# Introduction

The most famous and probably oldest example of a plasmonic material is the Lycurgus Cup (figure 1), which dates back to the $4^{th}$ century. This glass cup contains nanometer sized silver and gold clusters, in which localized plasmon resonances can be excited with visible light. When illuminated from the outside, the light reflected from the outer surface produces a green/yellow color but when it is illuminated from the inside the transmitted light produces a red color. This effect has been used for many centuries in the fabrication of stained glass windows in historical buildings but it was only in the beginning of the last century that people realized that these effects could be attributed to plasmonic resonances.

Figure 1: Roman nanotechnology: the Lycurgus gup shows up green/yellow (left) when light reflects from the outer surface and red (right) when it is illuminated from the inside

# Different types of plasmons

The first experimental observation of collective electron oscillations already dates back more than one century, when Wood reported on Wood’s anomalies: intensity drops in optical reflection spectra of metal gratings. It was only in the late 1960’s that this effect could be attributed to the excitation of surface plasmons, the collective oscillations of free electrons in the metal. The term “plasmon” was introduced shortly before by Pines. A plasmon can be defined as a quantum for the collective oscillation of free electrons, usually at the interface between (noble) metals and dielectrics. The term plasmon refers to the plasma-like behavior of the free electrons in a metal under the influence of electromagnetic radiation. Nowadays, due to ever improving nanofabrication methods, the field of plasmonics is more active than ever before.

Although the term plasmon covers any type of plasma-like oscillation of free electrons, we can distinguish between a few different types. The different types of plasmons that can be excited in metallic objects depend on their dimensions. In large three-dimensional metal structures volume plasmons can exist in the bulk of the metal. At the interface between metals and dielectrics propagating surface plasmon polaritons (SPPs) can be excited. Low-dimensional metal structures such as nanoparticles support a wide variety of localized surface plasmon resonances (LSPRs).

## Volume plasmons

Volume plasmons are the most fundamental and intrinsic type of plasmon resonance that can be supported by a metal. These resonances occur at the plasma frequency of metals $\omega_p$, which are transparent to radiation with higher frequencies and non-transparent to radiation with lower frequencies. The plasma frequency primarily depends on the electron density:

(1) $\omega_p^2 = \frac{N e^2}{m \epsilon_0}$

in which $N$ is the conduction electron density, $e$ the electron charge, $m$ the effective optical mass of the electron and $\epsilon_0$ the permittivity of free space. These volume plasmons are longitudinal modes which cannot be excited by an incident photon, but only by particle impact

## Surface Plasmon Polaritons (SPPs)

At the interface between a metal and a dielectric, propagating solutions of Maxwell’s equations exist, which are so-called surface plasmon polaritons (SPPs). These collective oscillations of the free electrons in the metal make up dispersive longitudinal waves that propagate along the interface and decay exponentially into both media (figure 2 (a and b)) with typical decay lengths of a few tens of nanometers in the metal and up to several hundreds of nanometers in the dielectric (depending on the resonant wavelength). The propagating solutions travel with an in-plane wave vector $k_{spp}$ which defines the dispersion relation in figure 2(c).

(2) $k_{spp} (\omega) = \frac{\omega}{c} \sqrt{\frac{\epsilon_d \epsilon_m}{\epsilon_d + \epsilon_m}}$

in which $\epsilon_m$ and $\epsilon_d$ are the permittivity of the metal and dielectric medium. Clearly the dispersion relation lies to the right of the light line in free space $\omega / c$, which implies that this SPP mode cannot directly be excited by an incident photon. Only when the dispersion relation of the incident wave and the propagating SPP mode coincide, an incident photon can excite the SPP mode. To do so, several coupling mechanisms can be applied, as described later.

Figure 2: Propagating surface plasmon polaritons (SPPs). (a) Charge density oscillations at the metal/dielectric interface with the associated EM fields. (b) The different decay lengths of the evanescent field component in the dielectric and metal, depending on the skin depth. (c) The dispersion relation of an SPP, illustrating its subwavelength confinement and the momentum mismatch that has to be overcome in order to excite them.

## Localized Surface Plasmon Resonances

Localized surface plasmon resonances (LSPRs) are the non-propagating counterpart of SPPs, which can be excited in nanometer-sized subwavelength metallic particles. The free electron cloud of the nanostructure can be resonantly excited by EM fields due to enhanced polarizabilities of the particles at certain frequencies. These enhanced polarizabilities give rise to strongly enhanced near fields close to the metal surface, which are often referred to as hot spots.

Figure 3: Illustration of the dipole polarizability of a spherical metal nanoparticle under the influence of a plane wave

The simplest structure in which LSPRs can be excited, and for which analytical solutions of Maxwell’s equations can be obtained is a metallic sphere which was already treated by Mie in 1908. This example is very instructional in order to understand more complex structures, so therefore we will describe the case of the dipolar plasmon resonance in a metal sphere with a radius $a$ much smaller than the wavelength of the incident field ($a \ll \lambda$). This allows us to treat this case in the quasistatic approximation: the electric field over the nanoparticle can be assumed to be constant, while the wavelength dependence of the permittivity of the metal $\epsilon_m$ and the surrounding medium $\epsilon_s$ is taken into account. To solve this problem, we have to look for solutions of the Laplace equation

(3) $\qquad \nabla^2 \Phi = 0$

from which the electric field can be calculated by

(4) ${\bf E} = - \nabla \Phi$

The boundary conditions at the interface between the sphere and the surroundings require that both the tangential component of the electric field and the normal component of the displacement are continuous. The obtained solution for the electric field consists of a superposition of the applied field ${\bf E_0}$ and an ideal electric dipole located at the center of the sphere with dipole moment

(5) $p = \epsilon_0 \epsilon_s \alpha E_0$

where $\alpha$ is the complex polarizability of the metal particle:

(6) $\alpha = 4 \pi a^3 \frac{\epsilon_m - \epsilon_s}{\epsilon_m + 2 \epsilon_s}$

From the shape of the denominator, one can see immediately that a resonant behavior in the polarizability is expected when $|\epsilon_m + 2 \epsilon_s|$ reaches a minimum. For (noble) metals at optical frequencies, the real part of the permittivity is negative, from which the Frölich resonance condition is obtained:

(7) $\epsilon_{m}' (\lambda) = -2 \epsilon_s$

When this condition is satisfied, the dipolar LSPR mode in the nanoparticle will be resonantly excited. The damping of the plasmon resonance depends on the magnitude of the imaginary permittivity, which is reflected in the value of $\alpha$. For this simple approximation in the quasistatic limit, the resonance position is independent of the size of the metal sphere, which is not generally true. An other important property of plasmon resonances is reflected in the Frölich resonance condition: the dependence of the resonance position on the dielectric properties of the surroundings. When the dielectric constant $\epsilon_s$ of the surroundings (and thus the refractive index $n_s$) increases, the resonance position shows a red shift. This principle is often applied in biochemical LSPR based sensors. On top of that, at the plasmon resonance there will be a tremendous near-field enhancement of the incident wave, dipole radiation of the excited dipole and increased scattering and absorption in the nanoparticle.

For larger and more complex nanoparticles, the quasistatic limit is not valid any more. For example, in larger structures also higher order modes (quadrupole, octopole, hexadecapole…) can be excited, given that the appropriate conditions are satisfied. These modes typically show much smaller line widths and much smaller scattering cross sections due to a largely reduced (or zero for symmetric particles) net dipole moment. We’ll discuss the excitation and tunability of some of these modes in the next sections.

# Plasmon excitation mechanisms

It was already pointed out in the previous sections that certain conditions apply to the possibility to excite plasmonic modes with EM fields. The excitation mechanisms are quite diverse for SPPs and LSPRs and largely depend on the geometry and sizes of the plasmonic (nano-) structures and the polarization state of the incident wave.

## Surface Plasmon Polaritons

From equation 2 and figure 2 it is clear that the dispersion relation of SPPs lies to the right of the light line of the dielectric, which implies that direct coupling to an incident wave is not possible. In order to excite SPPs on a metal film, the dispersion relation of the incident photon and the SPP should coincide, such that the quasi momentum $\hbar k$ and the energy $\hbar \omega$ are conserved, a condition which can only be fulfilled for incident waves with P-polarization. The matching of the dispersion relations can be achieved in different ways. In order to bridge the momentum gap 3 methods are used commonly: prism coupling by evanescent waves, diffraction grating coupling and coupling by surface corrugations (figure 4). Next to those, it is also possible to use focused high-energy beams or near-field coupling using a scanning near-field optical microscope (SNOM) tip.

In order to be able to excite propagating SPPs it is necessary to obtain phase matching between the in-plane wave vector of the incident wave along the interface and the wave vector of the propagating SPP (equation 8). We consider an interface between air and a metal surface, for which the in-plane component along the interface is determined by the incident angle $\theta$ with respect to the surface normal:

(8) $k_x = k \sin{\theta}$

If we consider a prism coupled to the system described above, the two most commonly used excitation mechanisms are the Kretschmann (figure 4(a)) and Otto (figure 4(b)) configurations. By directing the incident wave through a prism with a higher dielectric constant $\epsilon_p$, the wave vector along the interface is modified to

(9) $k_x = k \sqrt{\epsilon_p} \sin{\theta}$

which allows to couple to plasmons at the metal/air interface. In this way, SPPs can be excited with wave vectors in between the free space light line and the prisms dispersion relation (figure 4(e)). Note that the direct coupling to plasmons at the prism/metal interface can not be achieved in this way. The coupling is based on total internal reflection (TIR) at the prism interface, which results in tunneling of the evanescent fields, that can couple to propagating modes at the metal/air interface. The Kretschmann configuration is used most often, and in that case a thin metal layer is deposited on top of the prism, and the beam is incident with an angle larger than the critical angle, such that TIR occurs at the prism/metal interface. The evanescent fields of the TIR-wave tunnel through the metal layer and excite propagating SPPs at the metal/air interface. The Otto configuration is fairly similar, but in this case there is a small air gap between the metal layer and the prism, and the evanescent field of the TIR-wave tunnels through this air gap in order to excite an SPP mode at the air/metal interface. In practical applications, the reflected signals are monitored and show a minimum in the signal at the angle/wavelength where SPP excitation occurs.

Figure 4: Different coupling mechanisms for SPPs. (a) Kretschmann configuration. (b) Otto configuration. (c) Grating excitation. (d) Scattering by surface roughness. (e) Dispersion relations for excitation of plasmons at the air/metal interface by means of prism coupling.

If a grating structure is present on (a part of) the metal film (figure 4(c)), diffraction effects can be used to couple efficiently to propagating SPPs. In this case, the period of the grating $a$ determines the magnitude of the reciprocal vector of the grating $g$:

(10) $g = \frac{2 \pi}{a}$

Wave matching between the incident wave and the excited SPPs can than be achieved when the following condition is fulfilled

(11) $k_{SPP} = k \sin{\theta} \pm \nu g$

in which $\nu$ is an integer number. This general formula can be applied for any type of diffraction grating structure which is defined in or on top of the metal film. These can include many different features such as stripes, slits, dots, holes etc. In practical applications, the excitation of SPPs results in a (narrow) minimum in the intensity of the reflected light. Grating structures can be applied for the reverse process as well, in order to couple out SPPs into free space again.

Figure 4(d) illustrates the excitation of SPPs by means of surface corrugations, which can occur unintentionally due to undesired effects in sample fabrication, such as surface roughness and particle contamination. Alternatively, surface corrugations can be used as a controlled means of SPP excitation, for example by defining a structure which allows for excitation of LSPRs, which can also couple to propagating SPP waves on the metal film.

## Localized Surface Plasmon Resonances

In the previous section, we already showed how the dipolar plasmon resonance can be excited in small nanoparticles based on the quasistatic approximation. Although this model is very instructional for plasmon resonances in general, it doesn’t tell the full story. If we consider an asymmetric nanoparticle such as a nanodisk with radius a, and we assume that $a = \lambda / 4$, then the approximation of a uniform field over the volume of the particle is no longer generally valid. Therefore we consider two different polarization states of EM-waves propagating along two of the symmetry axes of the nanoparticle. For a plane wave that propagates perpendicular to the plane of the disk (figure 5(a)) it is only possible to excite a dipole resonance of which the dipole moment is aligned with the polarization vector of the incident wave. For a plane wave propagating in the plane of the disk with the polarization in the plane of the disk we can couple to the quadrupole (figure 5(b)) and the dipole mode of the disk.

Figure 5: Illustration of the effect of retardation for different polarization states in a disk particle with radius $a = \lambda / 4$. (a) A plane wave propagating perpendicular to the plane of the disk can only excite a dipole resonance. (b) A plane wave propagating in the plane of the disk with the polarization direction in the plane of the disk can couple to the quadrupole mode in the disk.

# Tunability of plasmon resonances

Plasmon resonances can be tuned in many different ways, depending on the intended application. As the excitation mechanisms for the different sorts of plasmon resonances differ, we also treat them separately in the next sections.

## Surface Plasmon Polaritons

SPP modes are propagating modes at the interface between a metal and a dielectric, and therefore they can be considered as modes propagating on a waveguide (in this case the metal/dielectric interface). Depending on the waveguide design, we can distinguish between long-range SPP waveguides (propagation distances of a few tens of microns in the VIS up to hundreds of microns in the NIR) and short-range SPP waveguides (propagation distances limited to tens of microns). The former typically show very low confinement and small damping, while the latter show very high confinement but strong damping. The properties of propagating SPPs depend strongly on the waveguide design and are determined by the dispersion relations of the materials used and their thicknesses. The propagation behavior can be characterized by the mode index $n_{eff}$ of the waveguide, which defines the degree of confinement and the wave vector of the SPP:

(12) $k_{spp} (\omega) = \frac{\omega}{c} \sqrt{\frac{\epsilon_d \epsilon_m}{\epsilon_d + \epsilon_m}} = \frac{\omega}{c} n_{eff}$

Clearly, the propagation behavior and mode index depend strongly on the permittivity of both the dielectric and the metal, and the propagation of SPPs for the overall system can be described using the effective mode index $n_{eff}$. As most dielectric materials show little or no damping, the propagation behavior is mainly dominated by the properties of the metal layer(s). Typical examples of long-range SPP waveguides are Insulator-Metal-Insulator waveguides in which the metal layer is thinner than the skin depth, resulting in very low confined modes with very long propagation distances. The aforementioned example of the Kretschmann configuration for SPP coupling also falls in the same class of waveguided modes. Typical examples of short-range SPP waveguides are Metal-Insulator-Metal (MIM) waveguides in which the metal thickness is larger than the skin depth, resulting in highly confined modes with shorter propagation distances.

## Localized Surface Plasmon Resonances

Localized surface plasmon resonances are highly tunable and allow to confine light down to deep sub-wavelength dimensions. With increasing dimensions of the nanoparticle, most plasmon resonances show a red shift. The shape of nanoparticles is an important design parameter which allows to favor the excitation of specific plasmonic modes. Plasmon resonances are also highly sensitive to the material properties of the metal and the surrounding dielectric medium. By playing with nanoparticle size, shape, material and the surrounding medium, localized surface plasmon resonances can be tuned from the ultraviolet part of the spectrum up to the NIR as illustrated in figure 6.

Figure 6: The tunability of LSPRs by playing with particle material, size, and shape

## Interactions between LSPRs

Next to the structural properties of the nanoparticles themselves, the LSPR wavelength also depends on the interactions with plasmon resonances in neighboring particles. These interactions can be devided into two sorts: near field coupling for particles which are separated by a distance smaller than the wavelength of the incident light and far field coupling for particles that are separated by distances larger than the wavelength of the incident light. The spectral shifts that are introduced due to these coupling phenomena depend mainly on phase coherence between the local mode in one particle and the scattered fields due to modes in the neighboring particles. In case of near-field coupling, the particles can be considered as simple point dipoles and longitudinal coupling results in a red-shifted resonance, while transverse coupling results in a blue shifted resonance. For longitudinal coupling, the scattered fields of the neighboring particles oppose the local resonance and decrease its energy, while in the transversal case, the scattered fields of the neighboring particles enhance the local resonance and increase its energy.

Figure 7: Near-field coupling of electric dipoles in linear arrays of nanoparticles with transversal (top) or longitudinal (bottom) coupling

For nanoparticles that are spaced further apart, also far-field coupling should be taken into account. In that case, the length scale on which the interactions contribute ranges up to several microns, and with increasing inter-particle spacing the plasmon resonance position and its line width show an oscillating behavior for both polarization states.

# Phase and amplitude of plasmon resonances

If we consider a simple mechanical system such as a pendulum, it will oscillate at its eigenfrequency when it is moved from its equilibrium position and released to move freely. If we consider the same pendulum and drive it with a harmonic force of which we scan the frequency, we observe that at the eigenfrequency of the pendulum the amplitude of the oscillation will be maximized. Moreover, if we measure the phase difference between the applied force and the oscillating pendulum, we observe a pronounced phase difference (approaching $180^{o}$) between in- and out-of-phase oscillation around the eigenfrequency of the system. The eigenfrequency of the system depends on the geometrical properties of the pendulum, similar to how the properties of (propagating or localized) plasmons depend on the size, shape and the material in which they are excited. In case of plasmon resonances a similar effect can be observed, where the driving force is the oscillating electric field of the incident EM wave, which triggers the oscillation of the free electrons on the metal surface. If the wavelength of the incident wave is scanned across the plasmon resonance wavelength, the electron cloud also makes the transition from in- to out-of-phase oscillation with respect to the incident wave, which can be experimentally observed.

## Surface Plasmon Polaritons

In the field of SPP-based sensing, substantial efforts have been made in order to measure the phase of propagating plasmons. The basic approach is to use a combination of P- and S-polarized waves for the excitation, and to measure the phase difference between the reflected waves using lock-in measurements. The P-polarized wave can couple to a propagating SPP mode very efficiently, while the S-polarized wave is reflected without picking up any substantial phase change. In that way, the transition between in and out-of-phase oscillation of the free electron cloud could be experimentally observed. The phase difference shows a much smaller spectral/angular footprint compared to the intensity based signals, as illustrated in figure 8.

Figure 8: Comparison of amplitude and phase signals for an SPR experiment

## Localized Surface Plasmon Resonances

For localized surface plasmon resonances, similar observations were made in recent years, inspired by earlier work in SPR-based sensing. For periodic arrays of gold nanoparticles, the amplitude and phase based reflection signals were recorded near the LSPR modes, which show a pronounced phase difference over the center wavelength of the plasmon resonance.

## Phase interference between plasmon resonances

Next to the phase behavior of individual plasmon resonances, it is also interesting to look at the interference between different localized modes which show spectral overlap. In that way it is possible to obtain coupling between bright dipolar modes and dark higher order modes, which allows to tune plasmon resonances at a higher level. Dark higher order modes typically have large quality factors, but are complex to excite, while bright dipolar modes have small quality factors and are easy to excite. When different nanoparticles are brought close together, it is possible to achieve interference between the plasmon modes in the individual cavities. In that way, by appropriate design the bright modes can be used to excite the dark modes, which gives rise to very interesting phenomena such as Fano-interference and sub-radiance, which allows to tune the plasmon line shapes.