# Optical interactions in plasmonic nanostructures

- Wounjhang Park
^{1}Email author

**Received: **21 December 2013

**Accepted: **29 January 2014

**Published: **1 April 2014

## Abstract

We present a review of plasmonic nanostructures in which the constituent materials are coupled together by optical interactions. The review first provides a comprehensive coverage of theoretical framework where the optical interactions are described by the multiple scattering among the nanostructures. We then discuss the two limiting cases of weak and strong interactions. The weak interaction regime is described well by the effective medium theory while the strong interaction regime requires theoretical tools that can describe the new eigenmodes delocalized over the entire system. Weakly interacting plasmonic nanostructures have been studied extensively in the metamaterials research, which has been a major research thrust in photonics during the past decade. This review covers some of the latest examples exhibiting perfect absorption and invisibility. Strongly coupled systems started to receive attention recently. As a representative example, plasmonic molecules exhibiting Fano resonance are discussed in detail. Plasmonic nanostructures offer an excellent platform to engineer nanoscale optical fields. With the recent progress in nanofabrication technologies, plasmonic nanostructures offer a highly promising pathway to discovering new phenomena and developing novel optical devices.

## Keywords

## Introduction

In the Drude theory, metal is modeled as a system of free electron gas undergoing forced oscillation with some damping. Other than some notable exceptions such as the strong interband absorption involving the *d* band in transition metals, the Drude model generally provides good descriptions of the optical properties of metals. It predicts that in the low frequency region the induced polarization oscillates 180° out-of-phase with the external field, resulting in negative permittivity and consequently high reflectivity. At certain frequencies, the electron gas can also undergo collective oscillations whose quantum is called plasmon. The collective oscillation may involve volume charge density (volume plasmon) or surface charge density (surface plasmon). Volume plasmons are longitudinal oscillations and cannot be excited by electromagnetic waves which are normally transverse. Surface plasmons, however, interact strongly with light. By storing a significant portion of its energy in oscillating electron gas, surface plasmon offers an effective means to localize light down to length scales much smaller than wavelength. For this reason, the past decades have seen an explosive growth in research on surface plasmon nanophotonics.

One of the topics of high interest is nanoscale waveguiding with potential applications in integrated photonics for which a large body of literature exists [1]. Another important class of plasmonic nanostructures more relevant to this review are those supporting localized surface plasmon (LSP) modes. These structures include metal nanoparticles of various shapes, nanoholes in metal films, and some variations of these. The LSP resonances typically manifest themselves with strong scattering and absorption. The characteristic color arising from the LSP resonances has enabled a wide range of applications from the medieval stained glasses to modern optical sensors. For example, colorimetric sensing of biomolecules and ions has been demonstrated [2, 3]. Also, strongly scattering nanoparticles have been used to enhance contrast in dark-field microscopy and optical coherence tomography [4] while thermal ablation sing strongly absorbing nanoparticles is being actively developed for targeted therapies of various diseases [5, 6].

LSP naturally provides a highly localized and intense optical field which consequently enhances a variety of optical processes. The best-known example is surface enhanced Raman scattering (SERS). Using a rough silver surface, Raman scattering by a single molecule has been observed with enhancements up to a factor of 10^{14}[7]. Much of the enhancement is believed to arise from the local field enhancement due to the highly localized hot spots produced by the silver nanostructure [8]. Surface plasmon resonance can also be used to enhance luminescence [9] and Förster energy transfer process [10, 11]. Since the optical processes depend on the local field strength, strong emphasis has been placed on achieving higher local field enhancement using novel nanostructure geometries. These structures are generally termed nano-antennas analogous to conventional antennas in the microwave frequency region and have been the subject of extensive research in the past decade [12]. Since hot spots are typically formed in the nano-gap between plasmonic nanoparticles, the nano-antenna research naturally directed attention to the interaction between LSP modes, which is the main subject of this review. An important recent development in this field is the concept of plasmon hybridization [13]. In this picture, the individual LSP modes in closely spaced nanoparticles hybridize to form a set of new modes delocalized over the entire structure, in much the same way atomic orbitals hybridize to form molecular orbitals in real molecules. For this reason, these plasmonic nanostructures are also called plasmonic molecules. The hybridized modes in plasmonic molecules have been extensively studied. The simplest form of plasmonic molecule is a dimer or a pair of nanoparticles in which the plasmon resonance of single nanoparticle hybridizes to produce symmetric and anti-symmetric dimer plasmons [14]. Hybridization in more complex structures has also been studied, e.g. trimer [15], quadrumer [16], hexamer [17] and most widely heptamer [18–22]. In these complex structures, symmetry properties of the hybridized plasmon modes should be described by the group representation theory [23]. In this formalism, how each plasmon mode transforms under the symmetry operations is described by the irreducible representations of the symmetry group. This allows for a systematic description of modes with complex field profiles and also their interaction with incident optical fields, providing a complete picture when combined with the numerical techniques that can precisely calculate the hybridized LSP modes. It is noted that the group theoretical approach classifying the symmetry properties of plasmon modes does not require any approximations. For example, in nanostructures with features sizes much smaller than the wavelength, the plasmon modes exhibit predominantly electric dipole nature and are often approximated as point dipoles. The group theoretical approach, however, does not require dipole approximation and is equally applicable to larger and more complex structures exhibiting higher order mulitpole components.

The strength and nature of the optical interaction between plasmonic nanostructures varies with the distance. For long distances, the interaction is mainly radiative and involves propagating electromagnetic waves undergoing multiple scattering among the plasmonic nanostructures. In this case, the original character of the individual LSP modes should largely be preserved. For short distances, evanescently decaying near-field components should be taken into account. In principle, the near-field interaction can be treated the same way as the far-field interaction in the general framework of multiple scattering theory. That is, the evanescent modes undergo multiple scattering between nanostructures as well as the propagating modes. In extreme short distances, where the individual LSP modes have significant overlaps with one another, the near-field interaction will dominate and hybridization resulting in fully delocalized modes would occur. This review is aimed at providing a unified view of plasmonic nanostructures in which the strength of the interaction among individual metallic components is varied. We will first present a theoretical framework followed by examples of nanostructures and their optical properties in the weak and strong coupling regimes.

## Review

### Theoretical framework

#### Classical description of surface plasmon resonance

*q*, is given by the dispersion relation [30],

_{1}and ε

_{2}are the permittivities of metal and dielectric medium, respectively, and often Drude model is used for the metal permittivity,${\mathit{\epsilon}}_{1}\left(\mathit{\omega}\right)=1+{\mathit{\omega}}_{\mathit{p}}^{2}/\left({\mathit{\omega}}^{2}+\mathit{i\gamma \omega}\right)$where ω

_{p}is the plasma frequency and γ is the damping parameter. This surface wave solution specifies the optical mode often called surface plasmon-polariton (SPP). For the lossless case where the permittivities are purely real, the dispersion relation has a singularity when ε

_{1}+ ε

_{2}= 0. This condition defines the surface plasmon frequency, ω

_{s}, which is given by${\mathit{\omega}}_{\mathit{s}}={\mathit{\omega}}_{\mathit{p}}/\sqrt{1+{\mathit{\epsilon}}_{2}}$, when using the Drude model for metal permittivity. Here ω

_{p}is the plasma frequency of the electron gas. When there is finite loss, the metal permittivity is complex and the resonance condition should be written as which defines the condition for maximum electric field amplitude. This theory can be straightforwardly extended to more complex geometries involving multiple interfaces to obtain SPP solutions [31–34]. The localized surface plasmon modes (LSPs) generally require numerical calculations as the surface plasmon modes are highly sensitive to the details of the given geometry. An important exception is the highly symmetric geometry of sphere and circular cylinder for which a complete analytical solution can be obtained by using the Mie theory where the scattering and absorption cross-sections are given by the scattering coefficients

*a*

_{ n }and

*b*

_{ n }corresponding to various multipole terms [35]. For small spheres, the scattering coefficients

*a*

_{ n }have vanishing denominators when

*m*

^{2}= − (

*n*+ 1)/

*n*, where

*m*is the relative refractive index of sphere relative to the embedded medium and

*n*is an integer. This leads to a sharp increase in both scattering and absorption. In the limit of extremely small sphere, the extinction is dominated by the leading term,

*a*

_{1}, resulting in the resonance condition,

*m*

^{2}= − 2 or

*ε*

_{1}= − 2

*ε*

_{2}when using the Drude model for the metal permittivity. For the lossless case where the permittivities are purely real, the resonance condition is satisfied when$\mathit{\omega}={\mathit{\omega}}_{\mathit{p}}/\sqrt{2+{\mathit{\epsilon}}_{2}}$, which is sometimes called the Fröhlich frequency. Thus, the LSPs supported by metallic spheres and cylinders may be considered special cases of Mie resonances.

#### Multiple scattering theory

**E**

_{ scat },

**E**

_{ int }and

**E**

_{ inc }represent scattered, internal and incident fields, respectively,

**M**

_{ mn }and

**N**

_{ mn }are the vector spherical harmonics. Similar expressions for the magnetic fields are readily obtained from the Maxwell’s equations. The coefficients

*p*

_{ mn }and

*q*

_{ mn }are naturally dependent on the exact form of the incident field. For plane wave incidence, all terms with

*m*≠ 1 vanish and the full expressions for the coefficients,

*a*

_{ n },

*b*

_{ n },

*c*

_{ n },

*d*

_{ n },

*p*

_{ n }and

*q*

_{ n }are available in the literature [39]. To describe multiple scattering in a cluster made of

*L*spheres, the coefficients,

*p*

_{ mn }and

*q*

_{ mn }, describing the incident field for a sphere needs to be rewritten to include the scattered fields by all other spheres. This in turn requires translating all vector harmonics functions separately obtained in the coordinate systems of individual spheres into the common coordinate system so they may be added. This is achieved by the addition theorem of vector spherical harmonics:

**M**

_{ mn }and

**N**

_{ mn }are the vector spherical harmonics about origin O,${\mathbf{M}}_{\mathit{mn}}^{\mathbf{\prime}}$and${\mathbf{N}}_{\mathit{mn}}^{\mathbf{\prime}}$are the vector spherical harmonics about origin O′, and$\mathit{A}{0}_{\mathit{\mu \nu}}^{\mathit{mn}}$ and are the additional coefficients whose complete expressions are available in the literature [36]. One can then express the scattered field by the

*l*th sphere in the coordinate system of

*j*th sphere using the new coefficients,

*p*

_{ mn }

^{ lj }and

*q*

_{mn}

^{ lj },

*B*0

_{ μv }

^{ mn }as given in Ref. [36]. Imposing the boundary conditions at the surfaces of the spheres finally leads to a set of linear equations

*j*

_{0}th sphere,

**r**=

**R**

_{n}differs only by a simple phase factor, exp(

*i*

**k**

_{||}⋅

**R**

_{ n }), from the wave scattered by a sphere at the origin. Here,

**k**

_{||}represents the tangential component of k-vector parallel to the plane of spheres. The scattered field can then be written as,

*b*

_{ mn }are coefficients to be determined,

**r**

_{n}=

**r**–

**R**

_{n},

*h*

_{ n }is the spherical Hankel function and

**X**

_{mn}is defined as$\sqrt{\mathit{n}\left(\mathit{n}+1\right)}{\mathbf{X}}_{\mathit{mn}}=-\mathit{i}\mathbf{r}\times \nabla {\mathit{Y}}_{\mathit{mn}}$where

*Y*

_{ mn }is the spherical harmonics function. While we try to keep the notation of original authors as much as possible, we note that the above equation is essentially in the same form as equation (6) where${\mathit{b}}_{\mathit{mn}}^{\mathit{E}}$and${\mathit{b}}_{\mathit{mn}}^{\mathit{H}}$in equation (9) are analogous to

*a*

_{ mn }and

*b*

_{ mn }coefficients in equation (6). The phase factor, exp(

*i*

**k**

_{||}⋅

**R**

_{ n }), in the above equation allows us to express the scattered field in terms of plane waves with wave vectors,

**k**

_{||}represents the reduced wave vector defined as

**k**

_{ tangential }=

**k**

_{||}+

**g**, where

**g**is the reciprocal lattice vectors of the 2D lattice, so that

**k**

_{||}now represents a vector within the unit cell of the 2D reciprocal lattice or the surface Brillouin zone. The sign in the superscript indicates forward and backward propagating waves in the direction perpendicular to the 2D plane. Now we construct the same plane wave expansion for the incident wave as well and obtain a matrix equation relating the coefficients of the incident wave and the total scattered wave,

*i*,

*i*′ indicate the Cartesian components. The explicit expression for the matrix

**M**′ may be found in Ref. [44]. Equation (11) fully describes the interaction of a 2D periodic array of spheres with an incident light but, since the summation is infinite, actual numerical evaluations will involve truncation which introduces numerical errors. Now we extend the theory to three-dimensional (3D) array composed of multiple layers of 2D arrays by constructing a transfer matrix describing the multiple scattering effects between the adjacent 2D planes. That is, the field between the

*n*th and (

*n*+ 1)th layers are determined by the scattering by the two neighboring planes of spheres. More specifically, the backward propagating wave in the region between

*n*th and (

*n*+ 1)th layer is the sum of the forward propagating wave in the same region back-scattered by (

*n*+ 1)th layer and the backward propagating wave between the (

*n*+ 1)th and (

*n*+ 2)th layers transmitted (or forward-scattered) by the (

*n*+ 1)th layer. The forward propagating wave is similarly defined. The transmission and reflection matrices are obtained from the matrix

**M**in equation (11) modified to account for the shift of origin between the two adjacent planes. This way, one obtains equations relating fields at adjacent layers which must also satisfy the Bloch theorem because the system is periodic in the 3rd dimension as well. Combining the multiple scattering equations and the Bloch theorem, one obtains an eigenvalue equation,

Here the matrices **Q** are the scattering matrices defined as${\mathit{Q}}_{\mathit{g}{\mathit{g}}^{\prime}}^{\mathit{s}{\mathit{s}}^{\prime}}={\mathit{M}}_{\mathit{g}{\mathit{g}}^{\prime}}^{\mathit{s}{\mathit{s}}^{\prime}}exp\left[\mathit{i}\left(\mathit{s}{\mathbf{K}}_{\mathit{g}}^{\mathit{s}}\cdot {\mathbf{d}}_{\mathit{r}}+{\mathit{s}}^{\prime}{\mathbf{K}}_{{\mathit{g}}^{\prime}}^{{\mathit{s}}^{\prime}}\cdot {\mathbf{d}}_{\mathit{l}}\right)\right]$where s and s′ are + or – signs, g and g′ index the 2D reciprocal lattice vectors, **d**
_{
r
} and **d**
_{
l
} are right and left translation vectors, the matrix **M** is given in equation (11), and **a**
_{3} is the primitive unit vector representing the periodicity in the 3rd dimension. It is noted that this last part is the standard transfer matrix formalism and can be used for non-spherical scatterers as well [45]. For given ω and **k**
_{||} in the surface Brillouin zone, the matrix in equation (12) is fully specified and thus can be solved using the standard matrix inversion techniques. The plane wave expansion over the reciprocal lattice vectors of course will need to be truncated for numerical evaluations.

#### Effective medium theory

**E**and

**P**are the macroscopic field and polarization of the medium. Relating the local field to the induced atomic dipole moment,

**p**=

*α*

**E**

_{ local }, we arrive at the Clausius-Mossotti equation,

*a*is the radius of the spherical object and ε

_{s}is the permittivity of the sphere. The Maxwell Garnett mixing rule is obtained by substituting equation (15) into (14),

_{s}is the volume fraction of the sphere. For spheres embedded in a dielectric matrix with permittivity ε

_{m}, we simply replace 1 and 2 in equation (16) with ε

_{m}and 2ε

_{m}, respectively.

*a*

_{ n }and

*b*

_{ n }are generally expressed as infinite series of spherical Hankel functions. Using the power series expansions of the spherical Hankel functions, it can be shown that the leading terms of these coefficients are

*a*

_{1}~

*x*

^{3},

*b*

_{1}~

*x*

^{5},

*a*

_{2}~

*x*

^{5},

*b*

_{2}~

*x*

^{7}, etc., where

*x*= 2

*πa*/

*λ*. Therefore, for small spheres, we may retain only the leading term in

*a*

_{1}. Since the scattering and absorption cross sections are proportional to |

*a*

_{1}|

^{2}and

**Im**(

*a*

_{1}) (imaginary part of

*a*

_{1}), respectively, the scattering and absorption efficiencies defined as the ratio of scattering and absorption cross sections to the physical cross section depend on

*x*

^{4}and

*x*, respectively, yielding the well-known Rayleigh scattering results. More specifically, the leading term of coefficient

*a*

_{1}is

*a*

_{1}coefficient and the polarizability is apparent,

This relationship reveals one of the key approximations involved in the Maxwell Garnet effective medium theory: retaining only the leading term of the leading coefficient *a*
_{1} in the general solution of the Mie theory. This is essentially the electrostatic approximation in which the polarizability of small spheres is modeled by the static polarizability given in equation (15). A modest improvement on this approximation may be achieved by the extended Maxwell Garnett theory in which the full expansion of *a*
_{1} is used to calculate the polarizability [47]. The extended Maxwell Garnett theory has been used successfully for metamaterials [48, 49]. However, if higher order terms in coefficient *a*
_{1} are to be included, it is logical to include *b*
_{1} and *a*
_{2} terms as well. These terms represent the contributions of the magnetic dipole and electric quadrupole terms, respectively. The inclusion of *b*
_{1} can be done straightforwardly by defining effective permeability in exactly the same way effective permittivity is calculated except that *a*
_{1} is replaced by *b*
_{1} to define magnetic polarizability. It has been shown that the optical properties of small particle composites may exhibit magnetic permeability even when the constituent materials are all non-magnetic [50]. Thus, the use of extended Maxwell Garnett theory is more appropriate when both effective permittivity and permeability are considered. It has also been attempted to improve the Maxwell Garnett theory using dressed polarizability to better account for the resonant effect [51]. In addition to neglecting higher multipole contributions to the light scattering and absorption, which is appropriate for small particles, the Maxwell Garnett effective medium theory also ignores multiple scattering among particles. This can be seen from the facts that the polarizability is defined by the single sphere Mie coefficient and that the effective permittivity is constructed based on the Lorentz local field concept. There are other issues related to the Maxwell Garnett effective medium theory such as the Lorentz catastrophe and non-symmetry and there exist other effective medium theories that improve on those shortcomings [52]. However, it can be said in general that the effective medium theory is appropriate for small particles with low volume fraction where both the higher order multipole contributions and inter-particle interaction by multiple scattering can be ignored.

#### Plasmon hybridization

_{0}and m are the electron density and mass, respectively, and σ is the surface charge density. This technique was shown to agree well with the full electrodynamic simulation results in the electrostatic limit where all of the involved length scales are small compared to the wavelength. It was successful in describing the plasmonic resonances in complex structures including nanoshell, nanoegg and nanorice [55]. The model has also been applied to multi-particle systems such as dimer, trimer, quadrumer, hexamer and heptamer [14, 20, 56]. A similar method has been proposed by Mayergoyz et al. who derived a boundary integral eigenvalue equation using the electrostatic approximation. In this approach, the surface plasmon resonance is described as an eigenstate of the homogeneous boundary integral equation [57],

**r**). The eigenvalue γ is related to the permittivity ε(ω) of the metal as

_{b}is the permittivity of the background material. It is thus possible to determine the resonance frequency from the permittivity of the metal. The eigenfunction σ(

**r**) describes the self-sustained surface charge distribution. This technique has been used to describe collective plasmon resonance supported by hexamers and heptamers made of gold nanoparticles and nanorods [17, 21, 22].

*k*th mode of$\mathit{\beta}$th nanoparticle. One can then use the bi-orthogonality between the surface charge distribution, σ(

**r**), and the surface dipole distribution, τ(

**r**), to derive,

The total surface charge distribution in equation (22) must satisfy the boundary integral eigenvalue equation in (20) where the integral is conducted over the surfaces of all nanoparticles. However, the problem now has been transformed to a matrix eigenvalue equation given in (24). Physically, the matrix **K** represents the interaction between the surface dipole density of the *m* th mode of$\mathit{\alpha}$th particle and the charge distribution of the *k* th mode of$\mathit{\beta}$th particle. Now suppose the arrangement of the nanoparticles is symmetric and certain symmetry operations keep the nanoparticle cluster unchanged. This means the symmetry operators leave the interaction matrix **K** invariant. Then, the symmetry operators and matrix **K** commute and consequently they share the same eigenvectors. It is now possible to classify all eigenvectors of matrix **K** based on how they transform under various symmetry operations. In the language of group theory, the eigenvectors are classified by the irreducible representations of the symmetry group. These eigenvectors possess well-defined transformation characteristics under all symmetry operators that leave the structure invariant and form the symmetry-adapted basis set. The symmetry properties of the plasmon modes proved powerful in analyzing the interaction between various plasmon modes and also between light and plasmons. For example, the incident light interacts with the surface plasmon mode via the induced dipole moment. For a given symmetry group, the dipole operator belongs to a certain irreducible representation. Thus, it naturally follows that only those surface plasmon modes belonging to the same irreducible representation as the dipole operator can interact with light. Other modes are the so-called dark modes that do not interact with incident light and thus cannot be detected by experiments. Furthermore, surface plasmon modes belonging to different irreducible representations do not mix together as their basis functions are mutually orthogonal. These properties provide powerful insights into how complex plasmon modes evolve and interact in complex nanostructures.

### Weakly coupled plasmonic nanostructures

As discussed in the previous section, when the plasmonic nanostructures are only weakly coupled, they may be described well by the effective medium theory. There is a large body of literature on metamaterials which are composed of deep sub-wavelength scale structures and are generally described well by the effective medium theory. The initial development of metamaterials was driven largely by the negative index materials for which excellent reviews are available [58, 59]. In this section, we focus on the latest development of plasmonic nanostructures in the weak coupling regime.

#### Perfect absorber

An interesting application of plasmonic nanostructures is the perfect absorber which absorbs all incident light with minimal reflection and transmission. Such a material would find a wide range of applications in, for example, coatings for optical instruments, sensors, and photothermal or photovoltaic energy conversion devices. An inspection of Fresnel’s coefficients immediately reveals that perfect absorption is achieved when the real part of the refractive index is matched with the incident medium and the imaginary part is small. This condition would result in negligibly small reflection and, if the material is thick enough, vanishing transmission as well. This is, in fact, how perfectly absorbing metamaterial was obtained using aligned carbon nanotube arrays [60]. This example illustrates clearly that a perfect absorber does not have to involve a plasmonic nanostructure [60, 61]. It is, however, of great technological importance to achieve perfect absorption in thin films and plasmonic nanostructures are ideally suited for that purpose.

*a*

_{ i }is the semi-axis along the

*i*-direction. For a sphere, L

_{x}= L

_{y}= L

_{z}= 1/3. For a cylinder oriented along the x-direction, L

_{x}= 0, L

_{y}= L

_{z}= 1/2. For a thin film in the yz plane, L

_{x}= 1, L

_{y}= L

_{z}= 0. We can now write a generalized Maxwell Garnett effective permittivity as follows [65].

_{s}and ε

_{m}are the permittivity of the ellipsoid and medium, respectively. Combining the two equations above yields

*i*. A similar approach can be made with nanoparticles instead of nanograting [66]. In this case, the polarizability of a sphere given in equation (15) in Section 2.3 can be used in the Maxwell Garnett effective medium theory. The major difference in the nanoparticle composite medium and the nanograting is the polarization dependence. The inherent isotropy of spherical geometry naturally leads to isotropic, polarization-independent medium in contrast to the strong polarization sensitivity in nanogratings.

*d*and refractive index

*m = n + i*$\mathit{\kappa}$on a semi-infinite substrate with refractive index

*n*

_{ s }and incident medium with refractive index

*n*

_{ i }(Figure 3a). The reflection and transmission coefficients for normal incidence can then be written in a general form as follows.

*= q*

_{ 0 }

*d*and

*q*

_{ 0 }is the vacuum wave number. Generalization to non-normal incidence is straightforwardly done by replacing q

_{0}with the normal component of incident wave vector. The absorptance is given by where

*C*=

*n*

_{ s }/

*n*

_{ i }for transverse-electric (TE) incident field and for TM. One can now survey the possible values of

*m*to find the maximum absorption. Figure 3(b ~ e) shows the absorptance at λ

_{0}= 900 nm for various film thicknesses of 2, 10, 50 and 150 nm. All cases show a maximum absorptance of approximately 50% along the line

*n =*$\mathit{\kappa}$. Thick films show additional maxima along the horizontal axis due to the Fabry-Perot resonances. In ultrathin films, only those maxima occurring along

*n =*$\mathit{\kappa}$line are achievable. Normally, for a fixed value of

*m*, the absorption vanishes with vanishing thickness as

*A*= O($\mathit{\delta}$). To go beyond this limit, Hägglund et al. sought a refractive index value with dependence where −1 < $\mathit{\nu}$ < 0 so that

*m*is large enough to produce substantial absorption while remains small [68]. The optimum value of

*m*

_{max}resulting maximum absorption was given as,

Equations (31) and (32) show that the maximum absorption occurs at *n =* $\mathit{\kappa}$and the maximum absorptance is close to 50%, both of which are shown in Figure 3.

*m*and thickness

*d*, separated from a perfect reflector by a spacer layer with refractive index

*n*

_{ s }and thickness

*h*. The incident medium index is

*n*

_{ i }. The reflection coefficient is then given as

_{c}is defined as the wavelength satisfying the quarter wavelength condition,

*λ*

_{ c }= 4

*n*

_{ s }

*h*Equation (33) has the same form as equation (30) and thus${\tilde{\mathit{n}}}_{\mathit{s}}$may be treated as the effective index of the spacer-reflector system. The maximum absorption condition is then found by requiring

*r*

_{ i }=

*r*

_{ d }

*γ*

^{2}in equation (33). Retaining only the leading term the optimum refractive index is given as [69],

_{c},${\tilde{\mathit{n}}}_{\mathit{s}}=0$and the optimum refractive index is given as$\mathit{m}=\left(1+\mathit{i}\right)\sqrt{{\mathit{n}}_{\mathit{i}}/2\mathit{\delta}}$. This represents the critical coupling condition where the required refractive index

*m*is minimum. On the other hand, when the spacer layer thickness tends to zero, n

_{ N }diverges and the optimum refractive index cannot be satisfied. Again, the implementation of this concept requires an artificial composite medium made of plasmonic nanostructures. Recently, strong absorption has been experimentally observed using a gold nanoparticle array fabricated by block copolymer lithography and overcoated with a dielectric layer by atomic layer deposition [70]. As shown in Figure 4, the ultrathin composite film with a thickness of ~25 nm exhibited near-perfect absorption at around 600 nm. Vast majority of absorption occurs in the gold nanoparticles, which is not desirable for photovoltaic applications where the absorption must take place in the semiconducting material for current extraction. It is nevertheless noteworthy that the effective absorption coefficient of the gold nanoparticles in this structure was an order of magnitude greater than solid metal. A similar critical coupling leading to perfect absorption has been reported on a gold nanodisk array deposited on a glass substrate [71]. In this case, the critical coupling condition was achieved by tuning the nanodisk density and incident angle of incoming light. When the critical coupling condition is satisfied, reflected light is annihilated by the destructive interference between the light reflected from the substrate and the gold nanodisk array layer.

#### Invisibility

In concluding the discussion on weakly coupled system that can be described well by the effective medium theory, it is worth noting the robustness of effective medium theory. As discussed in the previous section, the effective medium theory is valid only when the feature size is small compared to the wavelength and the interaction among the nanostructures is weak. When these two conditions are met, the optical properties of the composite structure are determined by the dipole moment of the individual nanostructure without having to include higher order resonances and any higher order interaction. These conditions certainly sound very restrictive but in reality the effective medium theory has been successfully applied to a wide variety of systems which at first glace do not seem to satisfy the conditions for validity. An excellent example is the silicon nanorod cloaked by an array of gold nanowires described above. The individual gold nanowire was 13 nm thick and 20 nm wide and the spacing between two adjacent gold nanowires was 63 nm. This system was modeled as a multilayer composed of 20 nm thick gold and 43 nm thick silica films. Despite this seemingly gross simplification, the effective medium theory provided a remarkably accurate result in good agreement with experiments. It can be argued that the effective medium theory, although oversimplified, has captured the essential part of the light-matter interaction, which in this case concerns the induced surface charge and resultant screening effect. The effective medium theory for multilayers resulting in effective permittivity given in equation [35] considers the induced polarization in each layer. When the incident electric field is perpendicular to the multilayer interfaces, it produces a net interface charge due to the discontinuity in permittivity at the interface. These interface charges then produce a polarization field that counteracts the incident electric field, resulting in a permittivity given as *ε*
_{⊥} in equation [35]. On the other hand, when the incident electric field is parallel to the interface, no interface charge is formed and thus the effective permittivity is simple given as an arithmetic mean of the constituent permittivities, as given for *ε*
_{||} in equation [35]. The experiments in the cloaked silicon nanorod were done for TM waveguide mode which has electric field parallel to the gold nanowires. This corresponds to the no interface charge case and the multilayer effective medium theory for no screening case is applicable. If the experiments were done for perpendicular polarization, it would have been important to accurately account for the interface charge density and thus the multilayer effective medium theory could have been inaccurate.

### Strongly coupled plasmonic nanostructures

#### Dimers

**μ**= α

**E**. In a dimer, the field,

**E**, contains both the applied field and the field produced by the neighboring particle. Since an individual nanoparticle is treated as a point dipole, we can write

_{m}is the permittivity of the background medium and g is the geometry dependent parameter given as g = 2 for end-to-end coupling and g = -1 for side-to-side coupling [93, 94]. The coupled equation in [36] can be solved to find the polarizability of the bonding mode,

_{o}, exhibits a universal scaling against the normalized gap width defined as gap/diameter, irrespective of the detailed geometrical parameters, as shown in Figure 8(b) [95]. This universal scaling was also found for dimers of non-circular shapes and nanoshells and also in trimers [96, 97]. The origin of the universal scaling behavior may be found in equation (39), which predicts that the fractional peak shift would behave as (

*d*/

*D*)

^{− 3}or [(

*s*+

*D*)/

*D*]

^{− 3}where

*s*=

*d*–

*D*is the gap width between the particles and

*D*is the diameter. When the gap width becomes extremely small, the simple dipole-dipole interaction model breaks down and one must take into account the higher order multipole terms, leading to an exponential dependence observed in Figure 8. This omission of multipole interaction terms is one of the key limitations in the dipole-dipole interaction model, which leads to a significant underestimation of plasmon coupling at extremely small gaps. Additionally, the dipole model is essentially based on the electrostatic approximation which ignores any retardation effect. To properly describe the interaction at small gaps including the retardation effects would require the use of numerical modeling techniques such as FDTD and FEM. Before moving on, it is worth mentioning that the universal exponential dependence has led to the derivation of the so-called plasmonic ruler equation, which enables precise determination of distances in biological systems, opening many opportunities for biosensing [95].

#### Heptamers

**Ax**

_{ n }=

*a*

_{ n }

**x**

_{ n }, we can define a group of symmetry operators that leaves the operator

**A**invariant. Note that the discussion here is generally applicable to any types of eigenvalue equations which could be the Schrödinger’s equation in quantum mechanics, Maxwell’s equation in optics and electromagnetics, or the fluid dynamics and surface charge density equations used in the plasmon hybridization model and boundary integral formalism, respectively, discussed in Section 2.4. If a symmetry operator,

**S**, leaves the operator

**A**invariant, it is straightforward to show that the two operators commute with each other and share the same eigenstates. If all symmetry operators that leave the operator

**A**invariant are found, then it is possible to classify all eigenstates of operator

**A**in terms of how they transform under the various symmetry operators. In the language of group theory, all eigenstates of operator

**A**can be indexed by the irreducible representations of the symmetry group. Now let us proceed to the plasmon modes in a trimer using the group representation theory. For a trimer composed of three identical nanoparticles placed at the corners of an equilateral triangle, the symmetry operators that leave the system invariant form the point group

*D*

_{ 3h }and thus the eigenmodes of a trimer can be classified by the irreducible representations of

*D*

_{ 3h }. For simplicity, we treat the plasmon mode of individual nanoparticle as a dipole mode, which is a good approximation for small particles. Then, the individual plasmon mode belongs to the

*D*

^{(l=1)}irreducible representation of the full rotation group. For a trimer, we should consider the triple direct product of

*D*

^{(l=1)}irreducible representations and their subsequent reduction into a combination of irreducible representations of point group

*D*

_{ 3h }, which yields [104],

*D*

_{ 3h }can be found in the character table available in the literature [23]. From the character table, one can then construct the projection operators and generate the symmetry-adapted basis functions, as shown in Figure 10, which transform according to the symmetry properties of the irreducible representations. The one-dimensional irreducible representations,${\mathit{A}}_{1}^{\prime},\phantom{\rule{0.5em}{0ex}}{\mathit{A}}_{2}^{\prime},\phantom{\rule{0.5em}{0ex}}{\mathit{A}}_{1}^{\u2033},\phantom{\rule{0.5em}{0ex}}{\mathit{A}}_{2}^{\u2033}$, have only one basis function each and the two-dimensional irreducible representations,

*E*′,

*E*″, have two which are degenerate. Figure 10 is a powerful illustration of the symmetry properties of the hybridized plasmon modes. First, we notice the planar geometry of trimer naturally separates the in-plane and out-of-plane modes, which belong to the primed and double-primed irreducible representations, respectively. For light incident normally to the plane of the trimer, the out-of-plane modes are orthogonal to the incident electric field oscillation. Therefore, only the in-plane modes would interact with the incident light and the out-of-plane modes would not respond. Furthermore, the two in-plane modes,${\mathit{A}}_{1}^{\prime},\phantom{\rule{0.5em}{0ex}}{\mathit{A}}_{2}^{\prime}$, represent perfectly symmetric combinations of the individual dipoles resulting in zero net dipole moment. These modes are therefore dark modes which do not interact with the incident light. The two in-plane

*E*′ modes possess net dipole moments and are the ones that will absorb or scatter light. It should be reminded that the actual plasmon modes excited by an incident light are dependent on the precise excitation conditions and would in general not be the

*E*′ modes shown in Figure 10. Rather, they may be expressed as a linear combination of the

*E*′ modes. A similar analysis has also been carried out for a nanorod trimer [17].

The group theory provides a powerful and effective way to describe more complex structures and extensive studies, both theoretical and experimental, have been conducted on quadrumers [56],[104–106], pentamers [20, 107], and hexamers [17, 18, 20]. The structure that received by far the most attention was the heptamer which is constructed by adding a central particle to a hexamer. The main reason for the interest has been the Fano resonance the heptamer supports. In most systems exhibiting resonance, the lineshape function describing the resonance is typically Lorentzian. However, when a discrete state interacts strongly with a continuum, a new resonance characterized by a distinctly asymmetric lineshape function may arise, as first discovered by Fano [108]. Since the origin of Fano resonance is the interference between the discrete and continuum quantum states, it is fundamentally a quantum phenomenon. However, the interference phenomenon is commonly observed in classical optics and Fano-like resonances have been observed in many systems. The oldest example is the Wood’s anomaly arising from the interference between the tangentially diffracted wave and the incident wave [109]. It has also been observed in coupled waveguide-cavity systems, photonic crystals and plasmonic nanostructures for which many reviews are available [110–112]. In addition to the rich physics it reveals, Fano resonance also has many technological applications thanks to its high sensitivity to the environmental parameters [19]. Naturally, this led to the demonstration of optical sensing based on Fano resonances [113, 114]. Here we review Fano resonances in heptamers in detail.

*D*

_{ 6h }. Since the central nanoparticle is not connected to any other nanoparticles by the symmetry operators of point group

*D*

_{ 6h }, the irreducible representations for a heptamer can be found by the union of those for the hexamer formed by the peripheral nanoparticles and the central nanoparticle, Γ

_{ hep }= Γ

_{ hex }+ Γ

_{ cent }. Similarly to the trimer case, considering only the dipole modes of the individual nanoparticles, the irreducible representations of the hexamer, Γ

_{ hex }, are found by taking the direct product of six

*l*= 1 irreducible representations, Γ

^{(l = 1)}, of the full rotation group and reducing it into the irreducible representations of the

*D*

_{ 6h }symmetry group. Likewise, the irreducible representation of the central nanoparticle is found by simply reducing Γ

^{(l = 1)}. The results are as follows

*E*irreducible representations are two-dimensional, they have two basis functions each. It is clearly seen that only the

*E*

_{ 1u }irreducible representation exhibits a net dipole moment and will thus interact with incident light. This means all plasmon modes observed in experiments would belong to the

*E*

_{ 1u }mode and the mode profiles can be represented by linear combinations of the basis functions shown in Figure 11(a). To probe the plasmon modes in a heptamer, Mirin et al. calculated the absorption and scattering spectra of a silver nanoparticle heptamer with various sizes. As shown in Figure 11(b) and (c), in the smallest heptamer where the individual nanoparticles have a radius of 10 nm and are separated by 1 nm, three lowest energy peaks are observed in agreement with the plasmon hybridization model which predicts two bright modes at 340 nm and 380 nm and a dark mode at 450 nm. The bright modes are prominent in both the absorption and scattering spectra while the dark mode is weak and only visible in the absorption spectrum. Numerical simulations showed that the bright mode has a field profile indicating that the dipole moment of the central nanoparticle is aligned to that of the hexamer while the dark mode has the two dipoles anti-parallel, resulting in a small net dipole moment. As the particle sizes and gaps are increased proportionally, all modes exhibit red shifts due to the retardation effect and thus deviate from the results of plasmon hybridization which is valid only in the electrostatic regime. The retardation effect also broadens the peaks, particularly the bright modes, leading to a spectral overlap between the lowest bright mode and the dark mode. This results in the Fano resonance with its distinct asymmetric lineshape clearly visible in the scattering spectra for silver nanoparticle radii of 20 ~ 40 nm. For radius of 50 nm, the energies of the bright and dark modes become almost the same. In the case, the Fano resonance presents itself as an antiresonance with a symmetric dip in the scattering spectrum.

_{6h}and, among the irreducible representations of D

_{6h}, E

_{1u}is the only one with a net dipole moment and thus optically active. Furthermore, we only consider two lowest energy E

_{1u}modes in the unstressed gold heptamer structure because all higher modes are masked by strong absorption by gold. In the first column of Figure 13, we show the charge distribution of the two lowest energy E

_{1u}modes in an unstressed gold heptamer structure composed of seven identical gold spheres where the sphere diameter is 150 nm and gap between the spheres is 25 nm. Here the modes shown in Figure 13(a) and (b) belong to the lowest energy E

_{1u}mode and (d) and (e) to the second lowest E

_{1u}mode. The two-diomensional E

_{1u}irreducible representation represetns a doubly degenerate mode with two orthogonal states having net dipole moment in the x and y directions, respectively. Accordingly, the four charge distributions shown in the first column of Figure 13 possess net dipole moment where (a) and (d) are x-dipoles and (b) and (e) are y-dipoles. Depending on the energy and the relative alignment of dipole moment of the center sphere to those of the six satelite spheres, the E

_{1u}modes can be classified as dark or bright modes. The lower energy E

_{1u}mode shown in Figure 13(a) and (b) is a dark mode where the dipole moment of center particle aligns against the dipole moments of satelite particles, making the total dipole moment small. On the other hand, the higher energy E

_{1u}mode shown in Figure 13(d) and (e) is a bright mode where the dipole moments align together and add. The energies of these two E

_{1u}modes were found to be 2.394 eV and 2.459 eV or 517.8 nm and 504.0 nm, respectively. These mode energy values would be accurate only for heptamers made of very small nanoparticles, as the boundary integral method is valid in the static limit only. For larger sizes, retardation effect will shift and broaden the modes. The resultant overlap and interference between the two modes lead to the Fano resonance. The bright mode will broaden much more significantly than the dark mode, resulting in Fano resonance which manifests itself in the form of a dip in the extinction spectrum as observed in Figure 12.

When the heptamer is subject to a uniaxial mechanical stress, the symmetry of the system is lowered to D_{2h}. The doubly degenerate E_{1u} mode splits into two non-degenerate modes belonging to B_{2u} and B_{3u} irreducible representations of the point group D_{2h}. Figure 13 shows the evolution of charge distribution as the mechanical stress is applied along the x direction. It clearly shows the original doubly degenerate modes split into x-dipole (B_{3u}) and y-dipole (B_{2u}) modes. Remarkably, the nature of the modes are preserved. That is, the bright E_{1u} mode splits into bright B_{2u} and B_{3u} modes while the dark E_{1u} mode spawns dark B_{2u} and B_{3u} modes. Also, all modes shift to shorter wavelengths with increasing mechanical strain values. However, the B_{3u} modes which have dipole moment along the direction of mechanical stress shift more than the B_{2u} modes with dipole moment perpendicular to the mechanical stress. This leads to the distinct polarization dependence observed in Figure 12 as the B_{3u} modes interact with x-polarized light and B_{2u} with y-polarized light. Therefore, as the heptamer is stretched along the x direction, x-polairzed light would show resonance features at shorter wavelengths than the y-polarized light. Even when the retardation effects are included and the resonance peaks broaden and shift, this general behavior survives and leads to the experimental observation in Figure 12: the dip in the extinction spectrum due to the Fano resonance blue shifts for polarization parallel to the direction of mechanical stress but red shifts for polarization perpendicular to it. The exact numerical simulations by FEM and generalized multiparticle Mie theory also confirmed this.

In addition to the splitting of E_{1u} modes leading to the poalrization dependence, the scattering spectra for x polarization also show an additional dip at shorter wavelengths. The additional dip is apparent in the spectra for 30% and 45% strain for x polarization but is clearly missing in all spectra for y polarization. The origin of this second dip can be found by the group theoretical analysis. Briefly reiterating, in the unstressed heptamer structure possessing D_{6h} symmetry, only the E_{1u} modes have non-zero dipole moments and thus are optically active. When the symmetry is lowered to D_{2h} by uniaxial stress, the E_{1u} modes split into B_{2u} and B_{3u} modes which interact with x- and y-polarized light, respectively. Since the unstressed heptamer has two E_{1u} modes in the frequency range of interest, we obtain two B_{2u} and two B_{3u} modes, producing Fano resonance just as in the original unstressed heptamer. However, what is missing in this narrative is that the optically inactive B_{1u} mode in the unstressed heptamer becomes optically active B_{3u} mode under uniaxial stress along the x direction. As shown in Figure 13(c), the charge distribution reveals that this B_{3u} mode is also a dark mode where the dipole moment of the center sphere aligns anti-parallel against those of the satelite spheres, thereby producing a second Fano dip in the scattering spectra. In constrast, there are no other modes of the unstressed heptamer evolving into B_{2u} mode within the frequency range we investigated and thus we do not see any additional dip for the y polarization.

*x*(horizontal) direction. Figure 15 shows the simulated extinction and absorption spectra at the induced mechanical strain values of 0%, 13%, 26%, and 42%. For the polarization parallel to the mechanical stress (

*x*direction), the Fano resonance exhibited a moderate blue shift while for the perpendicular polarization it showed a small blue shift. In addition to the spectral shifts with applied mechanical stress, the absorption spectra for the perpendicular polarization showed an additional feature at shorter wavelengths when the heptamer was under mechanical stress. This feature located near 620 nm was particularly noticeable at high strain values of 26% and 42%. The behavior of the spectral features in Figure 15 can be explained by analyzing the eigenmodes obtained by the boundary integral method. The evolution of charge distribution is dictated mainly by the symmetry reduction from D

_{6h}to D

_{2h}and is similar to that observed in the circular heptamer under mechanical stress described earlier. As was the case in the circular heptamer, the doubly degenerate E

_{1u}mode of the azimuthal nanorod heptamer splits into two non-degenerate modes belonging to B

_{2u}(

*y*-dipole) and B

_{3u}(

*x*-dipole) irreducible representations of the point group D

_{2h}. Furthermore, the bright E

_{1u}mode was found to split into bright B

_{2u}and B

_{3u}modes and the dark E

_{1u}mode into dark B

_{2u}and B

_{3u}modes. Both the dark and bright B

_{2u}and B

_{3u}modes shift to shorter wavelengths with increasing mechanical strain values. However, the B

_{3u}modes, which have dipole moment along the direction of mechanical stress, shifts more than the B

_{2u}modes whose dipole moment is perpendicular to the mechanical stress, leading to the polarization dependence observed in the extinction and absorption spectra in Figure 15. Once again, the retardation effect results in overlap and interference between the dark and bright modes thus causing the Fano resonance to be observed as a dip in the extinction spectrum.

The additional absorption peak observed under stress can be explained by noting that the optically inactive B_{2u} mode in the unstressed heptamer becomes an optically active B_{2u} mode under uniaxial stress along the *x* direction. Here we have an unfortunate coincidence of having the same label for two very different modes. The B_{2u} irreducible representation in D_{6h} symmetry has no net dipole moment and thus represents an optically inactive mode. However, the B_{2u} irreducible representation in D_{2h} symmetry has a finite dipole moment along the *y*-direction and thus represents an optically active mode that can interact with *y*-polarized light. The energy of this B_{2u} mode is higher than the lowest dark B_{2u} mode originating from the dark E_{1u} mode of unstressed heptamer, thereby producing an additional feature at a higher energy. The azimuthal nanorod heptamer differs from the circular heptamer in the behavior of this optically active mode. In the circular heptamer, the optically inactive B_{1u} mode was located between the two lowest energy E_{1u} modes in the unstressed heptamer and became optically active B_{3u} mode under uniaxial stress along the *x* direction. As a result, in the circular heptamer, the additional mode was observed for x-polarized light. The difference between the circular heptamer and azimuthal nanorod heptamer is readily understood by comparing the eigenmode charge distributions. In the B_{1u} mode observed in the unstressed circular heptamer, the satellite particles have their dipole moments aligned along the radial direction (Figure 13c). In the azimuthal nanorod heptamer, this configuration would have had a much higher energy because the radial dimensions of the nanorods are much smaller than the circular particles used in circular heptamer. On the other hand, the B_{2u} mode in which the satellite particles have their dipole moments aligned along the azimuthal direction has a lower energy because the nanorods have larger dimensions along that direction. For these reasons, we find the B_{2u} mode is located between the two lowest energy E_{1u} modes in the azimuthal nanorod heptamer structure. These behaviors predicted by the theoretical modeling studies have been confirmed by experiments on gold nanorod heptamers fabricated by electron-beam lithography [22].

## Conclusion and outlook

Plasmonics in mutually interacting nanostructures is an exciting field with rich physics and numerous potential applications. The recent progress clearly demonstrates that plasmonic nanostructures allow us to precisely control the nanoscale optical modes and how they interact with one another. These capabilities make the coupled plasmonic nanostructures an ideal platform to study nanoscale optical phenomena and build optical devices with novel functionalities. Though fundamentally a quantum mechanical entity, surface plasmons can be described well by classical electrodynamics in most cases. The theoretical framework presented in this review provides a firm foundation for designing interesting structures and analyzing their properties. Also, the strong field enhancement near the metal surface and the complex field profiles resulting from the interaction among the nanostructures makes the optical properties highly sensitive to the environment and consequently leads to exotic properties such as perfect absorption, invisibility and Fano resonance.

Some of the latest developments in the coupled plasmonic systems deal specifically with the quantum mechanical nature of the phenomenon. In surface enhanced Raman scattering and many other applications, it is generally advantageous to achieve high local field strength. In the simplest geometry of a dimer, a smaller gap supports higher field. However, when the two nanoparticles in a dimer are nearly touching, the electrons may tunnel through the gap, resulting in a dramatic decrease of local field strength and shift of plasmon energy [26, 116]. Furthermore, at such a small length scale, the abrupt interface between two materials is unrealistic and the non-locality of the dielectric function has to be considered [117]. These considerations have led to the development of a quantum-corrected model that could correctly reproduce the fully quantum mechanical calculations [118]. Unveiling the quantum mechanical nature of the gap plasmons in extremely small gaps remains an active research field with hopes of discovering new physics and also with potential for developing novel quantum devices.

Another interesting subject of fundamentally quantum nature is the coupling between plasmon and atomic or molecular exciton. The interaction between plasmon and exciton is of great fundamental interest with strong technological implications in organic electronics and photovoltaics. It has been shown that the surface plasmon can be used to control the relaxation pathways of molecular excitons [119]. Furthermore, in a fashion similar to plasmon hybridization, atomic or molecular exciton can hybridize with plasmon, forming a new quantum mechanically entity sometimes called plexciton [120, 121]. The strong coupling between plasmon and exciton has recently been observed in an organic photovoltaic material [122]. The strong interaction could result in dramatic changes in excitation and relaxation dynamics of the molecular excitons, opening new opportunities in controlling and engineering the optical processes in organic materials.

In closing, surface plasmons provide a powerful means to gain access to the near field components that could result in a wide range of unconventional properties. Controlling and manipulating the mutual interactions among the plasmonic nanostructures offer design flexibility and engineering freedom unparalleled by any other photonics technologies. Obtaining high quality materials with low loss in precise nanoscale geometry will continue to be a challenge but the remarkable progress in nanofabrication technologies in the past decades makes it a perfect time to invest in research on the plasmonic nanostructures.

## Declarations

## Authors’ Affiliations

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