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+\newcommand{\cns}{\!} %constant quantity superscript negative space adjust
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+%\usepackage[margin=1.0in]{geometry}
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+\usepackage{braket}
+%\usepackage{setspace}
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+\bibliographystyle{spiebook}
+
+%\doublespacing
+
+%% list of figures:
+% fig_circ_ell_hyp_dispersion.pdf
+% anisotropic_nr.pdf
+% wgstuff.pdf
+% cylindrical_scattering2.pdf
+% fig2_metacylinder.pdf
+% eff_medium_modes.pdf
+% image_resolution_b.pdf
+% fig1_spiral_2.pdf
+% semiclass_gaussian.pdf
+% fig8_imaging.pdf
+%
+\setcounter{secnumdepth}{5}
+\setcounter{tocdepth}{5}
+
+\begin{document}
+
+%% Chapter title
+\setcounter{chapter}{1}\addtocounter{chapter}{0}
+\chapter{{Optical Hyperspace}: Negative Refractive Index and Subwavelength Imaging}
+
+% Running heads
+\runningchapter{Chapter \thechapter} \runningtitle{Optical
+Hyperspace: Negative Refractive Index and Sub wavelength Imaging}
+
+%%% Authors and affiliations %%%
+
+% This style file does not automatically implement all the formatting
+% required for an SPIE manuscript. The user must make a few
+% manual adjustments For proper formatting, such as:
+% 1. Title, subsection, and subsubsection should be in lower case
+% except for the first letter and proper nouns or acronyms
+% 2. Format authors and their affiliations as follows
+% \author{author1\supscr{a}, author2\supscr{b}, and author3\supscr{c} )
+% \affiliation{\supscr{a}affiliation1\\
+% \supscr{b}affiliation2\\
+% \supscr{c}affiliation3
+% }
+%
+
+\begin{aug}
+\author{Leonid V. Alekseyev\supscr{a,b}, Zubin Jacob\supscr{b}, and Evgenii Narimanov\supscr{b}}
+\affilation{\supscr{a}Princeton University, Princeton, NJ\\
+\supscr{b}Purdue University, West Lafayette, IN
+}
+\end{aug}
+
+
+
+
+%%% Table of Contents %%%
+\tableofcontents
+
+
+%\title{ \textit{Optical hyperspace}: Negative refractive index and subwavelength imaging}
+
+%\author{Leonid V. Alekseyev, Zubin Jacob and Evgenii Narimanov}
+
+%\date{}
+
+
+\section{Introduction}
+
+The art and science of optics is centered upon our ability to
+control the refractive index of materials, thereby directing the
+flow of light. From the stained-glass windows of Gothic
+cathedrals to modern LCD projectors, from Galileo's telescope to
+terabit optical communication systems, devices made possible by
+skillful manipulation of the refractive index have resulted in
+countless technological and cultural breakthroughs. For
+centuries, the refractive index has been regarded as a strictly
+positive quantity --- such was the universal experience. Recent
+advances in fabrication and processing techniques, however, have
+enabled the creation of materials with a {\em negative} refractive index. This
+development opens many new chapters in the fields of optical
+physics and device engineering. Negative index greatly expands
+the parameter space accessible for manipulating light, opening the
+way for devices with unprecedented capabilities --- for example, imaging
+systems with subwavelength resolution and ultrasmall waveguides.
+The novel systems made possible by negative index materials (NIMs)
+may bring about revolutionary technological
+changes.\cite{Shalaev_NatPhot_2007}
+
+In the present chapter we describe a method to achieve negative refraction via
+materials with a {\em hyperbolic} dispersion relation. Both natural
+materials and metamaterials can exhibit this property. We show
+that in addition to providing a simple path to nonmagnetic
+negative refraction, the hyperbolic dispersion relation enables
+novel devices for waveguiding and subwavelength imaging.
+
+
+The present-day interest in NIMs started in the early
+2000s.\cite{pendry,PendrySmith2002,PhysTodayLHM} The origins of
+the subject, however, date back many decades. Indeed, as a general
+wave propagation phenomenon, negative refraction has been known
+since the early 20th century.\cite{Lamb1904,Schuster1904} It was
+noted, in particular, that negative refraction naturally occurs at
+the interface with a medium characterized by negative phase
+velocity. No such materials were known in the electromagnetic
+domain, and so the early discussions involved only mechanical
+oscillations. The first detailed treatment of negative refraction
+in electromagnetism was provided by Veselago in
+1968.\cite{veselago} He showed that to attain negative phase
+velocity for electromagnetic (EM) waves, the material response must be of the form
+$\epsilon < 0$, $\mu < 0$. When this condition is satisfied, the
+$\vecb{E}$, $\vecb{H}$ and $\vecb{k}$ vectors form a left-handed
+triplet. As a result, the wave vector $\vecb{k}$ and the Poynting
+vector $\vecb{S}$ are oriented in opposite directions; the system
+has negative phase velocity, which is the condition for negative
+refraction. Indeed, negative phase velocity serves as a
+definition of negative index
+materials.\cite{NarimanovVeselago2006} While mechanical and radio
+frequency devices exhibiting such effective negative indices were
+known at the time of Veselago's writing, bulk materials with
+negative phase velocity were not found in nature and were not readily
+attainable.\cite{NarimanovVeselago2006}
+
+The once-fledgling field of negative refraction has experienced a
+major surge in the past decade, owing to major theoretical and
+experimental advances. On the theoretical side, Pendry has
+proposed negative refractive media as a platform for subwavelength
+resolution and aberration-free imaging.\cite{pendry} In
+particular, Pendry showed that a slab of Veselago's ``left-handed''
+material with $\epsilon=\mu=-1$ acts as a perfect lens: it
+does not suffer from aberrations and is not subject to the
+diffraction limit. The proposed ``superlens'' stimulated enormous
+interest in NIMs, but generated some initial controversy regarding
+their experimental realizability. This controversy was soon
+resolved by Smith and colleagues, who fabricated a material with
+$\epsilon < 0$, $\mu < 0$ in the microwave band and explicitly
+demonstrated negative refraction.\cite{PendrySmith2002} The
+required response was attained by artificially structuring the
+material on a scale smaller than the operational wavelength,
+thereby creating a {\em metamaterial}. Utilizing the latest
+nanofabrication techniques, material patterning can be done on a
+submicron scale. This opens the way for NIMs operating at infrared
+and even visible wavelengths. Indeed, negative refraction was
+demonstrated experimentally with wavelengths as short as
+772~nm.\cite{Shalaev2007}
+
+Fabricating structures that exhibit negative refraction at such
+high frequencies presents many difficulties. The most challenging
+aspect of the engineered electromagnetic response is the required
+negative magnetic permeability. Negative permeability is a result
+of a resonant response by a miniature conductive structure. For an
+effective negative permeability response, these microresonators
+must reside in subwavelength unit cells. Thus, to attain negative
+permeability for THz and higher frequencies, one must resort to
+lithographic methods in structuring the materials. For the optical frequencies, fully three-dimensional subwavelength patterning is currently unfeasible.
+
+Aside from the manufacturing difficulties, negative magnetic
+response presents another significant challenge. The resonance in
+the real component of magnetic permeability which leads to
+negative values of $\mu$ is necessarily accompanied by a spike in
+its imaginary component. This leads to high absorption at the
+operating frequencies of magnetic negative index metamaterials,
+which can significantly impair NIM
+devices.\cite{PodolskiyNarimanovNSSL}
+
+In the quest to minimize losses, it becomes prudent to examine
+ways of obtaining negative refraction without resorting to optical
+magnetism. It was shown by several groups that negative
+refraction can arise for light in suitably designed photonic
+crystals.\cite{Notomi2000, Luo2002, Parimi2003, Schonbrun2006}
+From the standpoint of losses, photonic crystal materials are
+generally superior to magnetic NIMs.\cite{Schonbrun2006} However,
+photonic crystals present many of the same fabrication challenges
+as magnetic metamaterials, especially for 3D structures. While
+the characteristic features of photonic crystals are simpler and
+larger (and hence easier to produce), the photonic band behavior
+is strongly sensitive to disorder, necessitating high
+manufacturing precision.
+
+For an alternative approach to nonmagnetic negative refraction,
+we start with the observation that for appropriately cut surfaces
+of anisotropic crystals, negative refraction occurs (for a limited
+range of angles) due to Poynting vector
+walk-off.\cite{ZhangFluegelMascarenhas2003} This effect
+generalizes to {\em all-angle} negative refraction for a
+particular class of strongly anisotropic materials --- those in
+which the components of the dielectric tensor have opposite
+signs.\cite{Belov2003, Dumelow2005} Metamaterials designed to
+satisfy this condition are vastly simpler than typical magnetic
+metamaterials, and are therefore potentially more amenable to
+bulk fabrication. In addition, these metamaterials are not
+sensitive to disorder and operate far from resonances, thus
+helping minimize absorption losses. Finally, for certain
+frequencies, materials with the prescribed anisotropy can be found
+in nature.\cite{PodolskiyNarimanov2005}
+
+It should be noted that like all nonmagnetic negative refraction
+systems, this approach cannot be utilized to create the originally
+proposed superlens: the superlens relies on the excitation of
+high-wavenumber surface plasmon-polariton modes, which is only
+possible for simultaneously negative $\epsilon$ and $\mu$.
+However, negatively refracting materials based on strong
+anisotropy lead to an entirely new class of exciting devices.
+
+In subsequent sections we describe in detail the role of
+anisotropy in creating all-angle negative refraction and discuss
+natural and artificial materials that can be used to demonstrate
+this phenomenon. We then describe potential applications enabled
+by such materials. These applications include negative phase
+velocity waveguides, slow light waveguides, and {\em the
+hyperlens} --- a novel device that enables far-field
+subwavelength-resolved imaging.
+
+
+
+\section{Nonmagnetic Negative Refraction}
+
+For a plane wave with wave vector $\vecb{k}$, incident on some
+surface, translational invariance demands that $k_\parallel$, the
+component of $\vecb{k}$ along the surface, be preserved for the
+refracted wave. So long as the direction of the energy flow
+(given by the Poynting vector $\vecb{S}$) and the direction of the
+wave vector $\vecb{k}$ are the same, negative refraction cannot
+occur. Thus, negative refraction is only possible in media where
+the unit vectors $\hatb{k}$ and $\hatb{S}$ do not coincide. More
+specifically, for the transmitted wave we must have $S_\parallel <
+0$ when $k_\parallel > 0$ and vice versa. For a medium with
+negative phase velocity, $\hatb{S} = -\hatb{k}$ holds, and the
+condition $\{S_\parallel < 0$ and $k_\parallel > 0\}$ are then
+satisfied automatically. Material parameters $\epsilon<0$, $\mu<0$
+lead to exactly this scenario. More generally, however, we may
+inquire what material parameters lead to negative refraction
+without requiring negative phase velocity.
+
+
+The simplest answer to this question comes from considering wave
+propagation in anisotropic crystals and noting that the directions
+of $\vecb{S}$ and $\vecb{k}$ are, generally, different. To see how
+this comes about, we consider plane wave propagation in a uniaxial
+medium. Depending on polarization, the waves can be characterized
+as {\em ordinary} or {\em extraordinary}. For extraordinary
+waves, the electric field vector has a nonvanishing component
+along the optical axis; therefore, the different components of
+the electric field $\vecb{E}$ experience different dielectric
+constants. Furthermore, the relationship between $\vecb{E}$ and
+$\vecb{D}$ (the electric displacement vector) depends on the
+propagation direction. Ordinary waves, on the other hand, are not
+affected by the anisotropy and are of no special interest. For
+this reason, in the subsequent discussion we treat only the
+extraordinary polarization.
+
+Taking $\vecb{\hat{x}}$ as the direction of the optical axis, we
+may characterize the extraordinary wave in a uniaxial crystal by
+the dispersion relation\cite{landavshitz8}
+
+
+\begin{equation}
+\label{eq:dr}
+\frac{k_x^2}{\epsilon_z} + \frac{k_{y,z}^2}{\epsilon_x} =
+\frac{\omega^2}{c^2}.
+\end{equation}
+
+ For
+sufficiently weak absorption, the direction of the Poynting vector
+is identical to the direction of the group velocity vector
+$\vecb{v}_g =
+\nabla_{\vecb{k}}\omega(\vecb{k})$.\cite{landavshitz8} This means
+that $\vecb{S}$ is normal to the isofrequency curves given by
+Eq.~(\ref{eq:dr}).
+
+
+%-------------
+ \begin{figure}
+ \begin{center}
+ \begin{tabular}{c}
+ \centerline{\scalebox{.53}{\includegraphics{fig_circ_ell_hyp_dispersion.pdf}}}
+ \end{tabular}
+ \end{center}
+ \caption[]
+%>>>> use \label inside caption to get Fig. number with \ref{}
+ { \label{fig:dr}
+Isofrequency curve and relative direction of the wave vector
+$\vec{k}$ and the Poynting vector $\vec{S}$ for (a) isotropic
+material, (b) material with $\epsilon_x, \epsilon_z
+> 0$, and (c) material with $\epsilon_x
+< 0$, $\epsilon_z > 0$. [From
+Ref.~\inlinecite{AlekseyevNarimanov2006}.]}
+ \end{figure}
+
+
+
+What does this imply for the relative angle between $\vecb{S}$ and
+$\vecb{k}$? In the isotropic case, the wave vector surfaces are
+circles, and therefore $\vecb{S} \propto
+\nabla_{\vecb{k}}\omega(\vecb{k}) \propto \vecb{k}$, i.e.,
+$\vecb{S}$ and $\vecb{k}$ are collinear, as can be seen in
+Fig.~\ref{fig:dr}(a). Consider now the situation in Fig.~\ref{fig:dr}(b), where
+$\epsilon_x \neq \epsilon_z$ and $\epsilon_{x,z} > 0$. The wave
+vector surfaces become ellipsoidal; as a
+consequence, the angle between $\vecb{S}$ and $\vecb{k}$ is
+nonzero. (Its exact value depends on the direction of propagation
+and the degree of anisotropy.) This implies that it is possible to
+pick a coordinate system $x'z'$ in which $S_{z'} < 0$, $k_{z'}
+> 0$. If the material is cut such that $\vecb{\hat{x}'}$ defines the surface normal, negative refraction occurs.
+Note, however, that this situation is only realizable for a finite
+range of $\vecb{k}$ values, and hence a finite set of incidence
+angles.
+
+Finally, we consider the case shown in in Fig.~\ref{fig:dr}(c), where $\epsilon_x$ and
+$\epsilon_z$ are not only non-equal but also possess different {\em
+signs}. This drastically changes the nature of the dispersion
+relation in Eq.~(\ref{eq:dr}). More specifically, for a material with
+negative transverse dielectric permittivity ($\epsilon_x<0$) and
+positive in-plane permittivity ($\epsilon_z > 0$),
+Eq.~(\ref{eq:dr}) describes a hyperbola. Constructing the
+vectors $\vecb{S}$ and $\vecb{k}$, as before, we see that the
+signs of $S_z$ and $k_z$ are opposite for all admissible
+values of $\vecb{k}$.
+
+This result can also be obtained by examining the
+$\vecb{\hat{z}}$ component of the Poynting vector for the
+extraordinary wave:
+\begin{eqnarray} S_z & = & \frac{k_z}{\epsilon_x \, \omega/c}E_0^2, \label{eq:S} \end{eqnarray}
+where $E_0$ is the electric field amplitude (in CGS units).
+Evidently, if
+$\epsilon_x<0$, $S_z$ is negative, i.e., opposite to the direction
+of the wave vector component $k_z$.
+
+If we now consider a beam incident on an interface of a
+material exhibiting a hyperbolic dispersion relation, we find that
+the sign of the tangential component of the wave vector ($k_z$)
+is preserved as usual upon transmission through the boundary,
+while the Poynting vector (and thus the energy flux) undergoes
+negative refraction, as in Figs.~\ref{fig:anisotropic_nr}~(a) and (b).
+Furthermore, a slab of such material can function as a planar lens, as shown in
+Fig.~\ref{fig:anisotropic_nr}(c). In this sense, the
+$\epsilon_x < 0, \; \epsilon_z > 0$ material mimics the
+behavior of negative refraction systems with $\epsilon < 0$, $\mu < 0$. We should keep
+in mind, however, that the hyperbolic dispersion
+relation in Eq.~(\ref{eq:dr}) has a profound impact not only on
+refraction behavior at the interface, but also on the general
+properties of wave propagation. (Indeed, we shall see in a later
+section that this dispersion relation enables devices with
+negative phase velocity and near-zero group velocity.)
+
+One hyperbolic dispersion effect that is of particular interest in
+imaging applications involves directionality constraints on
+propagating radiation. Fig.~\ref{fig:dr}(c) shows that
+the allowed directions of the wave vector and the Poynting vector
+are restricted by the asymptotes of the hyperbola. The locus of
+the allowed $\vecb{S}$ vectors is a cone, with the half-angle $\theta_c$ given
+by
+
+\begin{equation}\label{eq:cone_angle}
+\tan\theta_c = \sqrt{\frac{\epsilon_z}{|\epsilon_x|}}.
+\end{equation}
+
+\noindent Such beam-like directional radiation patterns have
+indeed been observed for sources embedded in strongly anisotropic
+plasmas.\cite{resonance_cone_1,resonance_cone_2} It is
+interesting to note that in the case of an ideal point source and
+with zero losses, all of the energy is concentrated at the boundary of
+the propagation cone, since there are infinitely many wave vectors
+--- solutions of Eq.~(\ref{eq:dr}) --- that accumulate close to the
+asymptotes of the hyperbola, and therefore share the same
+direction. Furthermore, for $|\epsilon_x| \gg \epsilon_z$ the
+beam divergence angle approaches zero. Thus, in this so-called
+{\em channeling regime},\cite{Belov2006} subdiffraction-limited
+imaging can be performed.
+
+It should be emphasized that the absence of the conventional
+diffraction limit is a general feature for wave propagation in
+$\epsilon_x < 0, \; \epsilon_z > 0$ materials. For
+positive dielectric constants, Eq.~(\ref{eq:dr}) implies a finite
+spatial frequency bandwidth limit, which causes diffraction. For
+instance, in the isotropic case, we have
+\begin{equation}\label{eq:diff_cutoff}
+k_{x,y}^2 > \epsilon\frac{\omega^2}{c^2} \; \Longrightarrow \; k_z
+= i \kappa.
+\end{equation}
+
+\noindent In other words, for large values of $k_{x,y}$ the wave
+equation solutions $\exp(i \, \vecb{k}\cdot\vecb{r})$ are the
+evanescent waves that exponentially decay away from the source. At
+the same time, it is precisely these waves that carry information
+about structure on a subwavelength scale, as scattering from
+subwavelength features results in large $k_{x,y}$. For a hyperbolic
+ dispersion relation, however, Eq.~(\ref{eq:dr}) can be
+satisfied for {\em arbitrarily large} values of $k_{x,y}$ and
+$k_z$. These high spatial frequency waves propagate through the
+$\epsilon_x < 0, \; \epsilon_z > 0$ structure and enable
+subdiffraction-limited imaging.
+
+\begin{figure}
+\centerline{\scalebox{.238}{\includegraphics{anisotropic_nr.pdf}}}
+\caption{(a) The ray diagram and (b) the electric field for
+the refraction of a light beam at the boundary of air with an
+$\epsilon_x < 0$, $\epsilon_z > 0$ material. Note negative
+refraction of the beam and the direction of the wavefronts
+($\epsilon_z = 3$, $\epsilon_x = - 1.5$). (c) The intensity distribution of a beam
+ propagating through a slab made of
+such material. This slab functions as a planar lens. [Adapted from
+Ref.~\inlinecite{AlekseyevNarimanov2006}.]
+ }
+\label{fig:anisotropic_nr}
+\end{figure}
+
+
+
+\section{Hyperbolic Dispersion: Materials}
+
+Clearly, the special nature of the $\epsilon_x < 0, \;
+\epsilon_z > 0$ systems leads to a multitude of exotic effects.
+Here arises a natural question: how can we elicit such a response
+from physical materials?
+
+Perhaps surprisingly, the $\epsilon_x < 0, \; \epsilon_z
+> 0$ behavior is observed in a number of natural materials where
+structural anisotropy strongly affects the dielectric response.
+Examples of such materials can easily be found in the infrared and THz
+spectral bands. For instance, in the far infrared/low THz domain,
+this behavior can be found in triglycine sulfate (TGS), a compound
+widely used in fabricating infrared photodetectors. In TGS, a
+strong phonon anisotropy leads to a large anisotropy in the
+dielectric tensor. In particular, dielectric response for the
+field polarized along the crystal's monoclinic $C_2$ axis features
+a resonance at 268~\um, which is absent for light polarized
+transverse to $C_2$.\cite{tgs_00} Measured dielectric
+functions\cite{tgs_00,tgs_98,Dumelow2005} reveal that $\epsilon_x
+< 0$, while $\epsilon_z > 0$ in the region 250 $\le \lambda
+\le$ 268~\um\ (here and subsequently we let $\vecb{\hat{x}}$ lie
+along the appropriate crystallographic axis). Furthermore, the
+imaginary part of $\epsilon$ becomes small away from the
+resonance, minimizing absorption.
+
+
+Whereas the phonon anisotropy of TGS exists in the low THz domain,
+for other materials, it may occur in a different spectral band.
+The strong anisotropy of the dielectric response of sapphire
+(Al$_2$O$_3$) is also due to excitation of different phonon modes
+(polarized either parallel or perpendicular to the $c$ axis of the
+rhombohedral structure), but occurs around 20~\um. A region of
+$\epsilon_x < 0$, $\epsilon_z > 0$ for wavelengths of 19.5 to
+21~\um\ has been experimentally observed.\cite{sapphire}
+
+
+Anisotropic phonon excitation is not the only mechanism that can
+lead to strong dielectric anisotropy. Bismuth, a group V
+semimetal, exhibits such anisotropy due to a substantial
+difference in its effective electron masses along different
+directions in the crystal. Measurements of bismuth plasma
+frequencies\cite{bb_58, ke_74} can be used to reconstruct its
+dielectric tensor. The $\epsilon_x < 0, \; \epsilon_z
+> 0$ anisotropy is revealed between 54 and 63~\um. It should be
+noted that pure bismuth samples exhibit much lower absorption than
+most metals, due to long electron relaxation times (a conservative
+estimate is $\tau=0.1$~ns at 4~K\cite{ke_74}). The typical ratio
+of imaginary and real parts of the dielectric function in bismuth is
+thus expected to be on the order of 0.1\% in the frequency
+interval of interest.
+
+For spectral domains where natural effects do not result in
+differing signs of the dielectric tensor components, such
+anisotropy may be attained in metamaterials. To satisfy the
+requirement $\epsilon_x < 0$ and $\epsilon_z
+> 0$, the metamaterials must combine
+plasmonic or polar materials (with $\epsilon < 0$) with
+conventional dielectrics in an appropriate geometry.
+
+The $\epsilon < 0$ components of such nanocomposites may come from
+a variety of sources. For instance, these negative permittivity
+materials can be created artificially. One approach involves
+strongly doping a semiconductor, thereby creating a plasmon
+resonance. Another possible technique to induce negative
+permittivity is engineering quantum wells with appropriate
+intrasubband transitions. Negative permittivity is also quite
+common in naturally occurring materials. In the visible spectrum,
+plasmon resonances result in $\epsilon < 0$ for a number of
+metals. Silver is one example of a relatively low-loss plasmonic
+material. At longer wavelengths, phonon resonances can yield
+$\epsilon < 0$, with losses typically lower than those in silver.
+One such low-loss material, well-suited for studying
+negative-index phenomena in the mid-IR, is silicon
+carbide,\cite{Shvets2003,Shvets2006} with $\epsilon < 0$ between
+10.3 and 11 \um.
+
+
+The metamaterials can be structured in many different ways. For
+instance, the plasmonic inclusions can take the form of aligned
+nanowires. Alternatively, these inclusions can be anisotropically
+distributed in a dielectric host. The simplest arrangement that
+yields the desired dielectric properties is a layered medium with
+alternating permittivities in the $x$
+direction.\cite{PodolskiyNarimanov2005,Shvets2003,PodolskiyAlekseyevNarimanov2005}
+This medium consists of a sequence of ``dielectric'' layers
+($\epsilon_1 > 0$) and ``conductive'' layers ($\epsilon_2 <
+0$).\cite{PendryRamakrishna2003} The effective dielectric tensor
+of such a structure (with the volume fraction of the conducting
+layers $N_c$) is given by \cite{Wangberg2005}
+\begin{eqnarray}\label{eq:effectiveEpsilon}
+\epsilon_{x} & = &\frac{\epsilon_{1}\epsilon_2} {N_{c}\;\epsilon_1
++ (1- N_{c}) \epsilon_{2}} \\
+\epsilon_{z} & = & (1- N_{c})\epsilon_1 + N_{c}\; \epsilon_2.
+\nonumber
+\end{eqnarray} Provided that $\epsilon_1 > 0$ and $\epsilon_2 < 0$ in a certain frequency range,
+ these equations lead to a well-defined frequency interval with
+$\epsilon_x < 0, \; \epsilon_z > 0$ (the exact values of the interval are determined from the dispersive characteristics of $\epsilon_1$ and $\epsilon_2$). Such a layered system can be
+fabricated using epitaxial semiconductor growth, with selective
+doping used to attain $\epsilon_2 < 0$ in the ``metallic''
+regions.
+
+
+
+
+
+
+\section{Applications}
+\subsection{Waveguides}
+
+ As discussed above, the $\epsilon_x < 0, \; \epsilon_z
+> 0$ materials enable all-angle negative refraction for incident plane
+waves. However, for {\em guided modes}, this
+form of the dielectric tensor results in negative phase velocities
+and even negative group delays --- phenomena primarily associated
+with magnetic ($\epsilon_x < 0, \; \mu < 0$) negative index
+materials. To see how this comes about, let us consider guided
+mode solutions for a planar waveguide of thickness $d$ with
+perfectly conducting walls. Suppose that the boundaries of the
+waveguide lie at $x=0$ and $x=d$, and that guided modes propagate
+in the $z$ direction. We assume that the waveguide is filled with
+a uniaxial anisotropic material characterized by dielectric
+constants $\epsilon_x \equiv \epsilon_\perp$ (for field components
+transverse to the waveguide) and $\epsilon_{y,z} \equiv
+\epsilon_\parallel$. The solution for transverse magnetic (TM) modes propagating in
+such a waveguide is\cite{landavshitz8}
+
+\begin{equation}\label{eq:wg_solution}
+\vecb{E}(\vecb{r},t) =
+E_0\left[-i\frac{\beta}{\epsilon_\perp}\cos(\kappa x )\hatb{x} +
+\frac{\kappa}{\epsilon_\|}\sin(\kappa x)\hatb{z} \right]
+\exp[-i(\beta z - \omega t)],
+\end{equation}
+
+
+\noindent where $\kappa = m \pi/d$, and $\kappa$ and
+$\beta$ satisfy the dispersion relation similar to
+Eq.~(\ref{eq:dr}):
+
+\begin{equation}\label{eq:wg_dr}
+\frac{\beta^2}{\epsilon_\perp} +
+\frac{\kappa^2}{\epsilon_\parallel} = \frac{\omega^2}{c^2}.
+\end{equation}
+
+We note that in the isotropic case ($\epsilon_\parallel =
+\epsilon_\perp$), the above expressions reduce to the familiar
+solutions for a metallic waveguide with the maximum supported mode
+$m_\text{max}$ derived from the condition $\kappa \le
+\sqrt{\epsilon}\omega/c$:
+
+\begin{equation}\label{eq:m_max}
+m_\text{max} =
+\left\lfloor\frac{d\sqrt{\epsilon}\omega/c}{\pi}\right\rfloor
+\end{equation}
+\noindent ($\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denote floor and ceiling functions).
+
+When both $\epsilon_\parallel$ and $\epsilon_\perp$ $>$ 0, this
+expression generalizes readily to the anisotropic case [in fact,
+we only need to replace $\epsilon$ with $\epsilon_\parallel$ in
+(\ref{eq:m_max})]. However, if the {\it signs} of
+$\epsilon_\parallel$ and $\epsilon_\perp$ differ, the situation
+changes dramatically. Consider, for instance, the case
+$\{\epsilon_\perp < 0, \; \epsilon_\parallel > 0\}$. The
+condition for Eq.~(\ref{eq:wg_dr}) to be satisfied now reads
+$\kappa \ge \sqrt{\epsilon}\omega/c$, leading to
+
+\begin{equation}\label{eq:m_min}
+m_\text{min} =
+\left\lceil\frac{d\sqrt{\epsilon_\parallel}\omega/c}{\pi}\right\rceil.
+\end{equation}
+
+Rather than having a maximum mode cutoff, the guided modes are now
+bounded {\em from below}. By adjusting the values of $d$ and
+$\epsilon_\parallel$,it is possible to allow {\em all} modes to
+propagate in a waveguide, or to elevate the minimum cut-off
+threshold $m_\text{min}$ to admit only high-order modes.
+
+This result has interesting potential applications. First, the
+optical power in a given mode is proportional to $\beta$, which,
+asymptotically, is linear in the mode number $m$. Thus, it might
+be possible to concentrate unusually high fields in a
+subwavelength waveguide, an impossible feat with conventional
+materials. Such a capability would be of great interest in
+developing nonlinear devices.
+
+Secondly, it should be noted that mode profiles for high-$m$
+solutions exhibit rapid oscillations, i.e., correspond to high
+spatial frequencies. Such high-order modes would be able to
+couple to evanescent fields of finely structured objects, which
+are also characterized by high transverse spatial frequencies.
+These high spatial frequencies carry the information about the
+object's subwavelength features --- the information typically lost
+as a consequence of the diffraction limit. This ability to guide
+waves that would exponentially decay in an ordinary medium is of
+great interest in constructing subwavelength imaging devices, and
+will be discussed in more detail in a later section.
+
+Let us now consider the group velocity of the guided modes, $v_g =
+\partial \omega / \partial \beta$. Differentiating Eq.~(\ref{eq:wg_dr}) we obtain
+
+\begin{equation}\label{eq:vg}
+\frac{\partial \omega}{\partial \beta} =
+\frac{c^2}{\epsilon_\perp} \frac{1}{\omega/\beta} =
+\frac{c^2}{\epsilon_\perp} \frac{1}{v_\phi},
+\end{equation}
+\noindent where $v_\phi$ is the phase velocity. For
+$\epsilon_\perp < 0$ we see immediately that the phase velocity
+and the group velocity are of different signs. This implies that
+the Poynting vector $\vecb{S}$ is directed opposite the wave
+vector $\vecb{k}$.
+
+ It is worth noting that this conclusion can be
+made from the simple geometrical argument if we represent the mode
+of a metallic waveguide by a plane wave with wave vector
+$\vecb{k}$ bouncing between the two waveguide boundaries. Due to
+the $\{\epsilon_\perp < 0, \; \epsilon_\parallel > 0\}$
+anisotropy, the components of $\vecb{S}$ and $\vecb{k}$ along the
+waveguide, $S_z$ and $k_z$, differ in sign (as was shown in an
+earlier section). But in the process of constructing a waveguide
+mode out of the multiply reflecting plane wave, it can be seen that
+$S_z$ represents the net energy flow in the mode, while $k_z$
+coincides with the mode propagation constant $\beta$. We
+therefore arrive at the same conclusion --- that the direction of
+the phase fronts is opposite to the direction of the energy flow.
+
+
+
+
+
+\begin{figure}
+\centerline{\scalebox{.20}{\includegraphics{wgstuff.pdf}}}
+\caption{(a) Negative refraction exhibited by wavefronts in a 2D
+slab waveguide with metallic walls, filled with an isotropic
+dielectric on the left, and $\{\epsilon_\perp < 0, \;
+\epsilon_\parallel > 0\}$ material on the right. Arrows indicate the direction of the power flow. (b) Schematics
+of a waveguide supporting slow group velocity modes: dielectric
+cladding in regions 1, 3; $\{\epsilon_\perp < 0, \;
+\epsilon_\parallel > 0\}$ material in region 2. (c) Group
+velocity as a function of frequency for the waveguide in (b). Note
+that $v_g \lesssim 0.004\,c$ throughout the shaded region. [Adapted
+from
+Refs.~\inlinecite{PodolskiyAlekseyevNarimanov2005,AlekseyevNarimanov2006}.]
+ }
+\label{fig:wgstuff}
+\end{figure}
+
+
+
+The guided modes therefore mimic the refractive behavior of
+magnetic ($\epsilon < 0$, $\mu < 0$) negative-index materials. Indeed, if we
+consider the waveguide shown in Fig.~\ref{fig:wgstuff}(a), filled with a regular dielectric on the left
+and with an $\epsilon_\perp < 0$ anisotropic material on the
+right, and a mode with propagation vector $\vecb{\beta} = \beta_y
+\hatb{y} + \beta_z \hatb{z}$ incident on this boundary, the phase
+fronts of the mode reveal negative refraction.
+
+Yet another counterintuitive phenomenon is associated
+with propagation in ani\-so\-tropic waveguides. Recall that for a waveguide
+with perfectly conducting walls, as above, the energy flux in the
+core is antiparallel to the wave vector. The same is true if the
+core is bounded by a cladding made from a regular, isotropic
+dielectric. But for a dielectric waveguide, a portion of the
+energy flux exists in the cladding. In this region, the energy
+flux is, as usual, collinear with the wave vector
+[Fig.~\ref{fig:wgstuff}(b)]. For a particular value of the light
+frequency $\omega$ and the waveguide thickness $d$, the negative
+energy flux inside the waveguide can be nearly balanced by the
+positive energy flux outside. This leads to a dramatic reduction in the group velocity.
+
+
+ The frequency-dependent
+group velocity of a single slow mode is plotted in
+Fig.~\ref{fig:wgstuff}(c). It is evident that $v_g \lesssim
+0.004\,c$ is attainable over a 1.1 THz frequency range. Such a wide
+bandwidth suggests the possibility of using the proposed system as
+an optical buffer.
+
+
+\subsection{The hyperlens}
+\subsubsection{Theoretical description}
+
+We saw in an earlier section that a medium with a hyperbolic
+dispersion relation allows propagation of high spatial frequency
+waves which would decay in a conventional dielectric. This
+phenomenon, however, is of limited utility in stand-off
+subwavelength imaging, as the high-$k$ modes start exponentially
+decaying outside the material. It turns out, however, that
+hyperbolic dispersion implemented in curvilinear coordinates can
+yield devices that convert the high-$k$ modes to propagating waves
+by essentially magnifying subwavelength structures.
+
+A {\em hyperlens} is a hollow core cylinder (or half cylinder),
+made of a strongly anisotropic material, that can function as a
+far field subdiffraction
+lens.\cite{JacobAlekseyevNarimanov2006,EnghetaHyperlens2006,SmolyaninovHyperlens2007,ZhangHyperlens2007}
+To understand the origin of subwavelength resolution in the
+hyperlens, it is useful to consider the imaging problem in the
+context of detecting a wave, scattered by a subwavelength object.
+
+Waves scattered by the illuminated object can be examined in a
+monochromatic plane wave basis with a wide spectrum of spatial
+frequencies. The choice of basis, however, is dictated by the
+symmetry of the object under consideration and/or by convenience.
+Mathematically, the problem can be equivalently treated in a basis
+of cylindrical waves. In particular, any plane wave illuminating
+an object can be expanded in a basis of cylindrical waves as
+\begin{equation}
+\exp(ikx)=\sum_{m=-\infty}^{\infty}i^{m}J_{m}(kr)\exp(im\phi),
+\label{eq:besselExpansion}
+\end{equation}
+where $J_{m}(kr) $ denotes the Bessel function of the first kind
+and $m$ is the angular momentum mode number of the cylindrical
+wave. This decomposition is illustrated schematically in
+Fig.~\ref{fig:scattering}(a). In this representation,
+reconstructing an image is equivalent to retrieving the scattering
+amplitudes and phase shifts of the various constituent angular
+momentum modes. The resolution limit in the cylindrical wave basis
+can be restated as the limit to the number of retrieved angular
+momentum modes with appreciable amplitude or phase change after
+scattering from the object.
+
+\begin{figure}
+\centerline{\scalebox{.7}{\includegraphics{cylindrical_scattering2.pdf}}}
+\caption{(a) The scattering of an incident plane wave by a target
+(yellow object) can be represented as scattering of various
+angular momentum modes. The regions of high intensity are shown in
+black, and low intensity in white. (b) Higher-order modes are
+exponentially small at the center. (c) The attenuation of high-order
+modes results from an upper
+bound on values of $k_\theta$ and the formation of the caustic
+shown as a dashed circle in the panel. [From
+Ref.~\inlinecite{JacobAlekseyevNarimanov2006}.]
+ }
+\label{fig:scattering}
+\end{figure}
+
+
+We may think of the scattered angular momentum modes as distinct
+information channels through which the information about the
+object at the origin is conveyed to the far field. However, even
+though the number of these channels is infinite ($m$ is unbounded
+in Eq.~(\ref{eq:besselExpansion})), very little information
+is carried over the high-$m$ channels. Fig.~\ref{fig:scattering}(b) shows the exact radial profile
+of the electric field for $m$=1 and $m$=14. For high values of $m$
+the field exponentially decays toward the origin. This suggests that
+the interaction between a high-$m$ mode and an object placed at
+the origin is exponentially small; i.e., the scattering of such
+modes from the object is negligible. Classically, this corresponds
+to the parts of an illuminating beam that have a high {\em impact
+parameter} and therefore miss the scatterer and carry no
+information about the object into the far field.
+
+The high-$m$ modes are evanescent within a circle of critical
+radius called {\em the caustic}. This is because conservation of angular
+momentum implies that the tangential wave vector of a high-angular-momentum mode increases towards the center ($k_{\theta}r=m=\rm
+const$). In a medium such as vacuum characterized by a circular
+isofrequency curve (see Fig. \ref{fig:dr}(a)), this increase in the tangential component is
+not supported as both the tangential and radial wave vectors are
+bounded (see Eq.~(\ref{eq:diff_cutoff}) and related discussion).
+These incident high-angular-momentum modes simply reflect
+without ever reaching the scatterer. As such, they do not
+contribute to the retrieval of information about the object's
+structure. However, if there existed a way to drive these states
+to the center, whereupon they could interact with the object, then
+these high-angular-momentum states would act as extra information
+channels for subwavelength structure retrieval.
+
+It turns out that this scenario is possible for cylindrical
+systems with a hyperbolic dispersion relation. Consider wave
+propagation in a bulk medium with strong cylindrical
+anisotropy where dielectric permittivities have different signs
+in the tangential and radial directions ($ \epsilon_{\theta}>0$,
+$\epsilon_{r}<0$). Since there exist no natural materials with
+such an anisotropy, we assume that it could be implemented using
+metamaterials. In particular, the desired anisotropy may be
+attained in a cylinder composed of ``slices'' of metal and
+dielectric or alternating concentric layers of metal and
+dielectric (see Fig.~\ref{metacylinders}). The layer thickness $h$ in
+each of these structures is much less than the wavelength
+$\lambda$, and when $ h \ll \lambda \leq r$, the effective medium
+expressions in Eq.~(\ref{eq:effectiveEpsilon}) (with ${\epsilon_x,
+\epsilon_z} \rightarrow {\epsilon_r, \epsilon_\theta}$) can be
+used for dielectric permittivities. A low-loss cylindrically
+anisotropic material can also be achieved by metallic inclusions
+in a hollow core dielectric cylinder.
+
+\begin{figure}[t]
+\centering
+\scalebox{0.34}{\includegraphics{fig2_metacylinder.pdf}}\\
+ \medskip \caption{
+Possible realizations of metacylinders. (a) Concentric alternate
+metallic layers and dielectric layers or (b) radially symmetric
+``slices'' of metal and dielectric produce ($
+\epsilon_{\theta}>0$, $\epsilon_{r}<0$) anisotropy. This results
+in a hyperbolic dispersion relation necessary for penetration of
+the field close to the center. [From
+Ref.~\inlinecite{JacobAlekseyevNarimanov2006}.]}
+\label{metacylinders}
+\end{figure}
+
+
+
+It should be noted that the polar dielectric permittivities are
+ill defined at the center and any practical realization of
+cylindrical anisotropy, such as metamaterial structures, can only
+closely approximate the desired dielectric permittivities away
+from the center (when $r \geq \lambda $). However, numerical
+simulations show that the effective medium description is adequate
+and that the hyperlens functions even in the case where the inner radius is no greater than a wavelength.\cite{JacobAlekseyevNarimanov2006} The
+hyperlens functions in the channeling regime where a smaller inner
+radius aids in higher resolution.
+
+
+
+
+ As before, we focus on extraordinary waves (TM
+modes, with the magnetic field along the axis of the cylinder).
+These waves obey a hyperbolic dispersion relation similar to
+Eq.~(\ref{eq:dr}), namely,
+\begin{equation}\label{eq:drHyperbola}
+\frac{k_r^2}{\epsilon_\theta} - \frac{k_\theta^2}{|\epsilon_r|} =
+\frac{\omega^2}{c^2}, \end{equation} which allows for very high
+values of $k$, limited only by the patterning scale of the metamaterial medium.
+As the tangential component of the wave vector increases towards
+the center, the radial component also increases;
+Eq.~(\ref{eq:drHyperbola}) can be satisfied for any radius and any
+value of $m$. Thus, as long as the effective medium description
+is valid, the field of high-angular-momentum states has
+appreciable magnitude close to the center.
+
+
+This can be verified by solving Maxwell's equations for the TM
+mode in the cylindrical geometry for the ($ \epsilon_{\theta}>0$,
+$\epsilon_{r}<0$) anisotropy
+\begin{equation}
+\label{eq:effectivemedium}
+ B_{z}\propto
+J_{m\sqrt{\epsilon_{r}/\epsilon_{\theta}}}\left(\frac{\omega}{c}\sqrt{\epsilon_{\theta}} \ r \right)\exp(i
+m\phi).
+\end{equation}
+This mode is plotted in Fig.~\ref{fig:effMedium}(b). Note that the
+cylindrical anisotropy causes a high-angular-momentum state to
+penetrate toward the center --- in contrast to the behavior of
+high-$m$ modes in regular dielectrics [see Fig.
+\ref{fig:effMedium}(a)].
+
+\begin{figure}
+%\centerline{\scalebox{.85}{\includegraphics{wg_modes_1.pdf}}}
+\centerline{\scalebox{.73}{\includegraphics{eff_medium_modes.pdf}}}
+\caption{(a) high-angular-momentum states in an isotropic
+dielectric cylinder. (b) high-angular-momentum states in a
+cylinder made of $ \epsilon_{\theta}>0$, $\epsilon_{r}<0$
+metamaterial (in the effective medium approximation); note that
+the field penetrates to the center. [From
+Ref.~\inlinecite{JacobAlekseyevNarimanov2006}.]
+ }
+\label{fig:effMedium}
+\end{figure}
+
+
+
+ We now consider a hollow core cylinder of inner radius $R_{\rm_{inner}} \approx \lambda$
+and outer radius $R_{\rm outer}$, made of a cylindrically
+anisotropic homogeneous medium. The high-angular-momentum states
+with caustic radius $R_{c} \leq R_{\rm outer}$ are captured
+by the device and guided towards the core. In this case,
+cylindrical symmetry implies that the distance between the field
+nodes at the core is less than the vacuum wavelength (see
+Fig.~\ref{fig:effMedium}). Therefore, such high-angular-momentum
+states can act as a subwavelength probe for an object placed
+inside the core. Furthermore, since in the medium under
+consideration these states are propagating waves, they can carry
+information about the detailed structure of the object to the far
+field. The hyperlens thus enables extra information channels for
+retrieving the object's subwavelength structure. In the absence
+of the device, the high-angular-momentum modes representing these
+channels do not reach the core and, as such, carry no information
+about the object.
+
+\subsubsection{Imaging simulations}
+
+To confirm the subwavelength imaging capabilities of the hyperlens,
+we consider placing two point sources in the vicinity of the
+hollow cylinder's inner boundary. To improve the coupling of high
+spatial frequency Fourier components to the high-angular-momentum
+modes, we assume that the inner layer of the hyperlens has ${\rm
+Re}[\epsilon]\approx -1$. The two sources are placed a distance
+$\lambda/4.5$ apart (with $\lambda = 365$ nm), and we assume that the
+hyperlens is made of 160 alternating layers of silver ($\epsilon =
+-2.4012 + 0.2488i\,$) and dielectric ($\epsilon \approx 2.7$),
+each 10 nm thick. Exact numerical simulations can be used to
+study the imaging characteristics of this device. The resulting
+intensity pattern is shown in Fig.~\ref{fig:imagingSchematics}(a). The highly directional nature of the beams from
+the two sources allows for the resolution at the outer surface of
+the hyperlens. The separation between the two output beams at the
+boundary of the device is 5 times the distance between the sources
+and is bigger than the diffraction limit, thereby allowing for
+subsequent processing by conventional optics. This magnification
+corresponds to the ratio of the outer and inner radii, and is a
+consequence of cylindrical symmetry, together with the directional
+nature of the beams.
+
+
+The intensity distribution at the source is shown in
+Fig.~\ref{fig:imagingSchematics}(b), whereas the intensity
+distribution just outside the hyperlens is shown in
+Fig.~\ref{fig:imagingSchematics}(c). The two sources are clearly
+resolved, even though the distance between them is below
+the diffraction limit. It should be noted that realistic losses do
+not significantly affect the subdiffraction resolution
+capabilities of the hyperlens. Furthermore, due to the optical
+magnification in the hyperlens (by a factor of 5 in the simulation
+of Fig.~\ref{fig:imagingSchematics}), even for the subwavelength
+object, the scale of the image can be substantially larger than
+the wavelength, thus allowing for further optical processing of the
+image (e.g., further magnification) by conventional optics.
+
+
+
+
+
+\begin{figure}
+\centerline{\scalebox{0.35}{\includegraphics{image_resolution_b.pdf}}}
+\caption{(a) Schematics of imaging by the hyperlens. Two point
+sources separated by $\lambda/4.5$ are placed within the hollow core
+of the hyperlens. The hyperlens consists of 160 alternating layers of metal
+and dielectric, each of 10 nm thickness. Intensity plot in the region bounded by the rectangle shows the
+highly directional nature of the beams from the two point sources.
+The separation between the beams at the outer boundary of the device is
+greater than $\lambda$, due to magnification. (b) and (c)
+Demonstration of subwavelength resolution in the composite
+hyperlens containing two sources placed a distance $\lambda/4.5$
+apart inside the core: (b) field at the source; (c) field
+outside the hyperlens. [Adapted from
+Ref.~\inlinecite{JacobAlekseyevNarimanov2006}.]}
+ \label{fig:imagingSchematics}
+\end{figure}
+
+
+\subsubsection{Semiclassical treatment}
+
+\begin{figure}[tb]
+\centering \scalebox{0.54}{\includegraphics{fig1_spiral_2.pdf}}
+\caption{Trajectories of two rays incident on the hyperlens with
+different impact parameters, calculated using the analytical
+expression in Eqs.~(\ref{semiclassical_eq}--\ref{semiclassical_eq_theta0}). (a) $\eta$ = 0.1. (b) $\eta$ =0.5. Note the strong
+spiraling behavior. (c) ``Channeling regime'' for large $\eta$ ($\eta$ =100),
+where rays travel in straight lines radially. Note that all rays
+travel towards the center. [From
+Ref.~\inlinecite{JacobNarimanov2007}.]} \label{spiral}
+\end{figure}
+
+The above results were obtained by numerically propagating fields
+through the cylindrical layered structure. There exists, however,
+an analytic approach to analyzing light propagation in the
+hyperlens. Owing to the hyperbolic form of the dispersion
+relation in Eq.~(\ref{eq:drHyperbola}), the radial and tangential
+momentum of the fields increase as light approaches the core of
+the device. This leads to a substantial decrease of wavelength,
+which suggests a semiclassical description of field propagation
+using Hamiltonian ray optics.
+
+
+With the key assumption that the dielectric permittivity does not
+vary significantly over the scale of the wavelength, we can obtain
+the ray-optical Hamiltonian for a cylindrically anisotropic medium
+such as the hyperlens:
+
+\begin{equation}
+H=c\sqrt{\frac{p_{r}^{2}}{\epsilon_{\theta}}+
+\frac{p_{\theta}^{2}}{r^{2}\epsilon_{r}}},
+\end{equation}
+where $c$ is the velocty of light in vacuum, $p_{r}$ and $p_{\theta}$
+are the radial and angular momentum and $\epsilon_{\theta}$,
+$\epsilon_{r}$ are the tangential and radial dielectric
+permittivities.
+
+Solving for the ray dynamics, the equation of the ray trajectory
+inside the hyperlens is seen to be \cite{JacobNarimanov2007}
+
+
+\begin{equation}
+r(\theta)=\frac{p_{\theta}}{\xi\sqrt{|{\epsilon_{r}}|}\sinh[\eta(\theta-\theta_{0})]}.
+\end{equation}
+This is the equation of a spiral where $\theta_{0}$ is a
+parameter related to the initial conditions and
+\begin{equation}
+\eta=\sqrt{\frac{|\epsilon_{r}|}{\epsilon_{\theta}}}
+\end{equation}
+critically determines the ray dynamics inside the hyperlens. The implications of this analytical solution beyond the ray approach will be presented in the subsequent section.
+
+For a ray of light impinging on the hyperlens (outer radius
+$r_{\rm max}$) from vacuum with an impact parameter $\rho$, we can
+use the conservation of angular momentum
+($p_{\theta}=\rho\frac{\omega}{c}$) upon refraction to evaluate
+the constant $\theta_0$. The above equation then becomes
+\begin{equation}
+\label{semiclassical_eq}
+r(\theta)=\frac{\rho}{\sqrt{|{\epsilon_{r}}|}\sinh[\eta(\theta-\theta_{0})]},
+\end{equation}
+with
+\begin{equation}\label{semiclassical_eq_theta0}
+\theta_{0}=\sin^{-1}\left(\frac{\rho}{r_{\rm max}}\right)
+-\frac{1}{\eta}\sinh^{-1}\left(\frac{\rho}{r_{\rm
+max}\sqrt{|{\epsilon_{r}}|}}\right).
+\end{equation}
+
+ We plot the analytical result of Eq.~(\ref{semiclassical_eq}) in
+Figs.~\ref{spiral} (a) and (b) for small values of the parameter $\eta$
+which explicitly shows the spiralling behavior. The negative
+refraction of the ray is consistent with the known negative refraction
+of the Poynting vector in strongly anisotropic materials.
+For large values of the parameter $\eta$, we are in the
+channeling regime, where the ray moves in a straight line inside
+the hyperlens.
+
+
+
+
+If we visualize a Gaussian beam impinging on the layered hyperlens
+with impact parameter $\rho$ (Fig. \ref{gaussian_schematic}(a)) as a
+pencil of parallel rays, then Eq.~(\ref{semiclassical_eq}) predicts
+that the distance between the rays will decrease as it approaches
+the core, where the rays bounce off the inner hollow region. This is
+seen by plotting the analytical expression inside the hyperlens
+for $\eta = 1$, $\epsilon_{\theta}=1$, $\epsilon_{r}=-1$, and
+considering specular reflection at the inner radius, as shown in Fig.
+\ref{gaussian_schematic}(b). By choosing an appropriate metal
+($\epsilon_{m} \approx -0.4$) and dielectric ($\epsilon_{d}
+\approx 2.4$) we can achieve the layered hyperlens yielding the
+desired dielectric response, which is $\epsilon_{\theta}=1,
+\epsilon_{r}=-1$ according to Eq.~(\ref{eq:effectiveEpsilon}). We
+choose an inner radius of $\lambda$, outer radius $7\lambda$,
+thickness of layers $\lambda/100$, $N=600$ layers, and impact
+parameter $\rho=2.4\lambda$ at an operating wavelength of 700~nm.
+
+\begin{figure}[t]
+\centering
+\scalebox{0.4}{\includegraphics{semiclass_gaussian.pdf}}
+\caption{(a) Schematic of a Gaussian beam with impact parameter
+$\rho$ impinging on the layered hyperlens (top view) consisting of
+alternating layers of metal and dielectric. The inner hollow region
+and the region outside the hyperlens is vacuum. (b) Ray trajectories
+representing the Gaussian beam calculated for the effective medium
+parameters of the hyperlens using Eq.~(\ref{semiclassical_eq}). Note
+the narrowing of the Gaussian beam towards the core of the
+hyperlens, as predicted by the semiclassical theory. We consider
+specular reflection at the inner core. (c) Absolute value of the
+field for a Gaussian beam scattering from the
+layered hyperlens with parameters $rho \approx 4\lambda$, $r_{\rm min}
+\approx \lambda$, $r_{\rm max} \approx 7 \lambda$, $h \approx \lambda/100$,
+$\epsilon_{m} \approx -0.4$, $\epsilon_{d} \approx 2.4$. The ray
+trajectory shown in white is calculated using Eq.~(\ref{semiclassical_eq}) and specular reflection at the inner
+boundary. Note the narrowing of the Gaussian beam and also
+the motion of the center of the beam along the calculated ray trajectory.
+[Adapted from Ref.~\inlinecite{JacobNarimanov2007}.]
+}\label{gaussian_schematic}.
+\end{figure}
+
+
+The magnitude of the field is plotted in Fig.
+\ref{gaussian_schematic}(c), and the ray trajectory calculated from
+Eq.~(\ref{semiclassical_eq}) is in white, superimposed on the
+field plot. Black denotes
+regions of high intensity. The two circles denote the inner and
+outer boundaries of the device. The ray is clearly seen to move along
+the center of the Gaussian beam. The narrowing effect obtained from
+the ray equations is evident in
+the width of the Gaussian beam near the core. This validates the semiclassical
+description presented, as well as the adequacy of the effective medium
+approximation in describing the hyperlens. Note that the narrowing
+effect opens up the possibility of using the hyperlens for
+subdiffraction lithography in the channeling regime where the
+Gaussian beam is expected to travel radially to the core with
+reduced beam width.
+
+
+The semiclassical description can bring further insight into the
+hyperlens imaging setup of Fig.~\ref{fig:imagingSchematics}.
+Recall from earlier discussion that energy carried by waves in
+media with negative transverse permittivity is constrained to a
+cone. In the case of cylindrical anisotropy, the half-angle of
+the cone (see also Eq.~(\ref{eq:cone_angle})) is given by
+
+
+\begin{equation}\label{eq:cone_angle_cylindrical}
+\tan(\theta_{c})=\sqrt{\frac{\epsilon_{\theta}}{|\epsilon_{r}|}}=\frac{1}{\eta},
+\end{equation}
+where $\eta$ is the parameter entering the Eqs.~(\ref{semiclassical_eq}--\ref{semiclassical_eq_theta0}) that
+determines the pitch of the ray spirals. For large values of
+$\eta$ (the channeling regime), the energy cone divergence angle
+tends to zero, i.e., radiation from a point source propagates as a
+narrow beam. This is the condition that enables subdiffraction-limited imaging.
+We note from Fig. \ref{spiral}(c) that in the
+channeling regime, rays of light move in the hyperlens in straight
+lines, which is essential for a narrow beam divergence angle.
+
+
+
+
+We verify this fact using the analytical expression for rays
+inside the hyperlens in the case of two point sources kept inside
+the hyperlens. The point source is represented as a source of rays
+in all directions as shown in the inset of Fig. \ref{imaging}(a).
+Note that even though we have assumed isotropic emission in the
+core, the density of rays is high in two cone-like regions
+within the hyperlens. The rays of light are negatively refracted
+at the inner curved surface of the hyperlens, which helps in the
+formation of a beam. Inside the hyperlens, the rays then move in
+straight lines, almost radially, traveling to the outer interface.
+These rays arrive at normal incidence and the beam-like nature in
+the hyperlens is preserved as they emerge into vacuum. Thus, the
+two point sources give rise to two distinct beams in the far-field,
+even though they are separated by less than the diffraction limit inside
+the hyperlens. Furthermore, due to the cylindrical geometry and
+almost radial nature of propagation, the distance between the point
+sources is magnified and above the diffraction limit. We verify
+this behavior by considering a practical realization of the
+hyperlens made of alternating layers of metal
+($\epsilon_{m}\approx -1$) and dielectric ($\epsilon_{d} \approx
+1.1$) to achieve a dielectric response in the effective medium
+approximation ($\epsilon_{\theta}= 0.05$ ,$\epsilon_{r}=
+-22$). This gives a large value of $\eta \approx 20$, and
+hence we are in the channeling regime. The magnification due to
+the radial nature of light propagation is the ratio of the radii,
+which is approximately 5 in this case. The two beams emanating from
+the point sources that carry information to the far-field can
+clearly be seen in Fig.~\ref{imaging}(b), consistent with the plot
+obtained from the analytical expression for the rays in the hyperlens.
+
+
+\begin{figure}[t]
+\centering \scalebox{0.55}{\includegraphics{fig8_imaging.pdf}}
+\caption{Subdiffraction imaging in the hyperlens. (a) Beam-like
+radiation obtained from Eq. (\ref{semiclassical_eq}) for two point
+sources kept near the inner boundary of the hyperlens for large
+$\eta$ (channeling regime). The rays are negatively refracted at
+the inner surface and proceed radially outward, leading to
+magnification at the outer surface. The point source is
+represented as a source of rays in all directions (inset). (b)
+Numerical confirmation of the beam-like radiation using a layered
+metamaterial hyperlens made of alternating layers of metal
+($\epsilon_{m} \approx -1$) and dielectric ($\epsilon_{d} \approx
+1.1$) and two point sources near the inner boundary. The regions
+of high intensity are dark. [From
+Ref.~\inlinecite{JacobNarimanov2007}.]} \label{imaging}
+\end{figure}
+
+
+\section{Conclusion}
+
+Anisotropic metamaterials with hyperbolic dispersion relations were
+originally proposed as a simple alternative to negatively refractive
+media operating via magnetic resonances. We have seen, however,
+that this class of metamaterials goes far beyond geometric
+negative refraction. Its properties enable a multitude of novel
+systems with applications in imaging and wave guiding.
+
+In the coming years, we expect to see the emergence of many
+metamaterial-enabled devices. At radio frequencies, negative
+index metamaterials have already found applications in reflectors
+and radio antennas,\cite{Parazzoli2004} as well as in magnetic
+resonance imaging.\cite{Wiltshire2001} Optical domain
+metamaterials remain the subject of intense research. The goal of
+creating a medium with customized spatial and spectral variation
+of its dielectric tensor is ambitious but not far-fetched. After
+all, the ability to tailor electromagnetic response of materials
+by nanoscale patterning has become common in active optoelectronic
+devices (such as quantum cascade lasers), as well as in photonic
+crystal and plasmonic systems. Great opportunities exist for
+constructing and interfacing useful devices based on hyperbolic
+dispersion.
+
+
+\bibliography{anisotropic_nim_1}% Produces the bibliography via BibTeX.
+
+
+
+
+\end{document}
+
+% LocalWords: pdf
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