# Plasmonics | Enhancement

## Enhancing the Optical Field

Surface plasmons are coupled oscillations that arise from the interaction between light and the conduction electrons in a metal or semiconductor. Since the spatial variation of the charge density along the surface of a conductor can occur on a scale vastly smaller than the wavelength of light, a surface plasmon can effectively squeeze light into tiny, sub-wavelength volumes. Within these volumes, the optical fields can be strongly enhanced--well beyond that of the incident wave used to create the excitation--effectively magnifying the light-matter interaction. By leveraging surface enhancement, we can hope to increase the functionality and performance of optical materials.

In what follows, we provide an introduction to surface plasmons and examine the field enhancement for several different structures. In our group, we have made a concerted effort to study the enhancement associated with nanoparticles coupled to a planar metal surface, since this system provides significant fabrication advantages. We consider these film-coupled nanoparticles in the subsequent section.

## The Surface Plasmon

The surface plasmon is an excitation that travels along the interface between a conducting material and air. Bound to the surface of the conductor, the surface plasmon is an example of a surface mode, which does not radiate into space. Just as waves traveling in space are characterized by their wavelength and frequency, we can likewise characterize a surface plasmon by its spatial wavelength along the surface and its frequency of excitation. For a surface plasmon, the wavelength represents the distance over which the fields undergo one oscillation, as well as the distance over which the charge density on the surface undergoes one oscillation. The relationship between frequency and spatial wave number is the dispersion relation, which can be found as

${k_x} = {k_0}\sqrt {\frac{{{\varepsilon _0}{\varepsilon _m}\left( \omega \right)}}{{{\varepsilon _m}\left( \omega \right) + {\varepsilon _0}}}}$

In this equation, ${k_0}$ is the free space wave number (${k_0} = \omega / c$), ${\varepsilon_0}$ is the permittivity of free space, and $\varepsilon_m$ is the frequency-dependent permittivity of the metal or conductor. If $x$ is taken as the direction along the interface, then the wave vector $k_x$ lies in the $x$ direction. For a surface mode, $k_x$ must always be larger than $k_0$, so that the plasmon wavelength is always smaller than the free space wavelength. A plot of the surface plasmon dispersion equation, shown below, reveals solutions for values of $\omega < \omega_p / \sqrt{2}=\omega_{sp}$. For low frequencies, the optical fields of surface plasmons vary spatially on a scale similar to the free space wavelength and are delocalized; that is, the fields are not tightly bound to the surface. As the frequency approaches $\omega_p / \sqrt{2}$, the surface plasmon wavelength becomes smaller. In this regime, the surface plasmon is characterized by fields that are tightly bound to the metal surface, decaying rapidly into the free space region.

Near optical wavelengths, the dielectric function of many conductors (such as silver or gold) has the form

$\varepsilon \left( \omega \right) = 1 - \frac{{\omega _p^2}}{{{\omega ^2}}}$

In this equation material absorption has been ignored, and the values are always real. Below the plasma frequency $\omega_p$, the values of the dielectric function are negative. From the surface plasmon dispersion relation, it can be seen that solutions exist for $\varepsilon \left( \omega \right) < -1$. The negative dielectric function is what leads to the potential for surface modes, and the reason that metals and other conductors are often referred to as plasmonic materials.

Using the simple form of the dielectric function above, it is possible to visualize the fields of a surface plasmon. We show two possible surface plasmons, one at the frequency $\omega=\omega_{sp} / 1.2$ and the other at the frequency $\omega = \omega_{sp} / 1.05$. At the former frequency, the fields extend into the air region a bit more, and the surface plasmon wavelength is closer to the free space wavelength. Note the presence of an oscillating surface charges, shown by the red and blue regions, that serve to bind the optical fields to the metal surface. At the latter frequency, which is closer to the surface plasmon frequency $\omega_{sp}$, the fields are more tightly bound to the surface, with the surface plasmon wavelength significantly smaller than the free space wavelength. For comparison, the scale of the free space wavelength is indicated by the scale bar in the figures.

The streamline plots above provide an indication of the optical field structure and the fluctuating surface charge density associated with the surface plasmon. Another useful way to visualize the surface plasmon is to examine the field density, as shown in the plot below. In this plot, the brighter colors indicate stronger optical fields, and it can be seen that the fields decay exponentially on either side of the surface.

Because the surface plasmon does not couple to radiative modes (plane waves, for example), it does not make too much sense to talk about field enhancement with this structure. However, different geometries of metals, such as nanostructures, support surface plasmons while simultaneously coupling to the radiative field. When the optical fields present on such structures are induced by an incident wave, one can describe the field enhancement as the ratio of the local electric field to that of the incident electric field. This field enhancement effect has profound implications for many optical phenomena.

Plasmon Resonant Nanoparticles - When light interacts with a particle having a dimension much smaller than the wavelength of the light, the particle is polarized, with an induced charge distribution appearing over the particle surface. The appearance of positive and negative charges on the particle surface implies--just as it did for the surface plasmon described above--that a distribution of optical fields is present, localized around the particle surface. Since the scale of the particle and the induced charge density variation can be much smaller than the wavelength, so also can be the local field distribution that is bound to the surface charges. Since the wavelength of light is on the order of hundreds of nanometers (nm), a small particle has dimensions on the order of tens of nanometers and is thus considered nanoscale. For this reason, plasmonic nanoparticles and nanostructures are typically of interest.

A surface plasmon that exists on a closed surface, such as a sphere or other particle, does not possess a continuous dispersion as does the surface plasmon on an infinite planar interface. Instead, the plasmon mode structure changes into a discrete set of resonances terms surface plasmon resonances (SPRs). For a sphere, these modes have field patterns that correspond to the spherical harmonics, as would be expected from symmetry. In fact, the plasmonic nanosphere is one of the most studied plasmonic structures, and one of the easiest to visualize.

While the charge distribution and fields associated with a plasmon resonant sphere excited by a plane wave can be found exactly using either Mie theory or direct numerical simulations, the simpler solution of a dielectric sphere excited by a constant electrostatic field provides a very reasonable approximation to the exact solution for spheres with radius much smaller than the wavelength of the incident light. In this approximation, the effective induced dipole moment of the sphere can be found as

${\bf{p}} = 4\pi {\varepsilon _0}\left( {\frac{{{\varepsilon _m} - {\varepsilon _0}}}{{{\varepsilon _m} + 2{\varepsilon _0}}}} \right){a^3}{{\bf{E}}_0}$

From this equation, we can see that the polarization becomes very large (infinite, in fact) when the dielectric function of the metal (or conductor) reaches the value $\varepsilon_m \left(\omega_{spr}\right) = -2 \varepsilon_0$, which defines the surface plasmon resonance frequency $\omega_{spr}$. The vanishing of the denominator indicates that a resonance occurs. In reality, the polarization is not infinite, but is limited by material losses as well as radiation damping--mechanisms that are included in the more advanced treatments or numerical simulations. Nevertheless, the electrostatic solution provides a good picture of the field patterns, as shown in the figures below, which show the x-component (left) and y-component (right) of the electric field.

The field plots illustrate the nature of the field enhancement, which peaks on the surface of the sphere. For this calculation, the electric field is directed along the y-axis, so that the polarization of the sphere also lies along the y-axis. A positive charge density is induced on the upper surface of the sphere, giving rise the emanating field lines, while a negative charge density is induced on the lower surface of the sphere, where the field lines terminate. Because this is an electrostatic calculation, the effective wavelength of the incident field is infinite, such that the size of the sphere plays no important role in this calculation. In practice, the electrostatic approximation can be applied with just a few modifications to nanoparticles on the order of about 50 nm in diameter or smaller at optical wavelengths (above roughly 400 nm).

Because we are interested in the enhancement effect, it is reasonable to ask just how large the field enhancement can be for a sphere. In the electrostatic limit, a simple expression for the maximum field enhancement can be found:

$\frac{{\left| {{{\bf{E}}_{sphere}}} \right|}}{{\left| {{{\bf{E}}_0}} \right|}} = 1 + {\left| \frac{{{\varepsilon _m} - {\varepsilon _0}}}{{{\varepsilon _m} + 2{\varepsilon _0}}}\right|}$

The maximum field enhancement occurs when the real part of the denominator vanishes. At resonance, the losses dominate the enhancement, introduced as the imaginary part of the dielectric function. Using the dielectric function of metals such as gold, silver or aluminum, the maximum enhancement that can be expected is 6-10--a relatively modest field enhancement.

## Increasing Enhancement: Coupled Nanospheres

The enhancement of a nanosphere is limited by the nature of the charge distribution that can exist on its surface. Since dipolar excitation leads only to a fairly smooth variation of this charge density, the fields that are bound to the surface by the nanoparticle are relatively modest. Still, even with what we call here a modest enhancement, the plasmonic nanosphere is possibly the strongest scattering object known in optics; metal nanospheres are routinely used in microscopy for a variety of purposes. But, we can still ask if it is possible to enhance the fields even more with other types of structures.

It should be noted that the dipole resonance of the sphere is not the only surface plasmon resonance that exists. There are an infinite set of higher order modes, each mode having a more rapid variation of the charge density over the sphere surface. However, these modes occur at different frequencies, and do not inherently couple to an incident plane wave (or electrostatic field, as in the approximation above). Since these higher order modes do not radiate, even if they could be excited by a source, the result would be that all of the energy would just be lost to resistive currents in the nanosphere.

One way to unlock the higher order modes is to break the symmetry of the sphere geometry. There are many configurations that achieve this goal, but one that can be realized very conveniently is that of two coupled nanospheres. When two nanospheres are placed close together--separated by a gap that is on the nanometer scale--all modes of the individual spheres couple together resulting in a tightly localized region between the spheres where the charge density can be extremely large.

As with nanospheres, the properties of coupled nanospheres can be obtained through numerical simulations, as well as a variety of mathematical techniques involving expansions of special functions. However, within the spirit of discussion here, we find it convenient to consider the case of interacting cylinders, for which remarkably simple expressions have been found for the resonance frequencies and field distributions. These expressions were derived by Alexandre Aubry and colleagues using the powerful technique of transformation optics.

The surface plasmon resonance for an isolated cylinder is given by $\omega_{spr}=\omega_p/\sqrt{2}$. When two cylinders are brought close together, all of their modes begin to interact and the surface plasmon resonances couple, shifting the resonance frequencies and changing the field characteristics. The geometry consists of two plasmonic cylinders, each with diameter $d$, and separated by a gap of size $g$. The resonance frequencies for interacting cylinders can be found by solving the following equation:

${\left( {\sqrt \rho + \sqrt {1 + \rho } } \right)^4} = \frac{{{\varepsilon _p}\left( \omega \right) - {\varepsilon _0}}}{{{\varepsilon _p}\left( \omega \right) + {\varepsilon _0}}}$

In this equation, $\rho=g/{2d}$. We can use this equation to make a plot of the resonance frequency shift as a function of the gap distance, called an approach curve. The plot, shown below for several different nanocylinder diameters, shows that the resonance frequency (or wavelength, as plotted) continues to shift as the cylinders approach. This frequency or wavelength shift suggests that the interaction continues to increase without bound.

The interaction effect can be observed by plotting the electric field that exists around and inside of the cylinders, as shown below. Only the component of the field along the axis of the cylinder pair is plotted. When the gap between the cylinders is large (roughly 10 nm or larger for 10 nm diameter cylinders), the cylinders weakly interact and the field patterns are close to what would be expected for isolated cylinders. While there is some very minor field enhancement in between the cylinders, the effect is not large. When the cylinders are brought close together, however, the fields become very large and localized, as indicated in the field plots for cylinders with gaps below 3 nm. Although the color map is scaled for each image, the field localization is evident for the closely coupled cylinders, with localization volumes of dimension less than 1 nm.

As promised above, two interacting spheres or cylinders--often called dimers--provide field enhancement levels far beyond what a single sphere can provide. In fact, as the nanoparticles get closer and closer together, below a nanometer, the ratio of the local field to the incident field can be in the thousands, as shown in the plot below. Of course, to achieve these extreme values of enhancement, the nanoparticles must be almost touching, on the scale of a few tenths of a nanometer. At such a scale, the underlying assumption of the plasmonic material breaks down; that is, quantum effects become important and end up dominating the optical properties of the system. Effects such as electron-electron interactions and quantum tunneling place the ultimate limit on the enhancement, although those limits are only now in the process of being understood.

# Useful References

Surface plasmons on smooth and rough surfaces and on gratings
H. Raether
Springer Tracts in Modern Physics, Vol. 111 (Springer, 1988)

Plasmonics: Fundamentals and Applications
S. A. Maier
(Springer Verlag, 2007)

Interaction between plasmonic nanoparticles revisited with transformation optics
A. Aubry, D. Y. Lei, S. A. Maier, J. B. Pendry
Physical Review Letters 105, 233901 (2010)