\chapter{Introduction}
\label{ch:introduction}
\graphicspath{{figures/introduction/}}

The limits of technology are frequently determined by physical constraints.  For instance, the fundamental limit on the speed of information transfer is the speed of light in vacuum, a physical constant.  Transistors are limited in size by the sizes of atoms, and this imposes a constraint on the operational speed of conventional computers.  Revolutionary technologies arise when a new technique or a new material makes it possible to reach beyond constraints that had once been considered fundamental.

Another example of such a limit is the range of indices of refraction accessible in materials at optical frequencies.  The index of refraction is the ratio of the velocity of light in vacuum to its phase velocity in a particular material.  It is a measure of how light slows down in the material and is therefore related to how that material interacts with light.  Substances like glass and water have refractive indices greater than 1 because light travels more slowly in them than in vacuum.  Metals and other materials that do not transmit light have complex indices of refraction with substantial imaginary parts, which are related to absorption.  The ability to engineer a material with a refractive index close to or less than zero can significantly broaden the range of material properties available.  A negative index at optical frequencies could enable completely new optical devices, from wave guides to perfect lenses and perhaps even technologies that until now have sounded like science fiction, such as invisibility cloaks.

In this dissertation, I present experimental work aimed at reaching beyond conventional limits on refractive index in the visible.  In particular, I examine the possibility of isotropic negative refraction and demonstrate solutions to several of the experimental obstacles that have kept it beyond reach.

\section{Negative Refraction}

A material with a negative index of refraction would be quite different from the materials people encounter in everyday life.  Typically, the propagation direction of light changes at the interface between two dielectric media such that the light bends toward or away from the normal, as illustrated in Figure~\ref{fig:refraction}(a).  If instead light travels from a material with a positive index to one with a negative index, it bends so that its Poynting vector $\mathbf{S}$ points toward the opposite side of the normal, as shown in Figure~\ref{fig:refraction}(b).  In this case, the law of refraction (equation~\ref{eq:refraction_law}) is still satisfied at the interface,
\begin{equation} \label{eq:refraction_law}
n_1 \sin{\theta_1} = n_2 \sin{\theta_2},
\end{equation}
\noindent but the angle of refraction is negative.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm]{refraction}
\caption{\label{fig:refraction} Light refracts at the interface (blue) between two media.  (a) In conventional refraction, light bends toward or away from the normal as it crosses the interface.  (b) Changing the sign of the refractive index results in the new propagation direction pointing toward the opposite side of the normal. }
\end{center}
\end{figure}

For centuries after Ibn Sahl, Willebr{\o}rd Snell, and others independently discovered this law \cite{Kwan2002}, there were no materials shown to have a refractive index less than zero.  The idea of negative refractive index was considered physically impossible until Veselago showed otherwise, in a theoretical argument, nearly fifty years ago \cite{Veselago1968}.

\subsection{Theory}

Veselago began by pointing out how the refractive index arises in the dispersion equation.  For an isotropic substance and a monochromatic wave of frequency $\omega$, the wave vector $\mathbf{k}$ and material properties are related by
\begin{equation} \label{eq:dispersion}
k^2 = \frac{\omega^2}{c^2} \varepsilon_r \mu_r.
\end{equation}
\noindent Here $\varepsilon_r$ is the relative electric permittivity or dielectric constant of the material and $\mu_r$ is the relative magnetic permeability, defined by the usual constitutive relations for non-chiral materials \cite{Pendry2004}:
\begin{gather}
\mathbf{D} = \varepsilon \mathbf{E} = \varepsilon_0 \varepsilon_r \mathbf{E} \label{eq:varepsilon} \\
\mathbf{B} = \mu \mathbf{H} = \mu_0 \mu_r \mathbf{H}. \label{eq:mu}
\end{gather}
\noindent Their product defines the refractive index $n$:
\begin{equation} \label{eq:n_squared}
n^2 = \varepsilon_r \mu_r.
\end{equation}

The sign of the refractive index appears at first to be ambiguous if the product of $\varepsilon_r$ and $\mu_r$ is real:
\begin{equation} \label{eq:n_ambiguous}
n = \pm \sqrt{\varepsilon_r \mu_r}.
\end{equation}
\noindent Furthermore, according to this equation, the refractive index is real as long as $\varepsilon_r$ and $\mu_r$ are both positive or both negative.  Under what circumstances is $n$ negative and real?  To answer this question, one must reexamine several fundamental results of electromagnetism and carefully trace how $\varepsilon_r$ and $\mu_r$ propagate through them.

\subsubsection{Left-Handed Materials}

I begin with Maxwell's Equations in macroscopic media,
\begin{gather}
\nabla \cdot \mathbf{D} = \rho_{\mathrm{free}} \nonumber \\
\nabla \cdot \mathbf{B} = 0 \nonumber \\
\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} \label{eq:FaradayLaw} \\
\nabla \times \mathbf{H} = \mathbf{J}_{\mathrm{free}} + \frac{\partial \mathbf{D}}{\partial t}. \label{eq:AmpereLaw}
\end{gather}
\noindent If one assumes monochromatic plane waves, the electric and magnetic fields can be written in the form 
\begin{gather}
\mathbf{E} = \mathbf{E}_0 \mathrm{e}^{i \left( \mathrm{\mathbf{k}}\cdot\mathrm{\mathbf{r}} - \omega t\right)} \nonumber \\
\mathbf{B} = \mathbf{B}_0 \mathrm{e}^{i \left( \mathrm{\mathbf{k}}\cdot\mathrm{\mathbf{r}} - \omega t\right)}. \nonumber 
\end{gather}

Taking derivatives of these expressions and substituting into Faraday's Law (\ref{eq:FaradayLaw}), it can be shown that
\begin{equation}
\mathbf{k} \times \mathbf{E} = \omega \mathbf{B}. \nonumber 
\end{equation}
\noindent If one takes the cross product of $\mathbf{E}$ with each side of this equation and employs vector identities, one finds
\begin{equation} \label{eq:k_direction}
E^2 \mathbf{k} - \left( \mathbf{E} \cdot \mathbf{k} \right) \mathbf{E} = \omega \left( \mathbf{E} \times \mathbf{B} \right).
\end{equation}
\noindent Since $\mathbf{E} \cdot \mathbf{k} = 0$, the wave vector $\mathbf{k}$ must point in the same direction as $\mathbf{E} \times \mathbf{B}$.

Next I show that the wave vector does not necessarily point in the direction of energy flow.  Poynting's Theorem \cite{Poynting1884, Jackson1999} states that energy flow in the electromagnetic wave is determined by the Poynting vector $\mathbf{S}$:
\begin{equation} \label{eq:Poynting}
\mathbf{S} = \mathbf{E} \mathbf{\times} \mathbf{H}.
\end{equation}
\noindent The Poynting vector and the wave vector point in the same direction when $\mathbf{B}$ and $\mathbf{H}$ point in the same direction, but this holds only when $\mu$ is positive, per equation~\ref{eq:mu}.  When $\mu$ is negative, the energy flow and phase velocity are in opposite directions.  In a material where $\varepsilon < 0 $ and $\mu < 0$, the vectors $\mathbf{E}$, $\mathbf{H}$, and $\mathbf{k}$ therefore do not obey the usual right hand rule (which still applies to $\mathbf{S}$ but not to $\mathbf{k}$).  Veselago thus coined the term ``left-handed'' for substances with negative $\varepsilon$ and $\mu$ \cite{Veselago1968}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=10.0cm]{left-handed}
\caption{\label{fig:left-handed} Light of frequency $\omega$ can propagate through a material only if the product of $\varepsilon \left( \omega \right)$ and $\mu \left( \omega \right)$ is positive.  Common transparent dielectrics are in the upper right quadrant and metals occupy the upper left at optical frequencies.  Left-handed materials are in the lower left quadrant. This figure is based on Figure 7 in Veselago \cite{Veselago1968}, Figure 1 in Ramakrishna \cite{Ramakrishna2005}, and Figure 1.2 in Cai and Shalaev \cite{Cai2010}. }
\end{center}
\end{figure}

Figure~\ref{fig:left-handed} illustrates the space of conceivable materials in terms of permittivity and permeability \cite{Veselago1968, Ramakrishna2005, Cai2010}.  In general this picture is incomplete because $\varepsilon$ and $\mu$ can have substantial imaginary parts, which are related to absorption and impact the refractive index and attenuation loss in a material.  Here I assume they are negligible to illustrate key features of the parameter space.  Most materials have positive $\mu$, which confines them to the upper half of this diagram.  In the upper right quadrant, $\left( \varepsilon \mu \right) > 0$, so the refractive index is real, and the phase velocity and Poynting vector point in the same direction.  Dielectrics like plastic, glass, and water are in this quadrant.  In the upper left and lower right quadrants, $\left( \varepsilon \mu \right) < 0$, so the refractive index is imaginary, according to equation~\ref{eq:n_squared}, and fields decay exponentially inside the material.  Metals at optical frequencies are in the upper left quadrant.  Some natural magnetic materials up to GHz frequencies occupy the lower right \cite{Veselago1968, Ramakrishna2005, Cai2010}.  Left-handed materials are found in the lower left quadrant, where light can propagate but the phase velocity direction is reversed.

\subsubsection{Refractive Index of Left-Handed Materials}

I have not yet clarified whether one should choose the plus sign or minus sign for the refractive index of a left-handed material in equation~\ref{eq:n_ambiguous}.  To resolve this question, I consider refraction of light at an interface.  Two scenarios are illustrated in Figure~\ref{fig:neg_refraction_cases}.  In Figure~\ref{fig:neg_refraction_cases}(a), light impinges at normal incidence on a left-handed material from a right-handed one.  Causality and symmetry require that the Poynting vector points in the same direction in both media.  In the left-handed medium, however, $\mathbf{k}$ points opposite to $\mathbf{S}$, so it follows that the phase velocity is reversed and the wave vector points toward the interface in both media.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=11.0cm]{neg_refraction_cases}
\caption{\label{fig:neg_refraction_cases} Refraction at an interface between right- and left-handed media.  (a) At normal incidence, causality requires that the Poynting vector direction (black) remain constant across the interface, which means the wave vector (green) must change directions.  (b) When the angle of incidence is nonzero, the wave must be refracted at a negative angle.  Wave fronts are shown in gray.}
\end{center}
\end{figure}

Now I examine the case when the angle of incidence is nonzero, as shown in Figure~\ref{fig:neg_refraction_cases}(b).  Causality requires that light transmitted into the left-handed material must have the vertical component of its Poynting vector $\mathbf{S}$ pointing downward, just as in the right-handed medium.  The wave vector must therefore have an upward component, as in Figure~\ref{fig:neg_refraction_cases}(a).  Also, the phase must be continuous across the interface in order to satisfy the electromagnetic boundary conditions (assuming that the interface has not been engineered to introduce phase discontinuities \cite{Yu2011}).  Owing to the horizontal component of the wavevector in the right-handed medium, lines of constant phase travel horizontally along the interface.  The wave vector in the left-handed medium must therefore have a horizontal component in the same direction as its right-handed counterpart.  Accordingly, the Poynting vector points down and to the left.  The refracted ray must lie on the side of the normal opposite to where one would ordinarily expect to find it, and the angle of refraction is negative.\footnote{Veselago analyzed refraction at the interface between right- and left-handed media by invoking boundary conditions on $\mathbf{E}$ and $\mathbf{H}$, rather than causality and phase continuity, but reached the same conclusion \cite{Veselago1968}.}  Thus the left-handed material must have a negative refractive index to satisfy the law of refraction (equation~\ref{eq:refraction_law}).

\subsection{Potential Applications}

Negative refractive index remained a theoretical curiosity until John Pendry predicted a practical application \cite{Pendry2000}.  Pendry extended Veselago's work by considering image formation by a flat slab with isotropic $\varepsilon_r=-1$ and $\mu_r=-1$, surrounded by vacuum.  Veselago had pointed out that such a slab could behave like a lens because light from an object close to the slab is brought to a focus on the other side \cite{Veselago1968}, as illustrated in Figure~\ref{fig:perfect_lens}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=9.0cm]{perfect_lens}
\caption{\label{fig:perfect_lens} A flat slab of thickness $d$ with $n=-1$ can bring light from a nearby object to a focus on the opposite side.  Two example ``object planes'' $O_1$ and $O_2$ correspond to different ``image planes'' $I_1$ and $I_2$.  Each object is a distance $2d$ away from its image.}
\end{center}
\end{figure}
In a plane parallel to and close to the slab, Pendry argued, the electric field can be expressed in terms of a Fourier expansion,
\begin{equation} \label{eq:Fourier_expansion}
\mathbf{E} \left( \mathbf{r}, t \right) = \sum\limits_{k_x, k_y} \mathbf{E} \left( k_x, k_y\right) \mathrm{e}^{i \left( \mathbf{k} \cdot \mathbf{r} - \omega t \right)}.
\end{equation}
\noindent In vacuum, the dispersion equation (\ref{eq:dispersion}) requires that
\begin{equation} \label{eq:kz}
k_z = \sqrt{\omega^2 c^{-2} - \left( k_x^2 + k_y^2 \right)}.
\end{equation}
\noindent This expression gives a real value for the transverse wave vector $k_z$ as long as $k_x^2 + k_y^2 < \omega^2 c^{-2}$.  For high spatial frequencies corresponding to $k_x^2 + k_y^2 > \omega^2 c^{-2}$, however, $k_z$ is imaginary.  These Fourier components of the electric field  would normally be lost to the image plane as evanescent waves.

Pendry calculated the Fresnel coefficients for transmission through and reflection from this slab and concluded that the reflection coefficient is 0, and the transmission coefficient is
\begin{equation} \label{eq:amplify}
T = \mathrm{e}^{-ik_zd}.
\end{equation}
\noindent These evanescent waves are amplified in the negative-index slab.  Pendry thus came to the surprising conclusion that flat sheets of negative-index material can be used as ``perfect lenses,'' in that they recover all Fourier components of the electric field in the image plane.  The conventional Abbe diffraction limit \cite{Abbe1873} does not apply to such a lens because information about subwavelength structure, which is ordinarily carried in the evanescent waves, is propagated to the image plane.

It is important to note that Pendry's perfect lens does not magnify.  Instead, if an object is placed close to the slab, it forms a real image on the opposite side that is exactly the same size as the object.  Practical implementation of such a device would require a camera with subwavelength pixels.  Alternative ``hyperlens'' designs can magnify but have more complex geometry and material requirements \cite{Pendry2003, Liu2007_Science}.  Nevertheless, if researchers could fabricate a negative-index material and engineer a suitable way to record images, it would be a revolutionary technology.  Biological samples might be imaged at nanoscale resolution without damage by electron beams, and photolithography techniques could reach a new regime for nanofabrication \cite{Cai2010}.

More generally, however, the availability of negative-index materials would give scientists and engineers far greater control over the optical properties of materials.  Maxwell Garnett's theory \cite{Garnett1906} and Effective Medium Theory \cite{Bruggeman1935, Choy1999, Cai2010} explain how tunable effective optical properties emerge from combining materials with different refractive indices.  Negative-index materials would make it possible to create new classes of composites and optical devices and may lead to new designs for anisotropic materials as well.  For example, the technique of transformation optics \cite{Rahm2008} enables devices in which light travels along arbitrary, pre-programmed paths, but it requires precise control over $\varepsilon_r$ and $\mu_r$ anisotropically at all points.  The ability to engineer these material properties with precision could permit a practical implementation of this strategy for new technologies like invisibility cloaks \cite{Schurig2006} or light harvesters \cite{Narimanov2009, Pendry2012}.

\section{Metamaterials}

Metamaterials are artificial substances designed to exhibit physical properties that are not found in naturally occurring materials.  No materials are known to exhibit $\mu < 0$ naturally at optical frequencies \cite{Cai2010}, and thus there are no known natural substances that exhibit a negative index of refraction.  The design of metamaterials with negative index requires a deeper understanding of how the electric permittivity and magnetic permeability of a given material are determined and how they can be manipulated.

\subsection{Theory}

To understand how to make a material with negative $\mu_r$, one might first ask how to make a negative $\varepsilon_r$.  Negative $\varepsilon_r$ occurs in noble metals like gold and silver at optical frequencies \cite{Johnson1972}.  One can explain this phenomenon using the Drude model.

\subsubsection{Drude Model for the Permittivity of Metals}

I begin by noting that the relative dielectric function $\varepsilon_r$ is related to the electric field $\mathbf{E}$ and the polarization density $\mathbf{P}$ through the relationship
\begin{equation} \label{eq:polarization_constitutive}
\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} = \varepsilon_0 \varepsilon_r \mathbf{E} = \varepsilon \mathbf{E}.
\end{equation}
\noindent The polarization density $\mathbf{P}$ is the average electric dipole moment per unit volume, by definition \cite{Cai2010}.  In a metal, it can be expressed as 
\begin{equation} \label{eq:polarization_density}
\mathbf{P} = -n e \mathbf{r} \left( t \right),
\end{equation}
\noindent where $n$ is the free electron density, $-e$ is the electron charge, and $\mathbf{r} \left( t \right)$ is the change in position of free electrons as a function of time.  When monochromatic light impinges on a metal, the electric field varies sinusoidally in time and exerts a driving force on free electrons at the surface of the metal.  The equation of motion for an electron can be expressed as \cite{Moores2006, Cai2010}
\begin{equation} \label{eq:electron_EOM1}
m_e \frac{\partial^2 \mathbf{r} \left( t \right)}{\partial t^2}
+ m_e \Gamma \frac{\partial \mathbf{r} \left( t \right)}{\partial t}
= -e \mathbf{E}_0 \mathrm{e}^{-i \omega t},
\end{equation}
\noindent where $m_e$ is the mass of the electron and $\mathbf{E}_0 \mathrm{e}^{-i \omega t}$ is the incident electric field.  $\Gamma$ is a damping constant, related to viscous friction from inelastic collisions between electrons, electron-phonon coupling, and inelastic scattering from impurities \cite{Alvarez1997}.  This model assumes that all inelastic collisions may be treated with the same damping constant and it neglects long-range interactions between electrons or between the electron and bound ions \cite{Ashcroft1976}.  It also says nothing about electrons within the ions.

The equation of motion has the solution
\begin{equation} \label{eq:electron_amplitude1}
\mathbf{r} \left( t \right) = \frac{e \mathbf{E}_0}{m_e \left( \omega^2 + i \omega \Gamma \right)} \mathrm{e}^{-i \omega t}.
\end{equation}
\noindent This equation describes collective oscillations in the free electron gas in the metal, associated with quasiparticles called plasmons. If one uses this result to substitute for $\mathbf{r} \left( t \right)$ in equation~\ref{eq:polarization_density} and then substitute the resulting expression for $\mathbf{P}$ in equation~\ref{eq:polarization_constitutive}, one finds an expression for the relative dielectric function:
\begin{equation} \label{eq:epsilon_Drude1}
\varepsilon_r \left( \omega \right) =
1 - \frac{\omega_p^2}{\omega^2 + i \Gamma \omega},
\end{equation}
\noindent where $\omega_p$ is the metal's plasma frequency,
\begin{equation} \label{eq:plasma_frequency}
\omega_p = \sqrt{\frac{n e^2}{\varepsilon_0 m_e}}.
\end{equation}

\subsubsection{Impact of Electric Dipole Resonances on Permittivity}

Equation~\ref{eq:epsilon_Drude1} only accounts for free electrons, but bound electrons contribute to the dielectric functions of metals as well.  At optical frequencies, some bound electrons in noble metals can transition between bands.  For instance, electrons can transition from the 5d to the 6sp state in gold when excited by light \cite{Lasser1981}.  
% This occurs at a characteristic frequency $\omega_b$, corresponding to the oscillation frequency of these bound electrons under an applied potential \cite{Cai2010}.  
Since these electrons are bound, they are effectively subject to a restoring force $-k_b \mathbf{r}_b \left( t \right)$, and their equation of motion is
\begin{equation} \label{eq:electron_EOM2}
m_e \frac{\partial^2 \mathbf{r}_b \left( t \right)}{\partial t^2}
+ m_e \Gamma_b \frac{\partial \mathbf{r}_b \left( t \right)}{\partial t}
+ k_b \mathbf{r}_b \left( t \right)
= -e \mathbf{E}_0 \mathrm{e}^{-i \omega t}.
\end{equation}
\noindent Here $\Gamma_b$ is the damping coefficient for bound electrons.  The solution to this differential equation has a form similar to equation~\ref{eq:electron_amplitude1}, but with a characteristic frequency $\omega_b = \sqrt{k_b / m_e}$ due to the restoring force:
\begin{equation} \label{eq:electron_amplitude2}
\mathbf{r}_b \left( t \right) = \frac{e \mathbf{E}_0}{m_e \left( \omega^2 - \omega_b^2 + i \omega \Gamma_b \right)} \mathrm{e}^{-i \omega t}.
\end{equation}

If one includes the bound electrons in the expression for the polarization density (equation~\ref{eq:polarization_density}), the result for the dielectric function becomes
\begin{equation} \label{eq:epsilon_Drude2}
\varepsilon_r \left( \omega \right) =
\varepsilon_b \left( \omega \right)
- \frac{\omega_p^2}{\omega^2 + i \Gamma \omega},
\end{equation}
\noindent where
\begin{equation} \label{eq:epsilon_bound}
\varepsilon_b \left( \omega \right) =
1 - \frac{\omega_{pb}^2}{\omega^2 - \omega_b^2 + i \Gamma_b \omega}.
\end{equation}
\noindent Here $\Gamma_b$ is the damping coefficient for bound electrons and $\omega_{pb}$, the equivalent of a plasma frequency for bound electrons, is related to their number density in the same way that $\omega_p$ depends on the number density of free electrons (equation~\ref{eq:plasma_frequency}).  Because the relative dielectric function is complex, it can be written in terms of a real part and an imaginary part:
\begin{equation} \label{eq:epsilon_real_imag}
\varepsilon_r \left( \omega \right) = \varepsilon_r' + i \varepsilon_r'',
\end{equation}
\noindent where
\begin{gather}
\varepsilon_r' \left( \omega \right) = 1 
- \frac{\omega_{p}^2}{\omega^2 + \Gamma^2}
- \frac{\omega_{pb}^2 \left( \omega^2 - \omega_{b}^2 \right)}{\left( \omega^2 - \omega_{b}^2 \right)^2 + \omega^2 \Gamma_b^2} \label{eq:epsilon_real} \\
\varepsilon_r'' \left( \omega \right) = 
\frac{\omega_{p}^2 \Gamma}{\left( \omega^2 + \Gamma^2 \right) \omega}
+ \frac{\omega_{pb}^2 \omega \Gamma}{\left( \omega^2 - \omega_{b}^2 \right)^2 + \omega^2 \Gamma_b^2}. \label{eq:epsilon_imag}
\end{gather}

The last terms in equations~\ref{eq:epsilon_real} and \ref{eq:epsilon_imag} come from the bound electrons.  These electrons contribute a Lorentzian resonance to the dielectric function, evident from equation~\ref{eq:epsilon_bound} and shown in Figure~\ref{fig:epsilon}.  Here the real ($\varepsilon_b'$) and imaginary ($\varepsilon_b''$) parts have been plotted using the values $\hbar \omega_b = 2.8$ eV, $\hbar \omega_{pb} = 3.0$ eV, and $\hbar \Gamma_b = 0.6$ eV for gold \cite{Cai2010}, obtained by fitting to measured optical properties.  The peak in ($\varepsilon_b''$), centered at about 450 nm, corresponds to increased absorption for blue and green light, which gives gold its signature color.  Near this resonance, the real part ($\varepsilon_b'$) varies substantially, and even drops below zero.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm]{epsilon}
\caption{\label{fig:epsilon} Contribution of bound electrons to the real (green) and imaginary (blue) parts of the relative dielectric function of gold, calculated using equation~\ref{eq:epsilon_bound}. This plot is based on Figure 2.5 in Cai and Shalaev \cite{Cai2010}.}
\end{center}
\end{figure}

The free electron contribution to the dielectric function may be included as well to calculate the refractive index of gold using equation~\ref{eq:epsilon_Drude2} and assuming that $\mu_r = 1$ at optical frequencies.  The refractive index in this case has both real ($n$) and imaginary ($k$) components, both of which are plotted as a function of wavelength in figure~\ref{fig:gold_index}.  Here I use the values $\hbar \omega_p = 9.1$ eV and $\hbar \Gamma = 0.072$ eV, based on the experimental data of Johnson and Christy \cite{Johnson1972}, which are widely used \cite{Cai2010}.  The model and data agree well, particularly at wavelengths longer than 500 nm and far into the infrared.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm]{gold_index}
\caption{\label{fig:gold_index} Refractive index (blue) and extinction (green) for gold.  Solid curves are calculated using equation~\ref{eq:epsilon_Drude2}. Data (circles) are from Johnson and Christy \cite{Johnson1972}.}
\end{center}
\end{figure}

This analysis reveals that electric dipole resonances affect the bulk permittivity and refractive index of a material.  Moreover, because $\varepsilon_r$ is a material property that is meaningful only at length scales on the order of a wavelength or longer, subwavelength components of a composite material  \cite{Bruggeman1935, Choy1999, Cai2010} can also alter the effective permittivity of the material if they introduce electric dipole resonances.

How can one introduce electric dipole resonances into a transparent dielectric?  One way is to dope it with metallic nanoparticles.  If an electromagnetic wave illuminates a subwavelength metal particle, the free electrons at the surface of the metal are subject to the same forces as the free electrons described by equation~\ref{eq:electron_EOM1}.  However, the fact that they are constrained to a small volume introduces a restoring force $-K \mathbf{r} \left( t \right)$, where the spring constant $K$ depends on the size and geometry of the particle.
\begin{equation} \label{eq:electron_EOM3}
m_e \frac{\partial^2 \mathbf{r} \left( t \right)}{\partial t^2}
+ m_e \Gamma \frac{\partial \mathbf{r} \left( t \right)}{\partial t}
+ K \mathbf{r} \left( t \right)
= -e \mathbf{E}_0 \mathrm{e}^{-i \omega t}
\end{equation}
\noindent Equation~\ref{eq:electron_EOM3} has the same form as equation~\ref{eq:electron_EOM2}, the equation of motion for the bound electrons, a damped harmonic oscillator model.  It shows that the free electrons in a nanoparticle have a resonant frequency $\omega_0 = \sqrt{K / m_e}$.  The oscillation of the free electron gas in response to the external field is illustrated in Figure~\ref{fig:plasmon_nanoparticle}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.0cm]{plasmon_nanoparticle}
\caption{\label{fig:plasmon_nanoparticle} Electromagnetic waves drive oscillations in the free electrons at the surface of a metal nanoparticle, creating an electric dipole resonance.  This figure is based on Figure 1 of Moores and Goettmann \cite{Moores2006}.}
\end{center}
\end{figure}

Unlike the resonance of the bound electrons, the resonance of the free electrons in the nanoparticle can be tuned geometrically since the spring constant $K$ is related to the particle size.  The spring constant and resonance frequency both increase with decreasing particle radius.  Spectroscopy measurements have confirmed that the electric dipole resonance redshifts with increasing particle size \cite{Hurst2006, Liu2007_ColloidsSurfB}.  Effective Medium Theories \cite{Garnett1906, Bruggeman1935, Choy1999, Cai2010} relate the optical properties and volume fraction of metal nanoparticles to the macroscopic permittivity of the metamaterial.  Thus, adjusting the size and shape of subwavelength metal particles in a dielectric makes it possible to engineer a metamaterial with controlled $\varepsilon_r$ and index of refraction.

Achieving $\varepsilon_r < 0$ requires a narrow resonance.  This means that absorption must be small relative to scattering, or, equivalently, the scattering cross section $\sigma_{\mathrm{scatt}}$ of the nanoparticle must exceed its absorption cross section $\sigma_{\mathrm{abs}}$.  These parameters are different functions of the particle radius $R$ \cite{Hulst1981, vanDijk2006}:
\begin{equation}
\sigma_{\mathrm{scatt}} \propto R^6 \nonumber
\end{equation}
\noindent and
\begin{equation}
\sigma_{\mathrm{abs}} \propto R^3 \nonumber 
\end{equation}
\noindent for $R \ll \lambda$.  Thus, larger nanoparticles have sharper resonances.  For gold nanoparticles, the scattering cross section exceeds the absorption cross section when the particle diameter is larger than 80 nm \cite{Anker2008}.

\subsubsection{Manipulation of Permeability with Magnetic Dipole Resonance}

Since the electric permittivity of a metamaterial can be tuned with electric dipole resonances, it follows that the magnetic permeability can be tuned with magnetic dipole resonances.  The relative permeability $\mu_r$ is related to the magnetic field $\mathbf{H}$ and magnetization $\mathbf{M}$ in the same way that the relative permittivity $\varepsilon_r$ is related to the electric field $\mathbf{E}$ and polarization $\mathbf{P}$.  The relationship between $\mathbf{B}$, $\mathbf{H}$, $\mathbf{M}$, and $\mu$ \cite{Jackson1999, Cai2010},
\begin{equation} \label{eq:magnetization_constitutive}
\mathbf{B} = \mu_0 \left( \mathbf{H} + \mathbf{M} \right) = \mu_0 \mu_r \mathbf{H} = \mu \mathbf{H},
\end{equation}
\noindent parallels that between $\mathbf{D}$, $\mathbf{E}$, $\mathbf{P}$, and $\varepsilon$ (equation~\ref{eq:polarization_constitutive}).  However, magnetic dipole resonances are more difficult to engineer at optical frequencies than electric dipole resonances because, above GHz frequencies, the magnetic permeability of all known naturally occurring materials is always close to the permeability of vacuum.  This is a consequence of the fact that the magnetic field in light interacts with atoms much more weakly than the electric field does due to the absence of magnetic charge \cite{Alu2009}.  This asymmetry between electric and magnetic phenomena %is associated with the small value of the fine structure constant, $\alpha \approx 1/137$ \cite{Cai2010} and 
even led the authors of a widely used electromagnetism textbook to suggest that magnetic susceptibility is meaningless at optical frequencies and above \cite{Landau1984}.

Pendry and colleagues \cite{Pendry1999} found a solution to this problem: the split-ring resonator.  A split-ring resonator is a planar subwavelength device consisting of concentric, incomplete loops of metal embedded in a dielectric.  Incident electromagnetic waves can couple strongly to circulating current modes that can be engineered through the geometry of the device.  The concentric rings of metal give the device inductance $L$, while the gaps in the metal and the close proximity of concentric rings provide capacitance $C$.  Thus, a split-ring resonator is essentially an LC circuit [Figure~\ref{fig:split-ring}(a)].  The resonant frequency of the LC circuit, $\omega = \left( LC \right)^{-1/2}$, can be tuned by changing the rings and gaps.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=12.0cm]{split-ring}
\caption{\label{fig:split-ring} A split-ring resonator uses external electromagnetic waves to establish a circulating current resonance, like that found in an LC circuit (a), in order to manipulate $\mu_r$ in a metamaterial. Examples include (b) the original design by Pendry \emph{et al.} \cite{Pendry1999}, (c) the first design for which negative refraction was observed \cite{Shelby2001}, and (d) the resonator used in the first realization of electromagnetic cloaking \cite{Schurig2006}.}
\end{center}
\end{figure}

Because its current circulates in the plane, a split-ring resonator produces a magnetic dipole pointing out of the plane.  Thus, light can drive the structure to resonate magnetically, rather than just electrically.  A metamaterial with tunable $\mu_r$ can be constructed from dense arrays of split-ring resonators.  Figure~\ref{fig:split-ring}(b) shows the original proposed split-ring resonator \cite{Pendry1999}, and Figure~\ref{fig:split-ring}(c)-(d) show designs from some of the first successful implementations of these devices \cite{Shelby2001, Schurig2006}.

With this theoretical framework in place, the basic requirements for fabricating metamaterials with exotic optical properties are clear.  A metamaterial can be made from a positive-index material containing a high density of subwavelength structures whose geometry affords them both electric and magnetic dipole resonances.  The electric resonances control $\varepsilon_r$ and the magnetic resonances control $\mu_r$.  The geometry and number density of the structures, along with the materials from which they are made, determine the effective permittivity and permeability.

\subsection{Conventional Fabrication Routes}

Since metamaterials require subwavelength components, the first ones were designed for macroscopic wavelengths so that they could be built using printed circuit board technology.  Shelby \emph{et al.} \cite{Shelby2001} built a metamaterial by printing two-dimensional arrays of copper split-ring resonators and wires on interlocking circuit-board material.  They showed that resonators on the millimeter scale, with the design shown in Figure~\ref{fig:split-ring}(c), led to a negative $\varepsilon_r$ and $\mu_r$ for wavelengths on the order of centimeters.  The measured index of refraction of their metamaterial was $n = -2.7 \pm 0.1$.  Other examples of metamaterials at microwave frequencies, made using similar schemes, soon followed \cite{Schurig2006}.

Making metamaterials for optical frequencies, however, is a nanotechnological challenge, because the resonators must be on the order of 100 nm or smaller.  The first candidate techniques for making such structures were top-down methods such as electron-beam lithography \cite{Shalaev2005, Liu2008} and focused ion beam milling \cite{Enkrich2005, Valentine2008}.  In both methods, a beam of charged particles is focused to a small spot on a sheet of material in order to carve a nanostructure.  Although these methods can be used to build split-ring resonators, they have significant drawbacks.  First, they are serial techniques in which one nanostructure is made at a time, and thus manufacturing bulk quantities of metamaterial by these methods is slow and expensive.  Second, they are limited in precision and resolution by the spot size, which is typically about 10 nm \cite{Cai2010}.  While this size is much smaller than the resolution limit of techniques like photolithography, it limits the gap sizes and conductor spacing in the resonator, making it impossible to use certain designs.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=12.0cm]{SRR_anisotropic}
\caption{\label{fig:SRR_anisotropic} Planar split-ring resonator design used for optical metamaterials \cite{Enkrich2005, Liu2008}. (a) Light propagation within the plane can excite a strong magnetic dipole response for one polarization, since the electric field can drive currents $\mathbf{j}$ in the arms of the resonator.  (b) At a different orientation, the magnetic response is much weaker. }
\end{center}
\end{figure}

Most problematically, these methods produce nanostructures that are inherently two-dimensional.  Split-ring resonators are designed to couple to an electromagnetic wave coming from a specific direction and having specific polarization.  If the resonator rotates with respect to the source, its electromagnetic response can differ dramatically, as illustrated in Figure~\ref{fig:SRR_anisotropic}.  Thus, two-dimensional arrays of split-ring resonators are inherently anisotropic metamaterials.  While it is possible to build interlocking arrays of millimeter-scale split-ring resonators \cite{Shelby2001} to partially compensate for the anisotropy of a single two-dimensional array, this is not a straightforward process with their nanoscale counterparts.  It is possible to stack the arrays to create a three-dimensional lattice of resonators \cite{Liu2008, Valentine2008} but such lattices are still anisotropic.

\section{Metafluids}

Many applications, such as perfect lensing \cite{Pendry2000}, require metamaterials with an isotropic response.  Such materials are best fashioned from subwavelength elements whose resonances are themselves isotropic.  Isotropic resonances require nanostructures of a geometry different from that of planar split-ring resonators.

If isotropic subwavelength resonators could be made, their orientations relative to one another and relative to the incident light would have no effect on the optical properties of the metamaterial.  Even if the structures were dispersed randomly in a fluid, where they could translationally and rotationally diffuse, they would still respond to incident light.  Thus, a concentrated colloidal suspension of isotropic electromagnetic resonators would be a fluid metamaterial, also known as a metafluid \cite{Urzhumov2007}.

A metafluid could enable additional technologies that are impossible with solid metamaterials.  For instance, they could be used in microfluidic waveguides whose modes, unlike those of conventional waveguides, could be dynamically adjusted by tuning the fluid composition and flow rates \cite{Wolfe2004}.  Such waveguides have been demonstrated with aqueous solutions, but the availability of metafluids would expand the range of refractive indices available for these systems and would make different mode profiles possible.  Fluidic lenses are another potential application.  A lens made from a liquid has a smooth, spherical surface, and its radius of curvature and focal length can be dynamically adjusted using pressure \cite{Moran2006}.  As with microfluidic waveguides, metafluids could be used to increase the range of possible refractive indices for a fluidic lens.  Moreover, they may make it possible to dynamically tune both the curvature and the refractive index.

\subsection{Theory}

Urzhumov \emph{et al.} \cite{Urzhumov2007} used plasmon hybridization theory \cite{Prodan2003, Wang2007_ACR}, which relates the electromagnetic resonances of various structures to their symmetries, to show that some regular polyhedral clusters of metal nanoparticles exhibit isotropic electric and magnetic dipole resonances \cite{Urzhumov2007}.  Examples include Platonic solids such as the tetrahedron, octahedron, and icosahedron.

To understand this prediction, one can consider the tetrahedron, whose symmetry corresponds to the $T_d$ point group.  Irreducible representations of the $T_d$ group generate all possible arrangements of four electric dipoles situated at the vertices of a tetrahedron.  Because the representations corresponding to electric and magnetic dipoles are each three-fold degenerate, these modes are three-dimensionally isotropic.\footnote{The lowest-order electric multipole of the magnetic dipole resonance for the tetrahedron is an electric octupole ($2^3 = 8$) \cite{Urzhumov2007}.}

Another way to understand this prediction is to consider each face of the tetrahedron as a collection of three metal particles.  If there are gaps between these particles, they form a split ring, analogous to an LC circuit or a conventional split-ring resonator [Figure~\ref{fig:split-ring}].  Because there is a fourth particle in the cluster, one can think of the structure as consisting of four split-ring resonators with different orientations.  No matter how the structure is oriented with respect to incoming light, a magnetic dipole resonance can be excited in at least one face of the structure, or in at least three of the four particles, as illustrated in Figure~\ref{fig:metafluid}(a).  Thus, to make an optical metafluid, one needs only a dense suspension of tetrahedral clusters of metal nanospheres [Figure~\ref{fig:metafluid}(b)] \cite{Urzhumov2007}.  Negative index of refraction requires that the electric and magnetic dipole resonances overlap and these resonances are sensitive to the gaps between the particles.  The resonant frequencies can redshift by up to 10\% if the separation gaps decrease by just 1 nm \cite{Fan2010_Science}, so nanometer precision is required to ensure that all clusters have uniform resonances.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=9.8cm]{metafluid}
\caption{\label{fig:metafluid} Tetrahedral clusters of metal nanospheres are isotropic electromagnetic dipole resonators.  (a) Any set of three spheres is analogous to a split-ring resonator.  Incident light of an appropriate frequency will excite a magnetic dipole resonance in at least one face of the tetrahedron, regardless of its orientation. (b) A metafluid is a dense, disordered collection of these clusters. }
\end{center}
\end{figure}

Since the building blocks for metafluids are three-dimensional structures, they cannot be made using conventional fabrication routes, which yield two-dimensional structures.  A promising alternative is to make three-dimensional plasmonic nanoclusters ``from the bottom up,'' using self-assembly.  In this scheme, one controls the interactions between metal nanoparticles in a fluid such that specific clusters can assemble.  The clusters must be as symmetric and as uniform as possible so that their optical properties are isotropic and reproducible.

\subsection{Previous Experiments}

The simplest designs for optical metafluids call for symmetric clusters of metal nanoparticles \cite{Urzhumov2007, Alu2009}.  There are many methods for making spherical particles of noble metals.  A number of groups have developed mechanisms for making clusters on this length scale.

\subsubsection{Metal Nanoparticles and Nanoshells}

The study of gold nanoparticles has a rich history going back at least as far as Michael Faraday \cite{Faraday1857}.  Modern techniques for the preparation of commercial gold nanoparticles are based on citrate reduction of HAuCl$_4$ in water \cite{Turkevich1951, Frens1973}.  This method yields particles that are approximately spherical and monodisperse.  However, electron microscope images show that the particles have irregular, polyhedral morphologies, as shown in Figure~\ref{fig:gold_particles}(a).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=10.0cm]{gold_particles}
\caption{\label{fig:gold_particles} (a) Gold particles prepared by citrate reduction of HAuCl$_4$ are faceted, polyhedral crystals.  This sample is from a batch purchased from Nanopartz, Inc.  (b) Gold nanoshells with silica cores are more spherical and monodisperse but can have pinholes and the gold may dewet from the silica.  Scale bars are 200 nm.}
\end{center}
\end{figure}

Gold nanoshells are preferred for many applications because they are more spherical.  Oldenburg \emph{et al.} \cite{Oldenburg1998} developed a procedure for synthesizing these particles, which consist of a gold film grown on a spherical silica core.  The thickness of the gold layer can be controlled during synthesis, so that the ratio of core to shell diameter can be used to tune the electric dipole resonance \cite{Oldenburg1998, Prodan2003}.  However, pinholes can develop in the gold layer [Figure~\ref{fig:gold_particles}(b)] and this can lead to dewetting or other changes in nanoshell morphology \cite{Radloff2001}.  Smooth, spherical metal particles with superior thermal stability are needed for further progress in metafluid engineering.

\subsubsection{Cluster Self-Assembly}

Many groups have investigated colloidal self-assembly as a means of preparing small clusters.  One method involves attaching particles to the surfaces of emulsion droplets and then evaporating fluid from the droplets.  Capillary forces bring the particles togther in small clusters with well-defined, symmetric configurations such as tetrahedra and octahedra \cite{Manoharan2003, Wagner2008}.  However, the method generates a mixture of clusters, and the yield of any single type is low, unless the clusters are separated by size by using a density gradient \cite{Hinton1978}.  It is also possible to control the geometry of nanoparticle clusters by assembling ``patchy particles'' \cite{Sacanna2010, Wang2012}, but it is not clear how to make metal nanoparticles with well-defined nonspherical geometries.

It is possible to assemble small numbers of metal nanoparticle clusters.  Drying a colloidal suspension of gold nanoshells on a dielectric substrate results in the self-assembly of two-dimensional clusters.  Triangular clusters assembled this way have been shown to exhibit a magnetic dipole resonance \cite{Fan2010_Science}.  It is also possible to assemble plasmonic clusters in templated patterns on substrates \cite{Xia2003, Fan2012_NL_Plasmonic}, but because they must assemble on a surface, the yield is limited compared to that of self-assembly in bulk.

New routes to self-assembly of three-dimensional plasmonic nanoclusters have emerged recently.  For instance, Urban \emph{et al.} \cite{Urban2013} encapsulated small numbers of metal nanoparticles in polymer spheres.  They found evidence of isotropic response in a gold nanoparticle tetrahedron.  However, this method offers little control over interparticle separations, and the yield of any particular cluster geometry is limited.  An alternative is to assemble large clusters from metal nanoparticles on the surface of a much larger dielectric particle \cite{Muhlig2011, Sheikholeslami2013}.  This method is claimed to yield materials with a magnetic response at optical frequencies, but one cannot control the number of nanoparticles per structure or the gaps between the nanoparticles.  One can mitigate this problem by using icosahedral viruses as the dielectric particles and directing gold nanoparticles to bind at their vertices \cite{Fontana2014}.  A high yield of icosahedral clusters is possible in principle, but the virus must be genetically engineered to exhibit thiol groups at the desired sites on the capsid, and even then, many clusters are incomplete icosahedra.  Moreover, all of these approaches have used irregular nanoparticles or unstable nanoshells.

\section{Overview}

Our goal is develop a material exhibiting isotropic negative refraction at optical frequencies.  Metafluids are the most promising route to such properties, since they contain isotropic electromagnetic resonators, and the arrangement and orientation of the resonators need not be controlled.  Since a metafluid can be made from three-dimensional clusters of plasmonic nanospheres, colloidal self-assembly appears to be the most promising way, if not the only way, to make one.

In this dissertation, I present our progress towards this goal.  First, I show in Chapter~\ref{ch:gold_spheres} that spherical, smooth, monodisperse, and stable gold nanoparticles can be made by combining nanocrystal growth with slow chemical etching.  My collaborators and I demonstrate that the plasmonic behavior of these individual nanospheres is reproducible and uniform, unlike that of conventional gold nanoparticles.  In Chapter~\ref{ch:random_parking}, I show that self-assembly of tetramers in high yield is experimentally achievable.  We accomplish this by mixing spherical particles with larger spheres that can bind randomly and irreversibly to the smaller ones.  In Chapter~\ref{ch:FDTD_sim}, I discuss simulations of magnetic resonances in tetrahedral gold clusters and examine how key geometric parameters affect the resonance.  In Chapter~\ref{ch:gold_clusters}, I apply our nonequilibrium self-assembly method to our spherical gold particles and present methods for the self-assembly and purification of tetramers composed of gold nanospheres.  Finally, in Chapter~\ref{ch:conclusion}, I propose some experiments to build on these results and to ultimately demonstrate high-yield self-assembly of isotropic, reproducible electromagnetic resonators for optical frequencies.
