OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 26 — Dec. 21, 2009
  • pp: 23655–23663
« Show journal navigation

Effect of retardation on localized surface plasmon resonances in a metallic nanorod

Timothy J. Davis, Kristy C. Vernon, and Daniel E. Gómez  »View Author Affiliations


Optics Express, Vol. 17, Issue 26, pp. 23655-23663 (2009)
http://dx.doi.org/10.1364/OE.17.023655


View Full Text Article

Acrobat PDF (246 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The localized surface plasmon resonances in a metallic nanorod are determined using the “electrostatic approximation” and by a finite-difference time-domain numerical solution of Maxwell’s equations. The difference between the two methods is related to the effects of re-radiation, or retardation, which is not included in the electrostatic formulation. It is shown that high-order modes in a metallic nanorod can be modeled by both methods, even beyond the point where the electrostatic method is supposed to fail. This suggests that the simple analytical expressions derived from the electrostatic approximation are valid for describing the large range of resonant modes associated with metallic nanoparticles, including dark modes.

© 2009 OSA

1. Introduction

Localized surface plasmon resonances in metallic nanoparticles arise from an interaction between the surface charges and their associated electromagnetic fields. Although Maxwell’s equations provide an accurate description of these resonances, the equations are difficult to solve analytically for particles of geometries other than spheres, ellipsoids or infinite right cylinders. Alternatively, there are many numerical methods that can be used to solve the equations depending on the application, such as the Finite Difference Time Domain method [1

1. K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]

,2

2. A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, London, 1995).

], Rigorous Coupled Wave Analysis [3

3. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11(9), 2494–2502 (1994). [CrossRef]

5

5. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface relief gratings: enhanced transmission matrix approach,” J. Opt. Soc. Am. A 12(5), 1077–1086 (1995). [CrossRef]

], the Discrete Dipole Approximation [6

6. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]

] and the Boundary Element Method [7

7. F. J. García de Abajo and A. Howie, “Relativistic Electron Energy Loss and Electron-Induced Photon Emission in Inhomogeneous Dielectrics,” Phys. Rev. Lett. 80(23), 5180–5183 (1998). [CrossRef]

,8

8. F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” A., Phys. Rev. B 65(11), 115418 (2002). [CrossRef]

]. While these methods can provide accurate predictions of the properties of plasmonic systems, it is often difficult to extract parameter relationships from the numerical results. In this regard we have been pursuing analytical methods to model the underlying physics of interacting plasmonic systems, even if the methods give only approximate results. Through an understanding of the physics, it is then possible to design advanced systems of interacting plasmonic particles for applications in sensing and metamaterials.

Recently an approximate method was described for solving Maxwell’s equations for plasmonic systems in the case where the nanoparticle is much smaller than the wavelength of light [9

9. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B 72(15), 155412 (2005). [CrossRef]

]. In this size regime, the electric and magnetic fields decouple and Maxwell’s equations take the same form as in electrostatics. In effect, the nanoparticle experiences a spatially uniform but time varying electric field arising from the applied light field. In this “electrostatic approximation” it is a simple task to calculate the self-sustained surface-charge distributions, or eigenfunctions of the nanoparticle resonances, and obtain the eigenvalues that determine the resonant frequencies. The electrostatic theory has been extended to take account of the interactions between nanoparticles in an ensemble [10

10. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009). [CrossRef]

], providing a simple means by which complex nanoparticle systems can be designed to produce the required optical properties. A key aspect of this theory is that it yields relatively simple analytical expressions describing the near-field interactions between optically resonant nanoparticles of arbitrary shape. Such expressions enable us to identify the important terms that affect the resonances of coupled systems and that control the intensity and polarization of light scattering. For example, we were able to use the method to design a plasmonic circuit [11

11. T. J. Davis, K. C. Vernon, and D. E. Gómez, “A plasmonic “ac Wheatstone bridge” circuit for high-sensitivity phase measurement and single-molecule detection,” J. Appl. Phys. 106(4), 043502 (2009). [CrossRef]

], the equivalent of the Wheatstone bridge in electronics, which should enable the optical detection of single molecules, such as proteins, or provide a direct measure of phase shifts between two plasmonic nanoparticles. This circuit and similar subwavelength three-particle systems involve the formation of “dark modes” that interact weakly with the incident light [12

12. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef] [PubMed]

14

14. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]

]. The dark modes are associated with quadrupole and higher-order resonances that have zero dipole moments and therefore are not able to be excited directly by plane wave illumination. These three-particle systems can be analyzed algebraically using the electrostatic coupling theory [15

15. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems: applications to dark modes in nanoparticle pairs and triplets,” Proc. SPIE 7394, 739423 (2009). [CrossRef]

].

Since we wish to use the analytical expressions derived from the electrostatic theory for applications involving both dipole modes and higher-order resonance modes, it is important to determine whether or not the theory can model higher-order modes and at what point would we expect the errors to become large. In this paper we compare the results of electrostatic calculations of the resonant modes of a nanoparticle rod with the results of the finite difference time domain (FDTD) method that includes the effects of retardation. The advantage of the electrostatic approximation is that it provides all of the resonant modes of a nanoparticle or an ensemble of nanoparticles, within the limitation of the numerical grid used to represent the nanoparticle surface. Moreover, the resonances are determined by the eigenvalues and the electric permittivity of the nanoparticle according to a simple relation. This means that the eigenvalues need only be evaluated once and the resonances can then be determined for any material. As discussed above, it is possible then to deduce how one nanoparticle couples to another. A limitation with purely numerical techniques such as FDTD is that there is no guarantee that all the resonant modes will be found.

2. Electrostatic approximation and the resonances of a nanorod

The solution of the electrostatic eigenvalue problem involves a sum over the surface of the nanoparticle which, for numerical computation, is represented by a finite mesh. Here we consider a nanorod consisting of a cylindrical body 1.5 units long and 0.8 units in diameter with hemispherical ends 0.4 units high, giving a total rod length of 2.3 units. The surface of the rod was tessellated with 1856 triangles and the eigenvalues calculated using the method discussed by Mayergoyz et al [9

9. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B 72(15), 155412 (2005). [CrossRef]

]. The first ten modes of the rod are shown in Fig. 1
Fig. 1 The first ten localized surface plasmon modes of a nanorod. The color represents the relative strength of the surface-dipole distribution with blue positive and red negative. The surface-charge distributions are similar.
.

3. Resonances of the nanorod

3.1 Electrostatic resonances

To determine the resonant frequency of the nanorod, we assume that the rod is in a dielectric medium with a permittivityεb=1.75, which is approximately the same as water. It is convenient to represent the permittivity of the nanorod using the Drude formula that allows us to choose an appropriate damping factor. This is useful for the FDTD simulation discussed below. For the numerical simulation we use
ε(ω)=1(ωp2+Γ2)ω(ω+iΓ)
(4)
where the plasma frequency ωp=6.28×103 THz was chosen to occur at a vacuum wavelength of 300 nm and the damping factor is Γ=30THz. These values were selected to produce low damping and strong resonances in the visible spectrum. Other than that, the values are somewhat arbitrary but give a permittivity similar to silver. Using this model, the resonant energies E=ω and the equivalent vacuum wavelengths of each of the nanorod modes are evaluated. These are shown in Table 1

Table 1. Electrostatic Resonances of the Nanorod

table-icon
View This Table
, rounded to three significant figures.

3.2 Finite Difference Time Domain model

To compare the resonances obtained using the electrostatic approximation with those found by solving Maxwell’s equations including retardation, we have used an in-house FDTD numerical model based on the work of Taflove [2

2. A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, London, 1995).

]. The model uses a three-dimensional grid of points with the electric and magnetic fields placed a half step away from the centre of each point, following the Yee grid formalism [1

1. K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]

]. The grid consists of 75×75×75 points to represent the space around the nanorod, with the rod occupying a region 23 grid points in diameter and 67 grid points in length, including the hemispherical ends. The permittivity of the metal of the nanorod is represented by a second-order differential equation in time for the polarization [22

22. T. J. Davis, “Modelling and fabrication of tuned circuits for optical meta-materials,” Proc. SPIE 6038, Y380 (2005).

]. The coefficients of the differential equation are derived from the Drude model, Eq. (4). A small damping factor was used to promote strong resonances but to ensure some decay, particularly for non-radiating modes. The length scale is set independently of the number of grid points so that the same model can be run multiple times but with different scales to study the effects of size on the resonances.

The model simulates an electromagnetic field incident on the nanoparticle to excite the oscillations. However, a plane wave will only excite a nanoparticle mode if it has a dipole moment [10

10. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009). [CrossRef]

]. Since many of the modes do not have a dipole moment, this becomes problematic. The simulation was configured so that the excitation source was a rectangular aperture slightly wider than the diameter of the nanorod and about the length of the hemispherical end. The aperture was placed near one end of the rod and the incident electromagnetic wave was emitted through this aperture. The incident wave was a Gaussian-modulated sinusoidal pulse with a centre wavelength at 800 nm and a standard deviation (width) of the Gaussian of 200 nm. This enables us to apply a short “numerical” electromagnetic pulse to the top of the nanorod in the hope that it will excite a large number of modes. The x, y, and z components of the electric field were sampled just below the nanorod, at the end opposite to that of the excitation pulse. Data were sampled as functions of time and then Fourier transformed to obtain the spectra. This process was repeated for four different scales corresponding to nanorod lengths of 90 nm, 230 nm, 460 nm and 690 nm. The resonances were identified by the order of the peak positions in the spectra.

The results of the first two simulations are shown in Fig. 2
Fig. 2 The excitation spectra of 90 nm long and 230 nm long nanorods calculated using the FDTD method. The spectra were obtained from samples of the three electric field components at the end of the nanorod. Included on the spectra are the locations of the modes as calculated using the electrostatic approximation.
. For the 90 nm nanorod we find that there is a good agreement between the excitation resonances calculated using the FDTD method and the resonant modes as determined by the electrostatic approximation. In particular, the locations of the higher-order modes are in agreement. The condition Eq. (2) for the validity of the electrostatic approximation is εb(kd)2=0.4 for the 90 nm rod based on λ=1190nm for the fundamental resonance. If we use λ=750nm which is the resonance of mode 2, we find that εb(kd)2=1.0 so that we would not expect this mode to be accurately reproduced. However, Fig. 2 shows that the modes agree well, certainly within the 5% error we might expect for the FDTD model.

For the 230 nm long nanorod, the electrostatic approximation is much less accurate. For this nanorod, with λ=1190nm, we find that εb(kd)2=2.6 which is clearly violating the condition given by Eq. (2). The spectra associated with this rod are red-shifted with respect to the shorter nanorod, but note also that the longer wavelength resonances are broader. This is a consequence of radiation damping. The longer is the rod, the stronger the dipole moment and the more efficiently it will radiate. The electrostatic approximation does not include radiation as this requires a coupling between the electric and magnetic fields. For small dimensions, this coupling is weak (and is neglected in the electrostatic approximation) so there is little radiation from the small nanorod. The broadening of the spectral peaks by radiation damping can be included in the electrostatic approximation [10

10. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009). [CrossRef]

] which appears as an additional loss term. This can also lead to a small shift in the spectral peak position, depending on the electric permittivity of the nanorod.

The results of four nanorod lengths are summarized in Fig. 3
Fig. 3 The fractional error in the position of the resonance based on the difference between the FDTD calculation and the electrostatic approximation for four different length nanorods. Also shown is the location of each of the modes as determined by the electrostatic approximation and the validity condition for each rod.
. It is interesting to note that the lowest-order mode is the least accurately represented as the nanorod length is increased, showing the greatest fractional error. However, the errors in the higher-order modes increase more slowly. For the 230 nm long nanorod, the errors are below 8% for all modes except for mode 1 which has an error of about 18%. There is some fluctuation in the errors of the high-order modes for the longer nanorods which is probably due to the difficulty in identifying the individual modes, since some of them are not easily excited and the resonances can be broad.

For comparison with the results in Fig. 2, we show in Fig. 4
Fig. 4 The excitation spectra of 90 nm long and 230 nm long nanorods calculated using the FDTD method where the incident field was a plane wave. Many of the modes, notably those that have a zero dipole moment, are not excited in this case.
the resonant modes excited by a plane-wave electromagnetic pulse incident on the nanoparticle with the electric field polarized parallel to the long axis. The pulse was exactly the same as used previously except that it was applied to the entire rod instead of through the aperture. It is apparent from these spectra that many of the modes are missing. In particular, mode 2 is absent and in this regard can be considered a dark mode. From Fig. 1 it is clear that this mode has a strong linear-quadrupole moment with no dipole moment. As we discussed in section 2, modes that have no dipole moment cannot be excited by a plane wave when the nanoparticle is much smaller than the wavelength of the incident radiation. This highlights one of the limitations with this numerical method in determining the resonant modes of nanoparticle.

4. Discussion

The comparison between the two calculations clearly shows that the electrostatic method is capable of predicting the resonances of many of the modes of a nanorod, including the high-order modes. This is important as it allows us to make use of the analytical formulae associated with this method for predicting the resonant effects arising from interactions between metallic nanoparticles and, in particular, the resonances associated with dark modes.

5. Summary

In this paper we have modeled the localized surface plasmon resonances of a nanorod using the electrostatic approximation and the FDTD method. It was shown that the electrostatic formulation can accurately predict the resonances of both low and high-order modes, including dark modes, provided that the nanorod is not too large. This is important because we wish to use the simple analytical expressions provided by this method to design optically resonant nanoparticle systems. It was found that the simple criterion for the validity of the electrostatic approximation does not provide an adequate measure of the point of failure of the method. It was shown that the second-order correction to the resonance permittivity involves the eigenvalues of the resonances and should scale as the ratio of the surface area of the nanoparticle to the square of the wavelength of the radiation in the medium. This suggests that the modes of higher aspect ratio nanostructures can be predicted with higher accuracy than lower aspect ratio nanostructures with similar dimensions.

References and links

1.

K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]

2.

A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, London, 1995).

3.

E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11(9), 2494–2502 (1994). [CrossRef]

4.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]

5.

M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface relief gratings: enhanced transmission matrix approach,” J. Opt. Soc. Am. A 12(5), 1077–1086 (1995). [CrossRef]

6.

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]

7.

F. J. García de Abajo and A. Howie, “Relativistic Electron Energy Loss and Electron-Induced Photon Emission in Inhomogeneous Dielectrics,” Phys. Rev. Lett. 80(23), 5180–5183 (1998). [CrossRef]

8.

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” A., Phys. Rev. B 65(11), 115418 (2002). [CrossRef]

9.

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B 72(15), 155412 (2005). [CrossRef]

10.

T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009). [CrossRef]

11.

T. J. Davis, K. C. Vernon, and D. E. Gómez, “A plasmonic “ac Wheatstone bridge” circuit for high-sensitivity phase measurement and single-molecule detection,” J. Appl. Phys. 106(4), 043502 (2009). [CrossRef]

12.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef] [PubMed]

13.

N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. 9(4), 1663–1667 (2009). [CrossRef] [PubMed]

14.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]

15.

T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems: applications to dark modes in nanoparticle pairs and triplets,” Proc. SPIE 7394, 739423 (2009). [CrossRef]

16.

P. W. Barber, R. K. Chang, and H. Massoudi, “Electrodynamic calculations of the surface-enhanced electric intensities on large Ag spheroids,” Phys. Rev. B 27(12), 7251–7261 (1983). [CrossRef]

17.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37(9), 1792–1805 (2008). [CrossRef] [PubMed]

18.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004). [CrossRef]

19.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74(3), 033402 (2006). [CrossRef]

20.

C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 5.

21.

I. D. Mayergoyz, Z. Zhang, and G. Miano, “Analysis of dynamics of excitation and dephasing of plasmon resonance modes in nanoparticles,” Phys. Rev. Lett. 98(14), 147401 (2007). [CrossRef] [PubMed]

22.

T. J. Davis, “Modelling and fabrication of tuned circuits for optical meta-materials,” Proc. SPIE 6038, Y380 (2005).

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(240.6680) Optics at surfaces : Surface plasmons
(250.5403) Optoelectronics : Plasmonics
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Optics at Surfaces

History
Original Manuscript: October 23, 2009
Revised Manuscript: November 30, 2009
Manuscript Accepted: December 1, 2009
Published: December 10, 2009

Citation
Timothy J. Davis, Kristy C. Vernon, and Daniel E. Gómez, "Effect of retardation on localized surface plasmon resonances in a metallic nanorod," Opt. Express 17, 23655-23663 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23655


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]
  2. A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, London, 1995).
  3. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11(9), 2494–2502 (1994). [CrossRef]
  4. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]
  5. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface relief gratings: enhanced transmission matrix approach,” J. Opt. Soc. Am. A 12(5), 1077–1086 (1995). [CrossRef]
  6. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]
  7. F. J. García de Abajo and A. Howie, “Relativistic Electron Energy Loss and Electron-Induced Photon Emission in Inhomogeneous Dielectrics,” Phys. Rev. Lett. 80(23), 5180–5183 (1998). [CrossRef]
  8. F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” A., Phys. Rev. B 65(11), 115418 (2002). [CrossRef]
  9. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B 72(15), 155412 (2005). [CrossRef]
  10. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009). [CrossRef]
  11. T. J. Davis, K. C. Vernon, and D. E. Gómez, “A plasmonic “ac Wheatstone bridge” circuit for high-sensitivity phase measurement and single-molecule detection,” J. Appl. Phys. 106(4), 043502 (2009). [CrossRef]
  12. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef] [PubMed]
  13. N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. 9(4), 1663–1667 (2009). [CrossRef] [PubMed]
  14. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]
  15. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems: applications to dark modes in nanoparticle pairs and triplets,” Proc. SPIE 7394, 739423 (2009). [CrossRef]
  16. P. W. Barber, R. K. Chang, and H. Massoudi, “Electrodynamic calculations of the surface-enhanced electric intensities on large Ag spheroids,” Phys. Rev. B 27(12), 7251–7261 (1983). [CrossRef]
  17. V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37(9), 1792–1805 (2008). [CrossRef] [PubMed]
  18. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004). [CrossRef]
  19. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74(3), 033402 (2006). [CrossRef]
  20. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 5.
  21. I. D. Mayergoyz, Z. Zhang, and G. Miano, “Analysis of dynamics of excitation and dephasing of plasmon resonance modes in nanoparticles,” Phys. Rev. Lett. 98(14), 147401 (2007). [CrossRef] [PubMed]
  22. T. J. Davis, “Modelling and fabrication of tuned circuits for optical meta-materials,” Proc. SPIE 6038, Y380 (2005).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited