## Abstract

We investigate the propagation of surface plasmon polaritons through coupling of light to sub-radiant dipole modes in finite chains of Ag nanoparticles. End excitation of collections of closely spaced particles reveals a band of sub-radiant modes whereby the decay of surface plasmon polaritons due to radiative losses is minimized. We show that excitation of any of these sub-radiant modes results in the most efficient energy transfer throughout the optical spectrum, with smaller interparticle separations resulting in the longest propagation.

©2011 Optical Society of America

## 1. Introduction

Finite chains of metal nanoparticles have the potential to transport electromagnetic energy via the excitation of their localized surface plasmons [1,2]. The efficient coupling of light to surface plasmons provides a pathway to distribute and direct energy among an ensemble of particles [3–6]. In order to optimize energy propagation along linear chains, it is necessary to understand interparticle coupling as well as energy decay channels [7,8]. For a particle assembly, optical spectra provide a means to analyze these important factors [9–12]. In particular, the identification of super and sub-radiant modes provides essential information about propagating surface plasmon polaritons (SPPs) [13,14].

To date, SPP propagation in nanoparticle assemblies has mainly been investigated by determining the dispersion curves for infinite chains [3,15–20]. Using the coupled dipole approximation, it has been shown that the optical bandwidth is directly related to the energy splitting between the longitudinal, *L*, and transverse, *T*, polarized dipole resonances [3]. The largest SPP group velocity with minimal loss is predicted at the single particle plasmon resonance. Refined calculations that include far-field interactions as well as retardation and losses for chains consisting of particles with finite sizes give rise to more complex dispersion curves with larger bandwidths and greater SPP group velocities [18,21]. However, these studies have been limited to large interparticle distances on the order of the particle radius because only dipole interactions were considered [3,15–21]. On the other hand, the increasing importance of multipolar modes for decreasing particle separations has only been considered in the quasi-static limit [22].

In comparison to infinite chains, the use of finite nanoparticle chains as optical waveguides provides a more realistic system [21,23–25], as practical applications would involve well defined finite structures [26–29]. Here, non-degenerate modes appear as sub-radiant plasmon resonances that have suppressed radiative losses when compared to super-radiant plasmon modes [14,30–32]. Optical spectra of finite chains can be used to identify the spectral region where the sub-radiant modes occur, which are expected to support propagating SPPs with reduced losses. Thus, a theoretical investigation that considers the relationship between optical spectra and SPP damping in finite particle chains at close interparticle separations and hence includes multipolar plasmon interactions and retardation demands further attention.

In what follows, we investigate the effects of higher order plasmon modes of finite Ag nanoparticle chains on their collective optical properties. The exact solution to Maxwell's equations for spheres is used to demonstrate that the propagation of SPPs shows reduced decay with decreasing interparticle separations. More importantly, an excitation bandwidth formed from sub-radiant plasmon modes supporting low-loss SPPs is revealed in the optical spectra. The results are consistent with quasi-static solutions as the resonant energies of the sub-radiant modes follow calculated eigenvalue trends.

## 2. Optical characteristics of finite chains of Ag nanoparticles

First, as shown in Fig. 1
, we demonstrate that the energy splitting, ∆*E*, between the longitudinal and transverse polarized dipole resonances is highly dependent on both particle separations and numbers, *N*. Using the partial scattering solution to Mie's theory for multiple spherical particles as described by Gẽrardy and Ausloos [33], the optical spectra of finite linear chains are obtained to gain significant physical insight into the characteristics of propagating SPPs. Multipoles up to *l* = 20 were included in the calculations and all results have been independently tested for smaller chains of particles using the finite-element method.

In Fig. 1(a), we compare ∆*E* as a function of *N* for two interparticle separations σ = 2.1 and σ = 3.0, where σ is defined as the ratio of center-to-center distance, *d*, and nanoparticle radius, *a*, i.e. σ = *d*/*a*. The shift of the dipole resonance for longitudinal and transverse modes converges for particle chains of roughly *N* = 10, independent of σ [25]. The structure analyzed is shown in the lower corner of Fig. 1. The constituent particles have *a* = 25 nm and are surrounded by vacuum. In all cases, the optical cross-sections are normalized by *N*, while the incident wave vector, **k**, is fixed perpendicular to the chain of particles situated along the z-axis. The material response neglects small particle effects, and is described by a Drude fit to the dielectric function of Ag, where the bulk resonance frequency is ω_{b} = 9.5 eV, the non-radiative decay is given by Γ = 0.1 eV, and the high-frequency limit is ε_{∞} = 5.0 [34]. For all calculations the surface-to-surface separations are ≥ 1 nm so that non-local dielectric effects can be neglected [35].

The energy splitting, ∆*E*, which does not increase further upon addition of more particles, is a direct measure of the particle-particle interaction strength and alludes to an upper limit of the group velocity of SPPs [10]. If excited optically, chains with smaller interparticle separations should provide larger group velocities with enhanced propagation lengths, something not found in previous studies [1].

To determine the presence of higher order plasmon modes, we turn to the extinction spectra calculated for varying *N* at constant interparticle separation, σ = 2.1 (Fig. 1(b)). We first consider the longitudinal mode, which shows a drastic red-shift from the single particle response at 3.46 eV to the eventual saturation at 2.31 eV as *N* increases. The red-shift, due to increasing *N* and hybridization of higher order plasmon modes between particles at close separations, is accompanied by a broadening of the dipole peak. Radiative damping as well as additional dipole-like modes, seen as ripples to the blue of the main peak, contributes substantially to the width for *N* ≥ 5.

For comparison, Fig. 1(b) also shows the extinction for transverse polarization, resulting from the formation of anti-bonding hybridized modes which slightly blue-shift to 3.64 eV at *N* = 50. This much smaller shift in resonance energy is due to reduced plasmonic interactions of the anti-bonding modes compared to the bonding modes for longitudinal polarized excitation [34].

By studying the effect of interparticle separation on the scattering and absorption spectra separately, we can determine the contributions of super and sub-radiant plasmon modes to the collective optical properties of finite particle chains. This is shown in Fig. 1(c), where scattering and absorption cross-sections, Q_{scat} and Q_{abs}, of chains with *N* = 10 are compared for longitudinal polarization while varying σ. Important differences are seen for different σ in the scattering and absorption contributions to the overall extinction. Excluding the single particle Mie spectra at σ = ∞, it is clear that there are multiple peaks at increasingly distributed energies for smaller σ. Regardless of σ, however, the super-radiant dipole mode in the Q_{scat} plots, having all particle dipoles aligned parallel to the chain, is the most efficient at losing energy through far-field scattering.

At σ = 2.1, the secondary dipole peaks between the main super-radiant mode at 2.35 eV and the quadrupole resonance at 3.47 eV are sub-radiant in nature. As shown below (Fig. 2
), for sub-radiant modes, the surface charge density due to the induced polarization forms domains of dipole nature which alternate along the chain in standing wave configurations [13]. These optically active, or “bright”, sub-radiant plasmon modes have non-zero dipole strength at close interparticle separations and hence contribute to Q_{scat}, but always remain weaker than the super-radiant mode. Absorptive losses, as inferred from the Q_{abs} plots, exhibit a different trend. The super-radiant dipole resonance broadens and decreases in intensity due to dynamic depolarization [36], while the bright sub-radiant modes, like the one at 2.74 eV for σ = 2.1, display enhanced Q_{abs} compared to the main dipole peak. For small σ, the Q_{scat} and Q_{abs} plots illustrate that the most efficient coupling of light to the chain occurs at energies corresponding to sub-radiant and higher order modes, where losses due to far-field scattering are minimized.

We can identify super and sub-radiant plasmon modes by their distinctive surface charge densities. Analysis of the induced polarization charge density waves can characterize the resonant nature of optical features, like those discussed in Fig. 1. Shown in Fig. 2 are the real surface charge densities, Re[ρ_{pol}], for the super-radiant and two lowest energy, bright sub-radiant modes of the *N* = 10 particle chain at σ = 2.1, corresponding to Q_{abs} of Fig. 1(c). From this vantage point, the incident wave vector, **k**, is directed out of the page and is longitudinally polarized along the length of the chain.

The induced polarization charge density wave at 2.35 eV is characteristic of a super-radiant plasmon mode. A line segment (black line) taken along the surface of each particle through the chain axis, highlights the opposing concentrations of positive and negative charge. A standing wave pattern is visible in the surface charge density and is indicated by the magenta line, which identifies the plasmon mode as a collective dipole antenna, whose wavelength, λ_{0}, is equivalent to twice the length of the nanoparticle chain.

At 2.74 and 3.04 eV, the surface charge density also displays standing wave patterns due to their induced polarization (magenta lines). These standing waves have wavelengths that are equal to integer fractions of the wavelength for the super-radiant mode, i.e. λ_{0}/*n.* The overall dipole moment for these collective resonances is diminished as localized charge distributions form dipole-like domains which alternate along the chain. Bright sub-radiant modes, like those shown in Fig. 2 are identified by odd fractions of the fundamental charge density wave, *n* = 3, 5, …, and possess net dipole moments when excited in a collective manner. Dark sub-radiant modes, characterized by even integer fractions *n* = 2, 4, …, possess no net dipole moment when excited collectively, and therefore cannot be seen in the optical spectra in Fig. 1. Both bright and dark sub-radiant plasmon modes give rise to suppressed scattering, and can increase coupling to the incident field when excited asymmetrically (see section 4), as the surface charge density wave now forms nodal patterns which decay in magnitude along its length. Therefore it is expected that intrinsic sub-radiant eigenmodes should support propagation of low-loss SPPs along finite chains of metal nanoparticles [13].

## 3. SPP Propagation in finite chains of Ag nanoparticles

Thus far, we have demonstrated that when particle separations become small, higher order plasmon modes play a significant role in the optical characteristics of nanoparticle chains by contributing to super and sub-radiant dipole modes. Our next task is to apply these results to understand how sub-radiant plasmon modes contribute to the transport properties when chains are excited at one end, as would be performed in a waveguiding experiment. This was achieved by setting the coefficients for the incident field to zero for all but the first particle.

Figure 3
displays SPP near-field intensities along a chain of *N* = 50 Ag nanoparticles with *a* = 25 nm for different σ, and supports the notion that sub-radiant modes efficiently absorb incident radiation, sustain suppressed radiative losses, and thereby minimize SPP decay. The near-field intensity *I* = |**E _{s}**|

^{2}$\sqrt{{\epsilon}_{0}/{\mu}_{o}}$ was determined at the successive centers of the particle junctions along the z-axis, where

**E**is the total scattered electric field including near and far-field contributions, and ε

_{s}_{0}and μ

_{0}are the vacuum permittivity and permeability, respectively. The chain is end-excited with multiple wavelengths of longitudinally polarized plane waves, exciting both bright and dark sub-radiant modes [13].

*I*is then fit to an exponential decay,

*I*=

*I*

_{0}exp

*(-b*z

*)*with decay constant,

*b*. Figure 3 shows the intensity profiles for the excitation energies that yield maximum SPP propagation at each σ. For σ = 3.0, the decay follows previous calculations showing maximum SPP propagation for excitation energies near the single particle resonance [1]. However, smaller separations minimize SPP decay at excitation energies corresponding to sub-radiant modes, which is a direct result of plasmon interactions through higher order modes in finite particle chains.

The inset to Fig. 3 relates the energies at which maximum propagation takes place to each interparticle separation, σ. With decreasing σ, an increasingly broad band of energies is observed where exponential decay of *I* takes place, coinciding with the sub-radiant modes described in section 2 and section 4. Here the loss is mainly dominated by absorption as the intensity of the sub-radiant modes increases upon end-excitation (see section 4). The SPP amplitudes decay exponentially like a system of damped harmonic oscillators, consistent with a free electron response. Shown on top of Fig. 3 is the surface charge density, Re[ρ_{pol}], at an excitation energy of 2.62 eV and an interparticle spacing σ = 2.1. The induced polarization charge density alternates along the chain, and displays even symmetry with respect to the center of the nanoparticle chain. This is characteristic of exciting a dark sub-radiant plasmon mode [13], which supports maximum SPP propagation with minimal radiative decay [32].

In Fig. 4
we further verify the sub-radiant nature of SPP propagation in finite nanoparticle chains by correlating energies at which maximum propagation takes place to the optical extinction when the chain is excited collectively. Shown in the lower part of Fig. 4(a) are the 1/e SPP decay lengths as a function of excitation energy for the three interparticle separations highlighted in Figs. 1 and 3. Comparison to the corresponding optical extinction spectra (top part Fig. 4(a)) illustrates that maximum SPP propagation takes place at energies which correspond to sub-radiant plasmon modes for all σ considered. At energies which lie outside this energy band, the intensity fails to decay exponentially, uncharacteristic of pure propagating SPPs. This can be seen from the intensity profiles for the three analyzed separations at different excitation energies, as shown in Figs. 4(b)–4(d). In particular, at σ = 2.1, (Fig. 4(b)), an excitation energy of 2.62 eV results in a maximum 1/e decay length of 0.81 *μ*m. Excitation at energies outside the band corresponding to sub-radiant modes (and highlighted in Fig. 4(a)), e.g. at 3.14 and 2.2 eV, drastically reduces SPP propagation. The same trend is also observed for separations of σ = 2.4 and σ = 3.0, shown in Fig. 4(c) and Fig. 4(d). These results suggest that low-loss SPP propagation occurs via excitation of sub-radiant dipole modes.

Interestingly, for the smallest interparticle separations, the excitation band for maximum SPP propagation partially overlaps with the super-radiant plasmon mode, possibly due to enhanced near-field coupling at close distances. Although an exponential intensity decay is observed (which is the definition of the energy band in Fig. 4(a) and the inset of Fig. 3), the SPP decay length is significantly shorter for excitation at energies corresponding to the super-radiant mode.

## 4. Origin of sub-radiant modes

To further understand the physical nature of sub-radiant modes for particle chains, we performed quasi-static calculations, as illustrated in Fig. 5 . In the non-retarded limit, surface modes are defined by conservative polarizations which are irrotational $\nabla \times P=0$ [37].

Assuming a quasi-static response of the polarization **P** of each particle and insisting that no bound charge is induced except within an infinitesimal volume at the surface$\nabla \u2022P=0$, the macroscopic fields are described by the solutions to Laplace's equation [38]. Because of the time-harmonic nature of the material response, a polarization current, *J** _{pol}* = ∂

**P**/∂t with constant charge carrier density is included. Using the Hamilton relations, this leads to the equations of motion for the multipolar moments of each particle [39]. By solving for

**P**in spherical coordinates and neglecting the contribution of the energy contained within the magnetic fields, the multipolar moments

*q*are obtained by solving

_{lmi}*ω*'s are the resonant modes of the system of spheres and

_{l}*H*is a Hermitian matrix representing the interaction between multiple particles [40]. The dipole per unit volume of the

*ith*sphere with unit vector,

**n**

_{i}, normal to its surface is then given by

*r*

_{i}is the radial distance from the center of the particle and

*Y*are the spherical harmonics. By adding to the Hamiltonian the energy loss of a system of dielectric spheres placed in an alternating uniform field in vacuum, we can solve for the induced surface charge density, ρ

_{lm}_{pol,i}=

**P**

_{i}•

**n**

_{i}, and the fields attributed to it. In Fig. 5(a) we show the

**k**= 0 quasi-static results from collectively exciting a small chain with

*N*= 5 particles at a separation of σ = 2.1.

The largest resonant peaks in the extinction spectrum occur at the super-radiant dipole and quadrupole modes at 2.86 and 3.5 eV, respectively, as shown in the lower plot of Fig. 5(a). However, secondary peaks due to bright sub-radiant dipole modes are also visible in the optical cross-section at 3.19 and 3.53 eV. Although the energies are blue-shifted compared to generalized Mie theory calculations due to the lack of phase retardation, the energy trend for the different plasmon modes is in excellent agreement with the results discussed in section 2. The inset plots the energy of the *N* non-degenerate eigenvalues [21] as a function of σ. At large σ, all modes converge to the *l* = 1 mode of an isolated sphere. Only the super-radiant and the two bright sub-radiant modes marked by the colored boxes can be excited optically by a symmetric field, while the two dark sub-radiant modes are optically inactive as a result of the mirror symmetry of their charge distribution [13,41,42]. As σ decreases, the eigenvalues follow the same trend as the excitation energies that yield minimum SPP decay (Figs. 3 and 4).

The physical mechanism behind the sub-radiant mode at 3.18 eV is illustrated by plotting **P** • **n** in the upper part of Fig. 5(a). The charge density varies asymmetrically with respect to the center particle, which leads to near-fields that have odd parity upon negation of their coordinates [42]. Hence, they possess a finite dipole moment and can thus be optically excited. In contrast, dark sub-radiant modes, which lie close in energy to their bright counterparts (black lines in the eigenvalue plot in the inset of Fig. 5(a)) become optically active upon end-excitation [13]. The charge density obtained by quasi-static calculations shown in Fig. 5(a) agrees well with the results from generalized Mie theory (section 2).

Shown in Fig. 5(b) are the results of exciting the leading particle within chains of increasing *N* at a separation of σ = 2.1, which demonstrate that end-excitation creates a sub-radiant band of energies at which radiative decay is minimized and efficient coupling to particle chains occurs. The cumulative dipole moments along the z-axis are plotted for chains with varying *N*. For *N* = 5, the sub-radiant mode at 3.18 eV gains magnitude with respect to the lowest energy super-radiant dipole at 2.86 eV. This trend continues for larger *N* and is accompanied by a gradual red-shift as indicated by the red arrow. We can further conclude that for chains with larger *N* and at small σ, a broad band emerges consisting of low energy sub-radiant modes, in agreement with Fig. 4 and the inset of Fig. 3. This is evident from the appearance of additional peaks associated with both bright and dark sub-radiant modes, which at *N* = 50 become a broad featureless shoulder at the high energy side of the super-radiant dipole mode. The results of these quasi-static calculations should also apply to larger metallic nanoparticles with local dielectric properties that are governed by a free electron response.

## 5. Group velocity of SPPs in finite chains of Ag nanoparticles

The results presented so far have shown that SPP damping is strongly suppressed for sub-radiant modes. However, it is also necessary to examine the group velocity of the propagating SPPs, which can be obtained from the SPP dispersion [3]. Shown in Fig. 6
is a dispersion curve at energies corresponding to the sub-radiant plasmon modes for an *N* = 50 chain of Ag nanoparticles with *a* = 25 nm separated by σ = 2.1. The association between SPP propagation and sub-radiant plasmon modes in this finite chain is illustrated by the red colored region, corresponding to the same colored part in the optical extinction spectrum plotted in Fig. 4(a). The wave-number, k_{||}, for a particular incident excitation energy with longitudinal polarization was obtained by fitting the **E _{s}** •

**z**components along the particle chain to the equation of a damped harmonic oscillator (shown in inset). For excitation at 2.62 eV, corresponding to a sub-radiant mode, the decay constant,

*b*, of the damped harmonic oscillator equation is taken to be the exponential decay parameter in Fig. 4(a). For comparison, near the single particle plasmon resonance at 3.45 eV, excitation of localized plasmons leads to large radiative and intrinsic damping and therefore inefficient coupling along the Ag nanoparticle chain.

The slope of the dispersion curve determines the SPP group velocity *v _{g}* [22]. In the region of minimum SPP decay, ~2.62 eV, we find that the group velocity is

*v*/

_{g}*c*~0.3, where

*c*is the speed of light in vacuum. In addition to the small damping and large group velocity for SPPs excited via sub-radiant plasmon modes, it is important to point out that the dispersion curve in Fig. 6 lies below the light line within the error of the fitting analysis. This is consistent with low losses due to minimized radiative decay as observed from the optical scattering spectra. Furthermore, it should be pointed out that the dispersion relation for the sub-radiant plasmon modes in these Ag particle chains resembles the behavior of continuous plasmonic nanowires [43], which support long propagation distances due to reduced radiative decay [44–46].

In Table 1
, we present SPP propagation decay constants *b* [μm^{−1}] for linear chains of *N* = 50 Ag nanoparticles with sizes ranging from the quasi-static limit to *a* = 50 nm. For increasing σ and decreasing *a*, the decay constant increases, denoting smaller propagation lengths. Smaller σ results in reduced decay constants as multipolar contributions lead to suppressed radiative losses when excited at energies corresponding to the sub-radiant modes. For *a* = 50 nm and σ = 2.1 the 1/e propagation length is ~3 *μ*m when excited at 2.47 eV, nearly twice as large when compared to interparticle separations of σ = 3.0. For each particle size, the excitation energies that yield minimum SPP decay are easily correlated to the optical spectra, and lie between the main dipole resonance and the quadrupole mode. The width of this band broadens for close interparticle separations, providing a unique optical window for excitation of sub-radiant modes within which efficient energy transfer takes place. Particles with sizes near the quasi-static limit displayed decay profiles that were in very good agreement with electrostatic calculations. The energy propagation in chains of such small sizes is interpreted to be supported by the optical near-field and its dependence on higher order multipole modes, especially for small separations σ.

## 6. Conclusion

In conclusion, using generalized Mie theory we have shown that propagating SPPs along linear chains of Ag nanoparticles exhibit maximum propagation distances at small separations when exciting sub-radiant plasmon modes. As the number of particles increases, these discrete modes form an excitation band at which radiative losses are minimized, becoming increasingly broad at small interparticle separations. Electromagnetic energy transport by optically induced SPPs in collections of closely spaced nanoparticles offers distinct advantages for creating optical interconnects between other waveguides because SPPs can be transported around sharp corners or can be split, for example, into two SPPs in a ‘T’ – like nanoparticle geometry. The results presented here also point to the use of chemical methods to arrange nanoparticles in close proximity to each other, thereby maximizing SPP propagation as well as the excitation bandwidth formed by the sub-radiant modes.

## Acknowledgments

This work was supported by the Robert A. Welch Foundation (C-1664), a 3M Non-Tenured Faculty Grant, and the NSF (CHE-0955286). We acknowledge Wei-Shun Chang, Christy Landes, Liane Slaughter, Peter Nordlander, and Mark Stockman for fruitful discussions.

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