From 84f8e0916279ae65a4a0f3e9115410b60d021021 Mon Sep 17 00:00:00 2001 From: "U-LEO-FUJITSU-XP\\Leo" Date: Sun, 12 Jul 2009 00:13:43 -0400 Subject: [PATCH] Implemented the changes requested up to the hyperlens section --- anisotropic_nim_subsects_5.tex | 53 ++++++++++++++++++++-------------------- 1 file changed, 27 insertions(+), 26 deletions(-) diff --git a/anisotropic_nim_subsects_5.tex b/anisotropic_nim_subsects_5.tex index 903dde2..970d981 100755 --- a/anisotropic_nim_subsects_5.tex +++ b/anisotropic_nim_subsects_5.tex @@ -85,8 +85,8 @@ Hyperspace: Negative Refractive Index and Sub wavelength Imaging} \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} @@ -191,7 +191,8 @@ 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. +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 @@ -252,7 +253,7 @@ subwavelength-resolved imaging. -\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 @@ -268,7 +269,7 @@ 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 +inquire as to what material parameters lead to negative refraction without requiring negative phase velocity. @@ -378,15 +379,15 @@ behavior of negative refraction systems with $\epsilon < 0$, $\mu < 0$. We shou 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 @@ -416,7 +417,7 @@ 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 +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. @@ -457,7 +458,7 @@ 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, +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 @@ -473,7 +474,7 @@ imaginary part of $\epsilon$ becomes small away from 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 @@ -524,7 +525,7 @@ carbide,\cite{Shvets2003,Shvets2006} with $\epsilon < 0$ between 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 @@ -554,7 +555,7 @@ regions. \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 @@ -562,8 +563,8 @@ 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 +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 @@ -605,7 +606,7 @@ m_\text{max} = 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 @@ -619,7 +620,7 @@ m_\text{min} = 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. @@ -641,10 +642,10 @@ 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. +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} = @@ -652,7 +653,7 @@ Let us now consider the group velocity of the guided modes, $v_g = \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 $\vecb{k}$. @@ -664,7 +665,7 @@ $\vecb{k}$ 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 +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 @@ -707,7 +708,7 @@ 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 +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 @@ -719,17 +720,17 @@ positive energy flux outside. This leads to a dramatic reduction in the group v 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 -- 1.7.9.5