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
+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
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
+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. Figure~\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
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 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
+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
It should be noted that the polar dielectric permittivities are
-ill defined at the center and any practical realization of
-cylindrical anisotropy using metamaterials, can only
+ill defined at the center, and any practical realization of
+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
%\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
-dielectric cylinder. (b) high-angular-momentum states in a
+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
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
+$\epsilon_{m} \approx -0.4$, and $\epsilon_{d} \approx 2.4$. 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.
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
+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
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