## Abstract

The energy transport properties of plasmonic waveguides can be analyzed by solving the dispersion relation for surface plasmon-polaritons (SPPs). We use this approach to derive an approximate analytical expression for SPP propagation length when the waveguide is composed of linearly arranged metallic nanoparticles, while assuming that metal losses are small or partially compensated by gain. Applied to metal–dielectric (composite) nanospheres, the obtained expression allows us to optimize the performance of the waveguide and arrive at a number of practical design rules. Specifically, we show that SPP attenuation can be minimized at a certain interparticle distance for transverse modes, but gradually grows for both longitudinal and transverse modes with the increase of particle separation. We also show that the two basic methods of supplying gain to the system, i.e., embedding the particles into a gain medium or having a metal–gain composition for the particles, do not perform equally well and the former method is more efficient, but the way the two methods affect depends on the polarization of SPPs. To investigate the role of the nanoparticles’ arrangement in determining SPP characteristics, we follow a purely numerical approach and consider a two-segment bent waveguide as an example. Analyzing the waveguide’s transmission shows that it behaves in an oscillatory manner with respect to the angle between the two segments and is therefore higher for certain angles than for the others. This suggests that, in the design of waveguides with bends, careful attention needs to be paid in order to avoid bend angles that yield low transmission and to choose angles that give maximum transmission.

© 2011 Optical Society of America

## 1. Introduction

The demand for faster, smaller communication interconnects for on-chip information transport has resulted in enormous interest on nanoscale optical circuitry as their electronic counterparts are incapable of providing the required high level of miniaturization [1, 2]. A promising strategy to guide and manipulate light in the nanoscale is to employ surface plasmon-polaritons (SPPs), which are actually the electromagnetically excited surface charge waves that propagate in metal-dielectric (plasmonic) structures [3–5]. What makes SPPs ideal candidates for nanoscale guiding of light is that they are confined within the plasmonic structures in dimensions at least an order of magnitude smaller than the wavelength of light, as opposed to the half-wavelength limit in dielectrics [6, 7].

Various types of plasmonic structures are known to support SPPs. Thin metallic films embedded in a dielectric is one of them that is known for supporting long-distance SPP propagation of up to several millimeters, although the lateral confinement it provides (about several micrometers) is rather poor [8, 9]. Another type of one-dimensional plasmonic waveguides, which enable much stronger confinement of SPPs (up to tens of nanometers), is a metal–dielectric–metal heterostructure [10,11]. A nanometer-scale confinement can be also achieved by squeezing the planar metallic structure into a one-dimensional rod (or nanowire) [12, 13], but this increases the dissipation of SPPs and makes them highly sensitive to structural imperfections [14, 15]. The constraints associated with a nanowire can be somewhat relaxed by replacing it with a chain of closely spaced metallic nanoparticles, which offers deep subwavelength confinement and additional degrees of freedom in design, but at the expense of SPP decay lengths of a few micrometers [16]. A plethora of studies have investigated both theoretically and experimentally the characteristics of metallic nanoparticle chains to exploit the benefits they provide in guiding SPPs [17–22], though none of the proposed applications has reportedly been practically realized.

The strong damping of SPPs is possibly the main reason that prevents the practical use of a chain of metallic nanoparticles. As in all plasmonic structures, SPP damping in these chains is predominantly due to Ohmic losses in metal [23, 24]. Recent interest in addressing this issue has been to introduce optical gain to the system, either by embedding the nanoparticles into a gain medium [25, 26] or by using gain–metal composite particles [27]. The former method is proposed to be realized by having dye-doped silica as the host medium, while metal-coated quantum dots has been suggested for the latter. However, the relative advantages of these two approaches are not yet known and it seems timely that an in-depth analysis is carried out to investigate their efficiency in assisting low-loss SPP propagation. Another approach of suppressing SPP damping is by appropriately choosing the geometric parameters of the system. For example, it has been observed that SPPs exhibit improved propagation lengths when the particle separation is decreased [28] or when spheroidal nanoparticles are used instead of spherical ones [29, 30]. Although the dependance of particle shape on SPP propagation has been extensively studied, the effect of particle separation is less explored and needs further investigation.

The transmission characteristics of nanoparticle chains are usually analyzed through the SPPs’ dispersion relation of a linearly arranged chain. The dispersion relation is obtained by approximating the nanoparticles as point-like electric dipoles and numerically solving the characteristic equation that governs the electromagnetic coupling between them [31–34]. This method, referred to as the coupled-dipole method (CDM) in the literature, drastically reduces the complexity of the full solution of Maxwell’s equations for a nanoparticle assembly [35, 36]. Nevertheless, the dispersion equation does not lend itself to a closed-form analytical solution and requires to be solved numerically, making it difficult to intuitively evaluate the system’s parameters. To this end, compact analytical descriptors to characterize SPPs are clearly in demand, as they could be translated into straightforward recipes that allow parameters to be chosen optimally.

Although a linear waveguide serves as a convenient system to understand SPP characteristics, most practical waveguides would require SPPs to be guided in structures that are arranged in more complex layouts (e.g., L-bends or Y-junctions). Apart from the initial works of Brongersma *et al.* [37], who considered SPP propagation in an L-bend whilst restricting themselves to only near-field interparticle interactions, the effect of the waveguide’s layout on SPP propagation has not received much attention. Since there is no evidence that implies SPP propagation characteristics in linear waveguides are directly applicable for complex geometries, it is imperative to carry out further investigation.

In this paper, we thoroughly investigate how SPPs are affected by the geometric and material parameters of the nanoparticle system using analytical and numerical tools. We start our analysis in Section 2 by deriving analytical expressions based on the CDM to characterize the damping of SPPs. We use these expressions in Sections 3 and 4 to consider how interparticle distance and methods of supplying gain affect SPP attenuation, and arrive at useful guidelines on choosing the best parameters for the nanoparticle chain. In Section 5, we investigate the role of the nanoparticles’ arrangement on SPP propagation with the help of a purely numerical implementation of the CDM. We consider a waveguide constructed of two linear segments forming a simple bend (as an example) and analyze the behavior of SPPs with respect to the angle between the two segments and the polarization of the excitation source. In Section 6, we summarize our results and conclude the paper.

## 2. Analytical treatment of SPPs in a linear chain of metallic nanoparticles

The optical response of a chain of metallic nanoparticles is commonly investigated by the CDM, which treats the nanoparticles as dipoles and the SPPs as waves in the system of coupled dipoles [38]. This simplification is made possible by two approximations. One is due to the exceedingly small size of the particles compared to the wavelength of exciting light, which is usually justified for particle sizes of up to 100 nm at optical wavelengths [39]. The other is due to the assumption that the particles are separated far enough and their interactions are dominated by the dipolar electric fields emanating from them. When spherical nanoparticles with radii *R* are separated by a distance *d* ≥ 2.5*R*, CDM was shown to be in very good agreement with methods that consider higher-order multipoles in describing the extinction of a chain of nanoparticles. Even if the nanoparticles were almost touching each other (*d* ∼ 2*R*), error in the extinction calculated with the CDM is still less than 10% [40]. The CDM becomes more accurate when the nanoparticles become far apart, specifically, when *d* ≥ 3*R*, the dispersion of SPP modes determined by taking into account scattering by higher-order multipoles [38] and full-blown numerical experiments were shown to be very well approximated by the CDM [19].

Consider a linear chain of *N* identical and equally spaced metallic nanoparticles, each represented by a dipole with polarizability *α* and embedded in a homogeneous host medium of permittivity *ɛ*_{h}. Suppose also that the optical excitations in the system vary with time in proportion to *e*^{−iωt}. The polarizability of a nanoparticle in a radiation field is obtained from its quasistatic polarizability *α*_{0} and correction terms that allow for radiation damping and finite size of the particle, which reads

*R*; $q=\omega \sqrt{{\varepsilon}_{\text{h}}}/c$ and

*c*is the speed of light in vacuum [41, 42]. The quasistatic polarizability depends on the composition of the particle and its surrounding media and is given by

*α*

_{0}=

*R*

^{3}(

*ɛ*

_{np}–

*ɛ*

_{h})/(

*ɛ*

_{np}+ 2

*ɛ*

_{h}), when the nanoparticle’s permittivity is denoted by

*ɛ*

_{np}. The term with a

*q*

^{3}in the above expression accounts for the radiative decay due to spontaneous emission by the induced dipole, while the

*q*

^{2}term describes the radiation’s depolarization across the particle’s surface as a result of its finite size. For a composite metallic nanoparticle made of a dielectric core and metallic shell (see Fig. 1), the permittivity

*ɛ*

_{np}=

*ɛ*

_{m}

*ɛ*/

_{a}*ɛ*, where

_{b}*ɛ*=

_{a}*ɛ*

_{c}(3 – 2

*v*) + 2

*ɛ*

_{m}

*v*and

*ɛ*=

_{b}*ɛ*

_{c}

*v*+

*ɛ*

_{m}(3 –

*v*), is expressed through the complex permittivities

*ɛ*=

_{j}*ɛ*′

*+*

_{j}*iɛ*″

*of metal (*

_{j}*j*= m) and core medium (

*j*= c) and the volume fraction

*v*= 1 –

*x*

^{3}of metal, with

*x*=

*r*

_{c}/

*R*being the core-shell radii ratio [43].

Within the formalism of the CDM, the SPP modes that exist in the nanoparticle chain are given by the eigensolutions of a 2*N* × 2*N* matrix which governs the interparticle interactions [33]. When the chain has sufficiently large number of nanoparticles, it can be treated as infinitely long and, for SPPs propagating with a wave number *k*, the induced dipole moment on a *m*th particle can be taken as ∝ *e ^{ikmd}* [37, 44]. This assumption allows the matrix equation to be reduced to an analytical expression of the form

*S*= 2(Φ

_{3}–

*iqd*Φ

_{2}) for SPPs that are polarized along the chain’s axis (L modes) and

*S*= −Φ

_{3}+

*iqd*Φ

_{2}+ (

*qd*)

^{2}Φ

_{1}for transversely polarized SPPs (T modes); Φ

*= Li*

_{j}*[*

_{j}*e*

^{i(q+k)d}] + Li

*[*

_{j}*e*

^{i(q–k)d}] (

*j*= 0,1,2,3) with Li

*(*

_{j}*z*) being the polylogarithm function [26, 31, 32, 45]. The SPP modes in this case are given by the complex-valued solutions for

*k*that satisfy Eq. (1), with Re

*k*describing the dispersion of SPPs and Im

*k*representing the attenuation, usually defined in terms of the SPP propagation (or decay) length

*L*

_{SPP}= (2Im

*k*)

^{−1}. To determine

*k*, both the real and imaginary parts of

*F*should be equated to zero and the resulting two equations have to be solved simultaneously. But when none of the materials have losses or gains, the imaginary part of

*F*vanishes below the light line (for Re

*k*> Re

*q*) and only solving of equation Re

*F*= 0 is required, yielding real-valued solutions for

*k*[46]. Using these real-valued solutions and approximate analysis methods, the dispersion and attenuation of SPPs in a system having small amounts of material losses or gains can be determined, as it is shown below.

Assume that a real-valued solution *k*_{0}, which satisfies Re *F* = 0 when the losses and gains are absent in the system, is perturbed by a Δ*k* amount due to the presence of losses and gains, contributed by small imaginary parts for *ɛ*_{c}, *ɛ*_{m}, and *ɛ*_{h}. The imaginary parts of the permittivities are assumed to be small compared to the real parts, which is true for low-loss metals [51] and typical gain media [26]. In this case, we may expand Eq. (1) in a Taylor series and retain only the first-order terms, which gives

*∂*denotes the partial derivative

_{ξ}Y*∂Y/∂ξ*. This leads to an expression for Δ

*k*

*S*in Eq. (2) depend on the mode’s polarization and are given by

*= Li*

_{j}*[*

_{j}*e*

^{i(q+k)d}] − Li

*[*

_{j}*e*

^{i(q–k)d}] (

*j*= 0,1,2). Now, the propagation length of SPPs is given by

*L*

_{SPP}= [2Im (

*k*+ Δ

*k*)]

^{−1}= (2ImΔ

*k*)

^{−1}.

## 3. Effect of particle separation on SPP propagation length

It is apparent from the above expressions that the propagation length of SPPs can be optimized not only by introducing gain but also by modifying the system’s geometric parameters. For a given composition of the nanoparticles, a parameter that can be conveniently varied is the interparticle distance *d*. Varying *d* not only affects the propagation length of SPPs, but also strongly alters their dispersion. This is illustrated in Fig. 2, where the dispersion curves of three chains of nanospheres are plotted for *d* = *σ*, 4*σ*/3, and 5*σ*/3 with *σ* = 3*R*, assuming that metal is lossless. In a realistic system where metal has a small amount of loss (corresponding to *γ* ≪ *ω _{p}* in the Drude model), dispersion curves are slightly modified towards the end of the Brillouin zone edge. Specifically, the modes do not acquire a limiting value at

*k*=

*σ*/(2

*d*), but roll-over, resembling leaky modes [47]. It is seen from the figure that, when the particle separation is increased, the propagating bandwidth (shown by shaded regions in the Fig. 2) is drastically reduced. This is due to the weakening of the interparticle coupling. The bandwidth compression is stronger for L modes where it is reduced by about a factor of ten when

*d*is increased from

*σ*to 5

*σ*/3, whereas it is only halved for T modes with the same increase of

*d*.

Typically, the design problem associated with the particle separation is to find the optimum *d* that maximizes the SPP propagation length for a given excitation signal centered at a frequency *ω*_{0}. The frequency *ω*_{0} is not actually a constraint of the system, since a chain of nanoparticles may suitably be designed so that *ω*_{0} can be arbitrarily chosen. As we shall discuss below, this can be realized by varying *x*, which allows the SPPs’ bandwidth to be shifted in frequency. In the present example, we take *ω*_{0} = 0.168 for L modes and *ω*_{0} = 0.158 for T modes, so that at this frequency SPPs are sustained by the chains for the whole range of *d* values under consideration.

Figure 3(a) shows the exact solution for the propagation length determined by Eq. (1), and the approximate solution obtained with Eq. (2) for the *d* = *σ* chain when *γ* = 0.01*ω _{p}*. It is clearly seen that the approximate solution agrees well with the exact one in the regime of small metallic losses. In Fig. 3(b), we plot the propagation length as a function of

*d*. Although the accuracy of CDM worsens for

*d*≤

*σ*, we notice that

*L*

_{SPP}increases for both L and T modes when the particles become closer to each other, which could be anticipated as the near-field coupling is enhanced. However, an unexpected trend can be observed near

*d*= 1.6

*σ*for T modes:

*L*

_{SPP}reaches a maximum of about 75

*σ*before steeply descending for larger

*d*. Thus, keeping

*d*≈ 1.6

*σ*could be useful if the goal is to achieve long-range T modes, although for L modes the propagation distances reduce below 15

*σ*. Unlike for T modes, the propagations lengths for L modes gradually reduce when

*d*is increased. But, as can be observed from the figure, selecting

*d*between ∼

*σ*and

*∼*1.2

*σ*would support both L and T modes with sufficiently large propagation lengths.

It is also worth noting that the distance between nanospheres may affect the confinement of SPP mode in the chain, although previous works suggest that such a confinement mainly depends on the nanospheres’ radii [48]. To estimate the degree of mode confinement, the near fields of nanospheres need to be computed, which can be accomplished using full-blown numerical simulations (performed with, for example, FDTD method).

## 4. Effect of gain on SPP propagation length

It is well known that supplying gain to plasmonic nanostructures can compensate for SPP dissipation caused by Ohmic losses and other damping mechanisms [49–51]. To suppress the damping of SPPs in a metallic nanoparticle chain, recent works have considered embedding nanoparticles into active media [17,25,26] or putting gain materials inside metallic shells [27]. Since these are the two basically different ways to overcome material losses in the system, it is important to compare them in terms of their efficiency to compensate for losses.

To analyze the above two approaches in detail, we first focus on the dispersion of SPPs assuming that the material system is free of loss and gain. In Figs. 4(a) and 5(a), we plot the dispersion branches of longitudinal and transverse SPP modes for *d* = *σ*, *n*_{c} = 3.5, *n*_{h} = 1.5, and four values of *x* (0, 0.4, 0.6, and 0.8). As seen from the dispersion curves, both the L and T modes of the chain with *x* = 0 split into high- and low-frequency branches for *x* > 0. We denote them by *ω*_{+} and *ω*_{−} modes, respectively. These modes can be shifted in frequency by tuning the parameter *x*, which is a useful property of a composite nanosphere chain, as it gives flexibility to choose the excitation frequency [52,53]. One can see from the figure that when *x* is increased, SPP bandwidth of both the *ω*_{±} modes decreases. With transverse SPPs, the bandwidth of the *ω*_{+} mode for a given *x* is wider than that of the *ω*_{−} mode, while for the longitudinally polarized SPPs the situation is opposite. For example, the transverse *ω*_{+} mode for *x* = 0.8 has a larger bandwidth than the corresponding longitudinal mode, although the bandwidths of longitudinal modes are generally larger when the core’s thickness is small (cf. the dispersion curves for *x* = 0 and *x* = 0.4).

Referring to the inset of Fig. 5, it is observed that a peculiarity starts to appear in the transverse *ω*_{−} mode when *x* is increased further close to unity. Unlike other modes where the group velocity *v _{g}* =

*∂*tends to zero when Re

_{k}ω*k*→ 0.5,

*v*reaches a zero in the middle of the Brillouin zone before acquiring negative values and then finally vanishing at Re

_{g}*k*= 0.5. This behavior suggests that the transverse

*ω*

_{−}modes are backward propagating waves when

*x*→ 1. In the vicinity of critical frequency corresponding to

*v*= 0, one should expect enhanced interaction of SPPs with the environment [55].

_{g}With the dispersion relations computed for the chain in the loss- and gain-free regime, the effect of introducing gain to the core or host medium can be readily investigated with the help of the analytical expressions developed in Section 2. To do this, let us assume that the gain provided by the core or host media is characterized by small gain coefficients *κ*_{c} and *κ*_{h}, which contribute to the permittivities of the two regions as
${\varepsilon}_{\text{c}}\approx {n}_{\text{c}}^{2}-2i{\kappa}_{\text{c}}{n}_{\text{c}}$ and
${\varepsilon}_{\text{h}}\approx {n}_{\text{h}}^{2}-2i{\kappa}_{\text{h}}{n}_{\text{h}}$. A measure of the efficiency of the two approaches in assisting low-loss SPPs is the amount of gain required to fully compensate for losses. Since SPPs propagating along the chain without amplification or damping are characterized by a real wave number, the amount of gain in the core medium required to fully suppress the damping can be found by equating ImΔ*k* in Eq. (2) to zero. The result is

We plot *κ*_{c} as a function of frequency for different *x* in Figs. 4(b) and 5(b). Although the function *κ*_{c}(*ω*) extends over the whole frequency range due to being independent of *k*_{0} [cf. Eq. (3)], it has a physical meaning only within the SPP bands. Keeping this in mind, we notice that there are certain interesting features in *κ*_{c} that may aid in choosing the composition of the nanospheres as well as excitation conditions. One important observation for both L and T modes is that the required amount of gain to compensate for losses becomes larger for the *ω*_{+} bands with the increase of the core radius. In other words, the propagation losses for the *ω*_{+} bands are increased with the reduction of the shell thickness. Since we did not take into account the increase of the damping rate *γ* in the Drude model when the thickness of metal is decreased [54], it can be expected that these losses can be even larger for the *ω*_{+} band. However, one notices that the increase of losses in the *ω*_{+} band is contrasting to the variation of the bandwidths (recall that the bandwidth of longitudinal *ω*_{+} bands are decreased while their transverse counterparts increase). Also, for a given *ω*_{+} band, the required gains for the T mode are smaller than those for the L mode. This suggests that when gain is provided by the core medium, T-polarized SPPs are preferable to L-polarized SPPs at high frequencies. At the same time, it is seen that, for all *ω*_{−} bands, the required amount of gain is roughly the same (even when *x* becomes close to unity). Since the bandwidths of L modes are also appreciably large for *ω*_{−} bands, L polarized SPPs are more suitable for these frequencies.

A similar analysis can be conducted when gain is provided by the host medium. In this case, the gain required to compensate for damping acquires the form

This expression as a function of SPP frequency is plotted by solid curves in Figs. 4(b) and 5(b). The important thing to notice here is that the gains required from the host medium are always smaller than the gains required from the core medium. This is due to the amplification of the radiated electromagnetic fields by the nanoparticles, as opposed to the reduction of the effective damping rate within the particles themselves. Another consequence of the SPPs’ interaction with radiation fields is that for frequencies close to the light line, SPPs require smaller gains to suppress the losses. It is also observed that the gain requirement for *ω*_{+} bands remains approximately the same when *x* is increased, but for *ω*_{−} bands more gain is required to compensate for losses.

The absolute values of gains required to compensate for Ohmic losses in the above scenarios are readily obtained using the relation 2(*ω*/*c*)*κ _{j}* (

*j*= h, c). For example, the value

*κ*

_{h}≈ 0.005 corresponding to

*ω*= 0.15 in Fig. 5(a) translates to a gain of about 1250 cm

^{−1}for

*σ*= 75 nm. This value is close to the gain that is required to compensate for damping of localized plasmon resonance in a silver nanosphere [51]. It should be also recognized that, according to the expressions for

*κ*

_{c}and

*κ*

_{h}, the amount of gain varies in proportion to the losses in metal.

## 5. SPP propagation in a piecewise linear chain

In a practical scenario, transfer of optical energy using a chain of nanoparticles would necessarily include guiding SPPs in geometries other than straight lines. The analytical treatment employed in the previous sections becomes inappropriate in this case and requires the numerical form of the CDM to be invoked. The numerical form considers a finite-length chain, typically excited by a localized source such as a metallic nanotip, and determines the dipole moments induced on the nanoparticles by solving a matrix equation [17]. For a chain of *N* nanoparticles arranged in an arbitrary manner, this equation gives the dipole moments **p**^{(np)} induced on the nanoparticles as

**p**

^{(ext)}(see Appendix for the details).

With this formalism, we focus on a chain of 80 nanospheres constructed of two linear segments forming a *θ*-degree bend, as illustrated in Fig. 6. Considering the chain being illuminated by continuous-wave sources polarized in the coordinate directions, the transmission of differently polarized SPPs can be characterized by the magnitude of the dipole moment induced on the last nanosphere. Figure 6 shows the normalized magnitude of the last nanosphere’s dipole moment |**p**| obtained from Eq. (4) as a function of *θ*.

Referring to the figure, an immediate observation is that the presence of the junction results in inducing perpendicularly polarized SPPs in addition to SPPs polarized parallel to the source. For example, we see in Fig. 6(a) that both *x*- and *y*-polarized SPPs are induced with the increase of *θ*, even though the source is polarized in the *x* direction. It is also to be noticed that *y*-polarized SPPs dominate over *x*-polarized ones for both *x*- and *y*-polarized sources when the bend angle is large, specifically, when *θ* ≳ 40°, as seen from the blue curves in Figs. 6(a) and 6(b). This indicates that irrespective of the source polarization, SPPs polarized in the *y* direction propagate with less attenuation (in a linear chain this corresponds to T-polarized SPPs). Another interesting observation from the figure is that the variation of SPP transmission with respect to *θ* is oscillatory, implying that certain bend angles are preferable than others for SPP propagation. For instance, at *θ* ≈ 25° and *θ* ≈ 55°, the magnitude of the 80th dipole moment reaches its local minimum, while at *θ* ≈ 40° it reaches a local maximum for the *x*-polarized source. Similar behavior is displayed by SPPs polarized in the *x*, *y*, and *z* directions when the source is *y*-or *z*-polarized. The fluctuations exhibited by the transmission spectra in Fig. 6 are due to the constructive and destructive interference of the far fields emitted by different nanospheres. As may be seen from Fig. 6(c), these fluctuations exist regardless of the SPP frequency.

As a concluding remark, it is worthwhile to emphasize the importance of investigating waveguiding layouts other than straight lines, because the SPP characteristics in such structures cannot be elucidated by simple considerations. As we observed, the break of the symmetry in a linear chain by an irregularity such as a bend, causes the magnitudes and phases of the induced dipole moments to change abruptly (figures not shown) and results in completely different spectra than those correspond to a linear chain.

## 6. Conclusions

In summary, we have investigated how SPP propagation in metallic nanoparticle chains are affected by different parameters of the system such as interparticle distance, material gains and chain layout. Based on the CDM, we developed compact analytical expressions to describe SPP damping in a linear chain when the Ohmic losses are small or partially compensated by gain. Our analysis revealed that SPP propagation length for T modes acquires a local maximum at a certain interparticle distance and gradually decreases for both L and T modes when the particle separation is increased. The increase of interparticle distance also results in compressing the bandwidth of L modes at a higher rate than that for T modes. Considering SPPs along a chain of composite nanospheres where each of the L and T modes splits into high- and low-frequency modes, our analysis showed that embedding the chain into a gain medium is more effective in suppressing damping than having a gain medium inside a metallic shell regardless of the mode type. It was also noticed that when gain is provided by the latter method, T-polarized SPPs are preferable to L-polarized SPPs for the *ω*_{+} modes, because with L modes, in addition to the increased gain requirement, the mode bandwidth also gets reduced. We investigated the effect of the chain’s layout on SPP propagation by invoking the numerical form of the CDM and considering a chain constructed of two linear segments forming a bend. We observed that SPP transmission varies in an oscillatory manner with respect to the bend angle, implying that certain angles of the bend are more preferable for SPP propagation than others. It was also shown that regardless the in-plane polarization of the excitation source, the component of the induced dipole moment perpendicular to the propagating direction (corresponding to T polarization in a linear waveguide) has a larger magnitude than the parallel component.

## Appendix

Below, we briefly summarize the coupled-dipole method employed for the analysis of SPP propagation along an arbitrarily shaped chain of metallic nanoparticles. Consider a waveguide constructed of metallic nanoparticles that are located at sites *m* = 1,...,*N*. The electric field incident on a nanoparticle at the *n*th site, which is due to electric fields generated by external sources and scattered by other nanoparticles, can be represented in the form

**E**

*denotes the electric field at the*

_{nm}*n*th site due to electric field generated by a point-like scatterer or a source at site

*m*; without loss of generality, we assume that there is only a single external source in the system, located at site

*s*. This relationship can be rewritten in terms of the dipole moments

**p**

*induced on the nanoparticles and the dipole moment of the source*

_{m}**p**

*as*

_{s}**r**

*=*

_{uv}**r**

*–*

_{u}**r**

*,*

_{v}*r*= |

_{uv}**r**

*|,*

_{uv}**r̂**

*=*

_{uv}**r**

*/*

_{uv}*r*,

_{uv}*f*(

*r*) = (3 – 3

*iqr – q*

^{2}

*r*

^{2})

*e*/

^{iqr}*r*

^{3}, and

*g*(

*r*) = (−1+

*iqr*+

*q*

^{2}

*r*

^{2})

*e*/

^{iqr}*r*

^{3}. By writing this equation in a matrix form, the unknown dipole moments of the nanoparticles are found to be

**p**

^{(np)}= (

*p*

_{1,x}...

*p*

_{N,x}

*p*

_{1,y}...

*p*

_{N,y}

*p*

_{1,z}...

*p*

_{N,z})

^{T}, ${\mathbf{p}}^{(\text{ext})}={\left({p}_{s,x}^{(\text{ext})}{p}_{s,y}^{(\text{ext})}{p}_{s,z}^{(\text{ext})}\right)}^{\text{T}}$, the superscript T denotes matrix transpose, and Inv denotes matrix inversion; ${\mathcal{M}}_{\mathit{NN}}^{(\text{np})}$ is a 3

*N*×3

*N*block matrix comprising of nine

*N*×

*N*matrices

*A*(

_{uv}*u, v*= 1, 2, 3) with elements

*δ*= 1 − Δ

_{nm}*is the Kronecker delta (*

_{nm}*n*,

*m*= 1,...,

*N*), and Δ

*is implicitly defined. The matrix ${\mathcal{M}}_{\mathit{Ns}}^{(\text{ext})}$ is a 3*

_{nm}*N*×3 matrix composed of nine

*N*-dimensional vectors

*B*(

_{uv}*u, v*= 1, 2, 3)

*M*point sources,

**p**

^{(ext)}becomes a 3

*M*-dimensional vector and ${\mathcal{M}}_{Ns}^{(\text{ext})}$ should be replaced by the 3

*N*×3

*M*matrix ${\mathcal{M}}_{\mathit{NM}}^{(\text{ext})}$.

## Acknowledgments

The work of I. D. Rukhlenko and M. Premaratne was sponsored by the Australian Research Council’s (ARC’s) Discovery grant DP110100713.

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