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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, \;
+cladding in regions 1 and 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
\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
+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
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
+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 $m$
+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
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 ($k_{\theta}r=m=\rm
+momentum implies that the tangential wave vector of a high-angular-momentum mode increases toward the center ($k_{\theta}r=m=\rm
const$). In a medium such as vacuum characterized by a circular
isofrequency curve (see Fig. \ref{fig:dr}(a)), this increase in the tangential component is
-not supported as both the tangential and radial wave vectors are
+not supported, as both the tangential and radial wave vectors are
bounded (see Eq.~(\ref{eq:diff_cutoff}) and related discussion).
These incident high-angular-momentum modes simply reflect
without ever reaching the scatterer. As such, they do not
anisotropy where dielectric permittivities have different signs
in the tangential and radial directions ($ \epsilon_{\theta}>0$,
$\epsilon_{r}<0$). Since there exist no natural materials with
-such an anisotropy, we assume that it could be implemented using
+such an anisotropy, we assume that the required
+dielectric response 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
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
+cylindrical anisotropy using metamaterials, 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
\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
+As the tangential component of the wave vector increases toward
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
\begin{figure}
%\centerline{\scalebox{.85}{\includegraphics{wg_modes_1.pdf}}}
\centerline{\scalebox{.43}{\includegraphics{eff_medium_modes.pdf}}}
-\caption{(a) high-angular-momentum states in an isotropic
+\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
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,
+by the device and guided toward 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
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
+Re}[\epsilon]\approx -1$. The two sources are placed at a distance
+of $\lambda/4.5$ 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
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
+and is larger 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
If we visualize a Gaussian beam impinging on the layered hyperlens
-with impact parameter $\rho$ (Fig. \ref{gaussian_schematic}(a)) as a
+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
\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
+expression in Eqs.~(\ref{semiclassical_eq}) and (\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
+travel toward the center. [From
Ref.~\inlinecite{JacobNarimanov2007}.]} \label{spiral}
\end{figure}
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
+the narrowing of the Gaussian beam toward 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
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
+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
+where $\eta$ is the parameter entering the Eqs.~(\ref{semiclassical_eq}) and (\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
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 ($\epsilon_{\theta}= 0.05$ ,$\epsilon_{r}=
+approximation ($\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,
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