\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
+\affilation{\supscr{a}Princeton University, Princeton, NJ, USA\\
+\supscr{b}Purdue University, West Lafayette, IN, USA
}
\end{aug}
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.
+lithographic methods in structuring the materials. For the optical
+frequencies, fully three-dimensional (3D) subwavelength patterning is currently unfeasible.
Aside from the manufacturing difficulties, negative magnetic
response presents another significant challenge. The resonance in
-\section{Nonmagnetic Negative Refraction}
+\section{Nonmagnetic Negative Refraction}\label{sect:nonmagnetic_nr}
For a plane wave with wave vector $\vecb{k}$, incident on some
surface, translational invariance demands that $k_\parallel$, 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
+inquire as to what material parameters lead to negative refraction
without requiring negative phase velocity.
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
+properties of wave propagation. (Indeed, we shall see in
+Section~\ref{ssect:waveguides} that this dispersion relation enables devices with
negative phase velocity and near-zero group velocity.)
\begin{figure}[t]
\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
+$\epsilon_x < 0$, $\epsilon_z > 0$ material. Note the 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
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
+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.
> 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,
+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
resonance, minimizing absorption.
-Whereas the phonon anisotropy of TGS exists in the low THz domain,
+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
10.3 and 11 \um.
-The metamaterials can be structured in many different ways. For
+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
\section{Applications}
-\subsection{Waveguides}
+\subsection{Waveguides}\label{ssect:waveguides}
As discussed above, the $\epsilon_x < 0, \; \epsilon_z
> 0$ materials enable all-angle negative refraction for incident plane
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
+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
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
+Eq.~(\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
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
+$\epsilon_\parallel$, 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.
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.
+will be discussed in more detail in Section~\ref{ssect:hyperlens}.
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
+\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}{v_\phi},
\end{equation}
\noindent where $v_\phi$ is the phase velocity. For
-$\epsilon_\perp < 0$ we see immediately that the phase velocity
+$\epsilon_\perp < 0$, 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 $\vecb{k}$.
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
+earlier section). However, 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
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
+dielectric. However, 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
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
+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}
+\subsection{The hyperlens}\label{ssect:hyperlens}
\subsubsection{Theoretical description}
-We saw in an earlier section that a medium with a hyperbolic
+We saw in Section~\ref{sect:nonmagnetic_nr} that a medium with a hyperbolic
dispersion relation allows propagation of high spatial frequency
-waves which would decay in a conventional dielectric. This
+waves that 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
projects / spie_book.git / commitdiff