Introduction

The structuring of a metal at nanoscale dimensions results in novel optical properties that are not present for bulk metals. Metallic photonic crystals, metal-based structures with periodicities on the scale of the wavelength of light, have attracted particular attention due to their unique optical properties. Among the approaches taken to prepare a three-dimensional photonic crystal is to take advantage of the self-assembly of spheres from a colloidal solution. Spherical colloidal particles of polymers or silica with diameters ranging from 20 nm up to 1 μm and larger, with low coefficients of variation in their diameter, are readily available. The methods of producing monodispersive colloids are well discussed in ref. [1]. The importance and interest of these particles lies in the fact that it is possible to induce them into a close-packed structure analogous to an ordinary close-packed crystal. There are several methods for self-assembly of colloidal spheres, in particular, sedimentation, evaporation, and electrophoresis. These close-packed arrays of uniform particles offer an attractive and, in principle, simple means to template the three-dimensional structure of a variety of materials.

Generally, self-assembly is restricted to the formation of close-packed two-dimensional or three-dimensional assemblies of colloidal particles. However, the low cost and availability of a relatively easy protocol to obtain this type of photonic crystals, artificial opals, make the self-assembly technique very attractive and widely used. The next step in the development of this technique to prepare metallic photonic crystal is to infiltrate the sample with some appropriate material, removing the original structure, and obtaining in this way inverted opals.

Introduction

Surface plasmon polariton (SPP) modes have attracted much interest in recent years. Although known and studied for over 100 years [1–3], the dream of confining light to dimensions smaller than its propagating wavelength has led the way towards technological possibilities not previously addressed, such as optical circuitry within ultra small computer processors [4, 5], or small biochemical sensors [6, 7]. Confinement of light to sub-wavelength dimensions is also a possibility when one considers the field aspects of the electromagnetic waves near surfaces (near-field phenomena). Add to this the interest in materials and structures exhibiting a negative refractive index for the purpose of increasing the resolution of optical microscopy [8], and it is no wonder that the area of electromagnetic (EM) propagation in sub-wavelength structures is enjoying a renewed interest. Whether the far-field aspects of periodic resonating metallo-dielectric structures are the true manifestations of a negative refractive index or simply a unique, but already known, near-field dispersion phenomenon may be debated [9]. Nonetheless, the near-field aspects of periodic sub-wavelength metallo-dielectric structures, and especially recent advances in nano-fabrication of structures at dimensions smaller than optical wavelengths, deserve a closer look.

Artificial dielectrics (ADs) constitute a class of man-made materials: the effective permittivity and permeability of a given dielectric material may be altered by imbedding metallic or semiconductive structures on scales smaller than the propagating wavelength. For example, one may alter the equivalent capacitance and inductance of microwave waveguides by the addition of a pattern of fine metallic features along the waveguide axis.

Introduction

In recent years, increasing attention has been paid to the so-called correlated disorder in low-dimensional disordered systems. Interest in this subject is mainly due to two reasons. First, it was found that specific correlations in a disordered potential can result in quite unexpected anomalous properties of scattering. Second, it was shown that such correlations can be relatively easily constructed experimentally, at least in the one-dimensional Anderson model and in Kronig–Penney models of various types. Therefore, it seems to be feasible to fabricate random structures with desired scattering properties, in particular when one needs to suppress or enhance the localization in given frequency windows for scattering electrons or electromagnetic waves. In addition, it was understood that, in many real systems, correlated disorder is an intrinsic property of the underlying structures. One of the most important examples is a DNA chain, for which strong correlations in the potential have been shown to manifest themselves in an anomalous conductance. Thus, the subject of correlated disorder is important both from the theoretical viewpoint, and for various applications in physics.

The key point of the theory of correlated disorder is that the localization length for eigenstates in one-dimensional models absorbs the main effect of correlations in disordered potentials. This fact has been known since the earliest analytical studies of transport in continuous random potentials. However, until recently the main interest was in delta-correlated potentials, or in potentials with a Gaussian-type of correlation.

Introduction, general background, and history

Cloaking is the ability to make a region of space, and everything in it, invisible to an external observer. It has been the dream of fantasy writers for decades. In 2009, John Mullan [1] of The Guardian newspaper summarized the ten most important works that use the theme: The Invisible Man by H. G. Wells, The Republic by Plato, The Lord of the Rings by J. R. R. Tolkien, the Harry Potter books by J. K. Rowling, Theogony by Hesiod, Dr Faustus by Christopher Marlowe, The Tempest by William Shakespeare, The Voyage of the Dawn Treader by C. S. Lewis, The Emperor's New Clothes by Hans Christian Andersen, and The Hitchhiker's Guide to the Galaxy by Douglas Adams. A true cloak allows the clear observation of the space behind the cloaked region, and the cloaked region casts no shadow and produces no wavefront changes in the light that has passed through the cloaked region. It is not possible to build a perfect invisibility cloak, as was perceptively observed in the Star Trek series in which cloaked Romulan and Klingon spaceships could be detected by the subtle disturbances of space that the cloak produced.

Interest in making real cloaking devices can be traced to two seminal articles, one by John Pendry and his co-workers [2], and the other by Ulf Leonhardt [3]. Their approach can be called the transformational optics approach to cloaking, which will be discussed in more detail later.

A brief reminder of the history of grating anomalies and plasmon surface waves

The recent history of the research and development around plasmon surface waves that was initiated by the work published in Nature in 1998 by Ebbesen et al. [1] looks like a ten-fold compressed version of studies initiated more than a century ago by Robert Wood with his discovery of anomalies in the efficiency of metallic diffraction gratings, now known as Wood's anomalies [2]. In 1902, R. Wood wrote: “I was astounded to find that under certain conditions, the drop from maximum illumination to minimum, a drop certainly from 10 to 1, occurred within a range of wavelengths not greater than the distance between the sodium lines,” an observation that marked the discovery of grating anomalies.

The first period of the search for their explanation is marked by the attempt of Lord Rayleigh [3, 4] to link Wood's anomalies to the redistribution of the energy due to the passing-off (cut-off) of higher diffraction orders of the grating (transfer from propagating into evanescent type). As pointed out by Maystre [5], his prediction was all the more remarkable as the author first ignored the groove frequency of the grating used by Wood, and thus could not verify this assumption with experimental data.

Introduction

In recent years it has emerged that planar metamaterials offer a vast range of custom-designed electromagnetic functionalities. The best known are wire grid polarizers, which are established standard components for microwaves, terahertz waves, and the far-infrared. They are expected to be of increasing importance also for the near-infrared [1] and visible light [2]. Equally well developed are frequency selective surfaces [3–6], which are used as filters in radar systems, antenna technology [7], broadband communications, and terahertz technology [8, 9]. However, the range of optical effects observable in planar metamaterials and the variety of potential applications have only become clear since metamaterials research took off in 2000 [10]. Wave plate [11, 12] as well as polarization rotator and circular polarizer [13–15] functionalities have been demonstrated in metamaterials of essentially zero thickness. Traditionally, such components are large as they rely on integrating weak effects over thick functional materials. Polarization rotation has also been seen at planar chiral diffraction gratings [16, 17] and thin layered stereometamaterials [18, 19]. Electromagnetically induced transparency (EIT) [20–24] and high quality factor resonances [20] have been observed at planar structured interfaces. And finally, new fundamental electromagnetic effects leading to directionally asymmetric transmission of circularly [25–29] and linearly polarized waves have been discovered in planar metamaterials.

Planar metamaterials derive their properties from artificial structuring rather than atomic or molecular resonances, and therefore appropriately scaled versions of such structures will show similar properties for radio waves, microwaves, terahertz waves, and, to some extent, in the infrared and optical spectral regions where losses are becoming more important.

Introduction

All-angle negative refraction of electromagnetic waves [1, 2] has generated great interest because it provides the foundation for a wide range of new electromagnetic effects and applications, including subwavelength image formation [2] and a negative Doppler shift [1], as well as novel guiding, localization and nonlinear phenomena [3, 4]. There has been tremendous progress in achieving negative refraction in recent years using either dielectric photonic crystals [5–9] or metallic meta-materials [10–17]. For either approach, however, there is an underlying physical length scale that sets a fundamental limit [18]. Below such a length scale, the concept of an effective index no longer holds. For photonic crystals, it is the periodicity, which is smaller than but comparable to the operating wavelength of light [8]. For metallic meta-materials, it is the size of each individual resonant element. In the microwave wavelength range, constructing resonant elements that are far smaller than the operating wavelength is relatively straightforward. As one pushes towards shorter optical wavelengths, however, it becomes progressively more difficult to construct resonant elements at a deep subwavelength scale [15]. Moreover, in the optical wavelength range, the plasmonic effects of metals become prominent. The strong magnetic response of metallic structures, as observed in microwave and infrared wavelength ranges, may be fundamentally affected. It is therefore very desirable to accomplish all-angle negative refraction using structures that are flat at an atomic scale.

Introduction

Negative refraction (NR) has been theoretically predicted [1, 2] and experimentally realized [3–7] in three types of materials. One is a material with a simultaneously negative permittivity and permeability [8–12], leading to a negative refractive index for the medium. The second consists of a photonic crystal (PhC) [13–21], which is a periodic arrangement of scatterers in which the group and phase velocities can be in different directions leading to NR. The third is the indefinite medium [22–28], whose permittivity and/or permeability tensor is an indefinite matrix. In all cases, the bulk properties of the medium, which is inherently inhomogeneous at a subwavelength scale, can be described as having an effective negative refractive index. The active research in these artificial materials has opened doors to a plethora of unusual electromagnetic properties and new applications such as a perfect lens [29], subwavelength imaging [30], cloaking [31], slow light, and optical data storage [32, 33], that cannot be obtained with naturally occurring materials. The holy grail of manufacturing these artificial photonic metamaterial structures is to manipulate light at the nanoscale level for optical information processing and high-resolution imaging.

In order to achieve NR, engineering the bulk electromagnetic properties is normally needed such that the group velocity and phase velocity be at an obtuse angle or even anti-parallel to each other. However, refraction is a surface phenomenon. A bulk-engineered material will have certain inherent surface properties. Negative refraction can be realized in positive index materials by special orientation or by engineering the interface properties.

Introduction

The ability to localize electromagnetic energy below the diffraction limit of classical optics featured by surface plasmon polaritons (SPPs) (electromagnetic surface waves sustained at the interface between a conductor and a dielectric) is currently being exploited in numerous studies ranging from photonics, optoelectronics, and materials science to biological imaging and biomedicine [1]. While the basic physics of SPPs has been described in a number of seminal papers spanning the twentieth century [2, 3], the more recent emergence of powerful nanofabrication and characterization tools has catalyzed a vast interest in their study and exploitation. The dedicated field of plasmonics [4] brings together researchers and technologists from a variety of disciplines, with the common aim to take advantage of the subwavelength light confinement associated with the excitation of SPPs.

Most interest is focused on the optical regime, where SPPs are strongly confined to the respective metal/dielectric interface, i.e. where subwavelength mode localization is achieved in the direction perpendicular to the interface. These strongly confined SPPs occur at frequencies which are still an appreciable fraction of the intrinsic plasma frequency of the metal in question. In this regime, the motion of the conduction electrons at the interface is dephased with respect to the driving electromagnetic fields, leading to a reduction in both phase and group velocities of the SPP, and, therefore, to strong localization. A considerable fraction of the SPP field energy resides inside the conductor.

Introduction

Nanotechnology has seen enormous progress in recent years, and various techniques are now available for the realization of ordered periodic arrays of particles with nanoscale dimensions. Electron-beam [1] and interference lithography [2], polymer-based nanofabrication [3], and self-assembly techniques [4] indeed enable producing ordered one-dimensional (1-D), two-dimensional (2-D), and even three-dimensional (3-D) arrays of metallic or dielectric nanoparticles with sizes much smaller than the wavelength of operation. As is well established in the field of optical metamaterials, such arrays may interact with light in anomalous and exotic ways, provided that their unit cells are sufficiently close to the individual or collective resonance of these arrays.

The electromagnetic response of optical metamaterials and metasurfaces is very distinct from that of gratings and photonic crystals. In photonic crystals, for which lattice periods are comparable to the wavelength of operation, it is possible to tailor the optical interaction operating near the Bragg collective resonances and Wood's anomalies associated with their period, whereas in optical metamaterials and metasurfaces, we operate near the plasmonic resonances of the individual inclusions, leading to the advantage of a much broader response in terms of the angle of incidence, and the absence of grating lobes in the visible angular spectrum. On the other hand, unlike photonic crystals, optical metamaterials and metasurfaces require a much smaller scale for their unit cells. Moreover, plasmonic materials, required to support the required resonances at the nanoscale, are usually characterized by intrinsic non-negligible loss and absorption.

If a metamaterial can be defined as a deliberately structured material that possesses physical properties that are not possible in naturally occurring materials, then deliberately structured surfaces that possess desirable optical properties that planar surfaces do not posses can surely be considered to be optical metamaterials. The surface structures displaying these properties can be periodic, deterministic but not periodic, or random.

In recent years interest has arisen in optical science in the study of such surfaces and the optical phenomena to which they give rise. A wide variety of these phenomena have been predicted theoretically and observed experimentally. They can be divided roughly into those in which volume electromagnetic waves participate and those in which surface electromagnetic waves participate. Both types of optical phenomena and the surface structures that produce them are described in this volume.

The first several chapters are devoted to optical interactions of volume electromagnetic waves with structured surfaces. One of the earliest examples of a structured surface that acts as an optical metamaterial, and the one that today is perhaps the best known and most widely studied, is a metal film pierced by a two-dimensional periodic array of holes with subwavelength diameters. It was shown experimentally by Ebbesen et al. [1] that the transmission of p-polarized light through this structure can be extraordinarily high at the wavelengths of the surface plasmon polaritons supported by the film.

Introduction

A key weakness of the simple approach to nonlinear optics adopted in Chapter 1 was that the physical origin of nonlinearity in the interaction of light and matter was hidden inside the χ(n) coefficients of the polarisation expansion. This is such a fundamental issue that it is difficult to avoid some mention of how nonlinearity arises within a quantum mechanical framework, even in an introductory text. Unfortunately, the standard technique for calculating the nonlinear coefficients is based on time-dependent perturbation theory, and the expressions that emerge begin to get large and unwieldy even at second order. While every effort has been made in this chapter to provide a gentle lead-in to this aspect of the subject, this is almost impossible to achieve given the inherent complexity of the mathematical machinery.

From a mathematical point of view, Schrödinger's equation is linear in the wave function, but nonlinear in its response to perturbations. At a fundamental level, this is where nonlinear optics comes from. The perturbations of atoms and molecules referred to here arise from external electromagnetic fields. When the fields are relatively weak, the perturbations are relatively small, and the theoretical machinery of time-dependent perturbation theory can be deployed to quantify the effects. This is the regime where the traditional polarisation expansion of Eq. (1.24) applies, indeed the terms in the expansion correspond to successive orders of perturbation theory.

I set out to write this book in the firm belief that a truly introductory text on nonlinear optics was not only needed, but would also be quite easy to write. Over the years, I have frequently been asked by new graduate students to recommend an introductory book on nonlinear optics, but have found myself at a loss. There are of course a number of truly excellent books on the subject – Robert Boyd's Nonlinear Optics, now in its 3rd edition [1], is particularly noteworthy – but none of them seems to me to provide the gentle lead-in that the absolute beginner would appreciate.

In the event, I found it a lot harder to maintain an introductory flavour than I had expected. I quickly discovered that there are aspects of the subject that are hard to write about at all without going into depth. One of my aims at the outset was to cover as much of the subject as possible without getting bogged down in crystallography, the tensor structure of the nonlinear coefficients, and the massive perturbation theory formulae that result when one tries to calculate the coefficients quantum mechanically. This at least I largely managed to achieve in the final outcome. As far as possible, I have fenced off the ‘difficult’ bits of the subject, so that six of the ten chapters are virtually ‘tensor-free’.