Engineering plasmon dispersion relations: hybrid nanoparticle chain - substrate plasmon polaritons

We consider the dispersion relations of the optical excitations in a chain of silver nanoparticles situated above a metal substrate and show that they are hybrid plasmon polaritons, composed of localized surface plasmons and surface plasmon polaritons. We demonstrate a strong dependence of the system’s optical properties on the plasma frequency of the substrate and that choosing the appropriate plasma frequency allows one to engineer the modes to have a very high, very low or even negative group velocity. For the latter, Poynting vector calculations reveal opposite phase and energy propagation. We expect that our results will contribute to the design of nano-optical devices with specific transport properties. © 2015 Optical Society of America OCIS codes: (250.5403) Plasmonics; (240.6680) Surface plasmons; (240.5420) Polaritons; (160.3918) Metamaterials; (260.2030) Dispersion. References and links 1. W. Barnes, A. Dereux, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). 2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). 3. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nature Materials 9, 193–204 (2010). 4. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photonics 4, 83–91 (2010). 5. N. J. Halas, S. Lal, W.-S. Chang, S. Link, and P. Nordlander, “Plasmons in strongly coupled metallic nanostructures,” Chemical Reviews 111, 3913–3961 (2011). 6. S. A. Maier, Plasmonics: Fundamentals and Applications, (Springer, 2007). 7. L. Novotny and B. Hecht, Principles of Nano-Optics, (Cambridge University, 2006). 8. J. J. Mock, D. R. Smith, and S. Schultz, “Local Refractive Index Dependence of Plasmon Resonance Spectra from Individual Nanoparticles,” Nano Letters 3, 485–491 (2003). 9. M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chemical Reviews 108, 494–521 (2008). 10. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Letters 10, 2342–2348 (2010). 11. M. Piliarik, P. Kvasnička, N. Galler, J. R. Krenn, and J. Homola, “Local refractive index sensitivity of plasmonic nanoparticles,” Optics Express 19, 9213–9220 (2011). 12. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Optics Letters 23, 1331–1333 (1998). 13. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Physical Review B 67, 205402 (2003). 14. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials 2, 229–232 (2003). #223934 $15.00 USD Received 29 Sep 2014; revised 18 Nov 2014; accepted 19 Nov 2014; published 28 Jan 2015 © 2015 OSA 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002280 | OPTICS EXPRESS 2280 15. A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Physical Review B 74, 205436 (2006). 16. B. Willingham and S. Link, “Energy transport in metal nanoparticle chains via sub-radiant plasmon modes,” Optics Express 19, 6450–6461 (2011). 17. P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical Antennas,” Advances in Optics and Photonics 1, 438–483 (2009). 18. A. F. Koenderink, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Letters 9, 4228–4233 (2009). 19. L. Novotny and N. van Hulst, “Antennas for light,” Nature Photonics 5, 83–90 (2011). 20. R. S. Pavlov, A. G. Curto, and N. F. van Hulst, “Log-periodic optical antennas with broadband directivity,” Optics Communications 285, 3334–3340 (2012). 21. J. Munárriz, A. V. Malyshev, V. A. Malyshev, and J. Knoester, “Optical Nanoantennas with Tunable Radiation Patterns,” Nano Letters 13, 444–450 (2013). 22. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures.” Science 302, 419–422 (2003). 23. P. Nordlander and E. Prodan, “Plasmon Hybridization in Nanoparticles near Metallic Surfaces,” Nano Letters 4, 2209–2213 (2004). 24. G. Lévêque and O. J. F. Martin, “Optical interactions in a plasmonic particle coupled to a metallic film,” Optics Express 14, 9971–9981 (2006). 25. J. J. Mock, R. T. Hill, A. Degiron, S. Zauscher, A. Chilkoti, and D. R. Smith, “Distance-dependent plasmon resonant coupling between a gold nanoparticle and gold film,” Nano Letters 8, 2245–2252 (2008). 26. A. Farhang, N. Bigler, and O. J. F. Martin, “Coupling of multiple LSP and SPP resonances: interactions between an elongated nanoparticle and a thin metallic film,” Optics Letters 38, 4758–4761 (2013). 27. C. Lemke, T. Leiß ner, A. B. Evlyukhin, J. W. Radke, A. Klick, J. Fiutowski, J. Kjelstrup-Hansen, H.-G. Rubahn, B. N. Chichkov, C. Reinhardt, and M. Bauer, “The Interplay between Localized and Propagating Plasmonic Excitations tracked in Space and Time,” Nano Letters 14, 2431–2435 (2014). 28. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Physical Review B 62, R16356–R16359 (2000). 29. D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: the role of chain length unravelled,” Nano Letters 5, 985–989 (2005). 30. A. B. Evlyukhin and S. I. Bozhevolnyi, “Surface plasmon polariton guiding by chains of nanoparticles,” Laser Physics Letters 3, 396–400 (2006). 31. A. B. Evlyukhin, S. I. Bozhevolnyi, A. L. Stepanov, R. Kiyan, C. Reinhardt, S. Passinger, and B. N. Chichkov, “Focusing and directing of surface plasmon polaritons by curved chains of nanoparticles,” Optics Express 15, 16667–16680 (2007). 32. I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Surface plasmon polaritons in curved chains of metal nanoparticles,”, Physical Review B 90, 075405 (2014). 33. R. Tellez-Limon, M. Fevrier, A. Apuzzo, R. Salas-Montiel, S. Blaize, “Theoretical analysis of Bloch mode propagation in an integrated chain of gold nanowires,” Photonics Research 2, 24–30 (2007). 34. R. de Waele, A. F. Koenderink, and A. Polman, “Tunable Nanoscale Localization of Energy on Plasmon Particle Arrays,” Nano Letters 7, 2004–2008 (2007). 35. A. V. Malyshev, V. A. Malyshev, and J. Knoester, “Frequency-controlled localization of optical signals in graded plasmonic chains,” Nano Letters 8, 2369–2372 (2008). 36. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Physical Review B 70, 125429 (2004). 37. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Physical Review B 74, 033402 (2006). 38. V. Agranovich, Excitations in Organic Solids,, International Series of Monographs on Physics (Oxford University, 2008). 39. J.-W. Dong and Z.-L. Deng, “Direct eigenmode analysis of plasmonic modes in metal nanoparticle chain with layered medium,” Optics Letters 38, 2244–2246 (2013). 40. A. Farhang, S. A. Ramakrishna, and O. J. F. Martin, “Compound resonance-induced coupling effects in composite plasmonic metamaterials,” Optics Express 20, 29447–29456 (2012). 41. J. Cesario, M. U. Gonzalez, S. Cheylan, W. L. Barnes, S. Enoch, and R. Quidant, “Coupling localized and extended plasmons to improve the light extraction through metal films,” Optics Express 15, 10533–10539 (2007). 42. Y. Chu and K. B. Crozier, “Experimental study of the interaction between localized and propagating surface plasmons,” Optics Letters 34, 244–246 (2009). 43. A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Physical Review B 71, 134304 (2005). 44. P. J. Compaijen, V. A. Malyshev, and J. Knoester, “Surface-mediated light transmission in metal nanoparticle chains,” Physical Review B 87, 205437 (2013). 45. J.-Y. Yan, W. Zhang, S. Duan, X.-G. Zhao, A. O. Govorov, “Optical properties of coupled metal-semiconductor #223934 $15.00 USD Received 29 Sep 2014; revised 18 Nov 2014; accepted 19 Nov 2014; published 28 Jan 2015 © 2015 OSA 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002280 | OPTICS EXPRESS 2281 and metal-molecule nanocrystal complexes: Role of multipole effects,” Physical Review B 77, 165301 (2008). 46. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, 1998). 47. J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, 1998). 48. M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Optics Letters 8, 581–583 (1983). 49. A. F. Koenderink, R. de Waele, J. C. Prangsma, and A. Polman, “Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides,” Physical Review B 76, 201403 (2007). 50. W. C. Chew, Waves and Fields in Inhomogeneous Media, (IEEE, 1999). 51. M. Paulus, P. Gay-Balmaz, and O. Martin, “Accurate and efficient computation of the Greens tensor for stratified media,” Physical Review E 62, 5797–5807 (2000). 52. K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Optics Letters 32, 973–975 (2007). 53. K. B. Crozier, E. Togan, E. Simsek, and T. Yang, “Experimental measurement of the dispersion relations of the surface plasmon modes of metal nanoparticle chains,” Optics Express 15, 17482–17493 (2007). 54. V. M. Agranovich and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Physics-Uspekhi 49, 1029–1044 (2006)


Introduction
Over the past decade there has been considerable effort in understanding the optical properties of nano-structured materials and designing sub-wavelength optical waveguides [1][2][3][4][5].In these developments plasmons, collective free electron oscillations in metals, have proven to be of particular importance, since they allow localization and guiding of optical signals well below the diffraction limit.Usually, plasmons are divided in two different classes: Localized Surface Plasmons (LSPs) and Surface Plasmons Polaritons (SPPs), the former being non-propagating excitations in metal nanoparticles (MNPs) embedded in a dielectric environment, whereas the latter generally refer to propagating surface waves bound to the interface between a metal and a dielectric.Both of these modes have been studied in detail and are well understood [6,7].
Plasmons resonances of MNPs are very sensitive to their environment, which makes them highly valuable for sensor applications [8][9][10][11].Furthermore, they can couple strongly to nearby plasmonic nanoparticles, yielding collective plasmonic modes, a property which is often exploited in designing sub-wavelength optical waveguides [12][13][14][15][16] and nano-antennas [17][18][19][20][21]. Substrate SPP modes can only be excited if the appropriate phase matching conditions are met.A common method to supply the extra momentum is to make use of the electromagnetic near field of a local emitter.Therefore, the LSP mode of an MNP can excite and couple to the SPP mode of a metal substrate.These collective modes have been described very clearly and elegantly in the framework of plasmon hybridization [22,23].During the past years, the exploration of the properties of these modes has been the subject of several studies.It was shown that the optical properties of an MNP above a metal substrate are strongly dependent on the polarization and frequency of the excitation and their relative position [24][25][26].Furthermore, recent time-dependent studies revealed the relevance of different decay channels [27].
A thoroughly studied example of a plasmonic waveguide is a linear chain of closely spaced MNPs.Such chains can support propagating plasmon modes and have already been used to demonstrate guiding, splitting, bending [16,[28][29][30][31][32][33] and localization [34,35] of optical excitations, all well below the diffraction limit.The chain modes can be described as coupled LSPs [13,36], and actually, for chains longer than the resonance wavelength in the surrounding medium, represent polariton modes [37].The polaritons result from the strong coupling of the LSPs to electromagnetic radiation and will arise when the effects of retardation are important [38], otherwise the excitations are referred to as static chain modes.
Both substrate and chain SPPs have attracted considerable attention over the past years because they are promising candidates for applications in nano-optical waveguides, antennas and sensors.Many nano-optical devices include coupled MNPs as well as metal substrates, meaning that generally both types of SPPs will be present.Understanding and exploiting the full functionality of these compound devices requires a deep understanding of how these two modes interact with each other.The coupling between both types of SPPs has been studied recently by several groups, considering linear chains [39,40] and two dimensional nanoparticle arrays [41,42] above metal substrates.These system were investigated both experimentally [42] and theoretically [39,40,42].It has been shown that the coupling between the chain and substrate plasmon modes is strong, and that the dispersive properties of hybrid modes can be significantly different from the individual, uncoupled modes.
In this paper, we study the dispersion relations of the SPP modes of a compound system: a one dimensional chain of silver nanospheres and a metal substrate.In contrast to previous work, in which only one particular metal substrate has been studied [39], we show that the plasma frequency of the metal substrate plays a dominant role in determining the optical properties of this compound system.Choosing the appropriate plasma frequency, i.e., by selecting a specific substrate, the hybrid polaritons in this system can have very high, almost zero, or even negative group velocities.Furthermore, the dispersion relations of the compound system turn out to be highly polarization dependent.These insights open up a new possibility to engineer the optical response of coupled plasmonic systems.We present the dispersion relations for a wide range of plasma frequencies and discuss them with the aid of plasmon hybridization and dipole-dipole interactions.Finally, we use real material parameters to give an example of a system for which the negative index modes can be observed, and calculate the Poynting vector for one of these modes.In the conclusion, we will discuss the relevance of our findings for the development of nano-optical devices.

Formalism
We consider a one dimensional chain of identical and equally spaced silver nanospheres with radius a = 25 nm, center-to-center spacing d = 75 nm, embedded in glass (ε 1 = 2.25) at a height of h = 50 nm above a metal substrate, as shown in Fig. 1.Each nanosphere is treated as a point dipole.The MNP-MNP and MNP-substrate interactions are described using dyadic Green's functions which include retardation effects.For the configuration considered here, the dipole approximation provides an accurate description of the system at hand [43,44].As to including the contribution of higher order multipole effects, see Ref. [45].
Within the dipole-approximation, the dipole moment induced in the MNP by an electric field with amplitude E, is p = ε 1 αE.Here, ε 1 is the dielectric constant of the host medium, whereas the optical properties of the MNPs are described by the so-called polarizability α.For spheres of radius much smaller than the wavelength in the host medium, α is given by 1 where α (0) is the bare polarizability, as is derived from electrostatics [46,47].The last two terms, depending on the wavevector of light in medium 1, dispersion and radiative damping respectively: we are dealing with non-static fields [48].In terms of the frequency dependent permittivity of the sphere, ε(ω) and the radius of the sphere a, the bare polarizability has the form Since we consider silver nanoparticles, we use a generalized Drude response function which is fitted to experimental data [49], ε(ω) = 5.45 − 0.73 ω 2 p /(ω 2 + iωγ), with plasma frequency ω p = 17.2 rad fs −1 and damping coefficient γ = 0.0835 fs −1 .
The dipole moment induced in one of the MNPs is proportional to the electric field acting on that particle.This field consists of the external excitation field E ext , the field produced by all the other particles in the chain and the field reflected from the substrate.The dipole moment of each MNP in a chain of N particles can be found by solving the following system of equations 1 Here E ext i and p i are 3 × 1 vectors, representing x,y, and z components of the polarization, and Î is the 3 × 3 unit tensor.Ĝ0 (ω; r n , r m ) is the 3 × 3 Green's tensor describing the electric field produced at r n , by a unit dipole (oscillating with frequency ω) that is located at r m .In a homogeneous environment Ĝrefl is the Green's tensor that describes the field that is reflected from the substrate.The tensor components can be derived by expanding the spherical waves of the dipole field into plane waves with the appropriate Fresnel coefficients.This procedure, first applied by Sommerfeld, has been extensively studied in literature.For the expressions used in this research, we refer to Refs.[7,50,51].The optical properties of the substrate enter Ĝrefl through the frequency dependent permittivity of the metal, for which we will assume a standard Drude form: Throughout this paper, we will consider different values for the plasma frequency, keeping the damping coefficient fixed at γ = 0.0835 fs −1 , the value for silver.The properties of the plasmon mode of the substrate are contained in the Fresnel coefficient for p-polarized (TM) light.
In order to calculate the dispersion relation for the system under consideration, we need to find the eigenmodes of the set of coupled equations (3).These modes are obtained by solving the homogeneous system, i.e., setting the external excitation field E ext m = 0. Since both radiation and ohmic damping are accounted for, losses can be quite significant and therefore, Eq. (3) will only have solutions at complex frequencies, commonly referred to as normal modes frequencies.The imaginary parts of these frequencies represent the mode quality.Diagonalizing this system of equations, evaluated at the obtained frequencies, will give the corresponding modes and from this a dispersion relation can be constructed [36].
Although this method is exact and the only correct method if a short chain of nanoparticles is considered, it suffers from a major difficulty: the Green's tensors have to be evaluated at complex frequencies, which is numerically very challenging, especially, for the Sommerfeld integrals contained in Ĝrefl .A workaround for this problem is to consider the limit of an infinite chain and apply periodic boundary conditions, as is done in the so-called eigendecomposition method [52].It has been shown that for a chain of MNPs this limit can be reached for chains as short as 10 particles [29].This condition is easily met in practice for nanoparticle waveguides.Furthermore, studying polariton-polariton interactions requires a system that is much longer than the wavelength.
In the eigendecomposition method, Bloch's theorem is used to write p m = p exp (iqmd) and , where q is the quasi-wavevector of the Bloch mode and d the spacing between the nanoparticles.Substituting this in Eq. (3) reduces the system to three equations, one for each component of the polarization.For an infinite chain in the geometry under consideration, this results in [39] 1 The part of this equation that is between brackets, can be thought of as an inverted generalized polarizability of the system, α.Now, instead of finding the complex roots of this equation, calculation of the imaginary part of α as a function of the real frequency ω and the wavevector q gives dispersive properties of the plasmon modes of the system, as well as the quality of these modes [52].Physically, this approach corresponds to the calculation of an absorption spectrum, where one is not only able to tune the frequency of the excitation, but also the wavevector.Here, in contrast with the exact method, the mode quality, or lifetime, of the mode is reflected in the width of the spectrum: the wider the mode, the shorter the lifetime.

Results
Before discussing the dispersion relations of the compound system, it is insightful to investigate the dispersion relations of the chain and substrate surface plasmons, assuming them uncoupled.These relations have been published before and are discussed in detail in e.g.Refs.[36,37,53].In order for this work to be reasonably self-contained, we briefly discuss the most important results.
As was mentioned in the introductory part, plasmon polaritons result from the strong coupling between plasmons and electromagnetic radiation, which implies a characteristic anticrossing of the plasmon dispersion branch with the light line.In the MNP chain, this concerns the mixing of the LSPs with electromagnetic radiation.For the metal substrate it is in fact the non-propagating surface plasmon (SP), i.e. the large wave vector limit of the SPP, which is mixing with radiation.This gives rise to well-known substrate SPP.This is illustrated in Fig. 2, where the solutions to Eq. ( 3) are shown for different polarizations, setting Ĝrefl = 0 (uncoupled situation).In this case, the chain-substrate interaction is zero, and therefore, it is not necessary to compute the Green's tensor of the metal substrate and the exact method of finding the complex roots of Eq. ( 3) can be easily applied.In Fig. 2 only the real part of the frequency is depicted, the imaginary part, i.e. the mode quality, is not shown.The substrate is described by a Drude model with ω p = 6.00 rad/fs and γ = 0.0835 rad/fs.ω sp is the non-propagating surface plasmon (SP) of the substrate, given by ω p (1 + ε 1 ) −1/2 .The wavevector q runs from 0 to π/d, the edge of the first Brillouin zone.
Figure 2(a) displays the completely uncoupled situation, excluding retardation: the quasistatic longitudinal and transverse chain modes (green squares and red circles, respectively), the light line (black dashed) and the non-propagating surface plasmon of the substrate ω sp = ω p (1 + ε 1 ) −1/2 (black dotted).In the geometry under consideration (see Fig. 1), longitudinal modes consist of dipoles with x-polarization, while transverse modes contain y-or z-polarized dipoles.For an isolated chain these two transverse modes are degenerate.Comparing the longitudinal and the transverse modes, we see from the slope of the dispersion that the interparticle interactions for the two modes have opposite signs.From the energy difference between the high and low wavevector modes, we can deduce that longitudinal dipoles are interacting stronger than transverse ones.This is exactly what would be predicted from the near field interaction of point dipoles [47].The right plot shows the dispersion relations when retardation is included, implying the plasmon modes are interacting with the light line.The effect of radiative interaction on the chain plasmons can be seen by comparing Figs.2(a) and 2(b).It is clear that an avoided crossing with the light line occurs only for the chain modes with a polarization perpendicular to the chain axis (y-and z-polarization), indicating a strong coupling of the static chain modes to electromagnetic radiation.The reason for this is that a dipole emits mainly along the direction perpendicular to the dipole orientation.The branch of the substrate SPP is added manually, using the well-known dispersion relation From the figure it is clear that this branch arises from the anti-crossing of the non-propagating surface plasmon of the substrate and the light line.
We now turn to the discussion of the dispersion relation for an MNP chain close to a metallic substrate.If the distance between the MNP-chain and the substrate is smaller than the wavelength, the near-field of the MNPs can excite SPPs on the substrate and hence, will couple the chain and substrate plasmons, which are shown uncoupled in Fig. 2(b).In this case, the dispersion relations are easier to calculate using an eigendecomposition, i.e. by solving Eq. 5 [39,52].and c) shows the result for x-polarization for two specific plasma frequencies, the middle column for y-polarization and the right column for z-polarization.Animations presenting these dispersion relations for a wide range of plasma frequencies for x (Media 1), y (Media 2) and z-polarization (Media 3) can be found in the supplementary material.The solid white line gives the substrate surface plasmon polariton (SPP) dispersion.The steep dashed and dotted horizontal lines represent the light line and the substrate surface plasmon (SP), respectively.The ripples that occur close to the light line for the transverse mode of the isolated chain, result from the fact that a finite, but very long, chain is used for the calculations.Close to the light line the slowly decaying radiative interactions are very important and effects of the finiteness of chain can be seen.The labels AS, S, || and ⊥ refer to the sign of the coupling and the polarization of the hybrid polaritons, explained in more detail in Fig. 4. The wavevector q runs from 0 to π/d, the edge of the first Brillouin zone.
# To compare the two different methods, Fig. 3(a) also shows the dispersion relations for the isolated chain.It is clearly seen that they are very similar to the ones shown in Fig. 2. In addition, the dispersion relations of Fig. 3 also show the quality of the modes.For wavevectors q to the left of the light line, we see that the modes have much higher losses than those to the right of this line.The reason is that the chain modes will only radiate if their wavevector is smaller than that of light in the surrounding medium.As a reference for the strength and spectral position of the resonance, the polarizability of an isolated, single MNP is plotted as well.Figures 3(b) and 3(c) display the dispersion relations for the chain at a height of 50 nm above a substrate characterized by a plasma frequency of ω p = 6.00 and ω p = 7.40 rad/fs, respectively.Animations presenting the dispersion relations for a wide range of plasma frequencies can be found in the supplementary material.In these animations the plasma frequency of the substrate is stepwise increased, such that the substrate plasmon mode is scanned through the plasmon resonance of the chain.One has to be careful when the interaction between the chain and the substrate is taken into account.Not only will the longitudinal and transverse chain modes mix with the modes of the substrate, the latter also introduces a coupling between the x-and z-polarized chain modes, lifting the degeneracy between the transverse chain modes.Although the x-and z-polarized modes are coupled, it is still possible to separate them because in each branch the nanoparticles will have a dominant polarization along one direction, denoted by 'mainly x' and 'mainly z' in the figures.The coupling between x-and z-polarized is mediated by the substrate and therefore it has a phase difference with respect to the 'direct' MNP-MNP interaction.This implies that for large wavevectors it is more difficult to distinguish between the x-and z-polarized modes due to the strong difference in phase between neighboring particles.A consistent way to distinguish between the different modes is by separating them by the phase of the dipole moment rather than by magnitude.This subtlety only occurs at the edge of the Brillouin zone, for smaller wavevectors both methods of separation give the same results.Therefore the modes can still be referred to as 'mainly x' or 'mainly z'.
As is well known, two interacting modes will hybridize into two new modes, a symmetric (S) and an anti-symmetric (AS) mode.This is shown in Fig. 4, where the hybridization scheme for an MNP above a metal substrate is given [22,23].The arrows indicate the polarization of the charge distributions that are induced in the MNP and the substrate, respectively.From dipole-dipole interactions, one can calculate that the interaction between the MNP and the substrate is stronger when the polarization is perpendicular to the substrate.Furthermore, the sign of the interaction is opposite for parallel and perpendicular polarizations.This explains the interchange of the S and AS modes, as well as the energy separation between them.
The interaction strength between the two modes does not only depend on the polarization, but also on their spectral separation.In Fig. 3(b) the dispersion relations are shown for the chain close to a substrate with plasma frequency ω p = 6.00 rad/fs.For this ω p , the surface plasmon frequency ω sp is spectrally far away from the chain mode, and therefore, the interaction is small.Comparing these plots with those of the isolated chain, we see that the upper branches (S) of the dispersion relations for x-and y-polarization are indeed very similar to those of the isolated chain, in particular for y-polarization.For x-polarization the resemblance is smaller because in addition to the mixing of the chain and substrate modes, also the x-and z-polarized modes are coupled.As was mentioned before, distinguishing both polarizations is not so clear for large wavevectors.Comparing Figs.3(a) and 3(b) carefully, reveals that, for the 'mainly x' configuration, the part of the upper branch that is below the lightline actually is very similar to the transverse branch of the isolated chain.Similarly, the large wavevector part of the upper 'mainly z' polarized branch is comparable to the longitudinal mode of the isolated chain.
The lower frequency branch is always positioned below ω sp , between ω p and ω sp no fields can penetrate the metal substrate.Due to the large spectral separation in Fig. 3(b), the lower branch (AS) has very low intensity.Interestingly, for x-polarization the lower branch results from the hybridization of the chain SPP with the substrate SP, rather than the SPP.This shows that, similar to the isolated chain, the radiative interaction is not important for this polarization.The lower branch for y-polarization (AS) does show influence of the substrate SPP mode: once this branch has been crossed, the mode intensity goes to zero.This is explained by the fact that substrate SPPs only exist for TM polarization, and therefore, the SPPs excited by ypolarized dipoles, will not propagate along the chain, but instead perpendicular to it, channeling the energy away from the chain.
Contrary to x-and y-polarizations, the presence of the substrate strongly alters the dispersion relation for z-polarization.From this we can conclude that for z-polarization the interaction with the substrate is more important, as is also predicted from dipole-dipole interactions (see Fig. 4).The upper branch for this polarization is the AS mode, for which the interaction between the particles is partially canceled by the contribution from the substrate.This weak interaction gives rise to the almost flat dispersion that is observed.The opposite occurs for the lower branch (S): the surface contribution is in phase with the direct contribution, yielding a strong interaction and a steep dispersion.This mode is strongly related to the substrate SPP mode (white solid line), only with a small redshift due to the symmetric coupling with the chain.
Figure 3(c) shows the dispersion relations for the same system as before, but now with ω p = 7.40 rad/fs and hence, a smaller spectral separation.The features of the dispersion relations are very similar to those discussed above.However, since the substrate is now spectrally closer to the chain, the modes of the former carry more intensity.For x-polarization, we can now see a very clear anti-crossing between the chain SPPs and substrate SP modes, indicating strong hybridization of the chain-substrate modes.Again, a very strong influence of the substrate SPPs can be found for z-polarization, especially, at lower frequencies.Here, there are modes with a very steep dispersion, thus with a high group velocity, and a high mode quality, i.e., low losses.Therefore, these modes have a long propagation length, that is promising for sub-wavelength guiding applications [37,44].Interestingly, for z-polarization no qualitative difference is observed as a function of the plasma frequency, eventhough the particle-substrate interaction is the strongest for this polarization.The reason for this counterintuitive effect lies in the fact that the large interaction results in a large splitting.In this case the influence of the plasma frequency is less prominent, because the SP of the substrate is located in the gap between the two branches.Therefore no crossing or mixing with this line occurs and no qualitative difference arises.The low frequency branches of the x-polarized dispersion relations have another interesting feature: the slope of these modes is negative over a wide range of wavevectors.For lossless media, a negative slope is associated with a negative group and energy velocity, indicating that the direction of energy flux is opposite to the phase advance.These type of modes are commonly referred to as negative index modes [54,56].The animations in the supplementary material (Media 1) show that, for a large spectral overlap between the substrate surface plasmon and the chain SPPs, the upper and lower branch repel each other.The lower, anti-symmetric branch is pushed down by the upper, symmetric branch.
So far we have only considered artificial metal substrates, characterized by a certain plasma frequency ω p and a fixed damping coefficient γ = 0.0835 fs −1 .To observe the mode mixing effects that are presented in this paper, the most important parameter is the spectral overlap between the chain and substrate plasmon modes.This implies that depending on the size and material of the MNPs, different substrates can be selected.As an illustration, we will show that the negative index mode that is observed for x-polarization in Fig. 3(c), is also present when a realistic platinum substrate is considered.The permittivity of platinum is fitted with a simple Drude model, yielding ω p = 7.81 rad/fs and γ = 0.1051 fs −1 [55].Figure 5(a) shows the dispersion relation of the x-polarized modes for a chain of silver MNPs close to a platinum substrate.Due to the somewhat larger damping coefficient for platinum the mode quality is a bit smaller as compared to Fig. 3(c), but the key features are still there.To take a closer look at the direction of the phase and the energy flow in the negative index modes, the full first Brillouin zone is plotted.A dispersion relation shows the wavevector and energy of all the modes in the system, however not all these modes can be accessed by optical excitation.Due to causality the energy flux of a mode must be in the same direction as its excitation.Therefore, instead of considering negative group velocity (i.e.negative energy flux) and positive phase, one has to consider positive group velocity and negative phase.Thus the excited mode will be on the left side of the dispersion diagram, instead of on the right side.
In a system in which losses are present there is not a straightforward connection between group and energy velocity anymore [56].The clearest way to examine the direction of the energy flux is to explicitly calculate the direction of the Poynting vector with respect to the phase advance.In the Green's tensor formalism applied here, this vector can be easily calculated using [7] In the above equation Ĝ is the electromagnetic Green's tensor for a layered medium and k i and ε i are representing the wavevector of light and the dielectric constant in the i th medium, respectively.
Figure 5(b) displays the x-component of the Poynting vector, i.e., the component parallel to the chain axis, calculated through a plane perpendicular to the chain axis, bisecting the chain in between two particles.In general the energy flux can be in any direction.Since the flux directed along the y or z-axis implies energy flowing out of the chain, only the x-component of the flux is relevant for studying energy propagation in this system.The total energy that is propagating in the x-direction can be obtained by integrating the x-component of the Poynting vector over the defined plane.The energy flux has been calculated for two specific modes, one with negative phase (q = −0.02451/nm) and positive group velocity, the other with positive phase (q = 0.0245 1/nm) and positive group velocity.Furthermore, the former is on the lower branch, whereas the latter is on the upper.Note that for this calculation it is assumed that it is possible to excite these modes exclusively.The position of the modes on the dispersion relation is indicated with the letters A and B. As expected, the integrated value of the energy flux is positive for both modes.This implies that mode A indeed is a negative index mode with opposite energy flux and phase advance.The total energy flux of mode B is about 4 times as large as that of mode A. This is supported by the observation that mode B has as larger group velocity and a higher mode quality.Taking a closer look at the color profile of the Poynting vector, one can see that it resembles those in a Metal-Dielectric-Metal (MDM) waveguide, if the substrate and MNP chain are considered as the metal cladding layers, see Refs.[56][57][58][59].Comparing the profile of the two selected modes in between the substrate and the MNP, mode A has a continuous blue coloring, whereas mode B has two red areas separated by a white space.The white area represents a zero in the energy flux, similar to a node in that can be observed in higher order modes.This matches with the fact that mode B lies on the upper, high energy branch of the dispersion relation.
Interestingly, mode B has a negative energy flux in the dielectric gap between the chain and the substrate, even though the integrated energy flux is positive.In MDM waveguides, the negative contribution is usually found inside the metal layers.The fact that it occurs here in the dielectric, opens up the possibility to study this phenomenon in more detail, f.e. by placing small emitters in the gap between the chain and the substrate.

Summary
We found that the presence of a metal substrate strongly alters the dispersion relations of a chain of metal nanoparticles.The properties of the resulting hybrid chain-substrate SPPs depend to a large extent on the spectral overlap of the plasmon modes of the chain and the substrate.In particular, for a given nanoparticle chain, choosing the appropriate plasma frequency, i.e., selecting the appropriate substrate, results in a device in which the plasmons will either have a very high, very low, or negative group velocity.The latter is confirmed by considering a realistic device with a platinum substrate.Poynting vector calculations reveal that the negative group velocity corresponds to anti-parallel energy flux and phase advance and thereby confirm that these modes have a negative refractive index.In addition, the optical properties of the system strongly depend on the polarization and the frequency of the excitation.Our findings show that this simple system gives rise to a wide range of physical phenomena, all of which can be explained very well by considering interacting dipoles and the plasmon hybridization model.The model which we have applied is an easy tool to calculate the dispersion relations and can easily be extended to dipole emitters embedded in more general layered media, e.g.Metal-Dielectric-Metal or Dielectric-Metal-Dielectric geometries.

#Fig. 3 .
Fig.3.Dispersion relations of a chain of silver MNPs, with r = 25, d = 75, h = 50 nm, and ε 1 = 2.25, calculated using the eigendecomposition method.Plotted is Im[ α] on a logarithmic scale as a function of the frequency ω and quasi-wavevector q.The left column of b) and c) shows the result for x-polarization for two specific plasma frequencies, the middle column for y-polarization and the right column for z-polarization.Animations presenting these dispersion relations for a wide range of plasma frequencies for x (Media 1), y (Media 2) and z-polarization (Media 3) can be found in the supplementary material.The solid white line gives the substrate surface plasmon polariton (SPP) dispersion.The steep dashed and dotted horizontal lines represent the light line and the substrate surface plasmon (SP), respectively.The ripples that occur close to the light line for the transverse mode of the isolated chain, result from the fact that a finite, but very long, chain is used for the calculations.Close to the light line the slowly decaying radiative interactions are very important and effects of the finiteness of chain can be seen.The labels AS, S, || and ⊥ refer to the sign of the coupling and the polarization of the hybrid polaritons, explained in more detail in Fig.4.The wavevector q runs from 0 to π/d, the edge of the first Brillouin zone.

Fig. 5 .
Fig.5.a) Similar to Fig.3, but now for a Platinum substrate (ω p = 7.81 rad/fs, γ = 0.1051 1/fs[55]).The full first Brillouin zone is shown for the x-polarized mode.b) The xcomponent of the Poynting vector (i.e.parallel to the chain axis) is plotted over a surface perpendicular to the chain axis, bisecting the chain between two neighboring particles.Mode A corresponds to a point on the lower branch with a negative wavevector, mode B to a point on the upper branch with a positive wavevector (also indicated in a) ).The integrated value of the energy flux of mode B is about 4 times larger than that of mode A, and both values are positive.Mode A has opposite phase and energy flux, and therefore, it is a negative index mode.
Hybridization diagram for the plasmons of an MNP and a metal substrate.The black and blurred arrows indicate the polarization of the induced charge distributions in the MNP and substrate, respectively.It is shown that the LSP of the MNP hybridizes with the SP of the substrate into a symmetric (S) and anti-symmetric (AS) mode.The sign of the interaction is different for polarization parallel (||) or perpendicular (⊥) to the substrate.