Optical interactions in plasmonic nanostructures
© Park; licensee Springer 2014
Received: 21 December 2013
Accepted: 29 January 2014
Published: 1 April 2014
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.
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 . 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  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 1014. 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 . Surface plasmon resonance can also be used to enhance luminescence  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 . 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 . 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 . Hybridization in more complex structures has also been studied, e.g. trimer , quadrumer , hexamer  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 . 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.
Classical description of surface plasmon resonance
Multiple scattering theory
Here the matrices Q are the scattering matrices defined aswhere 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 . 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
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 . 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 . 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 . 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 . 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.
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 ofth particle and the charge distribution of the k th mode ofth 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.
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 . 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.
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  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 . 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 . 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
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 ,[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 . 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 . 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 . Naturally, this led to the demonstration of optical sensing based on Fano resonances [113, 114]. Here we review Fano resonances in heptamers in detail.
When the heptamer is subject to a uniaxial mechanical stress, the symmetry of the system is lowered to D2h. The doubly degenerate E1u mode splits into two non-degenerate modes belonging to B2u and B3u irreducible representations of the point group D2h. 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 (B3u) and y-dipole (B2u) modes. Remarkably, the nature of the modes are preserved. That is, the bright E1u mode splits into bright B2u and B3u modes while the dark E1u mode spawns dark B2u and B3u modes. Also, all modes shift to shorter wavelengths with increasing mechanical strain values. However, the B3u modes which have dipole moment along the direction of mechanical stress shift more than the B2u modes with dipole moment perpendicular to the mechanical stress. This leads to the distinct polarization dependence observed in Figure 12 as the B3u modes interact with x-polarized light and B2u 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 E1u 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 D6h symmetry, only the E1u modes have non-zero dipole moments and thus are optically active. When the symmetry is lowered to D2h by uniaxial stress, the E1u modes split into B2u and B3u modes which interact with x- and y-polarized light, respectively. Since the unstressed heptamer has two E1u modes in the frequency range of interest, we obtain two B2u and two B3u modes, producing Fano resonance just as in the original unstressed heptamer. However, what is missing in this narrative is that the optically inactive B1u mode in the unstressed heptamer becomes optically active B3u mode under uniaxial stress along the x direction. As shown in Figure 13(c), the charge distribution reveals that this B3u 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 B2u mode within the frequency range we investigated and thus we do not see any additional dip for the y polarization.
The additional absorption peak observed under stress can be explained by noting that the optically inactive B2u mode in the unstressed heptamer becomes an optically active B2u 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 B2u irreducible representation in D6h symmetry has no net dipole moment and thus represents an optically inactive mode. However, the B2u irreducible representation in D2h 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 B2u mode is higher than the lowest dark B2u mode originating from the dark E1u 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 B1u mode was located between the two lowest energy E1u modes in the unstressed heptamer and became optically active B3u 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 B1u 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 B2u 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 B2u mode is located between the two lowest energy E1u 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 .
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 . These considerations have led to the development of a quantum-corrected model that could correctly reproduce the fully quantum mechanical calculations . 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 . 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 . 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.
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