From 8d33b3b9bd67ede603b47f3806ae94c5cb23f75b Mon Sep 17 00:00:00 2001 From: "U-LEO-FUJITSU-XP\\Leo" Date: Wed, 24 Jun 2009 14:48:33 -0400 Subject: [PATCH 1/1] implemented all the textual changes requested (with the exception of suggestions in the reference section). --- anisotropic_nim_1.bib | 410 +++++++++++++ anisotropic_nim_subsects_5.tex | 1230 +++++++++++++++++++++++++++++++++++++++ anisotropic_nr.pdf | Bin 0 -> 214842 bytes cylindrical_scattering2.pdf | Bin 0 -> 167115 bytes eff_medium_modes.pdf | Bin 0 -> 59369 bytes fig1_spiral_2.pdf | Bin 0 -> 82642 bytes fig2_metacylinder.pdf | Bin 0 -> 382354 bytes fig8_imaging.pdf | Bin 0 -> 121247 bytes fig_circ_ell_hyp_dispersion.pdf | Bin 0 -> 73906 bytes image_resolution_b.pdf | Bin 0 -> 129936 bytes semiclass_gaussian.pdf | Bin 0 -> 685641 bytes wgstuff.pdf | Bin 0 -> 241613 bytes 12 files changed, 1640 insertions(+) create mode 100755 anisotropic_nim_1.bib create mode 100755 anisotropic_nim_subsects_5.tex create mode 100755 anisotropic_nr.pdf create mode 100755 cylindrical_scattering2.pdf create mode 100755 eff_medium_modes.pdf create mode 100755 fig1_spiral_2.pdf create mode 100755 fig2_metacylinder.pdf create mode 100755 fig8_imaging.pdf create mode 100755 fig_circ_ell_hyp_dispersion.pdf create mode 100755 image_resolution_b.pdf create mode 100755 semiclass_gaussian.pdf create mode 100755 wgstuff.pdf diff --git a/anisotropic_nim_1.bib b/anisotropic_nim_1.bib new file mode 100755 index 0000000..c3ce68f --- /dev/null +++ b/anisotropic_nim_1.bib @@ -0,0 +1,410 @@ +@Article{veselago, + author = "V.~G.~Veselago", + title = "The electrodynamics of substances with simultaneously negative value of $\epsilon$ and $\mu$", + journal = "Sov. 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Express", + volume = "14", + pages = "11184-11193", + year = "2006", +} + +@Article{JacobAlekseyevNarimanov2006, + author = "Z.~Jacob and L.~V.~Alekseyev and E.~Narimanov", + title = "Optical Hyperlens: Far-field imaging beyond the diffraction limit", + journal = "Opt. Express", + volume = "14", + pages = "8247-8256", + year = "2006", +} + +@ARTICLE{Wangberg2005, + author = "R. Wangberg and J. Elser and E. E. Narimanov and V. A. Podolskiy", + title = "Non-magnetic nano-composites for optical and infrared negative-refractive-index media", + journal = "J. Opt. Soc. Am. B.", + volume = "23", + pages = "498-505", + year = "2006", +} + + +@Article{Parazzoli2004, + author = "C.~G.~Parazzoli and R.~B.~Greegor and J.~A.~Nielsen and M.~A.~Thompson and K.~Li and A.~M.~Vetter and M.~H.~Tanielian and D.~C.~Vier", + title = "Performance of a negative index of refraction lens", + journal = "Appl. Phys. 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B", + volume = "74", + pages = "075103", + year = "2006", +} + +@Article{resonance_cone_1, + author = "R.~K.~Fisher and R.~W.~Gould", + title = "Resonance cones in the field pattern of a short antenna in an anisotropic plasma", + journal = "Phys. Rev. Lett.", + volume = "22", + pages = "1093-1095", + year = "1969", +} + + +@Inbook{resonance_cone_2, + author = "E. Arbel and L. B. Felsen", + title = "Electromagnetic Theory and Antennas", + editor = "E. C. Jordan", + edition = "", + publisher = "Pergamon Press", + address = "New York", + year = "1963", + ISBN = "", +} + +@Book{Schuster1904, + author = "Arthur Schuster", + title = "An Introduction to the Theory of Optics", + edition = "", + publisher = "Arnold", + address = "London", + year = "1904", + ISBN = "", +} + +@Article{Lamb1904, + author = "H. Lamb", + title = "On group-velocity", + journal = "Proc. Lond. Math. Soc.", + volume = "1", + pages = "473-479", + year = "1904", +} + + +@Article{JacobNarimanov2007, + author = "Z.~Jacob and L.~V.~Alekseyev and E.~Narimanov", + title = "Semiclassical theory of the hyperlens", + journal = "J. Opt. Soc. Am. A", + volume = "24", + pages = "A52-59", + year = "2007", \ No newline at end of file diff --git a/anisotropic_nim_subsects_5.tex b/anisotropic_nim_subsects_5.tex new file mode 100755 index 0000000..a14a29f --- /dev/null +++ b/anisotropic_nim_subsects_5.tex @@ -0,0 +1,1230 @@ +\documentclass{spiebook} + + + +\newcommand{\me}{\mathrm{e}} +\newcommand{\md}{\mathrm{d}} +\newcommand{\half}{\frac{1}{2}} +\newcommand{\thalf}{\tfrac{1}{2}} +\newcommand{\vecb}[1]{\mbox{\boldmath$#1$}} +\newcommand{\hatb}[1]{\mbox{\boldmath$\hat{#1}$}} +\newcommand{\vecbm}[1]{\boldmath#1} +\newcommand{\negquad}{\!\!\!\!\!\!} +\newcommand{\cns}{\!} %constant quantity superscript negative space adjust +\newcommand{\um}{$\mu$m} +\newcommand{\cube}{$^3$} +\newcommand{\mcube}{$^{-3}$} +\newcommand{\mthree}{$^{-3}$} +\newcommand{\squared}{$^2$} +\newcommand{\msquared}{$^{-2}$} +\newcommand{\mone}{$^{-1}$} +\newcommand{\inverse}{$^{-1}$} +%\newcommand{\minusone}{$^{-1}$} %pdflatex doesn't like this +\newcommand{\degc}{$\,^\circ$C} %command for the degree symbol (small space at the beginning) +\newcommand{\degree}{$^\circ$} +\newcommand{\halfhbar}{\frac{\hbar}{2}} +\newcommand{\thalfhbar}{\tfrac{\hbar}{2}} +\newcommand{\htwoo}{$\mathrm{H_2O}$} +\newcommand{\omegapl}{\omega_{\text{pl}}} + +%\newcommand{\choosenm}[1,2]{\begin{pmatrix}#1 \\ +%#2\end{pmatrix}} + +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{latexsym} +%\usepackage[margin=1.0in]{geometry} +\usepackage[]{graphicx} +\usepackage{braket} +%\usepackage{setspace} + +\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 diff --git a/anisotropic_nr.pdf b/anisotropic_nr.pdf new file mode 100755 index 0000000000000000000000000000000000000000..2aa8ce2f3c6d97aa0652a3b065c5602734dea644 GIT binary patch literal 214842 zcmeFZc{r5q`#(OV(k4%<7L1-2ZIWb(%w0)Rp+b~ul90lLEMeR&PpQO|7L}MvBHJV} zB!;9>(v*Fl7?~K1F=oH-d%oAy^E~h8{e6Ev$M5$&zJL6V-*J2 zZP}h}+YOA?ZqQiP!(r8HOx>WjPLFi#q~4}YhP!|NW&gH9W}6 zTQ!rZ#?*CxzQAoiKj{VheavKo`F|WUU1#ChM%Xn`|*PH8wIPY+7%!#mvNdy@`>Djfs)Xh7IQHO*XFEW@Bo+ z!E~#U%{ps>iaS$-&v*o#`NfHK8dH}ovvJ(6G4(%rwegocRO0Xo@Hulr&&0%VyRY9F zuOP$iejaDMwt0Dyj(Zu}>6w}vI_Vjk5(vR(g1kKZ0him39X)!~`{>a(wYaA;uJO_F z^Crz$s%t#Qc-@3`lZ{qbtQ&F5K6k~qqz~u2U5_x~x1+cxy%RoZOa+4bbBsSlsNw?< zC0K*K*BOuF9%nrC*8k+k9uFU{U_GOspPkR1_A;~v=ln=M|B^mn`JcC12a!&1C55}M 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