Preface

Example

<preface id="bk978-0-7503-3579-9ch0s2">
  <named-book-part-body>
    <p>This book is pedagogical. It is meant as an introduction to the ideas and
  phenomena that occur in the nonlinear optics of photonic crystals and
  metamaterials. At the same time it also provides the student with a basic
  introduction to the general ideas of nonlinear optics. The book should
  not, however, be viewed as a comprehensive review of the topics of
  photonic crystals and metamaterials and their nonlinear properties but
  rather as a survey of some of the most important fundamentals of the
  field. In this regard, it provides a strong background to a student new
  to the field. The phenomena discussed are illustrated within the context
  of simple models, offering an easy understanding of the physical
  phenomena that are important in these two rapidly developing areas of
  nanophotonic technology.</p>
    <p>The book first deals with nonlinear dielectric properties. A model based
  on classical physics is used to develop some of the ideas of the Kerr
  nonlinear dielectric and the generation of second and higher harmonics of
  radiation in a semi-quantitative way. The treatment is similar to that
  used in basic electrodynamics texts to provide an understanding of the
  dielectric properties of linear dielectric materials. An emphasis is on
  nonlinear effects as being small and on their treatment within the
  context of them entering the considerations as perturbations of otherwise
  linear systems. Consequently, along with the development of the
  dielectric properties, the theory of multiple scale perturbation theory
  is simultaneously formulated as an important technique in correctly
  dealing with the physics of systems with small nonlinearities. The
  reasons why multiple scale perturbation techniques are needed and a
  number of examples of the multiple scale treatment of nonlinear systems
  are given.</p>
    <p>The distinctions between the ideas of photonic crystals and metamaterials
  and the different types of applications they address are discussed with
  emphasis on their elementary properties. Photonic crystals have
  dielectric properties which exhibit periodicity in space. Due to their
  spatial periodicity the dispersion relation of light within the photonic
  crystal exhibits a sequence of frequency pass bands and stop bands. Light
  at pass band frequencies propagates in the photonic crystal and light at
  stop band frequencies does not propagate through the photonic crystal. On
  the other hand, metamaterials are composed as arrays of nanoscopic
  electromagnetic resonator units. They are engineered to appear to be
  homogeneous media on the scale of the wavelengths of light with which
  they are designed to interact. The frequency dependence of the resonators
  allow metamaterials to exhibit frequency dependent refractive indices not
  otherwise observed in nature. Of particular importance is their ability
  to display negative index of refraction.</p>
    <p>The theory of photonic crystals is greatly influenced by their spatial
  periodicity. In this regard, basic discussions of the dynamics of systems
  defined on a periodic lattice are made with a focus on the Maxwell
  equations expressed for periodic media. The direct and reciprocal
  lattices, Brillouin zones, Block functions, and the properties of stop
  and pass bands are introduced at an elementary level. Waveguides and
  single site impurities introduced into otherwise periodic bulk photonic
  crystal systems also are discussed at an introductory level, as are the
  techniques of Wannier functions and computer simulation treatments
  applied to these types of problems.</p>
    <p>The discussions of periodic systems are also extended to considerations
  of the basic theory of surface gratings and applications of Bragg
  gratings within fiber optics and general optical waveguides. The use of
  photonic crystal technology in the design of optical fibers is an
  additional consideration of new designs based on the ideas of periodic
  media. In this regard, periodicity is introduced both as a novel means of
  making fiber claddings and as a confining mechanism based on stop bands.
  In addition, discussions of soliton modes within optical fibers formed of
  nonlinear optical media is briefly presented, offering a comparison with
  the soliton-like intrinsic localized modes present in photonic crystal
  waveguides formed of nonlinear optical media.</p>
    <p>The application of metamaterials in the design of media with unusual
  refractive properties is discussed, with a particular focus on the
  development of materials exhibiting a negative refractive index.
  Metamaterials are typically designed as arrays of electromagnetic
  nanoresonators, and a commonly used resonator form is that of a split
  ring resonator (SRRs). In this regard, an understanding of the resonant
  properties of SRRs is developed in a detailed but elementary treatment.
  How these resonant properties are then employed to engineer a material
  exhibiting negative refraction is then explained in detailed.</p>
    <p>The refractive properties of plane waves traveling between semi-infinite
  media of positive and negative refractive index media are studied
  comprehensively. Differences in the radiative properties of antennas and
  radiating charges within both positive and negative index media are
  compared. The essential operating principles of perfect lenses are
  explained and a comparison is made with the properties of lenses formed
  only of positive refractive index materials. In addition, the application
  of metamaterials in the formulation of the ideas of optical cloaking
  mechanisms are developed.</p>
    <p>Both analytic and computer simulation methods are introduced at an
  elementary level for the treatment of basic problems in photonic crystals
  and metamaterials. In terms of computer solutions, the plane wave method,
  the finite difference time domain method, the method of moments, the
  conjugate gradient method, and the finite element method of computer
  simulation techniques are formulated and discussed in elementary and
  detailed manners. Various formulations for the boundary conditions of
  computer simulation studies needed to accurately approximate the
  interaction of radiation with dielectric features in infinite systems are
  presented.</p>
    <p>On the side of analytic models, a number of difference equation
  formulations are given for photonic crystal and metamaterial waveguides.
  These difference equation models are used to facilitate the understanding
  of the basic physics of photonic crystals and metamaterials. The
  difference equation models are also realizable in many experimentally
  accessible cases. They often provide simple exactly solvable models whose
  properties are easily made transparent and are also representative of
  more complex general systems studied using computer simulations. Analytic
  techniques for the solution of difference equation formulations are
  discussed and are related to the physics of many different types of
  physical systems that occur in physics and technology.</p>
    <p>The properties of photonic crystal and metamaterial systems with Kerr
  nonlinearity are discussed in terms of formulations based on difference
  equation models. Properties of optical bistability and resonant
  transmission anomalies are explained in detail based on elementary
  treatments in the difference equation formulations. The various
  properties are treated with discussions for their applications in devices
  and examples given in the context of the simple models.</p>
    <p>A chapter is dedicated to discussions of computer simulation studies of
  nonlinear properties of some recently considered systems and the
  properties with which these have dealt. One-dimensional photonic crystals
  formed of a periodic layering of Kerr slabs are discussed along with the
  properties of Kerr impurities in higher-dimensional photonic crystals. In
  another important treatment, studies of the applications of photonic
  crystals to enhance the generation of second harmonics are outlined as
  another application of the use of photonic crystals. Some recent computer
  simulation studies of nonlinear metamaterials and the properties these
  studies have treated are also summarized. In these discussions only some
  of the important basic results from computer simulations are presented,
  and the discussions are not meant to be a comprehensive review of the
  field of computer simulation applied to nonlinear systems.</p>
    <p>In the last chapter of the book a treatment is given of the effects of
  impurities and disorder introduced into photonic crystals and
  metamaterials. Single site impurities and finite clusters of impurities
  are discussed in the context of Green’s functions theories and with the
  applications of ideas of group theory. General random disorder is treated
  through the application of the coherent potentional approximation, and
  the theory of this methodology is explained in detail. The general theory
  of Anderson localization in random disordered optical media is discussed
  along with the ideas of a scaling theory approach to the effects of
  disorder on the transport properties of disordered systems. In this
  regard, optical systems have provided important systems for the
  discussion of the ideas of so-called strong and weak localization. As a
  final treatment of localization, a focus is given on the effects of weak
  localization observed in the general diffuse elastic scattering of light
  and the general diffusely generated second harmonics of light from a
  randomly rough surface which supports surface electromagnetic waves. In
  this context, an introductory treatment is presented to the theory of
  surface electromagnetic waves and the requirements for their existence on
  a rough interface.</p>
    <p>As a final note: both the Gaussian and Standard International Units
  formulations of the Maxwell equations are used at different points in the
  text. This is done as many of the results referenced from the literature
  occur in these sets of units, as will be indicated in the text. In
  addition, the author would like to thank the Department of Physics,
  University of California, Riverside, for allowing me the use of the
  University Library during the course of this project.</p>
    <p>Arthur R McGurn</p>
    <p>Rancho Mirage, California</p>
    <p>May, 2021</p>
  </named-book-part-body>
</preface>