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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 13070–13079
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Surface plasmon coupling in periodic metallic nanoparticle structures: a semi-analytical model

Tian Yang and Kenneth B. Crozier  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 13070-13079 (2008)
http://dx.doi.org/10.1364/OE.16.013070


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Abstract

We report a semi-analytical model for calculating the coupling effects between the dipolar surface plasmon nanoparticles of a periodic structure. This model involves real-valued frequencies only and is therefore applicable to periodic structures with arbitrary dipolar units and environments.

© 2008 Optical Society of America

1. Introduction

2. The semi-analytical model

2.1. Point dipole approximation

Fig. 1. Scanning electron micrograph of an array of gold nanoparticle chains on a glass substrate.

The values of α for a single nanoparticle for a range of real-valued frequencies are obtained by comparing the field scattered by the particle with the field generated by a point dipole, under pulsed conditions. The particle scatters a normally incident plane wave. Figure 2 is a schematic explanation of the modeling. α is defined as the equivalent point dipole moment divided by the incident electric field: α(ω)=[scattered field (ω)/point dipole generated field (ω) × point dipole moment (ω)]/incident electric field (ω). The scattered field and point dipole generated field values are sampled at (x,y,z)=(0, 0, 1 µm), i.e. above the nanoparticle and the point dipole. The point dipole is positioned at (0, 0, 20nm), a short distance from the air/glass interface in order to more closely approximate the surface plasmons in the 55nm tall cylinder shaped nanoparticle. The incident electric field is taken at the position (x,y,z)=(0, 0, 0), and is the sum of the incoming field and the field reflected by the air/glass interface. The scattered field is obtained by subtracting the incident field, i.e. the field in the absence of the particle, from the total field. Figure 3 shows α calculated using this approach for a range of real-valued frequencies for the structure of Fig. 1. It should be noted that we account for the periodicity of the structure by considering it a one-dimensional periodic arrangement of unit cells spaced by 140 nm along the x direction. Each unit cell consists of a line of particles spaced by 300 nm along the y direction, and, for the equivalent point dipole moment calculation, is equated to a line of identical point dipoles. Therefore, the α values obtained are the polarizability values of the unit cell. In the following part of this paper, we will describe our model as if the unit cell is one single nanoparticle, for the simplicity of description. It should be noted that the method we introduce is applicable to any types of unit cells that consist of uniformly excited nanoparticles. In Section 3, we will apply the method to a unit cell consisting of a single nanoparticle.

Fig. 2. Schematic explanation of the equivalent dipole modeling. (a) a normally incident field, including incoming, reflected and transmitted fields; (b) scattered field from a metallic nanoparticle in addition to the incident field; and (c) field generated by a point dipole. The metallic nanoparticle is modeled as a point dipole whose polarizability is chosen so that the field it generates is equal to the field scattered by the nanoparticle.
Fig. 3. Polarizability of a nanoparticle unit cell, consisting of a line of gold nanocylinders at an air/glass interface, obtained by FDTD. f/c is the frequency divided by the speed of light.

To confirm that the point dipole approximation is valid, we compare the field scattered by the nanoparticle to the field generated by the point dipole by normalizing the field values at two positions along the chain to the field at (x,y,z)=(0, 0, 1 µm). It is appropriate to normalize to the field at this point as the polarizability is found by equating the scattered field at this point to the field at this point generated by a point dipole. The two other positions are (x,y,z)=(140nm, 0, 0), which corresponds to the nearest nanoparticle in the chain, and (x,y,z)=(980 nm, 0, 0), which corresponds to seven periods away along the chain. As shown in Fig. 4, the nanoparticle scattered field profile and the point dipole generated field profile are in good agreement, which confirms the accuracy of the point dipole approximation for the frequency range of interest.

Fig. 4. Comparison of the profile of the transversely polarized scattered field from a chain of nanoparticles and the profile of the transversely polarized field generated by a chain of point dipoles. Nanoparticles and point dipoles are located near an air/glass interface. Red: scattered field value at (140 nm, 0, 0)/value at (0, 0, 1 µm)×1/3; magenta: point dipole field value at (140 nm, 0, 0)/value at (0, 0, 1 µm)×1/3; blue: scattered field value at (980nm, 0, 0)/value at (0, 0, 1 µm)×5; green: point dipole field value at (980nm, 0, 0)/value at (0, 0, 1 µm)×5. (a) amplitude ratio; (b) phase difference. f/c is the frequency divided by the speed of light.

2.2. The model

Let the equivalent point dipole moment of the plasmon oscillation in a single nanoparticle be denoted by P. Writing the plasmon oscillation in the form of the equation of motion of a harmonic oscillator, we have:

Md2Pdt2=K0P+Ei(ω,kx)Q2vdPdt+EoQ2+MGdPdt
(1)

where M is the total effective mass of all the electrons in the nanoparticle that participate in the plasmon oscillation, t is time, K0 is the Hooke’s coefficient for the restoring force and is a real number, Q is the total charge of all the electrons in the nanoparticle that participate in the plasmon oscillation, and Eo is the external incident electric field. ν represents all loss mechanisms including ohmic and radiation losses. Though ν should contain time derivatives of P to include the radiation loss, we will simply treat it as a real number under monochromatic condition. Coupling between particles is included in Equation (1) through the quantity Ei, which is a sum of electric fields generated by plasmons in all the other nanoparticles at the position of the particle being considered. Ei is a function of the oscillation angular frequency ω and the wave vector kx of the surface plasmon mode (kx is along the direction of chain: x). As we shall discuss, a real-valued artificial gain coefficient G for the dipole moment P: dP/dt=GP, is introduced for the purpose of defining resonance. Denoting every oscillating quantity X in Equation (1) by X0e-iωt, we rewrite Equation (1) in the frequency domain:

[(ω02ω2)MRe(K1Q2)]+i[ωvIm(K1Q2)+ωMG]=Q2Eo0P0
(2)

where ω0=K0M is the surface plasmon resonance frequency of an isolated single particle, and K1≡Ei0/P0. Later we will calculate the coupling term K1 and assume a linear relationship between Ei and P at any single (ω, kx) point. Thus in this model G is the gain coefficient for both the dipoles and the fields. Therefore G is the gain coefficient for the whole surface plasmon mode. Note that if we rewrite Equation (2) for an isolated single particle, i.e., with no coupling or artificial gain terms, we obtain the polarizability of a single metallic particle, P0/Eo0, which has the same form as the polarizability of a single metallic nanosphere in a uniform dielectric environment under Drude’s model as have been used in References [14

14. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004). [CrossRef]

,16

16. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006). [CrossRef]

]. For metallic nanospheres in a uniform dielectric environment, radiation loss is included in ν by including a ω2 term [14

14. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004). [CrossRef]

,16

16. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006). [CrossRef]

]. The left hand side of Equation (2) is separated into real and imaginary parts.

Let us now find the resonance conditions from Equation (2). Under zero external field Eo, the left hand side of Equation (2) is equal to zero. Thus the real part of K1 determines the shift in resonance frequency ωr; and the imaginary part of K1 changes the threshold gain Gth required for a non-decaying oscillation. Because the value of K1 is independent of the value of G, the value of ωr is determined by the condition that the real part of Eo/P be zero, regardless of the value of G. This can be physically explained by noting that when the real part of Eo/P is zero, Eo is π/2 out of phase with respect to P, meaning that the external field does not change the strength of the restoring force and the surface plasmon oscillates at its intrinsic frequency. For this reason, in our modeling, G is only incorporated to clarify the definition of resonance but not included in the actual calculation. Letting G=0, we obtain the value of ωr by finding when the real part of Eo/P equals zero. The imaginary part of Eo/P is proportional to ωGth under the assumption that the number of electrons participating in the plasmon oscillation is invariant. In section 4 it will be shown that Gth is twice the decay rate of the freely decaying surface plasmons. As an aside, the existence of Q2 terms in Equation (2) indicates larger coupling effects for larger nanoparticles [9

9. Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004). [CrossRef]

].

We now consider the calculation of Eo/P. Because P=α(Eo+Ei)=α(Eo+K1P), we have Eo/P=1/α-K1. The resonance frequency is therefore found by finding the frequency at which Re(1/α-K1)=0. The corresponding threshold gain Gth is found from Im(1/α-K1). The value of α is found as discussed in section 2.1. The value of K1 is found by a weighted sum of the fields generated by single nanoparticles, with the weights dependent on the wave vector kx [18

18. T. Yang and K. B. Crozier, “Dispersion and Extinction of Surface Plasmons in Gold Nanoparticle Chains: Influence of the Air/Glass Interface,” Opt. Express 16, 8570–8580. [PubMed]

]. Three FDTD simulations are performed in order to find α and K1, corresponding to the three plots in Fig. 2: the incident field, the sum of incident and scattered fields, and the point dipole field. For nanospheres in a uniform dielectric environment, it would be possible to replace the FDTD simulations by analytical calculations [14

14. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004). [CrossRef]

,16

16. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006). [CrossRef]

]. Figure 5 shows the values of K1 calculated for kx values spanning the first Brillouin Zone, for the 2-D metallic nanoparticle array at an air/glass interface shown in Fig. 1. The frequency is fixed at f/c=1.6 µm-1. It presents a fast phase change between the light lines of air and glass, which has been reported and attributed to the phase change of total internal reflection [18

18. T. Yang and K. B. Crozier, “Dispersion and Extinction of Surface Plasmons in Gold Nanoparticle Chains: Influence of the Air/Glass Interface,” Opt. Express 16, 8570–8580. [PubMed]

]. In Fig. 6(a), we present calculations of |Re(1/α-K1)| over a range of frequencies and wavevectors spanning the first Brillouin Zone. As discussed, the resonance frequencies are found from the zeros of |Re(1/α-K1)|. These are presented as the dispersion relation of Fig. 6(b), along with the threshold gain Gth.

Fig. 5. K1 (a.u.) of an array of point dipole chains at an air/glass interface, at the oscillation frequency of f/c=1.6 µm-1. c is speed of light. The point dipoles are polarized in y direction and spaced by d=140 nm in the chain direction x. kx spans the first Brillouin Zone from 0 to π/d at a fixed interval.
Fig. 6. Surface plasmon behaviors in an array of gold nanoparticle chains at an air/glass interface. (a) Magnitude of real part of the external electric field divided by excited equivalent point dipole moment in one particle of the array. (b) Resonance frequency versus wave vector along the chain direction, kx; and corresponding threshold gain at resonance. The two straight lines in each figure are the light lines of air and glass. f/c is frequency divided by the speed of light.

2.3. Comparison with modeling carried out completely by FDTD

Fig. 7. Extinction ratio of an array of gold nanoparticle chains at an air/glass interface. Plane waves illuminate the particles from the glass side at different angles to the chain direction x. ky=0. (a) Semi-Analytical modeling result; (b) FDTD-only simulation result. The two straight lines in each figure are the light lines of air and glass. f/c is the frequency divided by the speed of light.

From Fig. 7 it can be seen that the results of the semi-analytical model are in very good agreement with the modeling carried out using FDTD exclusively. There are some potential sources of inaccuracy, however. Inaccuracy of this model can come from the fact that we have employed a universal α value, which is obtained for the plane wave normal incidence case as in Fig. 2, while the actual α value depends on the incident field’s direction and profile. In addition, the point dipole approximation may not be sufficiently accurate. As frequency and wave vector change, the plasmon profile within the metallic particle is also different, meaning that the number of electrons participating in the plasmon oscillation is different, which has been assumed to be invariant when calculating Gth in Fig. 6(b).

3. Discussion of modeling results

Comparing Fig. 5 and Fig. 6, it is observed that the fast decrease of resonance frequency between the light lines of air and glass corresponds to the fast increase of the real part of the coupling coefficient K1. The fast decrease of Gth, or twice the free decay rate, in the same region corresponds to the fast decrease of the imaginary part of K1. Gth peaks up on the light line of air because of maximum coupling to the radiation mode; its fast decrease between the light lines of air and glass is because of decrease of the same coupling effect. The strength of this coupling depends on the amplitude of the radiation mode at the air/glass interface. This amplitude is determined by the plane wave transmission coefficient between glass and air, which peaks up on the light line of air and decreases to zero as kx increases to the light line of glass [18

18. T. Yang and K. B. Crozier, “Dispersion and Extinction of Surface Plasmons in Gold Nanoparticle Chains: Influence of the Air/Glass Interface,” Opt. Express 16, 8570–8580. [PubMed]

]. Below the light line of glass, radiation loss diminishes and Gth is completely due to the metal ohmic loss.

It is worth pointing out that the fast phase change of K1 between the light lines of air and glass does not necessarily correspond to fast changes in both of its real and imaginary parts. Fig. 8(a) presents K1 for a single chain of gold nanoparticles at an air/glass interface. This chain is the same as each chain in Fig. 1. Only the imaginary part of K1 has a fast change between the light lines of air and glass. Figure 8(b) shows the modeled dispersion and Gth behaviors. It is not obvious that the dispersion is significantly larger in the region between the light lines of air and glass.

Fig. 8. Surface plasmon behaviors in a single chain of gold nanoparticles at an air/glass interface. The equivalent point dipoles are polarized in y direction and spaced by d=140 nm in the chain direction x. (a) K1 (a.u.) at the oscillation frequency of f/c=1.6 µm-1. kx spans the first Brillouin Zone from 0 to π/d at a fixed interval. (b) Resonance frequency versus wave vector along the chain direction, kx; and corresponding threshold gain at resonance. f/c is the frequency divided by the speed of light.

In Fig. 6(b), the dispersion relation curve drops abruptly near the light line of air. This corresponds to the fast continuous increase of Real(K1) in Fig. 5, but not a jump in K1 or an anti-crossing split between the plasmon dominated and photon dominated polariton modes as in ref [16

16. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006). [CrossRef]

]. In our model, only the plasmon dominated mode will be observed because all the electric fields that have been considered are generated from plasmons. At this abrupt yet continuous drop, |r/dkx| is larger than the speed of light (fast light), which may seem paradoxical. This is because above the light line of glass the photon part of the polariton does not propagate along the x-axis and r/dkx is not a correct representation of the group velocity r/dk [20

20. Think of a plane wave propagating in z direction. It has an infinite dω/dkx value.

].

4. Decay rate of periodically coupled surface plasmons

Denoting every oscillating quantity X in Equation (1) by X0e-(iω+s)t, and ignoring the gain G and the external driving field Eo, we rewrite Equation (1) in the frequency domain with separated real and imaginary parts:

{(ω02ω2)M[Re(K1Q2)+(vMs)sM]}+i{ωvIm(K1Q2)+2ωMs}=0
(3)

The resonance frequency ω is obtained from equating the real part of Equation (3) to 0; the free decay rate s is obtained from equating the imaginary part of Equation (3) to 0. Compared to Equation (2), the resonance frequency thus defined is different from preceding sections by an amount that shifts with the decay rate s. By comparing Equations (2) and (3), it can be seen that the decay s=G/2. Here we have assumed that the mode profile has not changed, i.e., the parameters in Equation (2) and (3) are the same, which is not accurate if the decay rate is high. Another factor that we have ignored is that ν is no longer a real number due to radiation loss at complex frequencies, which will significantly change the result if the decay rate is high. As long as the decay rate, or the threshold gain, is small enough, the results obtained from the two different definitions are close to each other.

5. Summary

We have demonstrated a semi-analytical model to calculate the periodic coupling between dipolar surface plasmons of metallic nanoparticles. In this model the single nanoparticle polarizability values and the point dipole field values are obtained by FDTD simulations; then the periodic coupling effects is calculated analytically. The model only involves real-valued frequencies and is appropriate for general periodic structures composed of dipolar units, while it requires only a few number of FDTD simulations to obtain the whole dispersion diagram. The modeled dispersion and loss properties have been reported and physically explained. The modeled extinction properties agree well with full FDTD simulation results.

Acknowledgements

We thank the Defense Advanced Research Projects Agency (DARPA), the Charles Stark Draper Laboratory and the Harvard Nanoscale Science and Engineering Center (NSEC) for the financial support of this work. The Harvard NSEC is supported by the National Science Foundation (NSF).

References and links

1.

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, 1331–1333 (1998). [CrossRef]

2.

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, R16356–16359 (2000). [CrossRef]

3.

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 Mater. 2, 229–232 (2003). [CrossRef]

4.

N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095–3097 (2003). [CrossRef]

5.

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

6.

B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance,” Phys. Rev. Lett. 84, 4721–4724 (2000). [CrossRef] [PubMed]

7.

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]

8.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss,” Appl. Phys. Lett. 81, 1714–1716 (2002). [CrossRef]

9.

Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, “Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains,” Nano Lett. 4, 1067–1071 (2004). [CrossRef]

10.

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” Phys. Chem. B 107, 7337–7342 (2003). [CrossRef]

11.

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]

12.

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. V. Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Kall, “Controlling Plasmon Line Shapes through Diffractive Coupling in Linear Arrays of Cylindrical Nanoparticles Fabricated by Electron Beam Lithography,” Nano Lett. 5, 1065–1070 (2005). [CrossRef] [PubMed]

13.

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007). [CrossRef]

14.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004). [CrossRef]

15.

S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B 69, 125418 (2004). [CrossRef]

16.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006). [CrossRef]

17.

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,” Phys. Rev. B 76, 201403(R) (2007). [CrossRef]

18.

T. Yang and K. B. Crozier, “Dispersion and Extinction of Surface Plasmons in Gold Nanoparticle Chains: Influence of the Air/Glass Interface,” Opt. Express 16, 8570–8580. [PubMed]

19.

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,” Opt. Express 15, 17482–17493 (2007). [CrossRef] [PubMed]

20.

Think of a plane wave propagating in z direction. It has an infinite dω/dkx value.

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(230.4555) Optical devices : Coupled resonators

ToC Category:
Optics at Surfaces

History
Original Manuscript: July 21, 2008
Revised Manuscript: August 4, 2008
Manuscript Accepted: August 5, 2008
Published: August 11, 2008

Citation
Tian Yang and Kenneth B. Crozier, "Surface plasmon coupling in periodic metallic nanoparticle structures: a semi-analytical model," Opt. Express 16, 13070-13079 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13070


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References

  1. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, "Electromagnetic energy transport via linear chains of silver nanoparticles," Opt. Lett. 23, 1331-1333 (1998). [CrossRef]
  2. 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, R16356-16359 (2000). [CrossRef]
  3. 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 Mater. 2, 229-232 (2003). [CrossRef]
  4. N. Félidj, J. Aubard, G. Lévi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, "Optimized surface-enhanced Raman scattering on gold nanoparticle arrays," Appl. Phys. Lett. 82, 3095-3097 (2003). [CrossRef]
  5. J. R. Krenn, A. Dereux, J. C. Weeber, E. Bourillot, Y. Lacroute, and J. P. Goudonnet, "Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles," Phys. Rev. Lett. 82, 2590-2593 (1999). [CrossRef]
  6. B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, "Metal Nanoparticle Gratings: Influence of Dipolar Particle Interaction on the Plasmon Resonance," Phys. Rev. Lett. 84, 4721-4724 (2000). [CrossRef] [PubMed]
  7. 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]
  8. S. A. Maier, P. G. Kik, and H. A. Atwater, "Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss," Appl. Phys. Lett. 81, 1714-1716 (2002). [CrossRef]
  9. Q.-H. Wei, K.-H. Su, S. Durant, and X. Zhang, "Plasmon Resonance of Finite One-Dimensional Au Nanoparticle Chains," Nano Lett. 4, 1067-1071 (2004). [CrossRef]
  10. C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. J. Kall, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," Phys. Chem. B 107, 7337-7342 (2003). [CrossRef]
  11. 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]
  12. E. M. Hicks, S. Zou, G. C. Schatz; K. G. Spears, R. P. V. Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Kall, "Controlling Plasmon Line Shapes through Diffractive Coupling in Linear Arrays of Cylindrical Nanoparticles Fabricated by Electron Beam Lithography," Nano Lett. 5, 1065-1070 (2005). [CrossRef] [PubMed]
  13. F. J. García de Abajo, "Colloquium: Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007). [CrossRef]
  14. W. H. Weber, and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70, 125429 (2004). [CrossRef]
  15. S. Y. Park, and D. Stroud, "Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation," Phys. Rev. B 69, 125418 (2004). [CrossRef]
  16. A. F. Koenderink, and A. Polman, "Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains," Phys. Rev. B 74, 033402 (2006). [CrossRef]
  17. 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," Phys. Rev. B 76, 201403 (R) (2007). [CrossRef]
  18. T. Yang, and K. B. Crozier, "Dispersion and Extinction of Surface Plasmons in Gold Nanoparticle Chains: Influence of the Air/Glass Interface," Opt. Express 16, 8570-8580. [PubMed]
  19. 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," Opt. Express 15, 17482-17493 (2007). [CrossRef] [PubMed]
  20. Think of a plane wave propagating in z direction. It has an infinite d�?/dkx value.

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