From: U-LEO-FUJITSU-XP\Leo Date: Mon, 13 Jul 2009 08:29:00 +0000 (-0400) Subject: Zubin's changes to the hyperlens section + thereafter X-Git-Url: http://www.dnquark.com/git/?a=commitdiff_plain;h=19cf8efb38a494d67d322991b6f99687ec5005d6;p=spie_book.git Zubin's changes to the hyperlens section + thereafter --- diff --git a/anisotropic_nim_subsects_5.tex b/anisotropic_nim_subsects_5.tex index 970d981..5499145 100755 --- a/anisotropic_nim_subsects_5.tex +++ b/anisotropic_nim_subsects_5.tex @@ -6,9 +6,10 @@ \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{\vecb}[1]{\mathbf{#1}} +\newcommand{\hatb}[1]{\mathbf{\hat{#1}}} \newcommand{\vecbm}[1]{\boldmath#1} + \newcommand{\negquad}{\!\!\!\!\!\!} \newcommand{\cns}{\!} %constant quantity superscript negative space adjust \newcommand{\um}{$\mu$m} @@ -683,7 +684,7 @@ 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, \; +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 @@ -771,7 +772,7 @@ 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 +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 @@ -786,11 +787,11 @@ Ref.~\inlinecite{JacobAlekseyevNarimanov2006}.] 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 @@ -801,10 +802,10 @@ 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 ($k_{\theta}r=m=\rm +momentum implies that the tangential wave vector of a high-angular-momentum mode increases toward the center ($k_{\theta}r=m=\rm const$). In a medium such as vacuum characterized by a circular isofrequency curve (see Fig. \ref{fig:dr}(a)), this increase in the tangential component is -not supported as both the tangential and radial wave vectors are +not supported, as both the tangential and radial wave vectors are bounded (see Eq.~(\ref{eq:diff_cutoff}) and related discussion). These incident high-angular-momentum modes simply reflect without ever reaching the scatterer. As such, they do not @@ -820,7 +821,8 @@ propagation in a bulk medium with strong cylindrical anisotropy where dielectric permittivities have different signs in the tangential and radial directions ($ \epsilon_{\theta}>0$, $\epsilon_{r}<0$). Since there exist no natural materials with -such an anisotropy, we assume that it could be implemented using +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 @@ -836,7 +838,7 @@ in a hollow core dielectric cylinder. 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 @@ -870,7 +872,7 @@ Eq.~(\ref{eq:dr}), namely, \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 @@ -896,7 +898,7 @@ high-$m$ modes in regular dielectrics [see Fig. \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 @@ -912,7 +914,7 @@ Ref.~\inlinecite{JacobAlekseyevNarimanov2006}.] 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 @@ -933,8 +935,8 @@ 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 +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 @@ -943,7 +945,7 @@ intensity pattern is shown in Fig.~\ref{fig:imagingSchematics}(a). The highly d 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 @@ -1056,7 +1058,7 @@ the hyperlens. 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 @@ -1077,10 +1079,10 @@ parameter $\rho=2.4\lambda$ at an operating wavelength of 700~nm. \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} @@ -1094,7 +1096,7 @@ 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 +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 @@ -1131,13 +1133,13 @@ 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 +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 @@ -1170,7 +1172,7 @@ 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 ($\epsilon_{\theta}= 0.05$ ,$\epsilon_{r}= +approximation ($\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, diff --git a/spiebook.cls b/spiebook.cls index 45fd246..91935fb 100755 --- a/spiebook.cls +++ b/spiebook.cls @@ -397,7 +397,7 @@ \renewcommand*\l@subsection{\@dottedtocline{2}{\cw@section}{\cw@subsection}} \renewcommand*\l@subsubsection{\@dottedtocline{3}{4.85em}{3.45em}} % make subsubsection numbers align with subsection titles in toc -%\renewcommand*\l@subsubsection{\@dottedtocline{3}{1.0em}{10.1em}} +%\renewcommand*\l@subsubsection{\@dottedtocline{3}{7.0em}{10.1em}} %original line \renewcommand*\l@paragraph{\@dottedtocline{4}{10em}{5em}} \renewcommand*\l@subparagraph{\@dottedtocline{5}{12em}{6em}}