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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 16 — Aug. 11, 2014
  • pp: 18958–18965
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Exact solution for velocity of plasmon-polariton in metallic nano-chain

Witold A. Jacak  »View Author Affiliations


Optics Express, Vol. 22, Issue 16, pp. 18958-18965 (2014)
http://dx.doi.org/10.1364/OE.22.018958


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Abstract

In equidistant infinite chain of metallic nanospheres the collective mode of surface plasmons propagates without radiative losses, i.e., the Lorentz friction losses in each nanosphere are compensated by energy income in near-, medium- and far-field from the rest of the chain. Within an approximate approach including numerical studies in Green function framework it has been indicated superluminal propagation of some plasmon-polariton modes. By the exact solution of the nonlinear dynamic equation we demonstrate that the superluminal modes were an artifact of the perturbation solution type and we show that the group velocities for both polarizations are limited by light velocity, though vary in large range depending on chain parameters and are typically one order lower than the light velocity.

© 2014 Optical Society of America

1. Introduction

2. Equation for dispersion of plasmon-polariton in the chain

Assuming that in the sphere located in the point r we deal with the dipole D, then in the other place r0 (r0 is fixed to the end of r) this dipole causes electric field in the form as follows (including relativistic retardation of electromagnetic signals) [15

15. L. D. Landau and E. M. Lifshitz, Field Theory (Nauka, 1973).

, 16

16. J. D. Jackson, Classical Electrodynamics (John Willey and Sons Inc., 1998).

]:
E(r,r0,t)=1ε(2v2t21r0vt1r021r03)D(r,tr0/v)+1ε(2v2t21r0+vt3r02+3r03)n0(n0D(r,tr0/v)),
(1)
where, n0=r0r0 and v=cε. Above formula includes the terms corresponding to the near-field zone (denominator with r03), medium-field zone (denominator with r02) and far-field zone (denominator with r0) contributions to dipole field. This allows for writing out the dynamical equation for plasmon oscillations at each nano-sphere of the chain, which can be numbered by integer l (d denotes the separation between nano-spheres in the linear chain; vectors r and r0 are collinear if the origin is associated with one of nano-spheres in the chain). We assume d > 3a (a–the nano-sphere radius), when multipole corrections beyond dipole interaction model are negligible [1

1. D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: The role of chain length unravelled,” Nano Lett. 5, 985–989 (2005). [CrossRef] [PubMed]

, 3

3. S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. 120, 10871–10875 (2004). [CrossRef] [PubMed]

, 17

17. P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006). [CrossRef] [PubMed]

].

Taking advantage of the chain periodicity (in analogy to Bloch states in crystals with the reciprocal lattice of quasi-momentum) one can assume wave type mode of collective oscillations of dipoles in the chain,
Dα(ld,t)=Dα(k,t)eikld,0k2πd
(2)
(k is a number for 1D case, though we will call it traditionally as wave vector). This corresponds to the discrete Fourier transform component with respect to the positions similarly as for phonons in 1D crystal. The dynamics equation for time-position Fourier picture of dipole self-oscillations in the chain attains the form (for particularities of the derivation cf. [18

18. W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics 8, 1317–1333 (2013). [CrossRef] [PubMed]

]):
(ω2i2τ0ω+ω12)Dα(k,ω)=ω12a3d3Fα(k,ω)Dα(k,ω),
(3)
with
Fz(k,ω)=4m=1(cos(mkd)m3cos(mωd/v)+ωd/vcos(mkd)m2sin(mωd/v))+2i[13(ωd/v)3+2m=1(cos(mkd)m3sin(mωd/v)ωd/vcos(mkd)m2cos(mωd/v))],Fx(y)(k,ω)=2m=1(cos(mkd)m3cos(mωd/v)+ωd/vcos(mkd)m2sin(mωd/v)(ωd/v)2cos(mkd)mcos(mωd/v))i[23(ωd/v)3+2m=1(cos(mkd)m3sin(mωd/v)+ωd/vcos(mkd)m2cos(mωd/v)(ωd/v)2cos(mkd)msin(mωd/v))].
(4)
In Eq. (3) ω1 denotes the Mie dipole frequency of plasmons on single nanosphere. The damping ratio 1τ0 accounts for electron scattering losses [5

5. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000). [CrossRef]

],
1τ0vF2λb+CvF2a,
(5)
where, C is a constant of unity order, vF is the Fermi velocity in the metal, λb is the electron mean free path in bulk metal (including scattering of electrons on other electrons, on impurities and on phonons [5

5. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000). [CrossRef]

]); e.g., for Au, vF = 1.396 × 106 m/s and λb ≃ 53 nm (at room temperature); the latter term in the formula (5) accounts for scattering of electrons on the boundary of the nanoparticle, while the former one corresponds to scattering processes similar as in bulk. The other effects, as the so-called Landau damping (especially important in small clusters [19

19. C. Yannouleas, R. A. Broglia, M. Brack, and P. F. Bortignon, “Fragmentation of the photoabsorption strength in neutral and charged metal microclusters,” Phys. Rev. Lett. 63, 255–258 (1989). [CrossRef] [PubMed]

,20

20. W. Ekardt, “Anomalous inelastic electron scattering from small metal particles,” Phys. Rev. B 33, 8803–8805 (1986). [CrossRef]

]), corresponding to decay of plasmon for high energy particle-hole pair, are of lowering significance for nano-sphere radii larger than 2 – 3 nm [19

19. C. Yannouleas, R. A. Broglia, M. Brack, and P. F. Bortignon, “Fragmentation of the photoabsorption strength in neutral and charged metal microclusters,” Phys. Rev. Lett. 63, 255–258 (1989). [CrossRef] [PubMed]

] and are completely negligible for radii larger than 5 nm (we will consider here the large nanospheres with radii ≥ 10 nm). The contribution 23(ωd/v)3 to imaginary part of both formulae given by (4) describes attenuation caused by the Lorentz friction of surface plasmons [18

18. W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics 8, 1317–1333 (2013). [CrossRef] [PubMed]

]. Other terms in Eqs. (4) describe mutual impact of nano-spheres in the chain.

The imaginary part of the solution of Eq. (3) defines plasmon-polariton attenuation, while the real part of solving ω gives self-frequency of these oscillations.

3. Exact solution for group velocity of plasmon-polariton in the chain

The derivative of the self-frequency (the real part of solving ω of Eq. (3)) with respect to the wave vector k defines the group velocity of particular modes. Because of logarithmic singular term in the far-field transversal contribution to dipole interaction in the chain (taking advantage of the formula, m=1cos(mx)m=12ln(22cos(x)) [21

21. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic Press, Inc., 1994).

]),
m=1cos(m(kd+ωd/v))+cos(m(kdωd/v))m=12ln[(22cos(kd+ωd/v))(22cos(kdωd/v))],
(6)
one can not apply the perturbation method for solution of dynamical equation (3), at least in the region close to the singularity. Note that within the perturbation approach one substitutes ω with ω1 (Mie frequency) in functions F in r.h.s. of Eq. (3). This produces, however, the hyperbolic singularity in transversal group velocity resulted by virtue of Eq. (6). Moreover, within the perturbation approach the logarithmic singularity occurs for both polarizations, which is noticeable if one takes the derivative with respect to k from perturbation terms to the real parts of frequencies given by real parts of Eqs. (4) taken at ω = ω1. All this singularities occur at isolated points for which kd ± ω1d/v = (l–integer). Both hyperbolic and logarithmic divergences for perturbation formula for group velocities in this points would result in local exceeding c by corresponding group velocities. To resolve the problem of this unphysical divergence the exact solution of Eq. (3) must be found, because of divergence of the expression (6) the corresponding contribution cannot be treated still as perturbation. To avoid the perturbative method of solution of Eq. (3) resolving itself to substitution of ω1 instead of ω in the r.h.s. of this equation [18

18. W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics 8, 1317–1333 (2013). [CrossRef] [PubMed]

], we have applied the procedure FindRoot in Mathematica (employing combination of the damped Newton-Raphson, the secant and the Brent methods) to accurately solve the nonlinear Eq. (3) for each value of the wave vector k ∈ [0, 2π/d) (in 1000 points). Such a procedure can be treated as the accurate solution of Eq. (3) without deficiency caused by the perturbative approach. The exact solution of Eq. (3), found numerically, is plotted in Fig. 1 for both polarizations of plasmon-polaritons.

Fig. 1 Exact solution for self-frequency and damping rate of transversally (upper) and longitudinally (lower) polarized modes of plasmon-polariton in the nano-chain (ω in ω1 units)–there is presented solution of Eq. (3) in 1000 points on the sector kd ∈ [0, 2π); in plot for self-frequency of transversal mode a truncated logarithmic-type singularity of self-energy is marked; for longitudinal polarization the almost vertical local slop of self-frequency is marked

From Fig. 1 we notice by comparison with the corresponding plots obtained within the perturbation method [18

18. W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics 8, 1317–1333 (2013). [CrossRef] [PubMed]

], that for the longitudinal polarization the exact solutions for self-frequencies do not differ significantly from those obtained in the perturbation manner, but the change suffices to remove the logarithmic divergence from derivative of these self-frequencies. The difference is also not important for the attenuation plots for both polarizations. However, for the transversal polarization self-frequency in the case of exact solution we deal with quenching of the logarithmic divergence (6) in contrary to the approximated its version obtained within the perturbation approach. Instead of infinite singularity we observe in the exact plot for the transversal polarization self-frequencies only relatively small minimum resulting then conveniently in finite group velocity (not greater than c). This quenched logarithmic singularity into small local minimum is presented in Fig. 2.

Fig. 2 Exact solution for the group velocity of transversal mode of plasmon-polariton in the nano-chain for a = 10 nm and d = 4a; solution of nonlinear Eq. (3) removes the logarithmic singularity–the remaining local very narrow extremes are truncated exactly at value of c; in close vicinity of the singular point the solution was found in more than 1000 values for kd (in bottom: scale-magnification of marked fragments in upper panels); asymmetry in singular points is caused by imposition of hyperbolic and logarithmic truncated singularities of group velocity for the transversally polarized mode

Even though the real part of the function Fz and thus perturbative contribution to ωz(k) (Eq. (4)) is given by a continuous function, the corresponding group velocity will have logarithmic singularity as the derivative of ωz(k) with respect to k still will contain the sum of (cos(m(kd + ω1d/v)) − cos(m(kdω1d/v)))/m, arising from the term with factor cos(mkdm2 in Eq. (4) after taking the derivative. The similar term is present also in perturbation contribution to for ωx(y)(k) in corresponding function given by Eq. (4). The origin of these terms for both polarizations is the medium-field contribution to dipole interaction in the chain. In points kd ± ω1d/v = l2π, l–integer, the logarithmic singularity produces an artifact in the group velocities–exceeding c. This again precludes applicability of the perturbation approach, at least close to singularity points.

As it has been already mentioned above, the exact solution of nonlinear Eq. (3) can be found numerically and both real and imaginary parts of ω can be determined in the whole region kd ∈ [0, 2π). For the longitudinal polarization of plasmon-polariton its group velocity found by the exact solution is presented in Fig. 3. This solution for vz does not exhibit any singularities–the logarithmic singularity of perturbation term is quenched to only small local extrema limited by c, similarly as it was demonstrated above for the transversal polarization.

Fig. 3 Exact solution for the group velocity of longitudinally polarized mode of plasmon-polariton in the nano-chain for a = 10 nm and d = 3a; almost vertical slope of self-frequency (right panels) is limited by light velocity (in bottom: scale-magnification of fragments marked in upper panels)

The logarithmic-type singularity in self-energy of transversally polarized plasmon-polaritons in the chain is the feature which essentially differentiates these modes from the longitudinally polarized ones. This singularity is caused by the sum of far-field pieces of electric field of all nanoparticle dipoles which influence a charge oscillations in each component of the chain and produces hyperbolic-type discontinuity in group velocity exclusively for transversal polarization. Besides this discontinuity the medium-field component of electric interaction of dipoles additionally produces a logarithmic-type singularity in the group velocities for both polarizations (though without singularity in self-energies).

The truncated logarithmic-type or hyperbolic-type singularities distinguish the behavior of the group velocities obtained by the exact solution for self-energies for both polarizations. Generally they are sharpened and truncated at c visibly due to relativistic constraints imposed on the dynamic equation and manifesting themselves in the form of its solution. The retardation of electric signals prohibits the collective excitation group velocity to exceed the light velocity. This quenching concerns infinite singularities which occur in perturbation expressions for self-energy and in perturbation formulae for group velocities. The relativistic invariance of dynamic equation for collective dipole plasmon oscillations in the chain prevents, however, exceeding the light velocity by the group velocity of all plasmon-polariton modes. The exact solutions of the dynamical Eq. (3) inherently posses this property in opposition to perturbative its solutions. Exact self-energies have suitably regularized their dependence with respect to k, that their derivatives do not exceed c.

Demonstrated above distinction between truncated singularities in group velocities for transversal and longitudinal polarizations of plasmon-polariton in the chain, sheds the light on recent discussion of long range propagating plasmon-polariton modes in metallic nano-chains studied by numerical simulations in Green-function method for differential equations [11

11. I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett. 38, 4743–4746 (2013). [CrossRef] [PubMed]

14

14. V. A. Markel and A. K. Sarychev, “Comment on Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B 86, 037401 (2012). [CrossRef]

]. Indicated above truncated singularities in group velocities, i.e., the very narrow of logarithmic-type truncated divergence for longitudinal polarization and the wider hyperbolic-type mixed with also narrow logarithmic-type truncated singularity for transversal polarization give rise to understand the peculiarities of numerical studies. The long range propagation of narrow mode packets observed in numerical simulations might be linked with local enhancement of the group velocity of k-modes in singular regions. In the simulations [11

11. I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett. 38, 4743–4746 (2013). [CrossRef] [PubMed]

, 12

12. A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B 78, 035403 (2008). [CrossRef]

, 22

22. V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys. Rev. B 75, 085426 (2007). [CrossRef]

] it was assumed that the single selected nanosphere in the chain was initially excited and then it was observed the range of propagation for various modes of plasmon-polariton. The point-like initial excitation corresponds in Fourier picture to uniform excitations of all k modes including also those in singular regions. The different character of truncated singularities, originally infinite divergences in perturbation series, for longitudinal and transversal polarizations will result in different features of the corresponding modes (especially in relative narrowing of truncated singularities, stronger for longitudinal polarization). Though truncated at c level, the singular modes give the same long distance propagation for both polarizations (ca. two orders longer than for modes with other values of wave-vector k).

4. Damping of plasmon-polariton in the chain

The Ohmic losses (given by Eq. (5)) for plasmon-polariton are residual losses which remain for those wave vectors k for which the exact cancellation of the Lorentz friction takes place in the chain. They are still inconvenient and are of similar magnitude as in separate nanospheres (for the Au nanosphere with radius a = 10 nm, 1τ09×10131/s) [23

23. W. Jacak, J. Krasnyj, J. Jacak, R. Gonczarek, A. Chepok, L. Jacak, D. Hu, and D. Schaadt, “Radius dependent shift in surface plasmon frequency in large metallic nanospheres: Theory and experiment,” J. Appl. Phys. 107, 124317 (2010). [CrossRef]

, 24

24. T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998). [CrossRef]

]. The Ohmic losses limit the range of propagation of plasmon-polariton in the chain. These heat losses may be, however, compensated by adding active medium to the nanoparticle chain in the form of system of coupled quantum dots working as spasers [25

25. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B 85, 165419 (2012). [CrossRef]

], which is a promising way for development of ideal damping-less plasmon wave-guides.

5. Conclusions

In summary one can state that in infinite metallic nano-chain the long-range propagation of plasmon-polariton self-modes can be observed due to effective ideal compensation of the Lorentz friction losses in particular nanosphere by the energy income from other nanospheres in the chain, which takes place for both polarizations of the collective plasmon modes in relatively large part of the wave vector k domain (diminishing with d/a growth). Both the low (only Ohmic) attenuation of self-oscillations and the possibility to excite appropriately formed (by excitation of some part of the chain) wave packet of plasmon-polaritons modes in the chain propagating with suitably selected group velocity, may be responsible for experimentally observed of practically undamped propagation of collective plasmon signal over relatively large distances in several μm length nano-chains.

Acknowledgments

The present work was supported by the NCN project no. 2011/03/D/ST3/02643.

References and links

1.

D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: The role of chain length unravelled,” Nano Lett. 5, 985–989 (2005). [CrossRef] [PubMed]

2.

L. L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction spectra of silver nanoparticle arrays: Influence of array structure on plasmon resonance wavelength and width,” J. Phys. Chem. B 107, 7343–7350 (2003). [CrossRef]

3.

S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. 120, 10871–10875 (2004). [CrossRef] [PubMed]

4.

J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett. 82, 2590–2593 (1999). [CrossRef]

5.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000). [CrossRef]

6.

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65, 193408 (2002). [CrossRef]

7.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67, 205402 (2003). [CrossRef]

8.

S. A. Maier, P. G. Kik, L. A. Sweatlock, H. A. Atwater, J. J. Penninkhof, A. Polman, S. Meltzer, E. Harel, A. Requicha, and B. E. Koel, “Energy transport in metal nanoparticle plasmon waveguides,” Mat. Res. Soc. Symp. Proc.777, T7.1.1 (2003).

9.

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 inmetal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003). [CrossRef] [PubMed]

10.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005). [CrossRef]

11.

I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett. 38, 4743–4746 (2013). [CrossRef] [PubMed]

12.

A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B 78, 035403 (2008). [CrossRef]

13.

Y. Hadad and B. Z. Steinberg, “Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B 84, 125402 (2011). [CrossRef]

14.

V. A. Markel and A. K. Sarychev, “Comment on Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B 86, 037401 (2012). [CrossRef]

15.

L. D. Landau and E. M. Lifshitz, Field Theory (Nauka, 1973).

16.

J. D. Jackson, Classical Electrodynamics (John Willey and Sons Inc., 1998).

17.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006). [CrossRef] [PubMed]

18.

W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics 8, 1317–1333 (2013). [CrossRef] [PubMed]

19.

C. Yannouleas, R. A. Broglia, M. Brack, and P. F. Bortignon, “Fragmentation of the photoabsorption strength in neutral and charged metal microclusters,” Phys. Rev. Lett. 63, 255–258 (1989). [CrossRef] [PubMed]

20.

W. Ekardt, “Anomalous inelastic electron scattering from small metal particles,” Phys. Rev. B 33, 8803–8805 (1986). [CrossRef]

21.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products (Academic Press, Inc., 1994).

22.

V. A. Markel and A. K. Sarychev, “Propagation of surface plasmons in ordered and disordered chains of metal nanospheres,” Phys. Rev. B 75, 085426 (2007). [CrossRef]

23.

W. Jacak, J. Krasnyj, J. Jacak, R. Gonczarek, A. Chepok, L. Jacak, D. Hu, and D. Schaadt, “Radius dependent shift in surface plasmon frequency in large metallic nanospheres: Theory and experiment,” J. Appl. Phys. 107, 124317 (2010). [CrossRef]

24.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998). [CrossRef]

25.

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B 85, 165419 (2012). [CrossRef]

OCIS Codes
(240.5420) Optics at surfaces : Polaritons
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Plasmonics

History
Original Manuscript: May 26, 2014
Revised Manuscript: July 6, 2014
Manuscript Accepted: July 8, 2014
Published: July 29, 2014

Citation
Witold A. Jacak, "Exact solution for velocity of plasmon-polariton in metallic nano-chain," Opt. Express 22, 18958-18965 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-18958


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References

  1. D. S. Citrin, “Plasmon polaritons in finite-length metal-nanoparticle chains: The role of chain length unravelled,” Nano Lett.5, 985–989 (2005). [CrossRef] [PubMed]
  2. L. L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction spectra of silver nanoparticle arrays: Influence of array structure on plasmon resonance wavelength and width,” J. Phys. Chem. B107, 7343–7350 (2003). [CrossRef]
  3. S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120, 10871–10875 (2004). [CrossRef] [PubMed]
  4. J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, J. P. Goudonnet, G. Schider, W. Gotschy, A. Leitner, F. R. Aussenegg, and C. Girard, “Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles,” Phys. Rev. Lett.82, 2590–2593 (1999). [CrossRef]
  5. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B62, R16356 (2000). [CrossRef]
  6. S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B65, 193408 (2002). [CrossRef]
  7. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B67, 205402 (2003). [CrossRef]
  8. S. A. Maier, P. G. Kik, L. A. Sweatlock, H. A. Atwater, J. J. Penninkhof, A. Polman, S. Meltzer, E. Harel, A. Requicha, and B. E. Koel, “Energy transport in metal nanoparticle plasmon waveguides,” Mat. Res. Soc. Symp. Proc.777, T7.1.1 (2003).
  9. 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 inmetal nanoparticle plasmon waveguides,” Nat. Mater.2, 229–232 (2003). [CrossRef] [PubMed]
  10. S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005). [CrossRef]
  11. I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids,” Opt. Lett.38, 4743–4746 (2013). [CrossRef] [PubMed]
  12. A. A. Govyadinov and V. A. Markel, “From slow to superluminal propagation: Dispersive properties of surface plasmon polaritons in linear chains of metallic nanospheroids,” Phys. Rev. B78, 035403 (2008). [CrossRef]
  13. Y. Hadad and B. Z. Steinberg, “Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B84, 125402 (2011). [CrossRef]
  14. V. A. Markel and A. K. Sarychev, “Comment on Greens function theory for infinite and semi-infinite particle chains,” Phys. Rev. B86, 037401 (2012). [CrossRef]
  15. L. D. Landau and E. M. Lifshitz, Field Theory (Nauka, 1973).
  16. J. D. Jackson, Classical Electrodynamics (John Willey and Sons Inc., 1998).
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  18. W. A. Jacak, “On plasmon polariton propagation along metallic nano-chain,” Plasmonics8, 1317–1333 (2013). [CrossRef] [PubMed]
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