Plasmonic resonators: fundamental properties and applications

Resonators of surface plasmons are discussed in this review. Any material supporting the excitation of surface plasmons, either by light, or by electron beams can be used for designing a resonator. Despite the number of materials supporting surface plasmons being restricted to a small number, noble metals, some two-dimensional materials as graphene, transition metal oxides, and several highly doped semiconductors have been used in plasmonic applications. Isolated and coupled metal nanoparticles, arrays of particles on substrates, nanoparticle suspensions, alternated metal and dielectric thin films, dielectric cavities in surfaces of noble metals and sheets and stripes of two-dimensional materials are typical examples of systems used for excitation of plasmonic resonances. These resonances have been used in a fast increasing number of applications: transformation of light beam properties, selective optical induced heating, sensing of chemical or biological compounds, enhancement of non-linear optical excitation in nanomaterials, optical coherence applications in quantum optics. The operating spectral range, variety of sizes, shapes and geometrical configurations that can be tailored and the exceptional optical properties of surface plasmons promote their use in a wide variety of applications at the nanoscale, where other types of resonators cannot be used, or are too inefficient. We review a large variety of plasmonic resonators and their most relevant properties. We first discuss the fundamental properties of the plasmonic resonances, using simple physical models. Then we present a retrospective of applications evidencing the most relevant aspects of interaction of surface plasmons with matter and radiation. Finally we focus the most recent applications covering metamaterials, plasmonic materials with topological properties and resonators supporting plasmon-electron interaction.


Introduction
Electronic charge-density oscillations coupled with retarded electromagnetic field modes in metal surfaces are called surface plasmon-polaritons. They constitute a fundamental type of electromagnetic oscillation at the nanoscale with major impact in several fields of modern physics. These oscillations can be excited in all materials presenting negative real part of the dielectric function in a broad band of the electromagnetic spectrum, when interfacing with a dielectric medium. Surface plasmons can be excited in structures with a geometry of 1D, 2D, and 3D space dimensions. These structures are generally called plasmonic resonators. The dimensionality, size and morphology determine the kind of resonances that can be produced for a certain material, or combination of materials.
Key features of the surface plasmons are their excitation by electromagnetic waves, either propagating or evanescent, electron beams, and by electric polarisation in nanoscale junctions. However, propagating waves and electron beams require a phase matching condition to be verified. By other side, the extreme surface confinement of the surface plasmons is associated with an exponential decay with the distance of the nearfields [1].
Surface plasmons are distinct from other quasi-particle oscillations as bulk plasmons, phonons, magnons and excitons, as they are excited at the surface of metals. The wide range of resonators that can be conceived always relies on the strong confinement of the electromagnetic field, on the energy of the optical resonance and on the resulting scattering angular spectrum. The confinement of plasmons has other relevant implications, namely in the geometrical properties of several light quantities as spin, linear and angular momentum of near-fields, quite different than those of propagating fields.
Resonances on individual or coupled particles can be analysed using the equations of motion of classical mechanical or electrical resonators. Although this seems to be an oversimplification it provides a good qualitative and in some cases quantitative description of the interaction. However, a rigorous description of the far-and near-fields and derived quantities requires solving the Maxwell's equations. Almost all physical quantities related to plasmonic resonators can be described in the framework of classical or semi-classical physics. Only for very small particles, thin junctions and some matter-radiation interaction cases full quantum mechanical treatment is required.
Current investigations of plasmonic resonators do not focus only on the interaction between electromagnetic radiation, or electrons and plasmons, but make use of the already well known resonators to explore other physical effects as topological properties of materials, quantum states of light, metamaterials and transport phenomena.
In this review, after this introduction and a short historical note, we firstly give an overview of the fundamental properties of simple and coupled plasmonic resonators, describing their resonance properties using classical oscillators models. In some cases we compare the properties with the results obtained from rigorous calculations of the Maxwell's equations. We continue with a survey of already well known applications of plasmonic resonators in several levels of matter-radiation interaction, both classical and quantum. Simultaneouly we briefly discuss recent advances related to these applications. We finish with a bunch of the most recent research progresses relying on plasmonic resonators, including topological materials, metamaterials, and the revival of electron-plasmon interaction.

Historical note on plasmonic resonators
The discovery of surface waves called surface plasmon polaritons (SPP), is due to Ritchie [2]. SPPs are responsible for the angular dependent losses of fast electron beams crossing thin metal sheets. These surface excitations of the conduction electrons have lower energy than the bulk plasmons of the metal. Later, it was found that light could also excite surface plasmons, provided the phase matching condition between the wave vector of the incoming light and wave vector of the surface plasmon would be achieved [3,4]. It became then possible to relate the surface plasmons with the Mie resonances of metal colloids, well known since the beginning of the 20th century [5], but already experimentally investigated by Faraday [6]. These resonant modes arise by spatial confinement of the surface plasmons at the surface of a noble metal sphere, unlike the propagating modes in infinitely extended metal-dielectric interface.
Despite the fact that plasmons have been treated as an example of excitons of solid state physics, the progress of research on all phenomena related to surface plasmons, globally called plasmonics, evolved autonomously, often approaching classical optics and electrical engineering rather than other branches of solid state physics. This was mainly due to the fact that the number of materials suitable for plasmonic applications remained very small and new materials are difficult to engineer. Surface plasmons require materials with free electrons and low optical losses at the optical bandwidth, which restricts us to a few available in nature including gold, silver, and aluminium [7][8][9][10]. An easy way to check if a material has potential to be used in plasmonics is by analysing its dielectric function. Good plasmonic materials in the visible and near-IR must have a negative real part of the dielectric function and simultaneously a small imaginary part. More recently, other materials synthesised by different techniques extended the number of materials presenting plasmonic properties. This includes 2D materials as graphene, molybdenum disulphide, the so called MXenes [11], compound 3D materials as metal nitrides, highly doped semiconductors and transparent conducting oxides [12,13]. Limitation are the spectral region where these materials present adequate values of the dielectric function, which is mostly in mid-IR to far-IR and optical losses. However, in 2D materials as graphene surface plasmons are tunable.
As surface plasmons are electromagnetic surface waves, we can consider several ways to propagate and confine them. The most elementary geometrical structure, beyond the trivial metal-dielectric interface, supporting surface plasmon modes is the thin metal film surrounded by two dielectric semi-infinite media. This is usually called DMD (dielectricmetal-dielectric), also called IMI (insulator-metal-insulator) structure. Its counterpart forming a planar gap between semiinfinite metal regions is the MDM, or MIM. The properties of the plasmonic modes supported by these structures were extensively investigated elsewhere [14][15][16][17][18]. Planar structures supporting plasmon guided modes were among the first noble metal structures investigated. However, it became evident that surface corrugations, grooves and other geometrical shapes on a planar film would modify the propagation and confinement of surface plasmons, leading to localised resonances and modifying the dispersion relation [17,[19][20][21]. The first observation of an effect arising by plasmon confinement in metal gratings was reported by Wood in 1902, the so called Wood's anomaly [22]. Despite successive approaches, the first theoretical models provided only phenomenological explanations [23][24][25]. The nature of the surface waves leading to the Wood's anomaly remained not well understood. The discovery of the surface plasmons furnished the physical basis for a consistent explanation of the spectral anomaly observed.
In 1998, Ebbesen and collaborators found an intriguing large value of the zero-order optical transmission of white light through an array of sub-wavelength holes drilled on a silver film of 0.2 µm thickness [26]. The transmission was only remarkable for some spectral bands and not in the full visible spectrum. The transmission was considered extraordinary by comparing its maxima with the prediction of the Bethe theory of diffraction for a single sub-wavelength aperture on a perfectly absorbing screen [27]. The transmission value was also highly dependent on the angle of incidence. A satisfactory explanation was given in terms of the coupling of light with the plasmonic modes excited at the surface of the hole array. By other side, it was already known that 2D gratings on silver could present bandgaps [28]. This coupling between surface modes was already analysed by Fano in 1938, to explain the Wood's anomalies in 1D gratings [24]. Interestingly, Fano obtained a similar mathematical expression to describe the asymmetrical lineshapes arising in the ionisation spectra of noble gases, in the context of atom physics [29][30][31]. Since then these quite universal resonances are called Fano resonances.
Curved surfaces, or corrugations in metallic films where surface plasmons propagate, lead to reflections and eventually to localised plasmon modes. The curvature of the surface leading to reflections can be understood as a geometrical potential [32,33]. In metal films structured with deep grooves plasmons can easily couple to external light, unlike thin flats films. Deep grooves strongly confine plasmons and simultaneously produce narrow absorption resonances. Because of that, grooves of different shapes have been fabricated to produce structural colours [34][35][36].
Other way to increase the number of surface plasmon modes propagating along the metal-dielectric interfaces is by stacking alternating layers metal and dielectric media. The mode dispersion arising in such stacks was already investigated by Economou in 1969 [14]. In the last two decades, these stacks regained attention due to the discovery of the hyperbolic metamaterials [37][38][39]. The plasmonic modes become guided in volume of the stack. Some of the guided modes can have very large wavenumbers, which is required in sub-wavelength imaging. The stack itself does not form a cavity in the strict sense, but as some modes may have very large wavenumbers allows for radiative decay engineering [40][41][42][43].
Although the work of Mie established a milestone in the history of electromagnetics of metal colloidal particles, Mie resonances remained long time disconnected from other effects due to the excitation of surface plasmons. The intensive calculations required to solve the Maxwell's equations of arbitrary geometry and complex dielectric function only become possible with the advent of computational electromagnetics, after the 1960s. By the other side, a large amount of theoretical calculations based on approximations of the classical electromagnetic theory, over the full past century, permitted to achieve an increasingly detailed knowledge of the properties of small metal particles of various shapes, arrangements, and environments [44][45][46][47][48]. In the last decades new techniques of nanoscale fabrication [49][50][51][52], experimental and theoretical investigation of the optical properties of nanoparticles [47,53,54], and new microscopy and spectroscopy techniques [55][56][57][58][59][60] boosted a fast development of plasmonics extending the applications range.
The interaction of plasmonic nanostructures with matter in their immediate neighbourhood has been a subject of increasing attention, both in the classical and the quantum world. The incomparable near-field enhancement and confinement provides the basis for a large number of applications. Fundamentally, the plasmonic structure acts as a resonator with optical response in the near-and far-field. For a specific nanoparticle, the optical response of the resonator is mainly determined by the wavelength and polarisation state of the illuminating light. The near-field distribution of particles arranged in close proximity is also strongly modified, due to near-field coupling. The propagating far-fields present radiation patterns that depend on the polarisation direction and plasmonic mode excited. Particles of the size of the wavelength and larger have radiation patterns not only dependent on the polarisation direction of the incoming field but also on the number of multipoles excited. Single nanoparticles, or arrays of coupled nanoparticles can operate as optical antennas. On the other hand, when the excitation is done by an electron beam in vacuum, the polarisation state is not defined, but the point at which the electrons hit the metal determines the near-field pattern and the angular spectrum of radiation emitted. In most of applications light is used for the excitation of surface plasmons and many modern applications rely on hierarchical arrangements of optical nanoantennas.
Before we discuss applications of plasmonic resonators and the underlying morphology and materials, it is convenient to describe the fundamental characteristics of the plasmonic resonances using classical oscillators, as they furnish the basic elements to understand the physical mechanisms of the resonances.

General properties of plasmonic resonances
In this section the origin of plasmonic resonances is reviewed. Firstly, we address briefly the dielectric function of plasmonic materials. Then, we present a phenomenological treatment of plasmonic resonances in isolated and coupled systems of particles, as they form the basis for many effects based on plasmonic excitation. We give examples for the most relevant resonant effects.

Dielectric function of plasmonic materials
Plasmons are low energy excitations in an electron gas. In metallic solids where electrons are free and can be easily driven by external electric fields, oscillations of charge density arise due to the application of the external field and the restoring force of the positively charged material remaining. In a periodic potential the model describing the dynamics of electrons determines the effective mass m * , which depends on the band structure of the materials and is different than the mass of a free electron. The bulk plasma frequency is determined by the mean density of electrons n e , related to the Fermi energy of the material, and its effective mass. It reads ω p = (4πe 2 n e /m * ) 1/2 in cgs units, or ω p = [e 2 n e /(ϵ 0 m * )] 1/2 in SI units. This quantity is obtained from the equation of motion of an infinitesimal volume of electrons under an external electric field. Plasma frequencies range from 15 eV for aluminium down to THz frequencies for graphene.
The dielectric function of bulk plasmonic materials as noble metals according to the Drude-Sommerfeld electron model is given by ϵ ∞ ≥ 1 accounts for the contribution of the positive ionic cores to the dielectric function. For a perfect electron gas it takes the value of 1. The refractive index is also a complex quantity and related to the dielectric function byñ = n + ik = √ ϵ. The Drude-Sommerfeld model cannot describe the band structure of the metal and therefore, in calculations is often substituted by the experimental values of the dielectric function or refractive index [7]. More accurate models of the dielectric function include non-local response, which is dependent on the particle size, and other parameters [61][62][63].
Graphene is a 2D material with a more complicated optical response [64][65][66][67][68][69][70]. The optical conductivity of graphene is described by σ = σ I + σ D . The first term is a interband contribution expressed by σ I = σ ′ + iσ ′ ′ . The second is the Drude contribution, which dominates for heavily doped graphene and frequencies much below the Fermi level E F . Graphene is one of the most investigated materials today and new properties and applications in many fields beyond plasmonics have appeared.
In contrast to Drude metals graphene presents an extraordinary advantage: the optical response can be tuned by a chemical potential, or by applying an external voltage. The electrical tunability is not an exclusive property of graphene, in contrast with other bulk plasmonic materials, characterised by the Drude-Sommerfeld model. The dielectric properties of other 2D materials as the monolayered transition metal dichalcogenides (MoS2, WS2, WSe2) can also be tuned [71].
Graphene plasmons present as well very large wavenumbers and exhibit the strongest field confinement of any surface plasmons [68,72]. The only limitation is the frequency band where the tunability and field confinement are so large: the mid-IR to THz frequency range [73]. At visible wavelengths the optical absorption reaches 2.3%.
The search for better plasmonic materials in several directions as, lower loss, application at high temperature, wide bandwidth, tunability, low chemical reactivity, transparency, non-linear response and near-zero refractive index has been subject of intensive research and has been already addressed in many research articles and reviews [12,13,[74][75][76][77][78][79].

The driven damped oscillator and the plasmonic particle
The optical response of an isolated plasmonic particle where a plasmon is excited can be characterised by the phenomenological model of a driven damped oscillator. The spectrum of the harmonic oscillator with low damping (2ω 0 ≫ γ) is approximated by the Lorentzian lineshape [80]. We summarise the properties of such oscillator in order to analyse afterwards several examples of coupling between oscillators. The equation of motion of the harmonic damped oscillator of charge q and mass m, driven by an external harmonic force f (t) = f 0 exp(iωt), is given by We can normalise by the mass and obtain where a 0 = f 0 /m is the acceleration. A more general excitation in the Equation of motion includes the Abraham-Lorentz force term, F AL = (2q 2 /3c 3 ) d 3 x/dt 3 in cgs units, or F AL = (q 2 /6πϵ 0 c 3 ) d 3 x/dt 3 in SI units [81]. When the oscillator is driven the general solution in the steady-state is with The frequency-dependent amplitude A determines the shape of the resonance. γ turns out to be the half-width of the resonance at the half maximum (HWHM). The quality factor relating the amount of energy stored in the oscillator comparing to the excitation power is given by Q = ω/γ. When the external excitation is interrupted the oscillations decays with constant γ. The way the oscillations decays depends on the ratio r = γ/(2ω 0 ), which can be overdamped (r > 1), critically damped (r = 1), or underdamped (r < 1). Using this simplified model we can describe the time constant of damping τ = 1/γ. Estimations of the decay time from measurements of the extinction spectra of plasmonic nanoparticles indicate . For comparison, the extinction cross-section of a gold spheroid parallel to the electric field is also presented. Due to the symmetry conditions the plane wave cannot excite the quadrupole mode of a spheroid when its longest symmetry axis is parallel to the electric field. The dielectric functions used in the calculation were based on experimental values of the optical constants [7]. values of few to 10 fs [82]. However, the particle size and high pumping power in experiments leads to much longer relaxation times involving hot electron and lattice relaxations, in the order of ps [83]. The classical oscillator model is particularly useful to apply in spheroidal particles illuminated by plane wave fields in homogeneous environment. Metal nanoparticles as Mie spheres present multipolar excitations. Each multipole can be calculated individually, but the superposition of the modes leads to overlapping of the resonances in the optical extinction spectrum. Spheroidal particles and rods, by contrast, have well separated multipole resonances and thus, the extinction spectrum can be described as a sum of resonances with Lorentzian lineshape. The extinction spectra of elongated particles is dependent on the polarisation direction of the plane wave and the incoming field direction. Thus, some multipoles may be not excited for certain polarisation and illumination direction. In the left column of figure 1 is illustrated the multipolar extinction spectrum of a silver spheroid and its fitted spectrum with Lorentzian curves. In the right column are presented the extinction spectra of a gold spheroid for two distinct illumination directions.
Elongated particles as rods present narrower resonances than spherical particles of comparable volume and surface size. This is due to the different radiation damping rates [84][85][86][87][88][89] exhibited by the particles. In figure 2 the first multipoles of silver spheroids and their respective FWHM are shown, indicated by the error bars, as function of the particle aspect ratio. The shortest semi-axis is 30 nm and the longest 200 nm.
The spectral position of the longitudinal resonances scales up linearly with the aspect ratio of the particles. This has been found for elongated particles much smaller than the wavelength, but also for particles of the size of the wavelength. The scaling factors also depend on the particle shape, material, refractive index of environment and multipole order [71,[90][91][92][93]. This property is particularly useful for tailoring particles, or arrays of particles to be applied as optical antennas. Like in electrostatic charge distributions surface plasmons multipoles excited in single particles are associated with distributions of surface charge. These corresponding angular spectra of these plasmon modes are as well distinct. For instance, a dipolar resonance in a spheroid has an angular spectrum of scattering of toroidal shape. In figure 3 the radiation patterns associated with the first multipoles of a silver spheroid at an angle of 45 • to the incoming field direction are presented. The asymmetry of the radiation pattern is do to the particle inclination. However, for a plane wave illumination where the electric field is parallel to the particle long symmetry axis, no quadrupole is excited.
The highly localised surface charge density leads to strong near-field which is fundamental element in radiative decay engineering, non-linear optics, quantum optics and optomechanics and other levels of matter-radiation interaction. Several applications rely on enhanced near-fields, including optical nanoantennas, surface enhanced Raman scattering, plasmonic tweezers and photothermal heating and enhanced radiative decay of emitters.

Coupled oscillators
The interaction models and the discussion of the optical properties of single plasmonic particles of the previous section can extended to coupled particles. The most simple way to describe two interacting plasmonic particles is by using coupled equations of motion equations of two coupled oscillators. This elementary model is very powerful to explain a wide range of interactions, as we will see in the next sections. This model is mostly used to describe the far-field spectral features resulting from near-field interaction. The near-field patterns and corresponding surface charge distributions need a full electromagnetic calculation.
The equations of motion of two classical oscillators of free eigenfrequency ω 1 = √ k 1 /m, and ω 2 = √ k 2 /m with damping constants γ 1 and γ 2 respectively, read Only one oscillator is driven by an external harmonic force and the coupling strength is given by the Ω 12 = √ k 12 /m. The steady-state solutions, assuming the ansatz x i = c i exp(iωt) read [94][95][96] The complex amplitudes expressed by ] . This model can be used to describe coupling effects resulting in Fano resonances, classical analogue of strong coupling and the classical analogue of the electromagnetic induced transparency. All these effects have been investigated in many plasmonic resonators and form the building blocks of many applications. Notable is the fact that all these phenomena have been discovered in the context of quantum physics and have found many analogues in the classical world.

Fano resonance in plasmonic resonators
Fano found in 1961 a formula to describe quantitatively the asymmetric lineshape found in the ionisation spectra of helium [31]. It reads where q is the Fano asymmetric parameter and ε = (E − E ϕ )/(Γ/2) expresses the reduced energy at the ϕ discrete state of width Γ. σ represents the total cross-section due to the interference between the scattering amplitude in the continuum states with scattering amplitude in discrete states. The energy levels of the both continuum and discrete states have to be close to each other in order to produce interference. For q lim ∞ the lineshape approaches the Breit-Wigner or Lorentzian symmetric resonance lineshape. For q = 0 the lineshape has dip at ε = 0. Resonances presenting lineshapes of this kind arising from destructive interference between coupled resonant systems are called Fano resonances. These resonances occur in nature in all kind of physical systems either quantum, or classical. Plasmonic coupled resonators are no exception. Systems that can be described as classical oscillators benefit from the mathematical simplifications without loosing the physical insight. The Fano resonance in coupled classical oscillators was analysed by Joe et al [94]. This model has been very often applied to analyse the optical response in coupled plasmonic systems. Fano resonances have been extensively investigated for plasmonic systems in the last decades [95,[97][98][99][100][101][102][103][104][105][106][107][108][109][110][111]. Gallinet and Martin found a generalisation of the Fano formula for vectorial fields and lossy materials from first principles and more adequate to treat plasmonic systems [95].
Considering a non-radiative mode of frequency ω d and intrinsic damping γ d , the reduced frequency κ is defined by i q 2 a depend on the with Γ i of each resonance. For a non-radiative mode without losses Γ i = 0 and the classical Fano formula is obtained.  In order to underline how a Fano resonance arises we use the oscillation amplitudes for the coupled oscillators found in the previous section. We assume one of the oscillators having a narrow resonance (low damping) and the other with a wide (high damping) and a low coupling constant k 12 such that . The spectral position of the resonances should be proximate, but do not need to overlap. If we drive the first oscillator with an harmonic force, its amplitude presents at the resonance of the second oscillator an asymmetric lineshape 4. Simultaneously, the amplitude of second oscillator has a symmetric peak at the same resonance. This means that the excitation energy of the first oscillator is transferred to the second oscillator, resulting in a destructive interference in the motion of the driven oscillator. The corresponding phase also suffers a sudden shift of~π.
The same kind of interference occurs between bright (radiant) and dark (non-radiant) modes in coupled modes of plasmonic nanostructures. The Fano resonances are assessed from the extinction spectra. One of the first experimental systems evidencing a Fano resonance in the dark-field spectrum was the dolmen structure formed the three gold rods [99]. When the polarisation of the excitation field is parallel to the top rod and the rod lengths match in a way that the longitudinal dipole resonance of the top rod overlaps with the quadrupole resonance of the side rods a Fano resonance arises. Under these conditions the quadrupole resonance cannot be directly excited by the external field (dark mode), but it arises by near-field coupling. Thus, the dipolar resonance suffers a destructive interference from the quadrupole mode and presents a Fano dip.
Other example of plasmonic structure presenting Fano resonance are gold rods in H-like arrangement. The polarisation of the plane wave field in parallel to the middle rod and the lateral longer rods are not directly excited. However, due to the near-field coupling the quadrupole mode of each longer rod is excited absorbing energy from the dipole mode. The total cross-sections of the H-like structure are presented in figure 5. The corresponding radiation patterns and surface charge distribution for three different wavelengths are shown in figure 6. At the Fano dip the far-field radiation pattern is not the typical of a dipole excitation. Due to the near-field  interaction with the longer rods, it presents four lobes and simultaneously decreases the intensity. At the extinction spectrum peaks surrounding the Fano dip the corresponding radiation patterns resemble those of a dipole. Figure 7 presents the calculated reflectance, near-field maximum and near-field distribution of two dolmen structures made by rectangular rods of different sizes. The position of the Fano dip and the near-field maximum are not only sensitive to the inter-particle gap, but also on the particles dimensions. In contract with the H-like structure, the quadrupole is excited in both longer rods and not in each longer rod in a symmetric way. Thus, the near-field distributions of the Z-component of the electric field are antisymmetric (see figure 7(e)).

Strong coupling in plasmonic systems
When an emitter is in an excited state it decays spontaneously due to two pathways: a radiative and a non-radiative. Inside a cavity the radiative pathway can be strongly enhanced, due to the large increase of optical modes, and the decay rate becomes much faster than in free space. This is what happens when the lifetime of the optical mode is much shorter then the lifetime of the excited state of the emitter. This is the regime of weak coupling, arising between the emitters and cavity modes giving rise to the Purcell effect [112] for the spontaneous emission. However, if the optical mode lifetime in the cavity is longer than the lifetime of the excited state an interaction regime starts to arise between the optical mode and the emitter, resulting in an oscillatory decaying emission. This phenomenon is called Rabi oscillation. It was found by Rabi in the context of nuclear spin resonance [113]. The interaction regime is called strong-coupling, to distinguish it from the more common exponential decay of emission characteristic of the weak coupling. The Rabi oscillation in the timedependent emitted light is associated with a spectral splitting in the emitter resonance. The spectral splitting is an attribute of the strong coupling.
The strong coupling regime of radiation-matter interaction can be described at three levels [114]: (a) quantum optical, describing the interaction of a two-level quantum system interacting with a quantised electromagnetic field; (b) semiclassical, where the field electromagnetic field is treated classically; (c) classical Lorentzian oscillator and classical electromagnetic field. The classical case has been already introduced by Novotny [115] and more recently discussed by Törmä and Barnes [114]. The fully quantised case has been subject of thorough studies in cavity quantum electrodynamics [116][117][118].
In order to appreciate the importance of the strong coupling in plasmonic systems we analyse strong coupling using classical oscillators, as in the case of the Fano resonance. Following the method of Novotny [115], we first write the equations of the undriven coupled oscillators of equal mass To simplify the solution we set γ i = 0, Without coupling (k 12 = 0) the eigenmodes have frequencies ω 0,1 = √ k 1 /m and ω 0,2 = √ k 2 /m. Using the ansatz x i = A i exp(−iω ± t), we find the eigenfrequencies arising for nonzero coupling [115] We can make k 2 dependent on a parameter k and k 1 = k 0 having a fixed value. We have then k 2 = k 0 + ∆k. Thus, when k 2 = k 0 , both eigenfrequencies have the same value. We can now vary ∆k between −k 0 and k 0 and obtain the respective ω + and ω − . Assuming that both masses are identical m 1 = m 2 = m, it easy to see that the eigenfrequencies cross for k 12 = 0. However, for non-zero k 12 an anti-crossing arises in the values of ω + and ω − . The way we change the value of k 2 was purely arbitrary. It was chosen with the only purpose to make the eigenfrequencies dependent on a parameter that leads to crossing, in the absence of coupling. Figure 8 illustrates the anti-crossing of modes in coupled oscillators as function of the parameter ∆k.
Following the approach of Novotny [115], we first write the equations of the undriven coupled oscillators To simplify the solution we set γ i = 0, Without coupling (k 12 = 0) the eigenmodes have frequen- we find the eigenfrequencies arising for nonzero coupling [115] We make k 2 dependent on a parameter k and set k 1 = k 0 to a fixed value. The dependence of k 2 on the parameter k is arbitrary, but must allow a matching point between k 1 and k 2 . Thus, we set k 2 = k 0 + ∆k. When k 2 = k 0 , both eigenfrequencies have the same value. We can now vary ∆k between −k 0 and k 0 and obtain the respective ω + and ω − . Assuming that both masses are identical m 1 = m 2 = m, it easy to see that the eigenfrequencies cross for k 12 = 0. However, for non-zero k 12 an anti-crossing arises between ω + and ω − as function of ∆k. Figure 8 illustrates the anti-crossing of modes in coupled oscillators as function of the parameter ∆k.
We point out that the anti-crossing observed in the dispersion relation characteristic of strong coupling also arises for other coupled oscillators, for instance in photonic resonators, cavity quantum electrodynamics with single emitters and in many solid state systems. The first requirement is that their dispersion relations can cross at some point in the (k, ω) space, if considered uncoupled. The second is the requirement that the coupling must be stronger than the decay rates of the resonators, or the decay rate of the excited state of an emitter coupled with the resonator. Thus, the term 'strong coupling' applied in the present context should be distinguished from other contexts of physical theories.
A very useful and intuitive way to analyse the strong coupling regime in plasmonics was developed by Novotny and Törmä and Barnes [114,115]. The spectrum of a nanoscale emitter of frequency ω e and damping rate γ e is represented by a Lorentzian curve [80]. The Lorentzian expression can be found from the motion equation of the undriven damped oscillator model of a point dipole. As the emitter size is much smaller than the wavelength of light, we can assume that its dispersion is independent of the k-vector. It is represented by a horizontal line in the dispersion relation map (ω, k). A surface plasmon-polariton (SPP) by the other side is dispersive. The dispersion relation reads where ε m is the dielectric function of the metal and ε d is the dielectric function of the dielectric. We can assume for the sake of simplicity that the metal is a Drude metal. The dispersion curve of the SPP is always localised at the right of the light line. When the central frequency ω 0 of the emitter overlaps the SPP dispersion both lines would cross. Under certain conditions, however, there no crossing in the dispersion curves of the emitter-SPP system rather an anti-crossing. The condition is the following: the coupling constant between the emitter and the SPP must be larger than the sum of the decays rates of both [114,115]. More recently, other conditions for the strong coupling and the anti-crossing were discussed by Rodriguez [119]. When the coupling constant is smaller than the damping, only weak coupling occurs and leads to the Purcel effect. Strong coupling has been predicted and observed experimentally in many different optical systems. The classical strong coupling between an optical cavity and a plasmonic particle is illustrated in figure 9.
The strong coupling between a plasmonic resonator and a single emitter is difficult to achieve experimentally. In general only weak coupling is achieved. However, quantum emitters in small gap cavities can lead to strong coupling at room temperatures. Despite the low Q-factor of these cavities, the mode volumes are much smaller than for photonic resonators and thus, strong coupling based on a single molecule was measured (see figures 10 and 11).

Plasmonic analogy of the electromagnetic induced transparency
Electromagnetic induced transparency (EIT) is a term coined by Harris and coworkers to describe how an opaque medium may transmit radiation at its resonance [122][123][124][125]. It occurs in vapour cells of three level atoms. The description of the mechanism leading to EIT is based on transition between quantum states of three different energy levels. One of the schemes is based on a laser source tuned between the upper level |3 > and the lowest level |1 > (probe field of frequency ω p ) and a second laser is tuned around the middle level |2 > and the state |3 > (coupling laser field of frequency ω c ). The dipoletransition between the states |1 > and |2 > is forbidden. The coupling laser interacts with the upper state and is responsible for bringing large amount of atoms to the a steady state |2 >. This results in a sudden drop of intensity in the absorption spectrum at its resonance peak. This phenomenon has been extensively investigated for many quantum systems [125]. A related effect is the strong reduction of the speed of light propagating in cold atoms supporting EIT [126].
A classical analogy of the EIT based on two coupled oscillators was developed by Alzar et al [96]. It is similar to the Fano resonance, but the resonances of both oscillators overlap. In order to observe a narrow dip in the absorption spectrum of the driven oscillator the non driven oscillator must have a much smaller damping constant. The central dip in the absorption spectrum of the driven oscillator mimics the EIT observed in atomic vapours. Moreover, this simple model of EIT has been also applied in the analysis of coupled plasmonic resonators  leading to induced transparency by near-field coupling [127][128][129]. Figure 12 presents the power absorbed in the first oscillator (driven) and the corresponding dispersion, obtained by the real part of the oscillator amplitude c 1 , as function of the detuning frequency. Although the resonances arising in plasmonic resonators by coupling radiative dipole modes with dark quadrupole modes is usually classified as a Fano resonance, it can be seen as a plasmonic variant of the electromagnetic induced transparency. This is particularly true when the bright and dark modes overlap and the transmission increases at the Fano dip, because the destructive interaction strongly reduces the extinction. Figure 13 presents the transmittance and reflectance of arrays of stacked plasmonic rods fabricated by electron beam lithography, where the upper rod is progressively dislocated from he centre in each array. When the central rod is located above the ends of the lower parallel rods a maximum in the transmittance is reached. When it is in the middle position the transmittance drops to a minimum.

Parametric resonances
A parametric oscillator can be defined as an oscillator with a harmonic time-dependent oscillation frequency. Harmonic oscillators have constant frequency of oscillation ω 0 . In the parametric case the oscillation frequency is externally driven.  One of the most simple examples of parametric oscillators is the pumping of a swing [130][131][132]. A person sitting, or standing in a swing can pump the motion by changing the position of the centre of mass of the body. If this is done periodically, the oscillation amplitude can increase. Rayleigh was the first to present a general theory of the parametric oscillator [133]. An example of parametric oscillator based on a mechanical oscillators and two coupled RLC electric circuits is presented in [134]. The mechanical oscillator is coupled to the first electrical circuit by a capacitor which capacity changes periodically in time, and thus driving the first RLC circuit.
The equation of motion of a driven parametric oscillator readsẍ where we substitute the constant ω 2 0 by f (t) = f (t + T), a periodic function of time. The resulting equation for the period in time T = π is called Hill equation [135]. The specific case where f(t) = ω 2 0 [1 + η cos(2ωt)] corresponds to the Mathieu equation, with applications in dynamic problems as inverted pendulum, quadrupole mass spectrometers, quadrupole ion trap, and also in quantum physics. The amplitude η should be small, i.e. η ≪ 1 to achieve stability. Although the Equation is linear, the time-dependent coefficient hinders to find easily stable solutions [136].
The theory of the optical parametric amplifier (OPA) was developed just few years after the invention of the laser [137][138][139] accompanied by the first experimental demonstration [139]. Though, it took a long time until the first commercial OPAs were available. The main drawback was the quality necessary for the non-linear medium (beta barium borate, or BBO) required to operate the OPAs. Nowadays there are many applications in laser physics based on OPAs. The interaction of a strong laser field (pump laser of frequency ω p ) with the non-linear medium leads to the generation of two coherent laser fields by a process called wave mixing. The idler of frequency ω i < ω 2 /2 and the signal of frequency ω s > ω p /2, such that ω p = ω i + ω s , dictated by the conservation of energy. As well, the phase matching requires that the wavevectors verify k p = k i + k s . The interaction of the laser with the BBO results in a non-linear amplification process with tunable output frequencies.
Only few publications appeared so far dedicated to parametric oscillations in plasmonic systems [141][142][143]. However, the coupling between plasmonic systems and other kind of oscillators seems very promising to this end. For instance, the coupling between a plasmonic excitation on parallel thin gold stripes and the underlying silicon nitride film supporting acoustic vibrations can lead to a parametric resonance [144].

Survey of applications of plasmonic resonators
In this section we present several applications of plasmonic resonators and discuss their main characteristics. One of the earliest applications was in the detection of molecules by surface enhanced Raman scattering [145]. This effect has been subject of intensive research and there are numerous reviews available [56,[146][147][148][149][150][151][152]. Therefore, it is not treated in this article. Although not exhaustive, this section gives an overview of the major applications relying on plasmonic resonators interacting with electromagnetic radiation and electron beams, and covers several types of effects on the matter in the neighbourhood of the resonator.

Purcell effect
Purcell published a short abstract in the Proceedings of the American Physical Society, in 1946, showing how the spontaneous emission of nuclear magnetic moment transitions could be changed inside a resonant cavity [112]. The formula reads where Q is the quality factor of the cavity and V the volume of the resonator. An intrinsically related effect is the inhibition of the spontaneous emission [153]. Because no restrictions are made in the kind of spontaneous emission this result can be generalised for any spontaneous emission system inside a resonator. In optical systems a correction has to be done by introducing the refractive index of the cavity n, λ → λ/n. The mode volume V m deserves special attention. In non-dissipative resonators of very large Q-factor the normal modes are strongly confined inside the cavity and thus, the mode volume can be evaluated by where E is the electric field strength and ε the dielectric function. However, for dissipative and dispersive media and plasmonic materials the integrand would be negative, leading to an incorrect result for the mode volume. Maier and Koenderink proposed a redefinition of the electric energy stored in the cavity, using a former theory of the energy density in a dispersive and absorptive material by Ruppin, in order to evaluate the mode volume [154][155][156]. The relative electrical energy density reads then where ω is the angular frequency and Γ e the damping constant of the oscillator. ℜ(ϵ) and ℑ(ϵ) denote the real and imaginary part of the permittivity. Kristensen and Hughes proposed other method to evaluate the mode volume by calculation the density of quasi-normal modes [157,158].
Plasmonic resonators have small Q factors (range of 1 to tens), due to the intrinsic losses of the metal. Despite that, due to the much larger confinement of the modes than in photonic cavities, the mode volumes can be very small and Purcell factors of 100 s, or even larger then 1000 can be achieved [159][160][161][162][163]. Thus, the Purcell factor remains a measure of the efficiency of a resonator in terms of decay rate, which is fundamental in radiative decay engineering. Plasmonic resonators offer an relevant advantage over photonic cavities: their modes confinement is much larger than the diffraction limit of light and, therefore, can be integrated in devices using much smaller space.
In figure 14 experimental results recently achieved based on the Purcell effect of a nanosized emitter are presented. A nitrogen-vacancy (NV) in a nanodiamond is located under a cubic silver antenna, on top of silver surface coated by a polyelectrolyte. The presence of strong near-field due to the plasmonic nanoantenna on the NV leads to a much emission rate and a reduction of its fluorescence lifetime.

Extraordinary optical transmission
The enhanced optical transmission through sub-wavelength holes drilled in a thin metal film was reported by Ebbesen and collaborators in 1998 [26]. The term extraordinary was used to stress the enhanced transmission at some wavelengths in the visible and NIR, comparing to the theoretical prediction obtained from the theory of Bethe-Bouwkamp [27,165] (see figure 15). The Bethe theory of diffraction together with the improvements of Bouwkamp were taken as reference for a circular aperture in a perfectly electric conductor screen. Therefore, it was expected a reduced transmission in a real material with nonzero absorption and not larger. More recently, Yi and collaborators compared theoretically and experimentally the different regimes of diffraction arising for a single aperture in a real metallic screen of thickness several times larger than the skin depth of metal [166]. They have shown that the Bethe-Bouwkamp theory cannot account for surface plasmons and thus, cannot explain the enhanced transmission. Already in 2001, the importance of the contribution of plasmon tunnelling for the optical transmission was pointed out [167].
Generalising the coupling condition with light for onedimensional gratings [17,19,21], plasmons in periodic twodimensional gratings or arrays of holes observe the following phase matching condition with n, m being integers. We assume for sake of simplicity that k 0 is the free space k-vector and the reciprocal lattice constants are |G x | = 2π/Λ x and |G y | = 2π/Λ y , respectively. Contrarily The connection between the optical transmission in arrays of apertures and the Fano resonance was discussed by Genet et al [168]. The fact that plasmon can tunnel light in periodic metal-dielectric structures lead to the development of metamaterials supporting both electric and magnetic resonant modes (fishnet structure) [169,170], light beaming [171], optical imaging [39]. Some holes surrounded by nanostructured surfaces present optical beaming, This is a kind of light focusing in the perpendicular direction to the plane of the aperture. Examples of optical beaming are presented in figure 16.
Several reviews on the optical transmission by arrays of apertures have given a perspective of the importance of the topic in plasmonics [172,173]. The optical resonances in freestanding arrays of wires are also related those supported on arrays of holes in compact films. The comparative analysis was given by Collin [174].

Optical antennas
The anisotropic scattering of light by plasmonic nanoparticles has been intensively investigated over the last few decades. The dielectric function of the material of the particle, its size, shape, relative orientation comparing to the illumination field and the surrounding medium determine the angular spectrum of the scattered light. A simple dipole antenna based on subwavelength metal rod in homogeneous dielectric medium scatters light with toroidal angular dependence. This is the underlying mechanism of the optical antenna. If the supporting medium changes, or more particles of other size are placed close to the radiating dipole, the radiation pattern can be made more directional. One example of such antenna is the optical analogy of the classical Yagi-Uda antenna for radio waves. On the other hand, also the near-field is affected by the orientation of the electric field of the incoming light.
However, the sizes of the optical antenna cannot be tailored by scaling down the dimensions of the radio frequency antennas. Whereas metals for radio frequency antennas are almost perfect conducting, in the optical regime the surface plasmons matter and scaling laws are different. Novotny found a scaling law for small optical antenna and pointed out its differences comparing to the classical radio frequency antenna [91].
where n 1 and n 2 are constants dependent on the geometrical shape and material. The linear dependence of the optical resonances on the particle aspect ratio was verified for particles of spheroidal and rod shapes [93]. If a spheroidal particle has a diameter d and length L the multipolar resonance of order n reads [92,93] λ n ≃ A 0 + A e n (24) with e = L/d and the constants A 0 and A being independent of the resonance number. We could confirm the scaling for silver spheroids embedded in a medium of refractive index n = 1.45 (see figure 2). The importance of the geometrical factor, the environment and the shape on the optical resonances has been intensively investigated and modelled using different calculation methods [47,48,71,90,[175][176][177][178].
Some of first optical antennas were based on the radio frequency classical examples the dipole antenna and the Yagi-Uda antenna [179][180][181][182][183][184][185][186]. Yet, other shapes also proved to be efficient [187][188][189][190]. The main function of an optical antenna is to redirect the radiation pattern of an emitter located in its immediate neighbourhood. Hence, due to the near-field enhancement generated by the optical response of plasmonic material, the spontaneous emission decay rate is also modified by the Purcell effect. Thus, some optical antennas serve as cavities for radiative decay engineering as well [156,164,187,188]. In order to quantify the Purcell effect of an antenna on an emitter it turns out very important to evaluate accurately its quality factor, but also the mode volume. Due to the open nature of plasmonic cavity the mode volume is difficult to calculate [156,162,191,192]. Indeed, the quality factor of the plasmonic resonators is not very large and the main contribution to the Purcell enhancement comes from the tiny mode volume of the plasmons modes. More advanced designs of nanoantennas to enhance the Purcell factor, or to couple light to mechanical resonators include optical antenna-cavity coupling [190,193,194].
From the experimental point of view, the quality of plasmonic material used in a nanoantenna is of critical importance. The best optical antennas, but also the longest surface plasmon propagation length and the near-field confinement, are achieved with crystalline silver and gold particles [52,[195][196][197][198][199]. Nanoantennas and other nanostructures fabricated by lithography techniques usually have higher losses due to surface roughness, polycrystallinity [49,51,200] and additional absorption losses of the adhesion Cr, or Ti layers, required to hold the noble metal particles on the surface of dielectric substrates and commonly used in e-beam lithography [201].
Dark-field microscopy and spectroscopy can provide some information about the scattering properties of plasmonic particles [57,84]. However, due to the illumination conditions the directivity cannot be measured. Since 2004, a new experimental imaging method has been used to characterise the angular distribution of the scattered light of an optical antenna: the back focal plane imaging [184,186,[202][203][204][205][206][207][208]. The method consists in the introduction of additional lenses, in order to use the optical Fourier transform of the light at the back focal plane. The collected image in a CCD camera is that of the angular distribution of light scattered by the antenna. The collected light cannot have a larger in-plane k-vector than that allowed by the imaging objective. Several experimental setup for measuring the angular spectrum were proposed. Kurvits et al compared the different optical arrangements and their performance [208]. A related technique for the measurement of the orientation of fluorescent molecules using wide-field epifluorescence was demonstrated some years earlier [209].
The Fourier imaging using the back focal plane can be combined with spectroscopy in order to obtain a wavelength dependent angular spectrum of the scattered or emitted light. Thus, radiative modes associated with electric multipoles, or magnetic modes can be characterised by measuring the energy and momentum of light [186,210]. Moreover, radiation patterns can also be measured using cathodeluminescence [182,[211][212][213]. In figure 17 radiation patterns generated by optical antennas of various shapes are presented. In figure 18 the radiation pattern of a gold ridge antenna, measured using cathodeluminescence, is presented.
Plasmonic antennas have been used not only in the manipulation of the angular spectrum of radiation, but also in fluorescence enhancement by Purcell effect and as transducers of electromagnetic radiation to mechanical vibration in optomechanical systems. This is illustrated in figure 19. The plasmonic dipolar antenna mounted on top of two silicon nitride nanobeams is responsible for the optical coupling with incoming radiation. It increases the scattering for adequate polarisation and due to its small gap induces optical forces and photothermal heat to the underlying beams. The result is a mechanical oscillation of the nanobeams. Optomechanics is fast evolving research field connecting the macroscopic classical world with the quantum mechanical [214][215][216][217]. Plasmonic nanoantennas can be used to improve the coupling of light beams with the acoustic vibrations of microand nanomechanical systems [144,[218][219][220].

Magnetic response and chirality in plasmonics
According to Landau and Lifshitz, the magnetic susceptibility of matter at optical wavelengths is negligible [221]. This contradicts experimental results of split-ring resonators and other micro-and nanostructures used in engineered metamaterials. The argument of Landau and Lifshitz does not apply for metamaterials where the characteristic length is of the order of  Reprinted with permission from [190]. Copyright (2015) American Chemical Society. the wavelength and are characterised by a plasmonic dielectric function [222]. Indeed, there have been found many examples of materials where the magnetic resonance is associated with a relevant magnetic moment. This includes the split-ring resonators [223][224][225][226][227] and arrays of near-field coupled spherical particles, or lithography designed nanostructures [228][229][230].
The discovery of negative effective magnetic permeability at optical wavelengths in split-ring resonators, extending the well known properties at radio frequencies, led to the development of metamaterials with negative refraction index. The negative refraction index and its effect in optics was an idea worked out by Veselago already in 1968 [231]. For that, both the electric permittivity and magnetic permeability should be negative. Pure plasmonic materials have a real part of the dielectric function that is negative at optical frequencies, but its magnetic permeability is close to unity. Thus, an effective negative permeability must be engineered. The demonstration of the first basic building unit showing both negative electric permittivity and magnetic permeability at microwave frequencies, was presented by Smith and collaborators in 2000 [232].
In the following year, the negative refraction was experimentally verified, using an engineered material based on arrays of resonant elements [233].
The extension of the negative refraction to optical wavelengths followed in more recent years [169,170,[234][235][236]. The calculation of the effective permittivity ϵ eff and permeability µ eff of a metamaterial is usually based on the effective refractive index n and impedance Z. And these are obtained from measurement, or calculation of the S-parameters S 11 (reflectance) and S 21 (transmittance). Chen and collaborators developed a more general and robust method to obtain ϵ eff and µ eff [237]. The condition to achieve negative refraction is not so strict as before stated [238,239]. The refractive index n is more generally defined by whereε = ϵ 1 + iϵ 2 andμ = µ 1 + iµ 2 . The real part of the refractive index n 1 is negative if (ϵ 1 µ 2 + ϵ 2 µ 1 ) < 0. One concept related to the negative refraction is the perfect lens [240]. According to Pendry, a perfect image of an object could be achieved if a slab of a material of refractive index n =−1 would approach that object in near-field. This perfect lensing effect has been investigated by many researchers. It was demonstrated experimentally for a silver film in 2005 [241,242]. However, the perfect lensing by a slab has been subject of some controversy [243].
The excitation of a magnetic resonance in slot antennas and its effect on the optical reflectance is depicted in figures 20 and 21. Plasmonic nanostructures made of small colloidal gold spheres connected by DNA strands (DNA origami) is presented in figure 22. The geometrical configuration of the gold particles forming an helix leads to circular dichroism.
Although the real part of the dielectric function of silicon and other dielectric materials of large refractive index is not plasmonic at optical wavelengths, spheres, cylinders and other particle shapes present very bright Mie resonances. Characteristic are magnetic and electric resonant modes presenting strong scattering [46,[261][262][263][264]. This is in contrast with the plasmonic Mie modes, which are predominantly electric. This fact has been used for the generation of light based on multipolar excitation [265,266].
Many of the electric and magnetic resonances and their corresponding fields generated in sub-wavelength nanostructures have been incorporated in the fabrication of metasurfaces. An excellent review on plasmonic and photonic metasurfaces is given by Chen et al [267].

Toroidal electromagnetic modes in plasmonic materials
Textbooks on electrodynamics rarely treat other electromagnetic modes than electric and magnetic However, there is another class of multipoles, the toroidal multipoles [268,269], discovered by Zel'dovich in the context of the parity violation in the electroweak interaction [270]. The interest of such electromagnetic modes of toroidal topology by the plasmonics community started around 2007 [271]. Since then, several structures supporting toroidal modes have been investigated [272][273][274][275][276]. One of the properties of such structures is the generation of the anapole (the term was coined by Zel'dovich for the static case). The anapole is the resulting electromagnetic modes where no radiation is emitted in a narrow bandwidth [275,276]. It corresponds to the destructive interference by the superposition of a electric dipole and a toroidal dipole, both sharing the same radiation pattern [277][278][279]. The first experimental demonstration of a metamaterial presenting a dynamical anapole in the microwave regime, was in 2013, by Fedotov et al [280]. Anapoles have been investigated in in solid state physics, plasmonic and in dielectric particles [264,272,275,281].
The toroidal dipole is characterised by a toroidal magnetic field or equivalently, poloidal current density flowing at the surface of a torus. The toroidal dipole moment points out perpendicular to the plane containing the torus and it is defined by [277] Thus, for some wavelength when the magnitudes of both components equal in magnitude and are out of phase the farfield vanishes. Using the Mie theory the effect of the mode destructive interference was theoretically analysed. Anapoles were experimentally demonstrated for small particles of large refractive index (silicon) [264]. Figure 23 presents the excitation of an anapole in an array of circular cavities in a gold film. The COMSOL Multiphysics calculations show the spectral resonances of the cavities under plane wave p-polarisation at the vertical and an angle of 30 • .
At λ = 715 nm and incidence of 30 • a narrow dip in the 0order reflectance arises. This resonance is different from other resonances as for λ = 650 nm at vertical incident illumination.
The reason for this resonance is not a light confinement in the cavity, but the excitation of a toroidal dipole combined with a electric dipole. The typical topology of the electric (radial) and magnetic (toroidal) fields are shown in figures 23(c) and (d) respectively. Similar results were found for circular cavities of smaller size in silver [273].

Optical forces and thermal plasmonics
Plasmonic structures produce near-field gradients at the particle boundaries and between particles, which lead to mechanical effects on their neighbourhood. The mechanical effects due to the field gradients can be analysed using the Maxwell stress tensor. This tensor is the stress tensor of the electromagnetic field and is related to the Poynting vector and the momentum flux density [282]. The mechanical force exerted by the electromagnetic field on an object with finite boundary is given by the time-averaged surface integral of Maxwell stress tensor [80,283].
where ⟨. . . ⟩ T denotes time averaging and n is a unit vector perpendicular to a closed surface S. Light can also carry angular momentum. In that case a torque is exerted on the particle [80,284].
The momentum contribution of the electromagnetic field disappears in the equation due to the time averaging. Thus, if the fields E and B are known in all points of space where a particle is located the time averaged mechanical force exerted on the particle by the fields. For a particle much smaller then the wavelength the Rayleigh approximation applies and the mechanical force on the particle exerted by a monochromatic field of angular frequency ω can be divided into a scattering component and the gradient force of the field [80,285,286] where α = α ′ + iα ′ ′ is the polarisability of the particle. The first measurement of the mechanical force exerted by light, namely its radiation pressure, were done by Lebedev [287] and Nichols and Hull [288]. Ashkin showed how small suspension particles could be accelerated or trapped using focused laser light beams [289]. This is the fundamental mechanism of the later developed optical tweezers [286]. However, trapping of particles using plasmonics is not based on a strongly focused light beam, like in optical tweezers, but relies on the strong gradient fields near the plasmonic surfaces. Novotny estimated the trapping force on a small particle by a sharp tip under polarised light illumination [290]. Nieto-Vesperinas also studied theoretically optical forces and torques involving dielectric and plasmonic particles near surfaces [259,291,292]. More recently, the motion of dielectric particles trapped by an evanescent wave in dielectric materials was studied experimentally [293]. Volpe et al demonstrated experimentally how surface plasmons can trap polystyrene beads in water [294]. These results lead to experiments using nanostructured surfaces, instead of flat thin films [295][296][297]. The alignment and rotation of nanoparticles and nanowires on a surface was also achieved using polarised laser beams [293,298]. The optical trapping and binding of plasmonic particles in a fluid is possible as for dielectric particles in optical tweezers, but the dielectric function of material is negative and the material has non-zero absorption, which leads to heat generation [290,299,300].
The heat generation by plasmonic particles in liquid environment under intense light fields was investigated by several groups [301][302][303][304]. The fact that plasmonic particles can be heated by laser light lead to their application in sensing, diagnostic and therapy of cancer cells [301,[305][306][307]. An example of photothermal heating by excitation of a plasmonic resonance in spherical gold particle in water is presented in figure 24. Small particles of gold are more adequate for photothermal heating than silver particles, due to their larger optical absorption at wavelengths shorter than 550 nm, and in some applications due to the biocompatibility of gold. The equation governing the heat diffusion generated by a source in a medium without phase transformation reads [308,309] where T(r, t) is the time-dependent spatial distribution of temperature, ρ(r), c(r) and k(r) are the density, specific heat and thermal conductivity, respectively. The local heat source generated by light absorption is denoted by q(r, t). Thus, for time-harmonic plane wave external light field E(r, we obtain the photothernal heat by More generally we can also evaluate the heat power using the induced current and the electric field inside the particle [304] In the steady-state the maximum temperature increase in water surrounding a sphere of radius R is given by [303] where P is the absorbed power by the sphere and k w is the thermal conductivity of water. For non-spherical particles a geometry dependent effective radius substitutes R in the expression [303]. Under strong laser pulses, not only fast temperature rising, but also acoustic vibrations can be induced in plasmonic particles [310,311]. Depending on the normal modes of the particles, the spectral range of resonances lies between GHz and THz [312].

Quantum nature and quantum effects of plasmons
When in some extreme cases the classical theory fails quantum treatment of plasmons is recommended. However, the majority of phenomena related to the light interaction to plasmonic particles can be understood using classical electrodynamics. The optical properties of very small noble metal particles, or particle pairs of very small gaps, cannot be described reliably by the interaction of light with a local dielectric function and a semi-classical or pure quantum description of the interaction is required [61,[313][314][315]. In other cases a modification of the dielectric function from local to non-local is necessary for accounting the interaction effects [62,63,[316][317][318].

Nonlinear plasmonics
At low field intensities all effects involving plasmonic excitations are linear. This is due to the small value of the second order electrical susceptibility ξ (2) . The generalised polarisation reads [329] P(r, t) = ϵ 0 [ξ (1) E(r, t) + ξ (2) E(r, t)E(r, t)  or expressing each component of the polarisation (37) ξ (i) is the non-linear susceptibility tensor or order i. For example the wave mixing (WM), including the sum and difference of frequencies and the optical parametric oscillation processes, depend on the second harmonic generation are associated with ξ (2) , the Kerr effect and third harmonic generation depend on ξ (3) .
However, non-linear effects in plasmonic materials have been predicted and measured for both metal surfaces and metal nanostructures of several geometrical configurations. A short list includes, bare metal surfaces and films [330,331], sharp tips [332], tapered plasmonic waveguides [333], optical rods and nanoantennas [334][335][336], in graphene plasmons [327,337,338], in gratings [339,340] and in other geometrical arrangements of plasmonic particles [78,[341][342][343], including metamaterials and metasurfaces [344][345][346][347]. The main limitations of non-linear optical generation using plasmonic nanostructures are the small bulk non-linear susceptibilities of the metals, which are compensated by the strong near-field at  [43] and Barnes [376]. (Right) Hy field profiles of the Fabry-Perot and guided plasmonic modes in the structure for λ = 700 nm. The blue curve corresponds to a Fabry-Perot excited for a plane wave of small in-plane k-vector. The other three modes are guided plasmon modes labelled with increasing normalised k-vector u i . The field profiles were done using the program Moosh [377], based on scattering-matrix calculations. the plasmonic resonance, and the low damage threshold due to the unavoidable optical absorption of the metal.

Advanced applications of plasmonic resonators
In the previous sections we reviewed the application of plasmonic resonators distinguishing their interaction with radiation and matter by the physical mechanism. In the last 5 years new applications have been presented using plasmonic resonances as building blocks for more sophisticated matter-radiation interactions, only achieved in hierarchically engineered materials. This includes, metamaterials, topological materials, interaction of electrons and plasmons and fast nanoscale switching using plasmonic systems.

Plasmonic metamaterials
A metamaterial is an engineered material presenting properties not available in bulk materials of nature. This definition is not entirely correct and been revised, because many materials have been found in nature, either layered minerals, or resulting from the organic-inorganic growth processes leading to very complex optical, thermal and mechanical properties [348]. In this category belong biominerals and periodic and quasi-periodic organic structures of many living beings, as coloured feathers of birds, fish skin, chromatic, refringent and diffractive properties in the outer surface of many insect species. Plasmonic metamaterials are indeed a subset of metamaterials, not found in nature, due to the incorporation of noble metal elementary structures, or 2D materials.
The advent of plasmonic metamaterials began with the pioneering works of Pendry [240] and Smith [232], in 2000. Their achievement was the demonstration of the negative refraction, already theoretically postulated by Veselago [231], in 1968. Negative refraction requires in first approach a composite material presenting simultaneous negative dielectric permittivity and magnetic permeability. Plasmonic materials have negative real part of the permittivity, so the magnetic negative permeability, not found in bulk had to be engineered. At the same time optical effects as invisibility cloaking and transformation optics started to be investigated very actively [38,[349][350][351], as well as ε near-zero materials [79,352,353].
In the visible spectrum the experimental realisation of negative refraction is difficult to achieve, mainly due to the optical losses and tiny size of the nanostructures, but other effects attracted both the plasmonic and the electrical engineering community. Among them the hyperbolic metamaterials [354]. Bulk materials with isotropic electric permittivity have an optical propagation with the same wavelength in all space directions. Anisotropic materials as quartz and calcite have ellipsoidal dispersion because their isofrequency surfaces have ellipsoidal shape. The dispersion relation of an uniaxial medium, where the optical axis is parallel to the z-axis and two components of the permittivity tensor are equal, ϵ xx = ϵ yy ϵ ⊥ , The first term in the equation above corresponds to spherical surface for a fixed frequency (isofrequency surface) and the second to a ellipsoidal surface in the k-space. However, when one of the components of the permittivity tensor, either ϵ ⊥ , or ε zz has a negative value the surfaces of isofrequency of the dispersion equation become hyperbolic. On of the consequences of the hyperbolic dispersion is that plane waves of very large k-vector, corresponding to evanescent waves can propagate in the uniaxial medium, whereas they cannot propagate in isotropic media. For ϵ ⊥ > 0 and ϵ zz < 0 the isofrequency solutions are double hyperboloid surfaces mirrored against the XY-plane and the hyperbolic material is called dielectric, or type I. In the contrary case ( ϵ ⊥ < 0 and ε zz > 0) Reproduced from [405]. CC BY 4.0.
limit of very high density many very narrow layers, or small rod separation and diameter comparing to the wavelength, homogenisation can be achieved leading to effective hyperbolic properties. The most common homogenisation schemes of the uniaxial dielectric function is by an effective medium theory, e.g. using the Maxwell-Garnett and Bruggeman formulas for bi-isotropic media [356,357], or by the Rytov method [358].
Due to the propagation properties of hyperbolic materials plasmons can be excited and propagate in volume and permit a series of novel effects as, long range dipole-dipole interactions as in Förster resonance energy transfer (FRET) [359], super-Planckian thermal radiation emission in near-field [360,361], strong coupling [362], broadband Purcell effect [363] and its application in enhanced spontaneous emission and light extraction [364]. Figure 25 presents the dispersion and the field profiles of several guided plasmonic modes along a stack of silver films of 50 nm thickness separated by a 150 nm aluminium oxide layer. The outer regions (substrate and cover) are also alumina. The number of plasmonic modes is identical to the number of dielectric-metal interfaces. By replication of the number of bi-layers aluminium oxide-silver the number of plasmon modes increases and the dispersion curves move toward higher k-wavenumber. Thus, a very large number of bi-layers of narrow thickness leads to plasmons with very large k-vavenumber, a characteristic of the hyperbolic metamaterial.
Another class of materials were plasmonic nanostructures have been widely used to manipulate the propagation properties of light beams are metasurfaces. Metasurfaces are twodimensional arrangements or arrays of meta-atoms, as dielectric nanoparticles or plasmonic nanoantennas, which collectively can achieve a modification of the light properties, as in flat lensing [365,366], generation, conversion and detection of the angular momentum topological charge [367][368][369][370], holography [371,372] and polarimetry [373].
Metasurfaces can be designed to support hyperbolic propagation [374,375]. In figure 26 an example of metasurface based on nanostructured hexagonal boron nitride (hBN) showing hyperbolic propagation at middle-IR wavelengths is presented.

Plasmonic materials with topological properties
Topological insulators are a novel and very special class of electronic materials presenting insulator behaviour in bulk and conduction at the boundaries [378]. The conduction at the boundary is however robust, as its properties remain unperturbed by defects and disorder. The bulk material has a band structure presenting a gap like between the highest occupied band and the lowest empty band. The surfaces (or edges in 2D) of the material have gapless states which are protected by time-reversal symmetry. An example of 2D topological insulator is the quantum spin Hall insulator and also related to the integer quantum Hall effect [379].
The success obtained with topological insulators in electronic materials led to a pursuit towards photonic materials presenting similar topological properties, despite the nature of photons being bosonic instead of fermionic [380][381][382]. Both dielectric and plasmonic materials have been found to possess topological properties due to the optics of evanescent waves. Bliokh et al have shown that spin and momentum of light in evanescent waves may have geometrical and topological properties completely different than in propagating waves [383]. Moreover, for certain plasmonic nanostructures in surfaces spin-orbit coupling and its unusual light propagation is achieved [384]. Indeed, the spin-orbit coupling is an essential element of the topological materials. Thus, the search for new topological photonic materials relies on optical properties as chirality and Dirac states of light, and photonic structures mimicking the band structure of topological insulators have been investigated [385,386].
Plasmonics enters the realm of topological materials by several ways. It was found by Markel [387] and Zou et al [388] that arrays of nanospheres can produce remarkably sharp resonances, despite their individual extinction spectra being broad. This study was extended to different lattices of gold disks and coupled particles by Auguié, Humphrey and Barnes and others [389][390][391][392]. The near-field coupling between particles and the diffraction in the array leads to lattice plasmon modes distinct from those observed in individual particles. The lattice plasmon modes are responsible for new near-field distributions. Symmetries and a band structure that resembles that of graphene were found for the hexagonal lattice of silver nanoparticles at the Γ point [385,393]. Several lattices and particle shape have been investigated in lasing, strong coupling, Fano resonances and sensing applications [394][395][396][397][398]. More recently, the use of particle arrays in the investigation of topological effects of light has been pursued [399,400]. By other side, graphene is a Dirac material which presents not only topological properties for electrons but can also be tailored for photonic applications presenting topological properties [401,402]. Other 2D materials as transition metal dichalcogenide monolayers were also used for investigation of the spin-orbit coupling of light and its effect on the light propagation direction [403]. It was found that arrays of metallic nanoparticles arranged in hexagonal lattice can be used for engineering edge states [404]. This lattice is the same of graphene and thus, some eatures found for edge states of graphene can be expored in arrays of plasmonic particles. Topological metasurfaces allow a precise control of light propagation at certain edges. The design of edge states in arrays of nanoparticles in a surface are an example of potential topological metasurface for manipulation of the spin of light along the metasurface [405]. Figure 27 presents examples of metasurfaces based on arrays of metal nanoparticles and show how their edge states in the band structure be be modified by a symmetry breaking produced by the rotation of the array in the unit cell. This change in the edge state leads to a special light propagation at the boundary between two regions with different Brillouin zones, shown in figure 28. It should be stressed that different edge types lead to the different propagation properties. The edge between two lattices shown in figure 28(A) allows momentum flow (Poynting vector) and normalised spin angular momentum (T z ) in a complex way near the edge, whereas in (B), where the lower lattice is the mirror symmetry of the upper lattice, the flow is more linear.

New materials for electron-plasmon interaction
The discovery of surface plasmons is intimately related to electron beams [2], as it is the electron energy loss of electrons beams that produce the collective surface excitations called surface plasmons. However, it was found few years earlier in 1953, that non-relativistic electrons moving parallel to the surface and perpendicular to the grooves of a metal grating lead to emission of electromagnetic radiation at optical wavelengths, by a mechanism similar to Cherenkov radiation, the so called Smith-Purcell effect [406]. This phenomenon was later very intensively investigated in the field of electron accelerators and free electron lasers. More recently, due to its connection with surface plasmons in metal gratings and other periodic nanostructures, it was again investigated by the plasmonics community [407][408][409][410][411]. On the other hand, the interaction between electrons and plasmons underlies physical mechanisms used in several microscopy and spectroscopy techniques as electron energy loss spectroscospy (EELS), photoemission electron microscopy (PEEM) and cathodeluminescence (CL) [60,412,413]. EELS and cathodeluminescence are based on the direct interaction of electron beams with the plasmonic material. In the first a mapping of the beam energy loss is reconstructed in the plasmonic structure. In cathodeluminescence, the light emitted from the interaction of an electron beam with a nanostructure is used to obtain the angular spectrum and the energy spectrum of the plasmonic resonance. In PEEM, a pulsed laser beam generates electron emission mediated by the plasmonic excitation.
A second connection between surface plasmons and electrons arises in the fast emission of photo-electrons and nonlinear light pulses, by excitation of surface plasmons in a grating nanofabricated in a sharp metal tip [414,415]. In order to produce a strong near-field at the tip appex, the nanostructured grating is excited by femtosecond laser pulses, which leads to highly confined light and coherent pulsed emission of photons or electrons. Sharp tips under illumination of low power fs laser pulses produce high harmonic light emission, mediated by non-linear processes of the confined plasmons [416]. In parallel to light emission, illuminated but non-nanostructured tips can also be used as fs electron sources by photoinduced field emission [417]. These effects have triggered new research, combining highly non-linear photonics, ultra dynamics using fast pulsed electron sources, coherent spectroscopy and microscopy [418][419][420].
The third connection between electrons and plasmons arises in light emission by inelastic electron tunneling in junctions [421]. Light emission by inelastic electron tunneling has been reanalysed from the perspective of plasmonics [422], but its applications go beyond a pure side effect of tunneling. Tunneling is a very fast phenomenon and the control of tunnel currents by excitation of surface plasmons has a very high impact in fast switching and modulation devices [220,423] and high speed photodetection [424]. In figure 29 a high speed photodetector based on graphene and two-dimensional transitionmetal dichalcogenides (TMDCs) is presented, as is an MoTe 2 sandwiched between gold contacts. Efficient photoresponse at frequencies up to 30 GHz were demonstrated experimentally ( figure 30). Devices based either on plasmonic elements as gold, or 2D materials as graphene and two-dimensional transition-metal dichalcogenides, due to their electronic properties, allow for fast transducion mediated by tunneling [323,[425][426][427].
These examples demonstrate the importance of plasmonic systems in the new physics and new technological applications. Although many other applications could be listed, in sensing, energy harvesting and in nanolithography, these examples provide a perspective of the exceptional potential of plasmonic resonators.

Conclusions
In this review we discuss general properties of plasmonic resonators and their use in multiple applications. Two orders of interaction can be distinguished: interaction of plasmonic nanostructures with propagating electromagnetic radiation and electron beams, and near-field interaction of surface plasmons with other properties of matter. In the first level are included the Mie scatterer, the optical nanoantenna, Fano resonances, plasmonic analogue of the EIT, enhanced optical transmission and light beaming and the plasmonic metasurfaces, resonators exhibiting toroidal and anapole modes, and non-linear processes on plasmonic particles. Among the effects of the near-fields generated by plasmonic resonances on matter in the immediate neighbourhood are the surface enhanced Raman scattering, the Purcell effect affecting the spontaneous emission, the strong coupling with multiple, or single emitters, the optical forces and the photothermal heating.
Based on the elementary plasmonic resonator properties and interactions novel applications have been investigated on a higher hierarchical level, where the effects rely either on collective resonances, or in the plasmon-electron interaction. These include the metamaterials and metasurfaces, the topological plasmonic materials and the very fast switching devices based on inelastic electron tunneling coupled with plasmonic elements.
We covered the major application areas where plasmonic resonators have been employed. The number of research fields using plasmonic nanostructures has constantly increased and new effects relying on enhanced near-fields at small volumes cannot dispense the unsurpassed light confinement achieved by these resonators. This is valid either for classical sensing applications, or for the most recent quantum optical applications, as the coupling between plasmonic and optomechanical resonators.
We cite the most relevant literature of the field in the last 10 years. The reader can use this review article to compare the advantages and limitations of each type of resonator and their adequacy for a custom-made application, or for engineering new kinds of resonators.

Acknowledgments
M R G thanks M Hammer, M Müller, C Zeh, I Sfarcic, S Rappl, A Kyumbi, J Iduwe, J Tang, M Nielsen and V Rao for the collaboration over the last 10 years if the preparation and characterisation of nanoparticles and plasmonic nanostructures and technical support in the laboratory. Special thanks also deserve to go to G Neusser and C Kranz for the FIB facility and S Jenisch of the nanolithography facility of the Ulm University.
We acknowledge the Ministry of Education and Science of the Republic of Armenia (Grant 18 T-1C222) and the German Federal Ministry of Education and Research (BMBF Grant 01DK17040) for financial support provided in the last two years.
M R G also thanks W Barnes (University of Exeter, UK) and Antoine Moreau (Université Clermont-Auvergne, France) for discussions related to the guided modes in plasmonic multilayered materials.