to apply this method to the design of a transformation-optics modified optic. We show that even a highly-approximate implementation of such a lens would retain many of the key performance feautures that we would expect from a full material prescription.
However, the approximations made in the design of our lens may not be valid in other areas of transformation optical design. For instance, the high-frequency approximations of our lens design ignore the effects of impedance mismatch, and the approximation is not valid when the material parameters vary on the order of a wavelength. Therefore, in Chapter 4 we use other freedoms available to us to design a full-parameter cloak of invisibility. By tailoring the electromagnetic environment of our cloak, we are able to achieve three distinct material responses with a singe metamaterial unit cell. We show the power of our design by experimentally demonstrating a cloak of ten wavelengths in diameter at microwave frequencies.
In addition to these specific examples, we seek a general method to simulate transformation optics devices containing metamaterial inclusions. In Chapter 5, we examine the discrete-approximation, and we apply it to the design of an electromagnetic cloak. We show that the point-dipole description of metamaterial elements allows us to correct for some aberrations that appear when the limits of homogenization are violated.
Finally, we examine so-called ``complementary metamaterials'' and their utility in transformation optics devices. Complementary metamaterials exchange the void and metallized regions of conventional metamaterial elements, and thereby offer a dual response to the electromagnetic field. This duality is attractive because it provides a straightforward method of creating broadband, highly-anisotropic magnetics. We analyze these elements and show that they may be incorporated into our discrete-dipole model. However, we show that the unique characteristics of complementary elements limit their functionality when used as effective materials.
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