Spatial transformations: from fundamentals to applications

Transformation optics: a universal design tool

Sir John Pendry FRS, Imperial College London, UK

Abstract

Our intuitive understanding of light has its foundation in the ray approximation and is intimately connected with our vision: as far as our eyes are concerned light behaves like a stream of particles. Here we look inside the wavelength and show how the new concept of transformation optics that manipulates electric and magnetic field lines rather than rays can provide an equally intuitive understanding of sub wavelength phenomena and at the same time be an exact description at the level of Maxwell¹s equations. Examples will be given of applications to plasmonic structures with dimensions of just a few nanometres: a tenth or even a hundredth of the wavelength of visible light, and at the other extreme to cloaking of static magnetic fields. In both instance the ray picture fails utterly.

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Cosmology in the laboratory

Professor Ulf Leonhardt, Weizmann Institute of Science, Israel

Abstract

Transformation optics is based on the analogy between space-time geometries and electromagnetic media. In transformation optics, this analogy is primarily applied for finding the design of novel devices that are very difficult to conceive otherwise. Ideas from general relativity find practical uses in engineering applications. Here we go back to the basics of transformation optics and ask a different question: what can we learn for general relativity from the practical experience with electromagnetic materials? We report on our progress on laboratory analogues of the event horizon and on the regularisation of the Casimir force in spatial geometries.

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Optics of Metastructures

Professor Nader Engheta, University of Pennsylvania, USA

Abstract

Tailoring material parameters in metamaterials and transformation optics provides useful methodologies for engineering light-matter interactions at the marco-, meso-, micro-, and nanometer scales.  I will present an overview of some of our ongoing research efforts in the areas of optics of metastructures.  We will show how parameterization and functionalization of metamaterials and nanophotonics may lead to new optical features and applicable functionalities.  I plan to discuss:  (a) the extreme-parameter optics and some of the relevant optical features and potential applications, (b) nonreciprocal and unusual flow of photons and how they can be used for future devices and components, (c) materials that may function as optical metatronics and nanocircuitry, leading to nanoscale optical filters and switches, (d) digital metamaterials, in which desired permittivity values can be engineered by combinations of only two materials as metamaterial “bits”, forming metamaterial “bytes” act as building blocks, and providing simplicity in the design of functional materials; (e) nanostructures that provide platforms for mathematical operations and analog computing; and (f) classical and quantum features of “static optics” in which the electricity and magnetism are decoupled while they are dynamic fields.  In my group we are investigating new categories of phenomena and potential applications in engineering functional metastructures.  I will present some of our latest results, will discuss physical insights into these findings, and will forecast future directions and possibilities.

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Experimental model of topological defects in Minkowski spacetime based on disordered ferrofluid

Dr Igor Smolyaninov, University of Maryland, USA

Abstract

Cobalt nanoparticle-based ferrofluid in the presence of external magnetic field forms a self-assembled hyperbolic metamaterial. Wave equation which describes propagation of extraordinary light inside the ferrofluid exhibits 2+1 dimensional Lorentz symmetry. The role of time in the corresponding effective 3D Minkowski spacetime is played by the spatial coordinate directed along the periodic nanoparticle chains aligned by the magnetic field.  Here we present a microscopic study of point, linear, planar and volume defects of the nanoparticle chain structure and demonstrate that they may exhibit strong similarities with such Minkowski spacetime defects as magnetic monopoles, cosmic strings and the recently proposed spacetime cloaks. Experimental observations of such defects are described.

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Spatially variant periodic structures in electromagnetics

Professor Raymond C. Rumpf, University of Texas, USA

Abstract

Spatial transforms are a popular technique for designing periodic structures that are macroscopically inhomogeneous. Many times the structures are required to be anisotropic, provide a magnetic response, and to have extreme values for the constitutive parameters in Maxwell’s equations. Metamaterials and photonic crystals are capable of providing these, although sometimes only approximately. The problem still remains about how to generate the geometry of the final lattice when it is functionally graded, or spatially varied. This paper describes a simple numerical technique to spatially vary any periodic structure while minimizing deformations to the unit cells that would weaken or destroy the electromagnetic properties. New developments in this algorithm are disclosed that increases efficiency and improves the quality of the lattices. The ability to spatially vary a lattice in this manner enables new design paradigms that are not possible using spatial transforms, three of which are discussed here. First, spatially-variant self-collimating photonic crystals are shown to flow unguided waves around very tight bends using ordinary materials with low refractive index. Second, multi-mode waveguides in spatially variant band gap materials are shown to guide waves around bends without mixing power between the modes. Third, spatially-variant anisotropic materials are shown to sculpt the near-field around electric components. This can be used to improve electromagnetic compatibility between components in close proximity.

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Experiments on cloaking in optics, thermodynamics, and mechanics

Professor Martin Wegener, Karlsruhe Institute of Technology, Germany

Abstract

Spatial coordinate transformations can be used to transform boundaries, material parameters, or discrete lattices. We discuss fundamental constraints in regard to cloaking and review our corresponding experiments in optics, thermodynamics, and mechanics. For example, we emphasize three-dimensional broadband visible-frequency carpet cloaking, transient thermal cloaking, three-dimensional omnidirectional macroscopic broadband cloaking for diffuse light throughout the entire visible range, cloaking for flexural waves in thin plates, three-dimensional elasto-static core-shell cloaking using pentamode mechanical metamaterials, and elasto-static cloaking of voids.

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Manufacture of electrical and magnetic graded and anisotropic materials for novel manipulations of microwaves

Professor Patrick Grant FREng, University of Oxford, UK

Abstract

Spatial transformations (ST) provide a design framework to generate a required spatial distribution of electrical and magnetic properties of materials in order to effect novel manipulations of electromagnetic waves. To realise in practice the spatial variation of electromagnetic properties required by these designs, the most common materials approach has involved periodic arrays of metal-containing sub-wavelength elements. While aspects of ST theory have been confirmed using this type of approach, these structures have often been disadvantaged by relatively narrow-band operation, high losses and difficulties in implementation. An alternative approach involves spatial distributions of electrical and magnetic properties using arrangements of different materials of differing inherent electrical and magnetic response. Sometimes referred to as an “all-dielectric” approach, it involves generally weaker interactions with incident waves, but can offer more flexibility for practical implementation. This paper investigates a number of manufacturing approaches to produce composite materials that can then be conveniently arranged spatially according to ST-based designs. A key aim is to highlight both the limitations and possibilities of various manufacturing approaches, in order to constrain designs to those that may be achieved in practice. The focus is on polymer based nano- and micro-composites in which electrical and magnetic interactions with microwaves are achieved by loading the polymers with high permittivity and permeability particles, and manufacturing approaches based on spray deposition, extrusion, casting and 3D printing.

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Metasurfaces for general transformations of electromagnetic fields

Professor Sergei Tretyakov, Aalto University, Finland

Abstract

Electromagnetic fields can be controlled and transformed using engineered materials, often called metamaterials. The conventional paradigm of using metamaterials for transformations of electromagnetic fields implies that we engineer artificial materials in such a way that the polarization and conduction currents induced in the material, acting as secondary sources, create the desired fields outside or inside of the metamaterial sample. Huygens’ principle tells, however, that the same fields outside of the sample volume can be found as those generated by equivalent surface currents flowing only on the volume surface. Thus, it appears that the same field transformations can be achieved by engineering only surface currents of the volume surface, and there appears to be no reason why the volume enclosed by such an “engineered surface” could not be made negligibly small. In this review talk we will discuss electrically (optically) thin composite sheets with engineered and optimized properties: metasurfaces for general transformations of electromagnetic fields. The main motivation is a possibility to realize quite general functionalities (absorption, polarization transformations, control over reflection phase, focusing, etc.) using just single arrays of electrically  small engineered particles. In this talk we will explain what physical properties of metasurface unit cells are responsible for various field transformations, provide basic design equations, and illustrate the potentials of this technology by several examples from our experimental work.

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Metasurface Transformation Electromagnetics

Professor Stefano Maci, University of Sienna, Italy

Abstract

Metasurfaces constitute a class of thin metamaterials, able to support surface wave propagation. At microwave frequencies, they are constituted by sub-wavelength size patches printed on thin grounded or ungrounded dielectric substrates. By averaging the tangential fields, the metasurfaces may be characterized by homogenised isotropic or anisotropic boundary conditions, which can be approximated through a homogeneous equivalent impedance. In absence of losses, this impedance supports a surface-wave propagation. The impedance can be spatially modulated by locally changing the sizes/orientation of the local printed element. This allows for a deformation of the wavefront which addresses the local wavector along not-rectilinear paths. In fact, the modulated anisotropic impedance imposes a local modification of the dispersion equation and, at constant operating frequency, of the local wavevector. The effect of the metasurface-modulation can be analized in the framework of Transformation Optics This talk reviews theory and implementation of metasurface transformation optics in microwave devices.

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A new look at transformation electromagnetics approach for designing electromagnetic devices such as flat lenses, reflectarrays and blankets for radar cross section reduction of real-world objects

Professor Raj Mittra, Pennsylvania State University, USA

Abstract

In this paper we present an alternative approach to addressing the problem of designing a number of practical “microwave” devices such as: blankets serving as absorbers for radar targets; flat lenses; and reflectarrays. Recently, these design problems been dealt with by a number of researchers using the transformation optics (TO) algorithm, which is based upon transforming the geometry of an object from real space to virtual space while keeping the Maxwell’s field solutions from real space to virtual space intact. The TO algorithm typically leads to designs that call for anisotropic values in real space in order to preserve the field variations as we navigate from the real space to virtual space and vice versa. In contrast to the TO, the proposed algorithm is based on “Field Transformation (FT),” as opposed to geometry transformation. The FT algorithm has been designed to transform the electromagnetic field distribution in an input aperture, generated by a given source distribution, to a desired distribution in the exit aperture. We show how we can cast the design problem into a Scattering Matrix approach, wherein the case of RCS reduction problem the design is based on controlling only the Magnitude of S11, whereas for the Lens or Reflectarray problems, we specify only the desired Phase of S12 without being concerned about its magnitude. In contrast to this, the TO imposes strict conditions on both the magnitude and phase characteristics of S11 and S12, which in turn calls for anisotropic metamaterials. The Scattering Matrix/Field Transformation approach avoids these problems altogether and is able to work with only materials for the lens and reflectarray problems, and with realizable complex (materials that have wideband characteristics and do not suffer from the shortcomings of the MTMs.

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Spatial transformation enabled electromagnetic devices: From radio frequencies to optical wavelengths

Professor Douglas Werner, Pennsylvania State University, USA

Abstract

Transformation optics provides scientists and engineers with a powerful new design paradigm to tailor the material properties of a medium (i.e., the spatial distribution and/or anisotropy), thereby allowing for comprehensive manipulation of electromagnetic waves with unprecedented flexibility. Based on such a mathematical framework, various interesting electromagnetic wave phenomena have been revealed over the last decade and employed to develop a wide range of novel devices that target applications throughout the electromagnetic spectrum. This presentation will focus on both theoretical and experimental investigations into the design of transformation optics enabled devices for shaping or controlling electromagnetic waves (radiation and scattering), at radio frequencies as well as optical wavelengths. Several types of coordinate mappings, which are variations based on the originally proposed transformation optics design methodology, are exploited for different applications to provide expanded design flexibility, enhanced device performance, and reduced implementation complexity. These include the complex coordinate transformation for simultaneous amplitude and phase control, linear coordinate transformations for broadband operation, conformal mappings with electrically tunable transformed space geometry for microwave antenna systems, and quasi-conformal mappings for optical applications. The illustrated design examples will serve to demonstrate the comprehensive capability of transformation optics in controlling electromagnetic waves, while the associated devices will open up new pathways toward future synthesis and design of integrated electromagnetic components, ranging from the microwave to optical spectral regimes.

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Metasurfaces with spatial and temporal modulation to manipulate and control waves

Professor Andrea Alù, University of Texas, USA

Abstract

Metasurfaces have recently opened exciting opportunities to change the way we manipulate and control light at the nanoscale, by the means of their large field enhancement and localization. Gradient metasurfaces, or thin metamaterials with a suitably tailored transverse inhomogeneity, are able to transform the conventional boundary conditions for fields in space, and realize ultrathin optical components. A powerful addition to these tools is provided by local variations in time of the metasurface response. In this talk, we describe our recent theoretical and experimental advances in exploiting these effects to significantly boost nonreciprocal responses of subwavelength meta-molecules and arrays of them. We show large nonreciprocal response at the subwavelength scale by suitably tailored spatiotemporal modulation, realizing optical isolators, magnetic-free circulators, and advanced thermal management systems. We will also discuss the possibility of exploiting the largely uncharted scattering properties of Parity-Time (PT) symmetric systems to realize loss-compensated and broadband wave manipulation on a surface, including invisible sensors, ideal cloaks, and planar negative-index lenses. We will show that metasurfaces engineered to respect space-time inversion symmetries - i.e., that are invariant after taking their mirror image and running time backwards - can lead to exotic wave phenomena, without the bandwidth and loss-related issues typical of static and passive metamaterials. More broadly, in our talk, we will discuss how large light-matter interaction in suitably designed space-time gradient metasurfaces may open new important directions in science and technology.

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Transformation optics beyond the manipulation of light trajectories

Professor Philippe Tassin, Chalmers University, Sweden

Abstract

Transformation optics beyond the manipulation of light Since its inception in 2006, transformation optics has become an established tool to understand and design electromagnetic systems. It provides a geometrical perspective on the properties of light waves without the need for the ray approximation. Most studies have focused on modifying the trajectories of light rays, e.g., beam benders, lenses, invisibility cloaks, etc. In this contribution, we explore transformation optics beyond the manipulation of light trajectories. With a few well-chosen examples we demonstrate that transformation optics can be used to manipulate electromagnetic fields up to an unprecedented level. In the first example we introduce an electromagnetic cavity that allows for deep subwavelength confinement of light. The cavity is designed with transformation optics even though the concept of trajectory ceases to have any meaning in a structure as small as this cavity. In the second example we show that the properties of Cherenkov light emitted in a transformation-optical material can be understood and modified from simple geometric considerations. Third, we show that optical forces—quadratic functions of the fields—follows the rules of transformation optics too. By applying a folded coordinate transformation to a pair of waveguides, optical forces can be enhanced just as if the waveguides were closer together. Finally, we take a look at time-dependent transformations and we show that devices based on these metrics can change the frequency of light, but are otherwise linear in the electromagnetic fields. With these examples, we open up an entirely new spectrum of devices that can be conceived using transformation optics.

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Watching surface waves in phononic crystals

Professor Oliver Wright, Hokkaido University, Japan

Abstract

Phononic crystals—periodic structures that scatter acoustic waves—have been the subject of intensive investigation owing to their use in filtering and control of sound. In order to characterize phononic crystals and their derivative devices, the measurement of the acoustic field evolution in space is an attractive goal. This allows one to access fundamental properties of phononic crystals such as dispersion relations, phononic stop bands and their eigenstate field distributions, for example.  In this paper we review results obtained using such ultrafast time-domain imaging of the acoustic field on the surface of microscopic phononic crystals in two spatial dimensions in the range 100 MHz to 1 GHz.  The imaged area is typically 100 microns by 100 microns, and the lateral spatial resolution, limited by optical diffraction, is of the order of 1 micron. We first describe how the spatiotemporal acoustic displacement field can be Fourier analysed to yield the acoustic dispersion relations, in this case the phononic band structure. We then present applications for samples consisting of one- and two-dimensional surface phononic crystals exhibiting phononic stop bands, including phononic-crystal waveguides, and also demonstrate how time-domain acoustic imaging may be extended to kspace. Possible extension to acoustic metamaterial characterization are also discussed.

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Future directions

Professor Ian Youngs, Defence Science and Technology Laboratories, UK

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