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Transformation electromagnetics is a systematic design technique for optical and electromagnetic devices that enables novel wave-material interaction properties. The associated metamaterials technology for designing and realizing optical and electromagnetic devices can control the behavior of light and electromagnetic waves in ways that have not been conventionally possible. The technique is credited with numerous novel device designs, most notably the invisibility cloaks, perfect lenses and a host of other remarkable devices.

Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications presents a comprehensive treatment of the rapidly growing area of transformation electromagnetics and related metamaterial technology with contributions on the subject provided by a collection of leading experts from around the world. On the theoretical side, the following questions will be addressed: “Where does transformation electromagnetics come from?,” “What are the general material properties for different classes of coordinate transformations?,” “What are the limitations and challenges of device realizations?,” and “What theoretical tools are available to make the coordinate transformation-based designs more amenable to fabrication using currently available techniques?” The comprehensive theoretical treatment will be complemented by device designs and/or realizations in various frequency regimes and applications including acoustic, radio frequency, terahertz, infrared, and the visible spectrum. The applications encompass invisibility cloaks, gradient-index lenses in the microwave and optical regimes, negative-index superlenses for sub-wavelength resolution focusing, flat lenses that produce highly collimated beams from an embedded antenna or optical source, beam concentrators, polarization rotators and splitters, perfect electromagnetic absorbers, and many others.

This book will serve as the authoritative reference for students and researchers alike to the fast-evolving and exciting research area of transformation electromagnetics/optics, its application to the design of revolutionary new devices, and their associated metamaterial realizations.



Chapter 1. Quasi-Conformal Approaches for Two and Three-Dimensional Transformation Optical Media

Transformation optical design is generally complicated by the requirement for highly anisotropic and inhomogeneous constituent materials. Quasi-conformal mappings have appeared as an attractive subset of the general transformation optics method because they only require isotropic, dielectric-only materials. In this chapter, we examine the quasi-conformal method as it applies to transformation optics and show that while it does significantly ease the burden of material design and fabrication, it may also create severely aberrant behavior unless caution is taken. We also show how to extend the method to three dimensions, and examine the performance of an optic designed with the quasi-conformal method.

Nathan Landy, Yaroslav Urzhumov, David R. Smith

Chapter 2. Transformation Electromagnetics for Cloaking, Lensing, and Radiation Applications

The transformation electromagnetics technique provides a powerful tool to electromagnetic and optical designers by offering a blueprint for creating novel devices that feature a variety of unconventional wave-material interaction properties. Combined with the recent advances in metamaterials technology, the coordinate transformation-based design methodology paves the way to realizing devices that perform entirely new functions or traditional functions in different geometrical configurations that are advantageous in practical applications. Here, transformation electromagnetics techniques are applied to the design of invisibility cloaks, lenses, beam controllers, and antennas. Each device design is illustrated with an example, including the associated specifications and performance expectations. For the two-dimensional embedded antenna application, a microwave metamaterial design is also presented.

Do-Hoon Kwon, Qi Wu, Douglas H. Werner

Chapter 3. Metasurface Transformation Theory

Metasurfaces constitute a class of thin metamaterials, which are used from microwave to optical frequencies to create new antennas and microwave devices. This chapter describes how to use transformation optics (TO) to create anisotropic modulated-impedance metasurfaces able to transform planar surface waves (SW) into a predefined curved-wavefront surface wave. In fact, the modulated anisotropic impedance imposes a local modification of the dispersion equation and, at constant operating frequency, of the local wavevector. The general effects of metasurface modulation are similar to those obtained by TO in volumetric inhomogeneous metamaterials, namely readdressing the propagation path of an incident wave; however, significant technological simplicity is gained.

Enrica Martini, Stefano Maci

Chapter 4. Design for Simplified Materials in Transformation Electromagnetics

In this chapter we review some of the common methods used to simplify the design of transformation electromagnetics devices. One of the major challenges is the complexity of the needed material parameters, and a variety of analytical, numerical, and approximation techniques can be used to create devices that still perform well, although not ideally, but are much easier to fabricate. All of these techniques are a manifestation of the general concept that transformation electromagnetics designs are relatively robust to material perturbations.

Steven A. Cummer

Chapter 5. Creating Illusion Effects Using Transformation Optics

In this chapter, we show how to create illusion effects using metamaterials. We will see that a passive metamaterial device can be designed such that when it is placed next to or covering an object, the scattered fields of the object and the device together will be changed to be exactly the same as the scattered fields due to another object. Simply put, we can turn an object optically and stereoscopically into another one. For instance, an apple can be made to look like a banana. If we make a measurement of the electromagnetic fields at the designed working frequency, there is no way to distinguish optically between the true object and the illusion. The theory on realizing such an optical illusion effect is called

illusion optics

. Invisibility can be regarded as a special case of the illusion effect, in which the object is turned optically to a volume of free space. A metamaterial does not need to encircle the object to create the illusion effect. Furthermore, if we use this method to implement invisibility, the “cloaked” object will not be blinded by the cloak as in the cases of normal invisibility cloaks. The design of illusion optics is based on the replacement of optical spaces, where the material parameters are determined using the technique of transformation optics. One unique route to achieve illusion effects is to employ the idea of “complementary media”. Materials designed using “complementary media” typically contain negative refractive index components, and no extreme constitutive parameters values are needed. Slight variations of the scheme can create a variety of interesting illusion effects. For example, we can make an object appear larger in size, rotated, or located at other positions. Illusion optics may lead to some plausible applications, such as small or reduced form-factor optical devices that exhibit the same optical functions as much larger instruments or even “super-absorbers” that can absorb significantly more than their geometric cross-sections.

Yun Lai, Jack Ng, C. T. Chan

Chapter 6. Transformation-Based Cloak/Anti-Cloak Interactions: A Review

The intriguing concept of “anti-cloaking” was originally introduced within the framework of transformation optics (TO) as a “countermeasure” to invisibility-cloaking, i.e., to restore the scattering response of a cloaked target. More recently, its relevance was also suggested in applications to “sensor invisibility,” i.e., to strongly reduce the scattering response while maintaining the field-sensing capabilities. In this chapter, we review our recent studies on two-dimensional (cylindrical) and three-dimensional (spherical) canonical scenarios. More specifically, via generalized (coordinate-mapped) Bessel-Fourier and Mie-series approaches, we address the analytical study of plane-wave-excited configurations featuring a cylindrical or spherical object surrounded by a TO-based invisibility cloak coupled to an anti-cloak via a vacuum layer, and explore the various interactions of interest. Via a number of selected examples, we illustrate the cloaking and field-restoring capabilities of various configurations, with special emphasis on the scattering versus absorption tradeoff, as well as possible ideas for approximate implementations that do not require the use of





Giuseppe Castaldi, Vincenzo Galdi, Andrea Alù, Nader Engheta

Chapter 7. Transformation Electromagnetics Design of All-Dielectric Antennas

The discrete coordinate transformation is a practical implementation of transformation electromagnetics. It solves the transformation between coordinate systems in a discretized form. This method significantly relaxes the strict requirement for transformation media, and consequently leads to easily-realizable applications in antenna engineering. In this chapter, the discrete coordinate transformation is demonstrated and analyzed from the theory and is proved to provide an all-dielectric approach of device design under certain conditions. As examples, several antennas are presented, including a flat reflector, a flat lens, and a zone plate Fresnel lens. The Finite-Difference Time-Domain (FDTD) method is employed for numerical demonstration. Realization methods are also discussed, and a prototype of the carpet cloak composed of only a few dielectric blocks is fabricated and measured.

Wenxuan Tang, Yang Hao

Chapter 8. Transformation Electromagnetics Inspired Lens Designs and Associated Metamaterial Implementations for Highly Directive Radiation

In this chapter, the transformation electromagnetics (TE) approach for achieving highly directive radiation is introduced and demonstrated by both numerical simulations and experimental results obtained from laboratory prototypes. In addition to conventional approaches for designing directive antennas, the recently developed metamaterial-related techniques, such as the electromagnetic bandgap (EBG) structures, zero-index metamaterials, and transformation optics (TO), are reviewed. In particular, several coordinate transformations which can provide simplified material parameters are proposed, including the conformal mapping, quasi-conformal (QC) mapping, geometry-similar transformation, and the uniaxial media simplification method. All of these techniques are capable of achieving a certain degree of simplification in the transformed material parameters without sacrificing the device performance. The design and demonstration of various beam collimating devices illustrate their unique properties and suitability for different applications such as in compact wireless systems. In all, these TE-enabled lenses with simple material parameters are expected to find widespread applications in the fields of microwave antennas as well as optical nanoantennas.

Douglas H. Werner, Zhi Hao Jiang, Jeremiah P. Turpin, Qi Wu, Micah D. Gregory

Chapter 9. Transformation Electromagnetics for Antenna Applications

In recent years, transformation electromagnetics has found potential applications in propagation, waveguiding, scattering, and radiation. For antenna applications, using transformation techniques, one can transform bulky antennas to low profile ones. In general, the resulting medium will be both inhomogeneous and anisotropic. In this chapter, we proposed a spherical core-shell structure which can achieve arbitrarily large directivity. We investigated the problem by finding the transformed constitutive tensors and solving the equivalent problem in the core-shell configuration. Using the Ricatti-Bessel functions, we can represent the field components with Debye potentials and subsequently solve for the fields in all regions. We applied the formulation to several cases of dipole arrays within the shell, corresponding to both free-space and half-space problems in the virtual space. Overall, the calculation demonstrated that the formation of virtual aperture is indeed theoretically possible and the effects of loss on the number of available spherical harmonics and directivity are investigated.

Bae-Ian Wu

Chapter 10. Invisibility Cloak at Optical Frequencies

The invisibility cloak is an intensively studied topic in the fields of electromagnetism and optics. Since the first theoretical formulation of invisibility, growing attention has been paid to this rapidly growing field, both in theory and experiments. In this chapter, we review the recent progress in the invisibility cloak and transformation optics at optical frequencies.

Shuang Zhang

Chapter 11. Experimental Characterization of Electromagnetic Cloaking Devices at Microwaves

Electromagnetic cloaking has attracted much interest in recent years. In this chapter, we will describe experimental methods that can be applied to the analysis of electromagnetic cloaks in the microwave regime. Each method is explained and accompanied with real-life examples. At the end of the chapter, we also discuss experimental techniques for characterization of individual small particles (“artificial molecules”) which are useful in the design of various cloaks or low-scattering objects.

Pekka Alitalo, Sergei A. Tretyakov

Chapter 12. Broadening of Cloaking Bandwidth by Passive and Active Techniques

This chapter deals with the most serious drawback of transformation electromagnetic-based cloaking devices: narrow operating bandwidth (BW). It is shown that the maximal operating BW of every passive cloak is limited by the basic background physics (so-called energy-dispersion constraints). It is also shown possible to optimize the cloak parameters in order to achieve either the maximal BW or the maximal invisibility gain (IG). Finally, it is shown possible to go around the dispersion-energy constraints by inclusion of non-Foster active components, and to achieve a very broad operating BW that fairly exceeds the BW of passive cloaks.

Silvio Hrabar, Zvonimir Sipus, Iva Malcic

Chapter 13. Anisotropic Representation for Spatially Dispersive Periodic Metamaterial Arrays

A rigorous anisotropic (as opposed to bianisotropic) representation for three-dimensional (3D) spatially dispersive periodic arrays of passive inclusions separated in free space is developed, beginning with the microscopic Maxwell equations, that yields causal, macroscopic permittivities and inverse permeabilities for the fundamental Floquet modes of the arrays. (Macroscopic magnetoelectric coefficients are not required.) Reality conditions, reciprocity relations, passivity conditions, and causality relations are derived for these spatially dispersive macroscopic permittivity and permeability constitutive parameters. A significant feature of the formulation is that the macroscopic permittivities and permeabilities reduce to their anisotropic-continuum definitions in terms of ordinary macroscopic averages at the low spatial and temporal frequencies. In addition, diamagnetic metamaterial arrays require no special considerations or modifications to accommodate their unusual characteristics. Analytic and numerical examples of 3D arrays with dielectric-sphere inclusions and two-dimensional (2D) arrays with circular-cylinder inclusions are given that confirm the theoretical results for the macroscopic permittivities and permeabilities of these arrays which exhibit electric and magnetic or diamagnetic macroscopic polarizations. The realization of the potential innovations provided by transformation electromagnetics depends strongly upon the development of metamaterials. This chapter aspires to enhance the understanding of metamaterials and thus to facilitate their development.

Arthur D. Yaghjian, Andrea Alù, Mário G. Silveirinha

Chapter 14. Transformation Electromagnetics and Non-standard Devices

The use of transformation electromagnetics for microwave applications is presented. Implementation of non-standard devices such as microwave antennas and waveguide tapers proposed by the Institut d’Electronique Fondamentale at the University of Paris-Sud are reviewed. The operating principle and the respective coordinate transformation of each device is presented and numerical simulations are performed to verify the theoretical formulations. The method to obtain constitutive electromagnetic parameters mimicking the calculated transformed space is detailed and confirmed by full-wave simulations performed using discrete material parameter values and by measurements performed on fabricated metamaterial-based prototypes. The results show that transformation electromagnetics is very interesting for the design and realization of high-performance non-standard devices.

André de Lustrac, Shah Nawaz Burokur, Paul-Henri Tichit


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