Coupling and Guided Propagation along Parallel Chains of Plasmonic Nanoparticles

Here, extending our previous work on this topic, we derive a dynamic closed-form dispersion relation for a rigorous analysis of guided wave propagation along coupled parallel linear arrays of plasmonic nanoparticles, operating as optical 'two-line' waveguides. Compared to linear arrays of nanoparticles, our results suggest that these waveguides may support longer propagation lengths and more confined beams, operating analogously to transmission-line segments at lower frequencies. Our formulation fully takes into account the whole dynamic interaction among the infinite number of nanoparticles composing the parallel arrays, considering also realistic presence of losses and the frequency dispersion of the involved plasmonic materials, providing further physical insights into the guidance properties that characterize this geometry.


Introduction
Linear chains of plasmonic (silver or gold) nanoparticles have been suggested as optical waveguides in several recent papers [1]- [11]. Owing to design flexibility and relatively easy construction within current nanotechnology, the realization of such ultracompact waveguides has been thoroughly studied and analyzed in the past few years. However, the recent experimental realizations of such devices at the nanoscale have revealed challenges due to severe sensitivity to material absorption and to inherent disorder. The guided beam cannot usually travel longer than few nanoparticles before its amplitude is lost in the noise. This is mainly due to the fact that linear arrays of small nanoparticles have the property to concentrate the optical beam in a narrow region of space, in large part filled by lossy metal. If this is indeed appealing in terms of power concentration, it has the clear disadvantage of strong sensitivity to material and radiation losses.
As we have underlined in [12], a naked conducting wire at low frequencies has analogous limitations: although metals are much more conductive and less lossy in radio frequencies, connecting two points in a regular circuit with a single wire would still produce unwanted spurious radiation and sensitivity to metal absorption. This problem, which is much amplified at optical frequencies due to the poorer conductivity and higher loss of metals in the visible, is simply approached at low frequencies by closely pairing two parallel wires (or, which is the same, placing a ground plane underneath the conducting trace), forming the well known concept of a transmission-line that provides a return path for the conduction current. Analogously, applying the nanocircuit concepts [13]- [14], we have recently put forward ideas to realize optical nanotransmission-line waveguides in different geometries [15]- [16], which have been proven to be more robust to material and radiation losses and may provide wider bandwidth of operation. In particular, as we introduced in [12], one such idea consists in pairing together two parallel arrays of plasmonic nanoparticles, suggesting that the coupling among the guided modes may improve the guidance performance. In [12] we have shown that this is indeed the case: operating with the antisymmetric longitudinal mode, such parallel chains indeed may confine the beam in the background region between the chains, leading to confined propagation that is combined with robustness to material absorption and radiation losses. In particular, we have shown that operating with these modes near the light-line would, in many senses, lead to operation close to a regular transmission-line at low frequencies, but available in the visible regime.
Here, we extend our work in this area by deriving a closed-form full-wave dynamic solution for the dispersion of the eigenmodes supported by such parallel chains, fully taking into account the coupling among the infinite number of particles composing the two-chain array, even in the presence of material absorption, radiation losses and frequency dispersion. The results confirm the validity of this analogy, and they provide further insights into the operation and spectrum of modes guided by these paired arrays of nanoparticles. Applications for low-loss optical communications and sub-wavelength imaging devices are envisioned.

Dispersion Relations for Guided Propagation
Consider the geometry of Fig. 1, i.e., two identical linear arrays of plasmonic nanoparticles with radius a , period 2 d a  and interchain distance l d  . This geometry has been preliminarily analyzed in [12] for its longitudinally polarized guided modes, where it was shown that the coupling between the chains, limited in that analysis to its dominant contribution coming from the averaged current density on the chain axes, would generate the splitting of the regular longitudinal mode into two coexisting longitudinal modes, respectively, with symmetric and antisymmetric field distributions. The antisymmetric mode is the one corresponding to transmission-line operation [12], as outlined in the introduction, for which two antiparallel displacement current flows are supported by the parallel chains. A similar modal propagation has been analyzed in [9] for a related distinct geometry, consisting of longitudinal dipoles placed over a perfectly conducting plane. Also our analysis of quadrupolar chains [16] may, in the limit of 0 l  , have some analogies with this antisymmetric operation. In the following, we rigorously approach the general problem of modal dispersion along the parallel chains of Fig. 1, extending our general analysis in [10] that was valid for one isolated chain. Our formulation may fully take into account the whole coupling among the infinite nanoparticles composing the pair of arrays and the possible presence of material absorption, radiation losses and frequency dispersion. As we did in [12], here we model each nanoparticle as a polarizable dipole with polarizability  , an assumption that is valid as long as b a   , with b  being the wavelength of operation in the background material. For simplicity, we assume a scalar polarizability, implying that the particles are isotropic (nanospheres, easy to realize as colloidal metal particles), or for more general shapes focusing on one specific field polarization. In the following, we also assume an For a single isolated chain [10], the spectrum of supported eigenmodes may be split into longitudinal and transverse polarization with respect to the chain axis x .
In particular, for i x e  propagation, the corresponding guided wave number  satisfies the following closed-form dispersion relations, respectively, for longitudinal and transverse modes: can be applied also in the leaky-wave modal regime, for which Re 1       and the chain radiates as an antenna in the background region [18].
When l is finite in Fig. 1, the coupling between the two chains implies a modification of their guidance properties, which may be taken into account by considering the polarization fields induced by the electric field from each chain on the other. The fields radiated by each chain may be expanded into cylindrical waves, allowing us to write the general closed-form expressions for the coupling coefficients between the two chains.
Without loss of generality, we can assume that the particles composing the first chain, located at 0 y  , are polarized by an eigenmodal wave with dipole , where m   is the integer index for each nanoparticle of the chain. The equivalent current distribution on the x axis may be written as: where    an approximation that is consistent with the approach we used in [12]. The numerical results reported in the following sections have been obtained by considering the first ten terms in the summations (3), even though full convergence has been usually achieved after the first one or two terms. The other coupling coefficients not explicitly given in (3) or, in a more compact form: The left-hand side in Eqs. (4)-(5) consists of the product of two terms: the first determines the dispersion of the coupled modes polarized in the xy plane (among which the quasi-longitudinal antisymmetric modes that have been considered in [12]), whereas the second determines the purely-transverse modes polarized along z . It is noticed that this dispersion equation is completely general and it fully takes into account the whole dynamic interaction among the infinite particles composing the two parallel chains. Since the coupling coefficients (3) tend rapidly to zero for increased l , it is noticed that Eq. (5) represents the perturbation of the original transverse and longitudinal modes supported by the two linear chains independently given by 0 L  and 0 T  respectively [10], produced by the coupling coefficients C . In particular, it is seen that each of the three orthogonal polarizations (along , , x y z ) splits into two branches due to the coupling between the chains, one with symmetric and the other with antisymmetric properties, leading to six modal branches of guided modes, some of which supported at the same frequency. In particular, the modes in the xy plane are mixed together (i.e., the parallel chains do not support purely longitudinal or purely y  polarized modes).
In the limit of lossless particles, since L and T are real for any 1 inspecting Eq. (5) we notice that the parallel chains still support lossless guided In the following, we analyze in details the modal properties of this setup in its different regimes of operation.

Guided Modes of Parallel Chains of Silver Nanospheres
In this section we consider the different regimes of guided propagation supported by the parallel chains of Fig. 1 Since the guided modes are perturbations of the longitudinal and transverse modes supported by the isolated chains, it is of no need to analyze here again how variations in the chain geometry, i.e., in a , d and/or the involved materials, may affect the guidance of the parallel chains, since in [10] we have already studied in great details how these changes affect the guidance of isolated chains In the following, therefore, we focus on one specific realistic design of the chains and we employ the exact formulation developed in the previous section to characterize the modal properties of two of such parallel arrays coupled together. In particular, the geometry of interest is formed by colloidal silver nanospheres embedded in a glass background (

a) Quasi-longitudinal propagation (forward modes)
As we have shown in [10], an isolated linear chain of plasmonic nanoparticles supports forward-wave longitudinal guided modes ( x  polarized), satisfying the dispersion relation 0 L  , over the frequency regime for which: where   N Cl  are Clausen's functions [17]. For the case at hand (silver providing the following constraints on the polarization eigenvectors for the two chains: As an aside, it should be noted that in the leaky-wave region (blue lighter shadow in the left) the forward-wave modes are improper in nature [22], implying that the    operation, whose interesting properties we have already highlighted in details in [12], may lead to ultra-confined low-loss optical guidance in terms of optical nanotransmission-lines.  . At these frequencies, as seen in Fig. 3 Its field distribution (Fig. 5a) still shows strong confinement between the two chains, where a "quasi-uniform" magnetic field may propagate as if guided by a transmission-line. The wave is slower than in the case of Fig. 4 , once again provides worse field confinement, as expected. In this case (Fig. 5b) the field is spread around the chains and is very weak in the region between the two chains. Similar spreading is noticeable in the single isolated chain configuration of Fig. 5c, with 2.06 0.077 . We note that the field spreading in the region around the chains would also be more sensitive to radiation losses produced by disorder and technological imperfections. We predict, therefore, that the antisymmetric transmission-line operation of the parallel chains may produce more robust optical guidance confined in the region between the chains. arising from the coupling, which is nearly 90 out of phase with respect to the longitudinal polarization. In Fig. 6 , for the parallel chains of Fig. 2 and 3 we have calculated the level of transverse cross-polarization induced on the particles due to coupling, as a function of frequency. It is evident that its level increases for closer chains, as expected, and it is larger for antisymmetric modes. In the region of enhanced absorption that we have noticed in Fig. 3, the corresponding level of cross-polarization is also very high, at some frequencies even higher than the longitudinal polarization, noticeably affecting the chain guidance. The coupling is minimal near the light line and in the leaky-wave and stop-band regimes, whereas it hits its maximum somewhere inside the guidance region, whose position in frequency varies depending on the distance between the chains.

b) Quasi-transverse y -polarized propagation (backward modes)
We have reported in [10] that a single isolated linear chain may also support transversely polarized guided modes, satisfying the exact dispersion relation 0 T  . In this case, the condition on the particle polarizability is: where 1 min   is defined in [10]. In this regime the chain always supports two modes, both with the same transverse polarization: one is guided along the chain and has backward-wave properties, the other is weakly guided, with forwardwave properties and Re 1       (this eigenmode is basically a simple plane wave traveling in the background region, weakly polarizing the nanoparticles. This is not of interest for guidance purposes [10], but it is still reported here for sake of completeness). For the geometry at hand, transversely-polarized propagation is supported over the frequencies between 650THz and 800THz , in part overlapping with the longitudinally-polarized regime, as reported in Fig Due to the modal coupling in the xy plane, when the coupling between parallel chains is considered the quasi-transverse modes still satisfy the dispersion relations (8) given in the previous section and the polarization eigenvectors obey the same relations (9). It should be noticed, however, that in this regime the modes are quasi-transverse, and therefore the antisymmetric mode now corresponds to parallel y  polarized chains, whereas the symmetric mode supports anti-parallel polarization along y , consistent with (9).     ). Also in this case, the perturbation from the isolated chain is stronger and the bandwidth of backward operation may be substantially increased by using two parallel chains in the symmetric mode. Here leaky modes are proper in nature and therefore Eqs. (3) also apply to this regime in the way they are written. Both in Fig. 7 and 8, for completeness, we have also reported the modal branch associated with the weakly guided forward-wave transverse mode, which is located very close to the light line. Consistent with its forward-wave properties, Im 0       for this mode. As outlined above, this mode is of minor interest for guidance purposes, since it is a minor perturbation of a plane wave traveling in the background region, very weakly affected by the presence of the chains. It is noticed, as expected, that this second branch is present only for the antisymmetric modes, whose y  polarization is in the same direction for both chains. Figure 9 shows the magnetic field for these backward-wave modes as in Fig. 8  propagation (over one wavelength) is achievable using coupled parallel chains. Figure 10 reports the level of longitudinal cross-polarization for the chains of Figs. 7 and 8. In this scenario, the cross-polarization is in general lower than for quasi-longitudinal modes and it is stronger for symmetric modes. Once again, the cross-polarization is stronger for closely coupled chains and it has some resonant peaks in the middle of the guidance region, for which the damping is increased correspondingly.

c) Purely transverse z-polarized propagation (backward modes)
When the chains are polarized along ẑ the supported modes are purely transverse, consistent with (4). Due to the symmetry, the properties for isolated chains are identical to those described in the previous section, and therefore here we discuss how the coupling may affect differently the backward-wave guidance properties in this polarization. The coupling coefficient zz C splits the transverse modal branch of propagation into two modes, with dispersion relations: providing the following constraints on the polarization eigenvectors for the two chains: Also in this case, symmetric modes allow slightly longer propagation lengths near the light line, where the coupling is stronger. Increasing the coupling ( 30 l nm  ), as in Fig. 12, the perturbation is stronger, even if the trend is similar as in the previous scenario. Figure 11 -(Color online). Similar to Fig. 2 and Fig. 7, but for transverse z  polarized modes.
Here the interchain distance is 50 l nm  .
29 Figure 12 -(Color online). Similar to Fig. 11, but for 30 l nm  . Figure 13 reports the calculated orthogonal electric field distribution in the xy plane for the modes of Fig. 11 Fig. 12. The field confinement in this polarization is not drastically different from that of an isolated chain, as evident from the figure, and the main advantage of using parallel chains may reside in the longer propagation distance of symmetric modes near the light line.

Conclusions
We have presented here a fully general and complete theoretical formulation for the analysis of the dynamic coupling between two parallel linear chains of plasmonic nanoparticles operating as optical waveguides. These chains may support up to eight different guided modes with different polarization properties in the same range of frequencies, which we have fully analyzed here. We have shown that, compared to linear arrays, these waveguides may support longer propagation lengths and ultra-confined beams, operating analogously to transmission-line segments at lower frequencies. In particular, our results confirm that by operating near the light line with antisymmetric quasi-longitudinal modes we may achieve relatively long propagation lengths (of several wavelengths) and ultraconfined beam traveling, similar to a transmission-line, in the background region sandwiched between the two antisymmetric current flows guided by the chains. Our analysis has fully taken into account the whole dynamic interaction among the infinite number of nanoparticles, also considering presence of material and radiation losses and the frequency dispersion of the involved plasmonic materials.