Chapters
All of a books chapters must be captured in seperate <book-part
> elements within a <book-body> element.
Each <book-part > element must contain a
@book-part-type attribute with a value of chapter
.
Each <book-part > element must also contain a @id
attribute. The value of the attribute is based on the book primary ISSN with the format
bk[primary isbn]ch[chapter number]
.
Example
<book-part book-part-type="chapter" id="bk978-0-7503-3579-9ch1">
<book-part-meta>
<book-part-id book-part-id-type="doi">10.1088/978-0-7503-3579-9ch1</book-part-id>
<title-group>
<label>1</label>
<title>Introduction</title>
</title-group>
<contrib-group>
<contrib contrib-type="author" xlink:type="simple">
<name name-style="western">
<surname>McGurn</surname>
<given-names>Arthur R</given-names>
</name>
<xref ref-type="aff" rid="bk978-0-7503-3579-9ch1aff1"/>
</contrib>
<aff id="bk978-0-7503-3579-9ch1aff1">Department of Physics, <institution xlink:type="simple">Western Michigan University</institution>,
Kalamazoo, MI, <country>USA</country>
</aff>
</contrib-group>
<fpage>1-1</fpage>
<lpage>1-12</lpage>
<permissions>
<copyright-statement>© IOP Publishing Ltd 2021</copyright-statement>
<copyright-year>2021</copyright-year>
<copyright-holder>IOP Publishing Ltd</copyright-holder>
<license license-type="iop-standard-books" xlink:href="https://publishingsupport.iopscience.iop.org/iop-standard/books">
<license-p> This book is available under the terms of the
<ext-link ext-link-type="uri" xlink:href="https://publishingsupport.iopscience.iop.org/iop-standard/books">IOP-Standard Books License</ext-link>
</license-p>
<license-p> Permission to make use of IOP Publishing content other
than as set out above may be sought at <ext-link ext-link-type="email" xlink:type="simple">permissions@ioppublishing.org</ext-link> . </license-p>
<license-p>Arthur R McGurn has asserted his right to be identified
as the author of this work in accordance with sections 77 and 78
of the Copyright, Designs and Patents Act 1988.</license-p>
</license>
</permissions>
<self-uri content-type="pdf" xlink:href="bk978-0-7503-3579-9ch1.pdf"/>
<self-uri content-type="epub" xlink:href="bk978-0-7503-3579-9ch1.epub"/>
<abstract>
<title>Abstract</title>
<p>In this chapter the basic properties of photonic crystals and
metamaterials are qualitatively discussed. Photonic crystals are
periodic dielectric structures which modulate the flow of light
through application of the ideas of diffraction. Metamaterials are
designed as resonator arrays which appear as homogeneous media to the
light they refractively modulate. Metamaterials effect the flow of
light through application of the ideas of refraction. The refractive
properties of negative refractive index materials and their
applications are discussed. The ideas of soliton modes in continuous
and discrete media are described.</p>
</abstract>
</book-part-meta>
<body>
<p>In this book an introduction and discussion of some of the basic
principles of linear and nonlinear optical nano-systems are given. The
focus is on engineered optical systems that have been of recent interest
in physics, engineering, and applied mathematics for their
opto-electronic applications. These include photonic crystals and
metamaterials, and in the following discussions the operating principles
of photonic crystals and metamaterials are outlined. Photonic crystals,
which have been of great interest in opto-electronic designs, are
materials that exhibit a periodic dielectric variation in space and are
designed to manipulate light with wavelengths of order of the length
scales of the periodicity of the photonic crystal lattice [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib1">1</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib5">5</xref>]. The
manipulation of light is accomplished through the use of the Bragg
scattering properties of the periodic lattice of the dielectric which may
be periodic in one, two, or three dimensions. Metamaterials often have
periodic dielectric and magnetic properties but are designed to
manipulate light of wavelengths much greater than the periodicity of the
dielectric and magnetic lattice [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib1">1</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib5">5</xref>]. Metamaterials are designed
to exhibit homogeneous properties to the light traveling within them.
While photonic crystals are of interested for the diffractive effects
they have on light propagating within them, metamaterials are of interest
for the permittivity and permeability that they exhibit when they are
treated as a uniform medium.</p>
<p>For both systems a particular goal of our presentation is to describe
their behaviors as nonlinear optical systems [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib6">6</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib9">9</xref>]. To accomplish this, a
general review is also given of the properties of these systems
considered as linear optical systems. The treatments assume an
undergraduate background in electrodynamics and are developed starting
from an elementary level. The presentation provides an exposition of the
basic principles of the photonic crystal and metamaterial systems. The
wave excitations found in linear medium systems and their elementary
properties are treated in detail. A detailed development, requiring no
previous background in nonlinear materials, of the excitations in
nonlinear systems consisting of plane waves and bright, dark, and gray
solitons is also given.</p>
<sec id="bk978-0-7503-3579-9ch1s1-1">
<label>1.1</label>
<title>Photonic crystals</title>
<p>Photonic crystals are periodic arrays of different dielectric
materials [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib1">1</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib5">5</xref>]. They are literally crystals formed from dielectric
materials rather than the individual atoms of materials such as
crystalline NaCl, <inline-formula>
<tex-math><?CDATA ${\rm{CaC}}{{\rm{l}}}_{2}$?></tex-math>
<mml:math overflow="scroll">
<mml:mrow>
<mml:mi mathvariant="normal">CaC</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">l</mml:mi>
</mml:mrow>
<mml:mstyle fontsize="6.85pt">
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:mstyle>
</mml:msub>
</mml:mrow>
</mml:math>
<inline-graphic xlink:href="bk978-0-7503-3579-9ch1ieqn1.gif" xlink:type="simple"/>
</inline-formula>, etc The interest in the dielectric array is that it
exhibits a band structure for the propagation of electromagnetic modes
similar to the electronic band structure exhibited in metallic and
semiconductor materials used in electronic designs [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib10">10</xref>]. As
with the electron systems, the interest in the optical systems is in
modes that have wavelengths of order of the length scale of the
periodicity of the system. At these length scales the system displays
a variety of diffraction effects on the excitations in the system.
Various pass and stop frequency bands are opened in the frequency
spectrum of excitations that can exist in the photonic crystal. Light
at frequencies in a pass band will propagate through the photonic
crystal while light at frequencies in a stop band will not propagate
in a photonic crystal and are typically reflected from the system.</p>
<p>The existence of pass and stop frequency bands is the basis for
engineering applications of photonic crystals. A cavity resonator, for
example, can be formed as a cavity within the bulk of a photonic
crystal. An electromagnetic wave with a frequency in the stop band of
the photonic crystal cannot propagate out of the cavity through the
bulk photonic crystal so it will be confined to the cavity. Such
cavities offer high <italic>Q</italic> resonance cavities for laser
applications that are not available through other technologies [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib11">11</xref>].
Another application of photonic crystal cavities is in the suppression
and enhancement of atomic transitions. If the excited state of an atom
which radiates at a stop band frequency is put in the cavity it will
be suspended in the excited state due to its inability to radiate the
excitation energy though the bulk of the photonic crystal. Similarly,
the frequency mode density of states in a photonic crystal pass band
can be enhanced from those of free space. This enhancement of the
density of states increases the rate of decay of an excited atom
within a photonic crystal cavity into enhanced pass band frequency
modes over its free space decay rate [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib11">11</xref>].</p>
<p>The stop band effect can also be used in the design of waveguides
meant to channel the flow of light through space [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib1">1</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib5">5</xref>]. A
channel cut through the bulk of a photonic crystal will restrict light
at frequencies in the stop bands of the bulk photonic crystal from
propagating away from the waveguide, i.e., light at stop band
frequencies will only move along the channel of the waveguide.
Waveguides based on such photonic crystal designs can be more
effective than fiber optic technologies in forming optical circuits.
For example photonic crystal waveguides afford the possibility of
sharper bends in their guiding channels and lower losses in general
than are found in traditional fiber optics approaches [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib12">12</xref>]. In
addition, photonic crystals can be designed from a wider variety of
materials than are typically used in fiber optics.</p>
<p>One method of introducing a waveguide into a two-dimensional photonic
crystal formed as a periodic array of identical parallel axis
dielectric cylinders will be a focus in later considerations of guided
modes. For this system the interest is in light propagating in the
plane perpendicular the axes of the cylinders. A waveguide is
introduced into the cylinder array by replacing a periodic array of
dielectric cylinders of the photonic crystals by a set of identical
impurity cylinders with different dielectric properties from those of
the photonic crystal. This is done along a crystal axis of the
two-dimensional array of the photonic crystals, and by choosing the
impurity dielectric correctly a set of waveguide modes bound to and
propagating along the array of waveguide impurity cylinders can be
made to exist at stop band frequencies of the original photonic
crystal. Later it is shown that the modes of this kind of waveguide
system can be represented by a set of difference equations. This
representation provides a helpful means of understanding the physics
of photonic crystal waveguides and circuits.</p>
<p>One of the earliest applications of the ideas of photonic crystals,
occurring in the initial stages of the developments of laser
technology, was the application of one-dimensional or layered photonic
crystals in the design of laser mirrors [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib1">1</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib5">5</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib13">13</xref>]. Here the periodic
layering of different dielectric media can be used to create low loss,
highly reflective mirrors [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib13">13</xref>]. Similar layering
effects are observed in insects in which the metallic, mirrored
appearance of the shell of the animal is due to the layering of
dielectrics rather than to a presence of metallic reflecting elements
in the shell of the insect. In both of these systems the presence of
pass and stop bands of the layering leads to the functioning of the
reflecting surfaces. In addition to the insect example of photonic
crystals developed in nature, a number of periodic one-, two-, and
three-dimensional nano-systems that occur naturally in plant and
animal materials have been suggested as a basis for the design of
photonic crystals [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib1">1</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib5">5</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib13">13</xref>]. The interest in these
comes from the pre-existing nano-scale periodicity which is hard to
create in a laboratory. A number of laboratory techniques have been
developed for creating photonic crystals, but these are not discussed
here.</p>
<p>In addition to photonic crystals formed as arrays of slabs, or
cylinders, or three-dimensional periodic ordering of dielectric
features, surfaces with periodic surface profiles are also found to
exhibit important photonic crystal properties [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib1">1</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib5">5</xref>]. Periodic surfaces have
long been studied for their applications as diffraction gratings, and
a brief review of their basic theory is presented. In addition, under
certain well defined conditions periodic surfaces can support surface
electromagnetic waves. These are electromagnetic modes which are bound
to the interface between two media, representing excitations localized
about the interface and propagating parallel to the interface. These
surface waves are important excitations with applications in
electromagnetic scattering from the surface, in the design of various
sensors, and in surface enhanced Raman scattering (SER) [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib1">1</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib5">5</xref>]. They
feature prominently in the diffuse scattering of light at rough
surfaces and in the diffuse generation of second harmonics of light at
rough surfaces.</p>
<p>In regards to nonlinear effects in photonic crystals and waveguides, a
treatment of the basic properties of parametric oscillators and
parametric amplifiers is given [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib6">6</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib7">7</xref>]. These are discussed in
both bulk optical media and in the context of nonlinear fiber optical
waveguides [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib12">12</xref>]. Bragg grating technology in fiber optics is also a
topic. These systems are represented in many important technological
applications.</p>
</sec>
<sec id="bk978-0-7503-3579-9ch1s1-2">
<label>1.2</label>
<title>Metamaterials</title>
<p>Metamaterials are artificial materials that are designed as arrays of
nano-circuits known as split ring resonators (SRRs) [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib14">14</xref>]. SRR
nano-circuits are inductor–capacitor resonator circuits which in
metamaterial designs may be arrayed on a periodic nano-lattice or in a
more complex array arrangement. The resonator circuits are known as
SRRs because in the basic form of their design they are a metal ring
with a split gap interrupting the ring. The loop of the ring provides
self-inductance to the SRR, and the gap of the SRR is filled with a
dielectric material to form a capacitor. Composed in this way the SRR
is found to resonate at the frequency <inline-formula>
<tex-math><?CDATA ${\omega }_{{\rm{SRR}}}=\frac{1}{\sqrt{{LC}}},$?></tex-math>
<mml:math overflow="scroll">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>ω</mml:mi>
</mml:mrow>
<mml:mstyle fontsize="6.85pt">
<mml:mrow>
<mml:mi mathvariant="normal">SRR</mml:mi>
</mml:mrow>
</mml:mstyle>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:msqrt>
<mml:mrow>
<mml:mi mathvariant="italic">LC</mml:mi>
</mml:mrow>
</mml:msqrt>
</mml:mrow>
</mml:mfrac>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<inline-graphic xlink:href="bk978-0-7503-3579-9ch1ieqn2.gif" xlink:type="simple"/>
</inline-formula> where <italic>L</italic> is the SRR self-inductance
and <italic>C</italic> is its capacitance [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib15">15</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib21">21</xref>]. For practical
applications more complex SRR designs are used but all of the various
designs operate on the basic principles that are outlined above.
Within a particular metamaterial array the SRRs are coupled together
by mutual inductive couplings between the various SRRs. This gives
rise to many-body effects which also affect the optics of the
material.</p>
<p>The purpose behind the composition of metamaterials as arrays of SRRs
is to use their flexibility of design to create materials with
enlarged sets of permittivity, <italic>ε</italic>, and permeability,
<italic>μ</italic>, from those of naturally occurring materials
[<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib15">15</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib24">24</xref>]. In naturally occurring molecular solids the
permittivity and permeability of the bulk solid arises from those of
the molecules forming the systems. These properties are fundamentally
limited by the nature and interactions of the molecular constituents
of the materials. For example, in no naturally occurring material is
it found that <italic>ε</italic> < 0 and <italic>μ</italic> < 0
simultaneously at a single frequency of an applied electromagnetic
plane wave. Regions of frequency are often found for which
<italic>ε</italic> < 0 and regions of frequency are often found
for which <italic>μ</italic> < 0, but no natural materials have
been found for which these two conditions are simultaneously satisfied
at the same frequency. This limitation on <italic>ε</italic> and
<italic>μ</italic> comes from the size restrictions of the
molecular units and their effect on the magnetic response of molecular
solids. The development of an artificial nano-circuit array allows
these restrictions to be overcome.</p>
<p>At microwave and terahertz frequencies engineered SRR materials with
<italic>ε</italic> < 0 and <italic>μ</italic> < 0
simultaneously can be formulated at these frequencies through the use
of an SRR array to customize the magnetic response of the material.
The SRR ring geometry is arranged so that the resonant frequency <inline-formula>
<tex-math><?CDATA ${\omega }_{{\rm{SRR}}}=\frac{1}{\sqrt{{LC}}}$?></tex-math>
<mml:math overflow="scroll">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>ω</mml:mi>
</mml:mrow>
<mml:mstyle fontsize="6.85pt">
<mml:mrow>
<mml:mi mathvariant="normal">SRR</mml:mi>
</mml:mrow>
</mml:mstyle>
</mml:msub>
<mml:mo>=</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:msqrt>
<mml:mrow>
<mml:mi mathvariant="italic">LC</mml:mi>
</mml:mrow>
</mml:msqrt>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<inline-graphic xlink:href="bk978-0-7503-3579-9ch1ieqn3.gif" xlink:type="simple"/>
</inline-formula> of the ring is in the microwave or terahertz region.
Under this condition it is found that the size of the SRRs needed is
much greater than the molecules available in molecular solids. In
addition, at frequencies near <inline-formula>
<tex-math><?CDATA ${\omega }_{{\rm{SRR}}}$?></tex-math>
<mml:math overflow="scroll">
<mml:msub>
<mml:mrow>
<mml:mi>ω</mml:mi>
</mml:mrow>
<mml:mstyle fontsize="6.85pt">
<mml:mrow>
<mml:mi mathvariant="normal">SRR</mml:mi>
</mml:mrow>
</mml:mstyle>
</mml:msub>
</mml:math>
<inline-graphic xlink:href="bk978-0-7503-3579-9ch1ieqn4.gif" xlink:type="simple"/>
</inline-formula> the wavelength of the externally applied
electromagnetic waves is slowly varying over a volume containing many
SRRs so that the SRR material appears to the external wave to be a
continuous medium. As the frequency, ω, of an externally applied field
is changed so as to pass through the SRR’s resonance at <inline-formula>
<tex-math><?CDATA ${\omega }_{{\rm{SRR}}},$?></tex-math>
<mml:math overflow="scroll">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>ω</mml:mi>
</mml:mrow>
<mml:mstyle fontsize="6.85pt">
<mml:mrow>
<mml:mi mathvariant="normal">SRR</mml:mi>
</mml:mrow>
</mml:mstyle>
</mml:msub>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<inline-graphic xlink:href="bk978-0-7503-3579-9ch1ieqn5.gif" xlink:type="simple"/>
</inline-formula> the magnetic moment of the SRR changes sign.
Consequently, about the resonance of the SRR system the SRR can
exhibit a frequency region in which <italic>μ</italic> < 0. This
artificially induced <italic>μ</italic> < 0 is easily correlated
with a region of <italic>ε</italic> < 0 for the dielectric media in
which the rings are embedded. The resulting metamaterial displays a
set of frequencies for which <italic>ε</italic> < 0 and
<italic>μ</italic> < 0 at the same time.</p>
</sec>
<sec id="bk978-0-7503-3579-9ch1s1-3">
<label>1.3</label>
<title>Negative index of refraction</title>
<p>With the introduction of metamaterials a full range of
<italic>ε</italic> and <italic>μ</italic> values become available
for optical design. In particular, the case of <italic>ε</italic> <
0 and <italic>μ</italic> < 0 is now a practical possibility. This
is very important for refractive optics as materials with
<italic>ε</italic> < 0 and <italic>μ</italic> < 0 will
exhibit negative refractive indices [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib18">18</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib22">22</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib25">25</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib26">26</xref>]. The possibility of
negative indexed materials was theoretically considered early in the
twentieth century but was only treated as a curiosity [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib25">25</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib26">26</xref>]. In
naturally occurring positive index materials figure <xref ref-type="fig" rid="bk978-0-7503-3579-9ch1fig1">1.1</xref> shows
that light incident on a planar interface from a positive indexed
medium to an optically rarer or denser positive indexed medium is
refracted from the second to the fourth quadrants. In negative index
materials figure <xref ref-type="fig" rid="bk978-0-7503-3579-9ch1fig2">1.2</xref> shows that light incident on a planar interface from a
positive indexed medium to a negative indexed material is refracted
from the second to the third quadrants. Consequently, light incident
in the second quadrant can now be guided though any angle in the
forward direction of the third and fourth quadrants. This has found
applications in cloaking protocols in which light directed at an
object can be guided around it by using a continuous variation of
positive and negative refractive indices [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib14">14</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib18">18</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib27">27</xref>]. Once around the
object the light is sent along a path set to give the appearance that
the object is not present. The object is made to appear not present
even in its presence. Other such variations of positive and negative
indices have found applications in simulating optical effects from
general relativity. In noting these interesting properties of
metamaterials, it should also be pointed out that under special
circumstances the Bragg reflection properties of photonic crystals,
related to the periodicity of the dielectric medium, have been found
to mimic some of the properties of metamaterials.</p>
<fig id="bk978-0-7503-3579-9ch1fig1" position="float" orientation="portrait">
<label>Figure 1.1.</label>
<caption>
<p>Schematic for refraction of light at the interface between two
positive indexed media. The incident wave in medium 1 is
transformed at the surface into a reflected wave in medium 1 and
into a refracted wave in medium 2. The incident wave is in the
second quadrant and for the positive media the refracted wave is
in the fourth quadrant.</p>
</caption>
<graphic id="bk978-0-7503-3579-9ch1f1_tif" content-type="print" xlink:href="bk978-0-7503-3579-9ch1f1_pr.tif" position="float" orientation="portrait" xlink:type="simple"/>
<graphic id="bk978-0-7503-3579-9ch1f1_online" content-type="online" xlink:href="bk978-0-7503-3579-9ch1f1_online.jpg" position="float" orientation="portrait" xlink:type="simple"/>
<graphic id="bk978-0-7503-3579-9ch1f1_hr" content-type="high" xlink:href="bk978-0-7503-3579-9ch1f1_hr.jpg" position="float" orientation="portrait" xlink:type="simple"/>
</fig>
<fig id="bk978-0-7503-3579-9ch1fig2" position="float" orientation="portrait">
<label>Figure 1.2.</label>
<caption>
<p>Schematic for refraction of light at the interface between
positive indexed medium 1 and negative indexed medium 2. The
incident wave in medium 1 is transformed at the surface into a
reflected wave in medium 1 and into a refracted wave in medium
2. The incident wave is in the second quadrant and for the
negative index medium the refracted wave is in the third
quadrant.</p>
</caption>
<graphic id="bk978-0-7503-3579-9ch1f2_tif" content-type="print" xlink:href="bk978-0-7503-3579-9ch1f2_pr.tif" position="float" orientation="portrait" xlink:type="simple"/>
<graphic id="bk978-0-7503-3579-9ch1f2_online" content-type="online" xlink:href="bk978-0-7503-3579-9ch1f2_online.jpg" position="float" orientation="portrait" xlink:type="simple"/>
<graphic id="bk978-0-7503-3579-9ch1f2_hr" content-type="high" xlink:href="bk978-0-7503-3579-9ch1f2_hr.jpg" position="float" orientation="portrait" xlink:type="simple"/>
</fig>
<p>The positive and negative indexed materials are alternatively referred
to as right- and left-handed materials due to the different
relationship of the three orthogonal vectors of the electric vector,
magnetic vector, and wave vector between the two types of systems
[<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib26">26</xref>]. Indeed, it can be shown that waves in right-hand
materials propagate electromagnetic energy parallel to the wave vector
while waves in left-hand materials propagate electromagnetic energy
antiparallel to the wave vector. This can have effects for example on
the Cherenkov radiation in right- and left-handed materials,
respectively, as well as of course for the refractive properties and
the properties of antenna radiation [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib14">14</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib24">24</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib27">27</xref>].</p>
</sec>
<sec id="bk978-0-7503-3579-9ch1s1-4">
<label>1.4</label>
<title>Perfect lenses</title>
<p>One particularly interesting applications of metamaterials is in the
design of so-called perfect lenses [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib22">22</xref>]. The functioning of
the perfect lens is based on the new refractive property of negative
index of refraction. The bending of light at the interface of a
positive and negative indexed material from the second to the third
quadrant allows a planar surface to form a focused image as light
passes through planar surfaces. This is due to the increased bending
of the light as it goes between the two media over that found between
any two positive indexed media.</p>
<p>In the absence of negative refractive indexed materials, a curved
surface is needed between two positive indexed media to focus light as
it passes between the two media. This is the operating principle in
the design of telescope and microscope lenses. The design of lenses
with curved surfaces limits their size. Consequently, only light
passed through the aperture of the lens can be focused, and this
limits the wavelength of light which will pass through the lens and
become part of the image located at the focus of the lens [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib28">28</xref>]. Light
must have a wavelength less than the aperture of the lens to reach the
focus and, consequently, the image formed by the lens is not perfect
as the loss of some wavelength components of the image decreases the
resolution of the image.</p>
<p>With a perfect lens formed of negative refractive indexed material
only planar surfaces are needed to make a focusing lens. In addition,
the permittivity and permeability can be adjusted so that even the
evanescent components of light from the object are reassembled at the
image, giving a complete characterization in the image of the object.
In principle an image with perfect resolution can be formed by imaging
with an infinite slab of negative index medium. There are, however, a
number of technical difficulties in the practical application of the
ideas of a perfect lens. Due to the resonant nature of the materials
designed to exhibit negative refractive index, the frequencies over
which these properties are exhibited are limited. In addition,
resonant structures tend to exhibit losses, well known from the
Kramers–Kronig relations.</p>
</sec>
<sec id="bk978-0-7503-3579-9ch1s1-5">
<label>1.5</label>
<title>Periodicity</title>
<p>Both photonic crystals and metamaterials can be formed by the
repetition of mesoscopic features on a periodic lattice. In the case
of photonic crystals the interest is in the manipulation of
electromagnetic radiation with wavelengths of order of the lattice
constant, while metamaterials are used in the management of radiation
with wavelengths much larger than the lattice constant. Metamaterials
appear to be homogeneous while the important properties of the
photonic crystals arise from their detailed periodic structure. For
both photonic crystal and metamaterial systems formed from linear
dielectric media the light propagating in the systems is
electromagnetic waves satisfying linear wave equations. In
metamaterials the excitations are simple plane waves and the effect of
the media on the propagating waves is characterized by the effective
<italic>ε</italic> and <italic>μ</italic> of the metamaterial. For
the photonic crystal, on the other hand, a full account of the
periodicity of the dielectric media must be given so that the
excitations are more complex than simple plane waves, and a variety of
methods for the calculation of the electromagnetic band structure of
these materials are available. These methods are well known, having
initially been developed for the treatment of the motion of electrons
in the periodic ion background of semi-conductors and metals. Among
those methods discussed later are the plane wave expansion method and
the method of Wannier functions [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib2">2</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib4">4</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib10">10</xref>]. In addition, computer
simulations based on finite difference time domain methods [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib29">29</xref>] and
the method of moments [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib30">30</xref>] are useful in
determining the system properties.</p>
<p>The plane wave expansion technique for photonic crystals is based on
Fourier transforming the equations of motion of the systems and
studying their properties in frequency–wave vector space. These
methods give a good account of the properties of linear photonic
crystals and are quick and easy to implement. Computer simulation
methods in space and time or space and frequency can also be applied
to determine the dispersion relation and field properties of the modes
of the photonic crystal system. Such computer simulations are
particularly important in applications treating waveguides and
impurities in both photonic crystal and metamaterial systems. The
treatment of waveguides and impurities is also a strength of the more
analytic methods of Wannier functions [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib10">10</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib31">31</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib32">32</xref>]. In the method of
Wannier functions, an orthogonal set of basis functions which are
localized in space are generated. This set is then used to expand the
modes of photonic crystals and study their properties. All of these
methods will be discussed later.</p>
</sec>
<sec id="bk978-0-7503-3579-9ch1s1-6">
<label>1.6</label>
<title>Excitations in nonlinear media</title>
<p>While linear systems are easily handled numerically, nonlinear systems
in which the dielectric properties depend on the amplitude of the
excitations they support are much more challenging. In nonlinear
systems, a linear combination of the wave functions of two separate
solutions of the system is not a wave function solution of the system,
i.e., the principle of linear superposition of modes is no longer
valid. The violation of the principle of linear superposition is
easily seen in the wave-like modes of nonlinear systems [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib7">7</xref>]. Due to
the amplitude dependence of the dielectric, wave solutions of
nonlinear systems have dispersion relations that depend on the
amplitude of the wave. Consequently, increasing or decreasing the
amplitude of a wave changes the frequency of its oscillation, and this
change in frequency is not accounted for in a simple linear
combination of plane wave solutions.</p>
<p>In addition, nonlinear systems exhibit new types of excitations not
found in linear systems. Examples of these are solitons [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib9">9</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib33">33</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib36">36</xref>].
Solitons occur as pulses and kinks that propagate through nonlinear
optical systems. They exist due to the dependence of the dielectric
properties of the systems on the amplitude of the excitations. This
can be understood as follows: in a system composed solely of a linear
dielectric it is possible to bind a localized electromagnetic mode to
a finite region of space by appropriately changing the value of the
linear dielectric in that finite region. In line with these ideas,
introducing a localized electromagnetic pulse into a nonlinear
dielectric system can be done in such a way that the change in the
nonlinear dielectric constant caused by the changing pulse intensity
supports the localized pulse. The localized pulse in turn then
supports the change in the nonlinear dielectric. Such a pulse can be
stationary or it can travel through the system with a constant
velocity. The pulse is known as a bright soliton because of its
intensity maximum. These ideas also apply to kinks. For the case of
kink excitations the electric field intensity exhibits a dip rather
than an intensity peak. The kink intensity shapes the dielectric
response of the system which in turn supports the kink wave function.
As in the case of bright solitons, kinks can also be either stationary
or move through the system with constant velocities. If the intensity
of the dip goes to zero the kink is known as a dark soliton, and if
the intensity dip does not go to zero the kink is known as a gray
soliton. In addition to the single soliton solutions there are
solutions involving a number of solitons scattering from one
another.</p>
<p>The general types of solitons we will look at are bright, dark, and
gray solitons. The bright, dark, and gray solitons that are found in
our photonic crystals and metamaterials are further classified by the
nature of their wave functions [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib33">33</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib35">35</xref>] into simple pulsed
solitons and envelope solitons. Simple pulsed solitons occur as
smooth, slowly varying pulses or a smooth, slowly varying twisting of
the amplitudes of the electromagnetic fields in space. Simple pulses
do not exhibit any sinusoidal modulations of the basic wave function
envelopes of the excitations. Envelope solitons, on the other hand,
are more complex, being composed of plane waves modulated by an
envelope in the form of a pulse or kink. For these cases the solitons
may be obtained in the continuum limit of the system or in the
discrete lattice of the system. In discrete lattice systems they are
often referred to as intrinsic localized modes or discrete breathers
in the limit where they are stationary in the system. These
distinctions will be discussed later.</p>
<p>Later the theory of soliton-like modes is discussed for both the
discrete and continuum limits of photonic crystal and metamaterial
systems [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib35">35</xref>]. The solutions in the discrete lattice systems are
shown to be easily accessible to a variational treatment in which the
wave functions are expressed as linear expansions in an appropriate
set of basis states defined over the discrete lattice. Solutions of
the form of bright, dark, and gray soliton-like excitations are
obtained [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib33">33</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib35">35</xref>]. In the continuum limit of the lattice systems
descriptions in terms of the nonlinear Schrödinger and Klein–Gordon
equations are given for photonic crystal and metamaterial systems.
Both systems are found to exhibit closed form soliton-like solutions,
and the properties of the bright, dark, and the gray solitons found in
these systems are discussed. The relationships between the
soliton-like excitations in the nonlinear Schrödinger and Klein–Gordon
equations are also demonstrated.</p>
</sec>
<sec id="bk978-0-7503-3579-9ch1s1-7">
<label>1.7</label>
<title>Systems with defects and disorder</title>
<p>An important factor entering the study of optical components is the
effects on the system arising from the introduction of disorder into
the problem [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib37">37</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib49">49</xref>]. Disorder can enter the problem in the form of a mild
or small renormalization of properties or with increasing disorder as
a transition of the system to a whole new range of characteristic
behaviors [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib37">37</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib40">40</xref>]. This change of behavior is essentially a phase
transition similar to other magnetic or chemical phase transitions
[<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib10">10</xref>]. In the present case, however, the phase transition is
observed in a plot of the system transport properties as a function of
the intensity of disorder in the materials.</p>
<p>With the introduction into a system of mild disorder the studies of
its properties can be made using analytical or computer simulation
methods which treat the renormalized forms of the modal excitations in
the materials. As the disorder increases a point is reached in the
increasing degree of disorder of the system at which a fundamental
change occurs in the modal excitations of the materials [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib10">10</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib37">37</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib40">40</xref>]. In
this transition the functional forms of the wave function are
transformed to new types. An example of this is the introduction of
disorder into a homogeneous medium which supports plane wave modal
excitations.</p>
<p>With weak disorder the excitations are renormalized plane waves which
extend throughout the material. At a point of increased disorder in
the medium, however, a transition is made so that the modes in the
material become localized modes. Specifically, the localized modes are
restricted to a finite region of space in the materials. This modal
transition changes the transport properties of the material, and in
electrical conductors is observed as a metal–insulator transition
known as the Anderson transition [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib10">10</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib37">37</xref>]. Similar transitions
from conductive to non-conductive behaviors with the presence of
increasing disorder are observed in optical and acoustic
materials.</p>
<p>Related to random disorder, site impurities and lines or clusters of
impurities can also be purposely introduced into systems in order to
form various types of resonant structures and waveguides. These may
have various technological applications and may exist in engineered
materials in addition to design imperfections. They form an important
topic of technological interest and are introduced in the context of
the applications of group theory techniques in their study</p>
<p>Consequently, as a final topic in the book an introduction is given to
the study of disorder in optical media [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib34">34</xref>–<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib49">49</xref>]. This will include
some basic studies of single site and multiple site impurities in
photonic crystals and metamaterials. For this treatment, some elements
of group theory are introduced into the discussion of cluster
impurities. This is followed by an introduction of Anderson
localization. First a treatment of localization is given in the
context of conductivity in electronic systems [<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib10">10</xref>, <xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib37">37</xref>]. This is followed by
the occurrence of the localization transition in the treatments of
transport properties of optical systems. As a final point, we conclude
with the discussion of weak Anderson localization in the diffuse
reflection of light from randomly rough planar and periodic surfaces
[<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib44">44</xref>,
<xref ref-type="bibr" rid="bk978-0-7503-3579-9ch1bib49">49</xref>],
and in various effects in the presence of nonlinearity.</p>
<p>The order of the book is to first begin with a treatment of the
dielectric properties of materials. The properties and models of
linear photonic crystals and metamaterials are next discussed along
with the basic methods used to compute their properties. This is
followed by discussions of nonlinear photonic crystal and metamaterial
models and their theoretical treatments. Various soliton modes and
discrete breathers found in these systems are presented along with
some final comments on computer simulation studies. To conclude, a
treatment of sites, clusters, and waveguide impurities are given,
followed by a general discussion of Anderson localization.</p>
</sec>
</body>
<back>
<ref-list content-type="numerical">
<title>References</title>
<ref id="bk978-0-7503-3579-9ch1bib1">
<label>1</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Joannopoulos</surname>
<given-names>J D</given-names>
</name>
<name name-style="western">
<surname>Vilenueve</surname>
<given-names>P R</given-names>
</name>
<name name-style="western">
<surname>Fan</surname>
<given-names>S</given-names>
</name>
</person-group>
<year>1995</year>
<source>Photonic Crystals</source>
<publisher-loc>Princeton, NJ</publisher-loc>
<publisher-name>Princeton University Press</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib2">
<label>2</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Sakoda</surname>
<given-names>K</given-names>
</name>
</person-group>
<year>2001</year>
<source>Optical Properties of Photonic Crystals</source>
<publisher-loc>Berlin</publisher-loc>
<publisher-name>Springer</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib3">
<label>3</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Favennec</surname>
<given-names>P N</given-names>
</name>
</person-group>
<year>2005</year>
<source>Photonic Crystals: Towards Nanoscale Photonic Devices</source>
<publisher-loc>Berlin</publisher-loc>
<publisher-name>Springer</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib4">
<label>4</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
</person-group>
<year>2002</year>
<source>Survey of Semiconductor Physics</source>
<person-group person-group-type="editor">
<name name-style="western">
<surname>Boer</surname>
<given-names>W</given-names>
</name>
</person-group>
<publisher-loc>New York</publisher-loc>
<publisher-name>Wiley</publisher-name>
<comment>ch 13</comment>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib5">
<label>5</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="editor">
<name name-style="western">
<surname>Waser</surname>
<given-names>R</given-names>
</name>
</person-group>
<year>2005</year>
<source>Nanoelectronics and Information Technology: Advanced
Electronic Materials and Novel Devices</source>
<edition>2nd edn</edition>
<publisher-loc>Weinheim</publisher-loc>
<publisher-name>Wiley</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib6">
<label>6</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Boyd</surname>
<given-names>R W</given-names>
</name>
</person-group>
<year>2003</year>
<source>Nonlinear Optics</source>
<edition>2nd edn</edition>
<publisher-loc>Amsterdam</publisher-loc>
<publisher-name>Academic</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib7">
<label>7</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Mills</surname>
<given-names>D L</given-names>
</name>
</person-group>
<year>1998</year>
<source>Nonlinear Optics</source>
<publisher-loc>Berlin</publisher-loc>
<publisher-name>Springer</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib8">
<label>8</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Banerjee</surname>
<given-names>P P</given-names>
</name>
</person-group>
<year>2004</year>
<source>Nonlinear Optics</source>
<publisher-loc>New York</publisher-loc>
<publisher-name>Dekker</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib9">
<label>9</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Kivshar</surname>
<given-names>Y S</given-names>
</name>
<name name-style="western">
<surname>Agrawal</surname>
<given-names>G P</given-names>
</name>
</person-group>
<year>2003</year>
<source>Optical Solitons</source>
<publisher-loc>Amsterdam</publisher-loc>
<publisher-name>Academic</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib10">
<label>10</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Marder</surname>
<given-names>M P</given-names>
</name>
</person-group>
<year>2010</year>
<source>Condensed Matter Physics.</source>
<edition>2nd edn</edition>
<publisher-loc>Hoboken, NJ</publisher-loc>
<publisher-name>Wiley</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib11">
<label>11</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Noda</surname>
<given-names>S</given-names>
</name>
<name name-style="western">
<surname>Fujita</surname>
<given-names>M</given-names>
</name>
<name name-style="western">
<surname>Asano</surname>
<given-names>T</given-names>
</name>
</person-group>
<year>2007</year>
<article-title>Spontaneous-emission control by photonic crystals and
nanocavities</article-title>
<source>Nat. Photonics</source>
<volume>1</volume>
<fpage>449</fpage>
<pub-id pub-id-type="doi">10.1038/nphoton.2007.141</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib12">
<label>12</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Agrawal</surname>
<given-names>G P</given-names>
</name>
</person-group>
<year>2012</year>
<source>Fiber-Optics Communication Systems</source>
<edition>4th edn</edition>
<publisher-loc>New York</publisher-loc>
<publisher-name>Wiley</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib13">
<label>13</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Yeh</surname>
<given-names>P</given-names>
</name>
</person-group>
<year>2005</year>
<source>Optical Waves in Layered Media</source>
<publisher-loc>New York</publisher-loc>
<publisher-name>Wiley</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib14">
<label>14</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="editor">
<name name-style="western">
<surname>Engheta</surname>
<given-names>N</given-names>
</name>
<name name-style="western">
<surname>Ziolkowski</surname>
<given-names>R W</given-names>
</name>
</person-group>
<year>2006</year>
<source>Metamaterials: Physics and Engineering Explorations</source>
<publisher-loc>New York</publisher-loc>
<publisher-name>Wiley-IEEE</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib15">
<label>15</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Giri</surname>
<given-names>P</given-names>
</name>
<name name-style="western">
<surname>Choudhary</surname>
<given-names>K</given-names>
</name>
<name name-style="western">
<surname>Sen Gupta</surname>
<given-names>A</given-names>
</name>
<name name-style="western">
<surname>Bandyopadhyay</surname>
<given-names>A K</given-names>
</name>
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
</person-group>
<year>2011</year>
<article-title>Klein–Gordon equation approach to nonlinear split-ring
resonator based metamaterials: one-dimensional
systems</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">B</named-content>
</source>
<volume>84</volume>
<fpage>155429</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevB.84.155429</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib16">
<label>16</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Eleftheriou</surname>
<given-names>M</given-names>
</name>
<name name-style="western">
<surname>Lazarides</surname>
<given-names>N</given-names>
</name>
<name name-style="western">
<surname>Tsironis</surname>
<given-names>G P</given-names>
</name>
</person-group>
<year>2008</year>
<article-title>Magnetoinductive breathers in
metamaterials</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">E</named-content>
</source>
<volume>77</volume>
<fpage>036608</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevE.77.036608</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib17">
<label>17</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Eleftheriades</surname>
<given-names>G V</given-names>
</name>
</person-group>
<year>2009</year>
<article-title>EM transmission-line metamaterials</article-title>
<source>Mater. Today</source>
<volume>12</volume>
<fpage>30</fpage>
<pub-id pub-id-type="doi">10.1016/S1369-7021(09)70073-2</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib18">
<label>18</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Kourakis</surname>
<given-names>I</given-names>
</name>
<name name-style="western">
<surname>Lazarides</surname>
<given-names>N</given-names>
</name>
<name name-style="western">
<surname>Tsironis</surname>
<given-names>G P</given-names>
</name>
</person-group>
<year>2007</year>
<article-title>Self-focusing and envelope pulse generation in
nonlinear magnetic metamaterials</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">E</named-content>
</source>
<volume>75</volume>
<fpage>067601</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevE.75.067601</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib19">
<label>19</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Lapine</surname>
<given-names>M</given-names>
</name>
<name name-style="western">
<surname>Gorkunov</surname>
<given-names>M</given-names>
</name>
<name name-style="western">
<surname>Ringhofer</surname>
<given-names>K H</given-names>
</name>
</person-group>
<year>2003</year>
<article-title>Nonlinearity of a metamaterial arising from diode
inserts into resonant conductive elements</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">E</named-content>
</source>
<volume>67</volume>
<fpage>065601</fpage>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib20">
<label>20</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Shadrivov</surname>
<given-names>I V</given-names>
</name>
<name name-style="western">
<surname>Morrison</surname>
<given-names>S K</given-names>
</name>
<name name-style="western">
<surname>Kivshar</surname>
<given-names>Y</given-names>
</name>
</person-group>
<year>2006</year>
<article-title>Tunable split-ring resonators for nonlinear
negative-index metamaterials</article-title>
<source>Opt. Express</source>
<volume>14</volume>
<fpage>9344</fpage>
<pub-id pub-id-type="doi">10.1364/OE.14.009344</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib21">
<label>21</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Tamayama</surname>
<given-names>Y</given-names>
</name>
<name name-style="western">
<surname>Nakanishi</surname>
<given-names>T</given-names>
</name>
<name name-style="western">
<surname>Kitano</surname>
<given-names>M</given-names>
</name>
</person-group>
<year>2013</year>
<article-title>Observation of modulation instability in a nonlinear
magnetoinductive waveguide</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">B</named-content>
</source>
<volume>87</volume>
<fpage>195123</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevB.87.195123</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib22">
<label>22</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Pendry</surname>
<given-names>J B</given-names>
</name>
</person-group>
<year>2000</year>
<article-title>Negative refraction makes a perfect
lens</article-title>
<source>Phys. Rev. Lett.</source>
<volume>85</volume>
<fpage>3966</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevLett.85.3966</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib23">
<label>23</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Agranovich</surname>
<given-names>V M</given-names>
</name>
<name name-style="western">
<surname>Gartstein</surname>
<given-names>Y N</given-names>
</name>
</person-group>
<year>2006</year>
<article-title>Spatial dispersion and negative refraction of
light</article-title>
<source>Phys.–Usp.</source>
<volume>49</volume>
<fpage>1029</fpage>
<pub-id pub-id-type="doi">10.1070/PU2006v049n10ABEH006067</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib24">
<label>24</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Agranovich</surname>
<given-names>V M</given-names>
</name>
</person-group>
<year>2006</year>
<article-title>Hybrid organic–inorganic nanostructures and
light–matter interaction</article-title>
<source>Problems of Condensed Matter Physics</source>
<person-group person-group-type="editor">
<name name-style="western">
<surname>Ivanov</surname>
<given-names>A L</given-names>
</name>
<name name-style="western">
<surname>Tikhodeev</surname>
<given-names>S G</given-names>
</name>
</person-group>
<publisher-loc>Oxford</publisher-loc>
<publisher-name>Clarendon</publisher-name>
<comment>ch 2</comment>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib25">
<label>25</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Ramakrishna</surname>
<given-names>S A</given-names>
</name>
</person-group>
<year>2005</year>
<article-title>Physics of negative index materials</article-title>
<source>Rep. Prog. Phys.</source>
<volume>68</volume>
<fpage>449</fpage>
<pub-id pub-id-type="doi">10.1088/0034-4885/68/2/R06</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib26">
<label>26</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Veselago</surname>
<given-names>V G</given-names>
</name>
</person-group>
<year>1968</year>
<article-title>The electrodynamics of substances with simultaneously
negative values of <italic>ε</italic> and μ</article-title>
<source>Sov. Phys.–Usp.</source>
<volume>10</volume>
<fpage>509</fpage>
<pub-id pub-id-type="doi">10.1070/PU1968v010n04ABEH003699</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib27">
<label>27</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Cai</surname>
<given-names>W</given-names>
</name>
<name name-style="western">
<surname>Shalaev</surname>
<given-names>V</given-names>
</name>
</person-group>
<year>2001</year>
<source>Optical Metamaterials: Fundamental and Applications</source>
<publisher-loc>New York</publisher-loc>
<publisher-name>Springer</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib28">
<label>28</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Smith</surname>
<given-names>W J</given-names>
</name>
</person-group>
<year>2007</year>
<source>Modern Optical Engineering</source>
<edition>4th edn</edition>
<publisher-loc>New York</publisher-loc>
<publisher-name>McGraw-Hill</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib29">
<label>29</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Taflove</surname>
<given-names>A</given-names>
</name>
</person-group>
<year>1995</year>
<source>Computational Electrodynamics: The Finite-Difference
Time-Domain Method</source>
<publisher-loc>Boston, MA</publisher-loc>
<publisher-name>Artech House</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib30">
<label>30</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>El Misilmani</surname>
<given-names>H M</given-names>
</name>
<name name-style="western">
<surname>Kabalan</surname>
<given-names>K Y</given-names>
</name>
<name name-style="western">
<surname>Abou-Shahine</surname>
<given-names>M Y</given-names>
</name>
<name name-style="western">
<surname>Al-Husseini</surname>
<given-names>M</given-names>
</name>
</person-group>
<year>2015</year>
<article-title>A method of moment approach in solving boundary value
problems</article-title>
<source>J. Electromagn. Anal. Appl.</source>
<volume>7</volume>
<fpage>61</fpage>
<pub-id pub-id-type="doi">10.4236/jemaa.2015.73007</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib31">
<label>31</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
</person-group>
<year>2005</year>
<article-title>Impurity mode techniques applied to the study of light
sources</article-title>
<source>J. Phys.</source>
<volume>38</volume>
<fpage>2338</fpage>
<pub-id pub-id-type="doi">10.1088/0022-3727/38/14/008</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib32">
<label>32</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
</person-group>
<year>1996</year>
<article-title>Green’s-function theory for row and periodic defect
arrays in photonic band structures</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">B</named-content>
</source>
<volume>53</volume>
<fpage>7059</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevB.53.7059</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib33">
<label>33</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Dauxois</surname>
<given-names>T</given-names>
</name>
<name name-style="western">
<surname>Peyrard</surname>
<given-names>M</given-names>
</name>
</person-group>
<year>2006</year>
<source>Physics of Solitons</source>
<publisher-loc>Cambridge</publisher-loc>
<publisher-name>Cambridge University Press</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib34">
<label>34</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Sievers</surname>
<given-names>A J</given-names>
</name>
<name name-style="western">
<surname>Page</surname>
<given-names>J B</given-names>
</name>
</person-group>
<year>1995</year>
<source>Dynamical Properties of Solids</source>
<person-group person-group-type="editor">
<name name-style="western">
<surname>Horton</surname>
<given-names>G K</given-names>
</name>
<name name-style="western">
<surname>Maradudin</surname>
<given-names>A A</given-names>
</name>
</person-group>
<publisher-loc>Amsterdam</publisher-loc>
<publisher-name>North-Holland</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib35">
<label>35</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
</person-group>
<year>2013</year>
<article-title>Barrier transmission map of one-dimensional nonlinear
split-ring-resonator-based metamaterials: bright, dark, and gray
soliton resonances</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">B</named-content>
</source>
<volume>88</volume>
<fpage>155110</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevB.88.155110</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib36">
<label>36</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
</person-group>
<year>2008</year>
<article-title>Transmission through nonlinear barriers</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">B</named-content>
</source>
<volume>77</volume>
<fpage>115105</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevB.77.115105</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib37">
<label>37</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Lee</surname>
<given-names>P A</given-names>
</name>
<name name-style="western">
<surname>Ramakrishnan</surname>
<given-names>T V</given-names>
</name>
</person-group>
<year>1985</year>
<article-title>Disordered electron systems</article-title>
<source>Rev. Mod. Phys.</source>
<volume>57</volume>
<fpage>287</fpage>
<pub-id pub-id-type="doi">10.1103/RevModPhys.57.287</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib38">
<label>38</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Segev</surname>
<given-names>M</given-names>
</name>
<name name-style="western">
<surname>Silverberg</surname>
<given-names>Y</given-names>
</name>
<name name-style="western">
<surname>Christodoulides</surname>
<given-names>D</given-names>
</name>
</person-group>
<year>2013</year>
<article-title>Anderson localization of light</article-title>
<source>Nat. Photonics</source>
<volume>7</volume>
<fpage>197</fpage>
<pub-id pub-id-type="doi">10.1038/nphoton.2013.30</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib39">
<label>39</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A</given-names>
</name>
</person-group>
<year>2020</year>
<source>Introduction to Photonic and Phononic Crystals and
Metamaterials</source>
<publisher-loc>San Rafael, CA</publisher-loc>
<publisher-name>Morgan & Claypool</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib40">
<label>40</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="editor">
<name name-style="western">
<surname>Shing</surname>
<given-names>P</given-names>
</name>
</person-group>
<year>1990</year>
<source>Scattering and Localization of Classical Waves in Random
Media</source>
<publisher-loc>Singapore</publisher-loc>
<publisher-name>World Scientific</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib41">
<label>41</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
</person-group>
<year>2018</year>
<source>Nanophotonics</source>
<publisher-loc>Cham</publisher-loc>
<publisher-name>Springer</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib42">
<label>42</label>
<element-citation publication-type="book" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Cai</surname>
<given-names>W</given-names>
</name>
<name name-style="western">
<surname>Shalaev</surname>
<given-names>V</given-names>
</name>
</person-group>
<year>2009</year>
<source>Optical Metamaterials</source>
<publisher-loc>Berlin</publisher-loc>
<publisher-name>Springer</publisher-name>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib43">
<label>43</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
<name name-style="western">
<surname>Maradudin</surname>
<given-names>A A</given-names>
</name>
<name name-style="western">
<surname>Christensen</surname>
<given-names>K</given-names>
</name>
<name name-style="western">
<surname>Mueller</surname>
<given-names>F</given-names>
</name>
</person-group>
<year>1993</year>
<article-title>Anderson localization in one-dimensional randomly
disordered optical systems that are periodic on
average</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">B</named-content>
</source>
<volume>47</volume>
<fpage>13120</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevB.47.13120</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib44">
<label>44</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
</person-group>
<year>1990</year>
<article-title>Enhanced retroreflectance effects in the reflection of
light from randomly rough surfaces</article-title>
<source>Surf. Sci. Rep.</source>
<volume>10</volume>
<fpage>357</fpage>
<pub-id pub-id-type="doi">10.1016/0167-5729(90)90006-Y</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib45">
<label>45</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
<name name-style="western">
<surname>Maradudin</surname>
<given-names>A A</given-names>
</name>
</person-group>
<year>1994</year>
<article-title>Localization of light in linear and non-linear random
layered dielectric systems that are periodic on average in the
proceeding of Third Int. Conf. on Electrical Transport and Optical
Properties of Inhomogeneous Media</article-title>
<source>Physica <named-content content-type="jnl-part" xlink:type="simple">A</named-content>
</source>
<volume>207</volume>
<fpage>435</fpage>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib46">
<label>46</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>Devillard</surname>
<given-names>P</given-names>
</name>
<name name-style="western">
<surname>Souillard</surname>
<given-names>B</given-names>
</name>
</person-group>
<year>1986</year>
<article-title>Polynomial decaying transmission for the nonlinear
Schrodinger equation in a random medium</article-title>
<source>J. Stat. Phys.</source>
<volume>43</volume>
<fpage>423</fpage>
<pub-id pub-id-type="doi">10.1007/BF01020646</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib47">
<label>47</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
<name name-style="western">
<surname>Sheng</surname>
<given-names>P</given-names>
</name>
<name name-style="western">
<surname>Maradudin</surname>
</name>
</person-group>
<year>1992</year>
<article-title>Strong localization of light in two-dimensional
disordered dielectric media</article-title>
<source>Opt. Commun.</source>
<volume>91</volume>
<fpage>175</fpage>
<pub-id pub-id-type="doi">10.1016/0030-4018(92)90433-R</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib48">
<label>48</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>de Raedt</surname>
<given-names>H</given-names>
</name>
<name name-style="western">
<surname>Lagemdijk</surname>
<given-names>A</given-names>
</name>
<name name-style="western">
<surname>de Vries</surname>
<given-names>P</given-names>
</name>
</person-group>
<year>1989</year>
<article-title>Transverse localization of light</article-title>
<source>Phys. Rev. Lett.</source>
<volume>62</volume>
<fpage>47</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevLett.62.47</pub-id>
</element-citation>
</ref>
<ref id="bk978-0-7503-3579-9ch1bib49">
<label>49</label>
<element-citation publication-type="journal" xlink:type="simple">
<person-group person-group-type="author">
<name name-style="western">
<surname>McGurn</surname>
<given-names>A R</given-names>
</name>
<name name-style="western">
<surname>Leskova</surname>
<given-names>T A</given-names>
</name>
<name name-style="western">
<surname>Agranovich</surname>
<given-names>V M</given-names>
</name>
</person-group>
<year>1991</year>
<article-title>Weak localization effects in the generation of second
harmonics of light at a randomly rough vacuum-metal
grating</article-title>
<source>Phys. Rev. <named-content content-type="jnl-part" xlink:type="simple">B</named-content>
</source>
<volume>44</volume>
<fpage>11441</fpage>
<pub-id pub-id-type="doi">10.1103/PhysRevB.44.11441</pub-id>
</element-citation>
</ref>
</ref-list>
</back>
</book-part>