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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 7 — Jun. 25, 2012
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Simulating light scattering from supported plasmonic nanowires

Vladimir D. Miljković, Timur Shegai, Peter Johansson, and Mikael Käll  »View Author Affiliations


Optics Express, Vol. 20, Issue 10, pp. 10816-10826 (2012)
http://dx.doi.org/10.1364/OE.20.010816


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Abstract

We present a method for calculating the differential scattering cross sections from nanostructures close to an interface separating two semi-infinitive dielectric media. The method combines a fast finite element software (Comsol multiphysics), used for calculations of the fields around and inside the structure, and the Green's functions method, which is used to find the far field distribution from the calculated total fields inside the nanostructure. We apply the method to calculations of scattering spectra from silver nanowires supported by an air-glass interface, a system that is of high current interest in relation to various nanophotonics applications. The results are analyzed in relation to analytical models and compared to experimentally measured spectra, to which we find a good agreement.

© 2012 OSA

1. Introduction

Surface plasmons in metal nanostructures [1

1. E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Zeitschrift Fur Naturforschung Part A-Astrophysik Physik Und Physikalische Chemie A 23, 2135–2136 (1968).

, 2

2. U. Kreibig and L. Genzel, “Optical absorption of small metallic particles,” Surf. Sci. 156, 678–700 (1985). [CrossRef]

] have attracted enormous recent interest because of a wide range of potential applications in areas such as biochemical sensing [3

3. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). [CrossRef] [PubMed]

], photovoltaics [4

4. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

], nanophotonics [5

5. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef] [PubMed]

] and metamaterials [6

6. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

]. However, much of this development would most likely not have occurred without the continuous development of sophisticated methods and software for electrodynamics simulations of nanooptical phenomena. In this paper, we present a method for calculating the differential scattering cross section of supported nanostructures using a combination of the Green's function method and a commercial finite element method (FEM) (Comsol multiphysics 3.5a). We believe that the methodology fills an important gap among the large number of existing computational techniques in that it makes it possible to simulate comparatively large metal structures that are optically close to an interface. Indeed, this is a situation that is encountered in practice in a very large number of experimental plasmonics studies. Here we use the method to study the near-field and far-field properties of supported silver nanowires, but the method can in principle be applied to any nanostructure shape and material.

Progress in colloidal synthesis over the last two decades has allowed for the fabrication of a large diversity of metal nanostructures with high crystalline quality, in particular various types of nanowires [7

7. Y. G. Sun, B. Gates, B. Mayers, and Y. N. Xia, “Crystalline silver nanowires by soft solution processing,” Nano Lett. 2(2), 165–168 (2002). [CrossRef]

, 8

8. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]

]. Metal nanowires are interesting from a photonics point-of-view because they can support surface plasmon polaritons (SPPs) that are propagating along one direction while being localized in the other two [8

8. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]

16

16. A. W. Sanders, D. A. Routenberg, B. J. Wiley, Y. N. Xia, E. R. Dufresne, and M. A. Reed, “Observation of plasmon propagation, redirection, and fan-out in silver nanowires,” Nano Lett. 6(8), 1822–1826 (2006). [CrossRef] [PubMed]

], a phenomenon that could be interesting for a number of applications. Recent reports pointing in this direction include investigations of coupling between plasmonic and photonic nanowires [17

17. R. X. Yan, P. Pausauskie, J. X. Huang, and P. D. Yang, “Direct photonic-plasmonic coupling and routing in single nanowires,” Proc. Natl. Acad. Sci. U.S.A. 106(50), 21045–21050 (2009). [CrossRef] [PubMed]

19

19. X. Guo, M. Qiu, J. M. Bao, B. J. Wiley, Q. Yang, X. N. Zhang, Y. G. Ma, H. K. Yu, and L. M. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett. 9(12), 4515–4519 (2009). [CrossRef] [PubMed]

] and between nanoparticles and wires [20

20. F. Hao and P. Nordlander, “Plasmonic coupling between a metallic nanosphere and a thin metallic wire,” Appl. Phys. Lett. 89(10), 103101 (2006). [CrossRef]

, 21

21. M. W. Knight, N. K. Grady, R. Bardhan, F. Hao, P. Nordlander, and N. J. Halas, “Nanoparticle-mediated coupling of light into a nanowire,” Nano Lett. 7(8), 2346–2350 (2007). [CrossRef] [PubMed]

], demonstrations of plasmon routing [22

22. Y. R. Fang, Z. P. Li, Y. Z. Huang, S. P. Zhang, P. Nordlander, N. J. Halas, and H. X. Xu, “Branched silver nanowires as controllable plasmon routers,” Nano Lett. 10(5), 1950–1954 (2010). [CrossRef] [PubMed]

] and interferometric logics [23

23. H. Wei, Z. Li, X. Tian, Z. Wang, F. Cong, N. Liu, S. Zhang, P. Nordlander, N. J. Halas, and H. Xu, “Quantum dot-based local field imaging reveals plasmon-based interferometric logic in silver nanowire networks,” Nano Lett. 11(2), 471–475 (2011). [CrossRef] [PubMed]

] and studies of how single photon emitters, such as quantum dots and fluorophores, couple to nanowires [24

24. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007). [CrossRef] [PubMed]

28

28. H. Wei, D. Ratchford, X. E. Li, H. X. Xu, and C. K. Shih, “Propagating surface plasmon induced photon emission from quantum dots,” Nano Lett. 9(12), 4168–4171 (2009). [CrossRef] [PubMed]

]. More fundamental studies on plasmonic nanowires include research on near field [29

29. J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9(6), 2372–2377 (2009). [CrossRef] [PubMed]

] and far field [30

30. J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. 10(9), 3596–3603 (2010). [CrossRef] [PubMed]

, 31

31. E. R. Encina and E. A. Coronado, “Plasmonic nanoantennas: Angular scattering properties of multipole resonances in noble metal nanorods,” J. Phys. Chem. C 112(26), 9586–9594 (2008). [CrossRef]

] properties, emission directionality [32

32. T. Shegai, V. D. Miljković, K. Bao, H. Xu, P. Nordlander, P. Johansson, and M. Käll, “Unidirectional broadband light emission from supported plasmonic nanowires,” Nano Lett. 11(2), 706–711 (2011). [CrossRef] [PubMed]

34

34. Z. P. Li, K. Bao, Y. R. Fang, Y. Z. Huang, P. Nordlander, and H. X. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett. 10(5), 1831–1835 (2010). [CrossRef] [PubMed]

] and optical rotation [35

35. L. M. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10(1), 268–273 (2010). [CrossRef] [PubMed]

].

Most of the experimental studies mentioned above deal with nanowires supported by an optically mismatched interface, and this is therefore the focus of the present study. We are particularly interested in understanding how plasmonic nanowires behave as Fabry Perot (FP) resonators and in characterizing the near-field and far-field properties of such systems. We also calculate the plasmon dispersion relations for supported nanowires of different diameters and compare the results with analytical models valid for homogeneous environments. Some of the calculations are compared with experimental results and a good agreement is found.

2. Methods

Our calculations combine the use of a commercial FEM package with the Green’s function method and are carried out in two steps. First the total fields inside the scatterer are calculated using the finite element method and afterwards, as a second step, the far field distribution of the fields is calculated using the Green's functions for the two-layer background and the field inside the scatterer found in the first step as input.

For the FEM calculation, we use the RF module in the scattering formulation to find the electric field distribution inside, and in the near proximity of, a single silver nanowire, i.e. a cylinder with the radius R (we consider the cases R = 40 nm, R = 80 nm, and R = 160 nm in this work) and length L = 5 µm. Figure 1(a)
Fig. 1 Schematic of the calculation methods. (a) Comsol calculation domain for the calculations of the fields inside and in near surrounding of the nanowire. (b) Calculations of the differential scattering cross sections using the Green's function method and meshing of the nanowire.
shows the Comsol simulation domain, with the interface between air and glass at z = 0 and the surroundings of the wire have refractive index n1=1for z>0, andn2=1.5 for z<0. The wire has a refractive index of silver taken from Johnson and Christy [36

36. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

]. In the FEM calculation the surroundings of the nanowire are delimited by two finite cylinders, the smaller with radius 600 nm and length 7 µm and the bigger with radius 900 nm and the length 8 µm. The part of space in between these cylinders (blue color in the Fig. 1(a)) are perfectly matched layers (PML's), which absorb all the scattered light with minimal reflections. We apply scattering boundary conditions at the surface of the larger cylinder, i.e. that interface is transparent to the scattered light, and perfect magnetic conductor boundary conditions at y = 0 because the magnetic field is only directed along the y axis, which allows us to use the symmetry and simulate only half the wire.

The plane wave driving the system is incident along the z axis from the air side and is, using the Fresnel formulas, E=(Ex,0,0)where Exis
Ex={E0eik0n1z(1re2ik0n1z),z>0E0teik0n2z,z<=0,
(1)
andE0is the magnitude of the field, k0is the wave number in vacuum, r=n1n2n1+n2is the reflection magnitude and t=2n1n1+n2is the transmission magnitude for normal incidence at the air-glass interface. Here we use a time dependence eiωt, whereas Comsol uses the electrical engineering convention with a time dependence eiωt,which means that that the arguments of the exponential functions in Eq. (1) should have an opposite sign when the equation is implemented in that software.

After solving the model, the field distribution inside and in the proximity of the nanowire is known. In order to find differential scattering cross sections of the nanowire, we then use the Green's function method [37

37. P. Johansson, “Electromagnetic Green’s function for layered systems: Applications to nanohole interactions in thin metal films,” Phys. Rev. B 83(19), 195408 (2011). [CrossRef]

]. Here, we derive an analytical formula of the asymptotic Green's function for the case of a single interface, which is easy to implement. It provides a generalization to the case of a layered background of the Stratton-Chu formula [38

38. J. A. Stratton, Electromagnetic Theory (McGraw-Hill: New York, 1941).

] that is built into the Comsol software.

The differential scattering cross section can be calculated as,
dσdΩ(θ,φ)=r2SfarSin,
(2)
whereSfaris the radial component of the Poynting vector at a large distance r, Sinis the Poynting vector magnitude of the incident field, and the scattering direction is defined by the spherical coordinate anglesθand φ. In our case the field is incident from the air side, while the scattered fields propagate in both air (labeled with index 1) and glass (labeled with index 2), so that the corresponding Poynting vector magnitudes can be calculated as,
Sfar1(2)=12cε0ε1(2)|E(r)|2,
(3)
Sin1=12cε0ε1|E0|2,
(4)
where E(r) is the scattered field from the nanowire (in a direction defined with angles θand φ) and E0is the magnitude of the incident field.

The scattered electric fields from a nanostructure can be calculated by discretizing the electric fields on a mesh with equally sized cubic elements (see Fig. 1(b)), summing contributions from each mesh element [37

37. P. Johansson, “Electromagnetic Green’s function for layered systems: Applications to nanohole interactions in thin metal films,” Phys. Rev. B 83(19), 195408 (2011). [CrossRef]

],
E(r)=i=1NG(r,ri')k02ΔεiEiVi,
(5)
wherek0is the wave vector in vacuum, Δεi=εiε1,εiis the relative dielectric permittivity of the scatterer (in our case εiis dielectric permittivity of silver),ε1 is dielectric permittivity of the surrounding medium (in our case the wire is in air, thus ε1=1), Vi=aM3is the volume of the cubic mesh element with size aM(we use aM=5nm),Ei=(Ex,i,Ey,i,Ez,i) is the total field at the position of ith mesh element (in our case, those fields were calculated using FEM), and G(r,ri') is the Green's function, which describes field propagation in a direction specified by the angles θand φ.

In the following, we present expressions for the asymptotic Green's functions at a single interface positioned at z = 0 as shown in Fig. 1. The asymptotic Green's function G1 on the air side (i.e. for z > 0 or 0 < θ < π/2 in spherical coordinates) and G2 on the glass side (i.e. for z < 0, or π/2 < θ < π) can be calculated as follows,
G1(r,ri')=eik1r4πreik||ri'eikz1z'[p^1+p^1++s^s^+e2ikz1z'(rpp^1+p^1+rss^s^)],
(6)
G2(r,ri')=eik2r4πreik||ri'kz2kz1eikz1z'[tpp^2p^1+tss^s^],
(7)
where k1(2)=ε1(2)k0is the magnitude of the wave vector in air (glass), ri'=(xi,yi,zi) are coordinates of the mesh elements, is dyadic product that makes the Green's function a 3×3 matrix, and the in-plane wave vector is k||=k1(2)(sinθcosφ,sinθsinφ,0) when the field point is in air (glass). The z component of the wave vector in air (glass) is kz1(2)=k1(2)2k||2, while the unit polarization vectors for s and p polarizations are given by

s^=(sinφ,cosφ,0)
(8)
p^1±=(±kz1k1cosφ,±kz1k1sinφ,k||k1),
(9)
p^2=(kz2k2cosφ,kz2k2sinφ,k||k2).
(10)

The reflection and transmission amplitudes for s and p polarizations can be calculated as,

rs=kz1kz2kz1+kz2,ts=1+rs,
(11)
rp=ε2kz1ε1kz2ε2kz1+ε1kz2,tp=k2kz1k1kz2(1rs).
(12)

From a physical point of view the Green's function G1(r,ri'), from Eq. (6), describes the fields emitted from a mesh element at the position (xi,yi,zi) to the far field on the air side, while G2(r,ri') in Eq. (7) describes fields transmitted into the glass side. In Eq. (6) the first two terms inside the brackets describe p and s polarized waves that are propagating into the far field directly and the last two terms describe waves propagating in the same direction after having been reflected off the air-glass interface, while in Eq. (7) the Green's function G2(r,ri') only involves waves that are transmitted through the air-glass interface.

3. Results and discussion

Figure 2
Fig. 2 (a-c) Total scattering spectra for thin (R = 40 nm), intermediate (R = 80 nm) and thick (R = 160 nm) nanowires calculated for three different surrounding media: (a) Wires surrounded by air; (b) Wires surrounded by glass; (c) Wires supported at an air-glass interface (the total spectra are calculated by integration of scattering in glass only). The corresponding mode orders l are shown above the each resonance.
shows total scattering spectra for nanowires calculated in the three different environments air, glass and air-glass, respectively (for the air-glass case, spectra represent the total scattering to the glass side). The scattering from thin nanowires (R = 40 nm) displays a series of pronounced resonance peaks and, as we will see, the nanowire acts like a Fabry-Perot cavity for plasmons. Upon an increase of the wire thickness (green and blue curves), regardless of the environment, the resonances are weakened and almost completely vanish for R = 160 nm (blue dash-dotted lines).

Experimental scattering spectra for supported silver wires of different dimensions are summarized in Fig. 3
Fig. 3 Experimental scattering spectra for silver wires of different length and diameters supported by an air-glass interface. Similar to the calculations, thin wires show pronounced standing wave profiles while thick wires are poor resonators. Inset shows representative SEM images of D = 70 nm and D = 197 nm wires.
. Propagating plasmons were in this case excited by focusing a fiber-coupled white-light source onto one end of a wire, which results in excitation of plasmon resonances of both even and odd symmetry. The scattering from the output end was then collected by the same objective (60 × NA = 1.49 oil immersion) and sent to a fiber-coupled spectrometer, more details can be found elsewhere [32

32. T. Shegai, V. D. Miljković, K. Bao, H. Xu, P. Nordlander, P. Johansson, and M. Käll, “Unidirectional broadband light emission from supported plasmonic nanowires,” Nano Lett. 11(2), 706–711 (2011). [CrossRef] [PubMed]

]. Also here, in good agreement with the calculated results, the standing wave resonances become more and more damped with increasing wire diameter, in spite of the somewhat different scattering geometries.

In order to understand the nanowire resonances it is useful to turn to the dispersion relation for plasmons propagating along an infinite cylinder. Figures 4(a)
Fig. 4 (a-c) Dispersion relations calculated by extracting plasmon wavelengths from Fabry Perot model (crosses), analytical solution of infinite metal cylinder (solid lines), and from the experimental spectra (squares) for thin (a), intermediate (b), and thick (c) nanowires, respectively. The dispersions are calculated for wires in air (red), glass (green), and at an air-glass interface (blue). In addition, the light line in air (black dashed line) and the light line in glass (black dash-dotted line) are shown.
4(c) show such dispersion relations for Ag cylinders with radius R = 40 nm, R = 80 nm and R = 160 nm, respectively. The solid lines represent the dispersion relation for TM0 modes of a cylinder in a homogeneous environment [15

15. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98(26), 266802 (2007). [CrossRef] [PubMed]

] found from the solution of
εmκ1J1(κ1R)J0(κ1R)εdκ2H1(1)(κ2R)H0(1)(κ2R)=0,
(13)
where Jn and Hn(1)are cylindrical Bessel and Hankel functions, εm is the real part of the dielectric constant of silver, εd = n2 is the dielectric constant of the surrounding medium, and κ1(2)=k0ε(d)(kspp/k0)2. In our case the surrounding refractive index is n = 1 in air (red lines), n = 1.5 in glass (green lines), and n = 1.25 at the air-glass interface (blue lines). The crosses are calculated from the Fabry-Perot resonance condition [8

8. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]

, 29

29. J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9(6), 2372–2377 (2009). [CrossRef] [PubMed]

]
kspp=lπϕrLlπL,
(14)
where l is the mode order found from the near-field patterns (ør is the phase shift upon reflection at the wire ends which here can be neglected because l >> 1).

For the thicker nanowires (Figs. 4(b), 4(c)), the plasmon dispersions on wires in the homogeneous media fall closer to the air and glass light lines, respectively. Effectively, we have a higher plasmon energy for thicker wires as a result of stronger restoring forces. However, the dispersion relation for the supported nanowires no longer follow the analytical solution calculated with an average refractive index, n = 1.25. As we will see, not only charge oscillations along the wire play a role as the wire gets thicker. Instead, other degrees of freedom gain importance and the fields at the top and the bottom of the wire becomes more and more decoupled. To roughly distinguish between thin and thick wires in this context, we compare the wire circumference (which is about 250 nm, 500 nm, and 1000 nm, respectively, for the wire radii we consider here) with the plasmon wavelength (2π/k||), which here lies in the interval 400-600 nm.

Squares in Figs. 4(a), 4(b) show the dispersion relations for the two thinnest wires calculated from the experimental standing wave profiles spectra in Fig. 3 using Eq. (14), where we have to assume the resonance order for those wires (as shown in the Fig. 3). The agreement between experimental and calculated dispersions is very good for thin wires.

Turning to the calculated near-field patterns, Figs. 5(a)
Fig. 5 Electric field intensity along xz plane for thin nanowires in air (a), glass (b), and at an air-glass interface (c). The fields are calculated for corresponding resonance peaks at vacuum wavelength ~750 nm (see Fig. 2). (d) Corresponding electric field distribution inside the supported nanowire at x = 2500 nm. (e) Fourier image of radiation in glass substrate for thin nanowire calculated at 763 nm.
5(c) show the intensity of the total fields calculated for thin nanowires (R = 40 nm) in air, glass and at the air-glass interface at wavelengths 745 nm, 751 nm and 763 nm, respectively. These field patterns show in a very clear way the existence of the Fabry-Perot resonances discussed above, and it is possible to find the mode order l of each of the peaks seen in the spectra by counting the number of nodes in the near field intensity (the mode order is shown above each resonance peak in Fig. 2). At approximately equal photon energies, the nanowires embedded in glass yields the highest mode order l = 27, the nanowires in air yields the lowest mode order, l = 17, while the order of the resonance for supported nanowire is l = 21. In terms of SPP wavelengths we thus haveλsppair>λsppair-glass>λsppglass. We see that the plasmon fields at the top (air) and bottom (glass) edges of the nanowire are in phase with each other. This means that in the case of a thin supported nanowire it is mainly the cylindrically symmetric plasmons (with an azimuthal quantum number m = 0) that are excited. Comparing the different dielectric environments in Fig. 5, the nanowires embedded in air gives the largest field enhancement. The fields penetrate more easily into the metal for a denser surrounding medium leading to a decrease in the field enhancement for the wires embedded in glass. For supported nanowires the strongest fields appear near the glass interface due to the well known image dipole interaction.

Figure 5(e) shows calculated results for the far field scattering to the glass side on the Fourier plane for the supported nanowire at free-space wavelength λ = 763 nm, where kx = kglass sin(θ) cos(φ), ky = kglass sin(θ) sin(φ) and kglass = 2πn / λ is the wave number in glass. The inner green dashed line represents the border (k|| = kair) between propagating and evanescent waves in air, while the green solid line represents the same border (k|| = kglass) for waves in glass. The light that is scattered inside the dashed circle is often termed “allowed” light, while the light between the dashed and the solid circle is termed “forbidden”. The scattering is concentrated to in-plane directions that are perpendicular to the nanowire and peaks at angles near the critical angle for the air-glass interface, i.e. 41.8 degrees. This radiation pattern is in agreement with previously published results on emission from nanowires in homogeneous media involving higher order resonance modes [30

30. J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. 10(9), 3596–3603 (2010). [CrossRef] [PubMed]

, 31

31. E. R. Encina and E. A. Coronado, “Plasmonic nanoantennas: Angular scattering properties of multipole resonances in noble metal nanorods,” J. Phys. Chem. C 112(26), 9586–9594 (2008). [CrossRef]

, 42

42. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: evolution of sub- and super-radiant modes,” Nano Lett. 11(3), 1020–1024 (2011). [CrossRef] [PubMed]

]. As is known from these publications, the higher order odd resonances in the nanowires radiate mostly perpendicular to the wire orientation, with some weaker side lobes. The currents in the nanowire largely give a radiation pattern similar to that occurring from a chain of dipoles oscillating in phase. In case the incident field hits the wire at an angle, there will be phase differences between the currents along the wire that moves the scattering maximum to a finite kx given by the momentum of the incident photons along the wire. The width of the central lobe (or stripe) along the kx direction is determined by the length of the wire, since the first minimum, according to multi-slit diffraction theory occurs when kx ≈2π/L.

Figure 6(a)
Fig. 6 Electric field intensity along xz plane for intermediate (a) and thick (b) nanowires supported at an air-glass interface. The corresponding vacuum wavelengths are 787 nm and 785 nm, respectively. (c, e) Corresponding electric field distribution inside intermediate and thick nanowires at x = 2500 nm. (d, f) Corresponding Fourier images of the radiation scattered into the glass substrate for the intermediate and thick nanowires shown in (a,b).
, 6(b) show near field intensities of the total field around supported nanowires with radius R = 80 nm and R = 160 nm, respectively. These wires can be characterized as thick (for the photon energy in question) since, as pointed out above, now the wire circumference reaches values, approximately 500 nm and 1000 nm, respectively, that are comparable to, or larger than the plasmon wavelength. Compared to the thin nanowire (Fig. 5(c)), the near field intensity is much weaker here. The field patterns along the top and bottom edges of the wire differ quite a lot in this case, indicating a decoupling between the plasmons at opposite edges of the wire. Consequently, more plasmon modes are involved in forming the field in and around a thicker nanowire than was the case for R = 40 nm. The plasmons appearing here are hybridized modes, primarily between m = 0 and m = 1 modes of a wire in a homogeneous environment.

4. Summary and conclusion

In conclusion, we have investigated far field and near field properties of silver nanowires supported by a glass substrate using a method which calculates differential scattering cross sections based on the total fields inside the wire obtained using a commercial finite element software (Comsol multiphysics 3.5a). For supported thin wires, which are much better resonators than thick wires, we find that the fields on the air and glass sides are in phase with each other. The spectral properties of such wires can be then be understood from the dispersion of cylinder plasmons in a homogeneous environment with a refractive index intermediate between air and glass. However, the dispersion starts to deviate from the average refractive index approximation as the wire diameter increases beyond the plasmon wavelength, because the fields along the different edges of the wire then decouple from each other. Looking at the fields inside the wires, which ultimately are the sources for the scattered light, there are also clear differences between thin and thick wires. In particular, the fields in the thicker wires are mainly concentrated to the top rim of the wire while they are more evenly distributed in a thin wire. Finally, we compared the simulation results with experimentally measured spectra and found an excellent agreement. Further issues that could be addressed using our method include how the shape of the wire-tip influences the optical properties, the question on optical coupling and plasmon propagation between two wires of different shape, length or diameter, and the role of crystal facets and wire imperfections.

Acknowledgment

This work was financially supported by the Swedish Research Council and the Göran Gustafsson Foundation.

References and links

1.

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Zeitschrift Fur Naturforschung Part A-Astrophysik Physik Und Physikalische Chemie A 23, 2135–2136 (1968).

2.

U. Kreibig and L. Genzel, “Optical absorption of small metallic particles,” Surf. Sci. 156, 678–700 (1985). [CrossRef]

3.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). [CrossRef] [PubMed]

4.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

5.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef] [PubMed]

6.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

7.

Y. G. Sun, B. Gates, B. Mayers, and Y. N. Xia, “Crystalline silver nanowires by soft solution processing,” Nano Lett. 2(2), 165–168 (2002). [CrossRef]

8.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005). [CrossRef] [PubMed]

9.

J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef] [PubMed]

10.

R. M. Dickson and L. A. Lyon, “Unidirectional plasmon propagation in metallic nanowires,” J. Phys. Chem. B 104(26), 6095–6098 (2000). [CrossRef]

11.

R. Gordon, “Reflection of cylindrical surface waves,” Opt. Express 17(21), 18621–18629 (2009). [CrossRef] [PubMed]

12.

R. Zia, J. A. Schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides,” Phys. Rev. B 74(16), 165415 (2006). [CrossRef]

13.

J. C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J. P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B 60(12), 9061–9068 (1999). [CrossRef]

14.

J. R. Krenn, B. Lamprecht, H. Ditlbacher, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, “Non diffraction-limited light transport by gold nanowires,” Europhys. Lett. 60(5), 663–669 (2002). [CrossRef]

15.

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98(26), 266802 (2007). [CrossRef] [PubMed]

16.

A. W. Sanders, D. A. Routenberg, B. J. Wiley, Y. N. Xia, E. R. Dufresne, and M. A. Reed, “Observation of plasmon propagation, redirection, and fan-out in silver nanowires,” Nano Lett. 6(8), 1822–1826 (2006). [CrossRef] [PubMed]

17.

R. X. Yan, P. Pausauskie, J. X. Huang, and P. D. Yang, “Direct photonic-plasmonic coupling and routing in single nanowires,” Proc. Natl. Acad. Sci. U.S.A. 106(50), 21045–21050 (2009). [CrossRef] [PubMed]

18.

X. W. Chen, V. Sandoghdar, and M. Agio, “Highly efficient interfacing of guided plasmons and photons in nanowires,” Nano Lett. 9(11), 3756–3761 (2009). [CrossRef] [PubMed]

19.

X. Guo, M. Qiu, J. M. Bao, B. J. Wiley, Q. Yang, X. N. Zhang, Y. G. Ma, H. K. Yu, and L. M. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett. 9(12), 4515–4519 (2009). [CrossRef] [PubMed]

20.

F. Hao and P. Nordlander, “Plasmonic coupling between a metallic nanosphere and a thin metallic wire,” Appl. Phys. Lett. 89(10), 103101 (2006). [CrossRef]

21.

M. W. Knight, N. K. Grady, R. Bardhan, F. Hao, P. Nordlander, and N. J. Halas, “Nanoparticle-mediated coupling of light into a nanowire,” Nano Lett. 7(8), 2346–2350 (2007). [CrossRef] [PubMed]

22.

Y. R. Fang, Z. P. Li, Y. Z. Huang, S. P. Zhang, P. Nordlander, N. J. Halas, and H. X. Xu, “Branched silver nanowires as controllable plasmon routers,” Nano Lett. 10(5), 1950–1954 (2010). [CrossRef] [PubMed]

23.

H. Wei, Z. Li, X. Tian, Z. Wang, F. Cong, N. Liu, S. Zhang, P. Nordlander, N. J. Halas, and H. Xu, “Quantum dot-based local field imaging reveals plasmon-based interferometric logic in silver nanowire networks,” Nano Lett. 11(2), 471–475 (2011). [CrossRef] [PubMed]

24.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007). [CrossRef] [PubMed]

25.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97(5), 053002 (2006). [CrossRef] [PubMed]

26.

T. Shegai, Y. Z. Huang, H. X. Xu, and M. Käll, “Coloring fluorescence emission with silver nanowires,” Appl. Phys. Lett. 96(10), 103114 (2010). [CrossRef]

27.

A. L. Falk, F. H. L. Koppens, C. L. Yu, K. Kang, N. de Leon Snapp, A. V. Akimov, M.-H. Jo, M. D. Lukin, and H. Park, “Near-field electrical detection of optical plasmons and single-plasmon sources,” Nat. Phys. 5(7), 475–479 (2009). [CrossRef]

28.

H. Wei, D. Ratchford, X. E. Li, H. X. Xu, and C. K. Shih, “Propagating surface plasmon induced photon emission from quantum dots,” Nano Lett. 9(12), 4168–4171 (2009). [CrossRef] [PubMed]

29.

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9(6), 2372–2377 (2009). [CrossRef] [PubMed]

30.

J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. 10(9), 3596–3603 (2010). [CrossRef] [PubMed]

31.

E. R. Encina and E. A. Coronado, “Plasmonic nanoantennas: Angular scattering properties of multipole resonances in noble metal nanorods,” J. Phys. Chem. C 112(26), 9586–9594 (2008). [CrossRef]

32.

T. Shegai, V. D. Miljković, K. Bao, H. Xu, P. Nordlander, P. Johansson, and M. Käll, “Unidirectional broadband light emission from supported plasmonic nanowires,” Nano Lett. 11(2), 706–711 (2011). [CrossRef] [PubMed]

33.

Z. P. Li, F. Hao, Y. Z. Huang, Y. R. Fang, P. Nordlander, and H. X. Xu, “Directional light emission from propagating surface plasmons of silver nanowires,” Nano Lett. 9(12), 4383–4386 (2009). [CrossRef] [PubMed]

34.

Z. P. Li, K. Bao, Y. R. Fang, Y. Z. Huang, P. Nordlander, and H. X. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett. 10(5), 1831–1835 (2010). [CrossRef] [PubMed]

35.

L. M. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10(1), 268–273 (2010). [CrossRef] [PubMed]

36.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

37.

P. Johansson, “Electromagnetic Green’s function for layered systems: Applications to nanohole interactions in thin metal films,” Phys. Rev. B 83(19), 195408 (2011). [CrossRef]

38.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill: New York, 1941).

39.

P. Hanarp, M. Kall, and D. Sutherland, “Optical properties of short range ordered arrays of nanometer gold disks prepared by colloidal lithography,” J. Phys. Chem. B 107(24), 5768–5772 (2003). [CrossRef]

40.

F. Neubrech, T. Kolb, R. Lovrincic, G. Fahsold, A. Pucci, J. Aizpurua, T. Cornelius, M. Toimil-Molares, R. Neumann, and S. Karim, “Resonances of individual metal nanowires in the infrared,” Appl. Phys. Lett. 89(25), 253104 (2006). [CrossRef]

41.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101(15), 157403 (2008). [CrossRef] [PubMed]

42.

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: evolution of sub- and super-radiant modes,” Nano Lett. 11(3), 1020–1024 (2011). [CrossRef] [PubMed]

OCIS Codes
(050.2230) Diffraction and gratings : Fabry-Perot
(140.4780) Lasers and laser optics : Optical resonators
(230.7370) Optical devices : Waveguides
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: February 17, 2012
Revised Manuscript: April 4, 2012
Manuscript Accepted: April 15, 2012
Published: April 25, 2012

Virtual Issues
Vol. 7, Iss. 7 Virtual Journal for Biomedical Optics

Citation
Vladimir D. Miljković, Timur Shegai, Peter Johansson, and Mikael Käll, "Simulating light scattering from supported plasmonic nanowires," Opt. Express 20, 10816-10826 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-10-10816


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References

  1. E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Zeitschrift Fur Naturforschung Part A-Astrophysik Physik Und Physikalische Chemie A23, 2135–2136 (1968).
  2. U. Kreibig and L. Genzel, “Optical absorption of small metallic particles,” Surf. Sci.156, 678–700 (1985). [CrossRef]
  3. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater.7(6), 442–453 (2008). [CrossRef] [PubMed]
  4. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9(3), 205–213 (2010). [CrossRef] [PubMed]
  5. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science311(5758), 189–193 (2006). [CrossRef] [PubMed]
  6. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
  7. Y. G. Sun, B. Gates, B. Mayers, and Y. N. Xia, “Crystalline silver nanowires by soft solution processing,” Nano Lett.2(2), 165–168 (2002). [CrossRef]
  8. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett.95(25), 257403 (2005). [CrossRef] [PubMed]
  9. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett.22(7), 475–477 (1997). [CrossRef] [PubMed]
  10. R. M. Dickson and L. A. Lyon, “Unidirectional plasmon propagation in metallic nanowires,” J. Phys. Chem. B104(26), 6095–6098 (2000). [CrossRef]
  11. R. Gordon, “Reflection of cylindrical surface waves,” Opt. Express17(21), 18621–18629 (2009). [CrossRef] [PubMed]
  12. R. Zia, J. A. Schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides,” Phys. Rev. B74(16), 165415 (2006). [CrossRef]
  13. J. C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J. P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B60(12), 9061–9068 (1999). [CrossRef]
  14. J. R. Krenn, B. Lamprecht, H. Ditlbacher, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, “Non diffraction-limited light transport by gold nanowires,” Europhys. Lett.60(5), 663–669 (2002). [CrossRef]
  15. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98(26), 266802 (2007). [CrossRef] [PubMed]
  16. A. W. Sanders, D. A. Routenberg, B. J. Wiley, Y. N. Xia, E. R. Dufresne, and M. A. Reed, “Observation of plasmon propagation, redirection, and fan-out in silver nanowires,” Nano Lett.6(8), 1822–1826 (2006). [CrossRef] [PubMed]
  17. R. X. Yan, P. Pausauskie, J. X. Huang, and P. D. Yang, “Direct photonic-plasmonic coupling and routing in single nanowires,” Proc. Natl. Acad. Sci. U.S.A.106(50), 21045–21050 (2009). [CrossRef] [PubMed]
  18. X. W. Chen, V. Sandoghdar, and M. Agio, “Highly efficient interfacing of guided plasmons and photons in nanowires,” Nano Lett.9(11), 3756–3761 (2009). [CrossRef] [PubMed]
  19. X. Guo, M. Qiu, J. M. Bao, B. J. Wiley, Q. Yang, X. N. Zhang, Y. G. Ma, H. K. Yu, and L. M. Tong, “Direct coupling of plasmonic and photonic nanowires for hybrid nanophotonic components and circuits,” Nano Lett.9(12), 4515–4519 (2009). [CrossRef] [PubMed]
  20. F. Hao and P. Nordlander, “Plasmonic coupling between a metallic nanosphere and a thin metallic wire,” Appl. Phys. Lett.89(10), 103101 (2006). [CrossRef]
  21. M. W. Knight, N. K. Grady, R. Bardhan, F. Hao, P. Nordlander, and N. J. Halas, “Nanoparticle-mediated coupling of light into a nanowire,” Nano Lett.7(8), 2346–2350 (2007). [CrossRef] [PubMed]
  22. Y. R. Fang, Z. P. Li, Y. Z. Huang, S. P. Zhang, P. Nordlander, N. J. Halas, and H. X. Xu, “Branched silver nanowires as controllable plasmon routers,” Nano Lett.10(5), 1950–1954 (2010). [CrossRef] [PubMed]
  23. H. Wei, Z. Li, X. Tian, Z. Wang, F. Cong, N. Liu, S. Zhang, P. Nordlander, N. J. Halas, and H. Xu, “Quantum dot-based local field imaging reveals plasmon-based interferometric logic in silver nanowire networks,” Nano Lett.11(2), 471–475 (2011). [CrossRef] [PubMed]
  24. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature450(7168), 402–406 (2007). [CrossRef] [PubMed]
  25. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett.97(5), 053002 (2006). [CrossRef] [PubMed]
  26. T. Shegai, Y. Z. Huang, H. X. Xu, and M. Käll, “Coloring fluorescence emission with silver nanowires,” Appl. Phys. Lett.96(10), 103114 (2010). [CrossRef]
  27. A. L. Falk, F. H. L. Koppens, C. L. Yu, K. Kang, N. de Leon Snapp, A. V. Akimov, M.-H. Jo, M. D. Lukin, and H. Park, “Near-field electrical detection of optical plasmons and single-plasmon sources,” Nat. Phys.5(7), 475–479 (2009). [CrossRef]
  28. H. Wei, D. Ratchford, X. E. Li, H. X. Xu, and C. K. Shih, “Propagating surface plasmon induced photon emission from quantum dots,” Nano Lett.9(12), 4168–4171 (2009). [CrossRef] [PubMed]
  29. J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett.9(6), 2372–2377 (2009). [CrossRef] [PubMed]
  30. J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett.10(9), 3596–3603 (2010). [CrossRef] [PubMed]
  31. E. R. Encina and E. A. Coronado, “Plasmonic nanoantennas: Angular scattering properties of multipole resonances in noble metal nanorods,” J. Phys. Chem. C112(26), 9586–9594 (2008). [CrossRef]
  32. T. Shegai, V. D. Miljković, K. Bao, H. Xu, P. Nordlander, P. Johansson, and M. Käll, “Unidirectional broadband light emission from supported plasmonic nanowires,” Nano Lett.11(2), 706–711 (2011). [CrossRef] [PubMed]
  33. Z. P. Li, F. Hao, Y. Z. Huang, Y. R. Fang, P. Nordlander, and H. X. Xu, “Directional light emission from propagating surface plasmons of silver nanowires,” Nano Lett.9(12), 4383–4386 (2009). [CrossRef] [PubMed]
  34. Z. P. Li, K. Bao, Y. R. Fang, Y. Z. Huang, P. Nordlander, and H. X. Xu, “Correlation between incident and emission polarization in nanowire surface plasmon waveguides,” Nano Lett.10(5), 1831–1835 (2010). [CrossRef] [PubMed]
  35. L. M. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett.10(1), 268–273 (2010). [CrossRef] [PubMed]
  36. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
  37. P. Johansson, “Electromagnetic Green’s function for layered systems: Applications to nanohole interactions in thin metal films,” Phys. Rev. B83(19), 195408 (2011). [CrossRef]
  38. J. A. Stratton, Electromagnetic Theory (McGraw-Hill: New York, 1941).
  39. P. Hanarp, M. Kall, and D. Sutherland, “Optical properties of short range ordered arrays of nanometer gold disks prepared by colloidal lithography,” J. Phys. Chem. B107(24), 5768–5772 (2003). [CrossRef]
  40. F. Neubrech, T. Kolb, R. Lovrincic, G. Fahsold, A. Pucci, J. Aizpurua, T. Cornelius, M. Toimil-Molares, R. Neumann, and S. Karim, “Resonances of individual metal nanowires in the infrared,” Appl. Phys. Lett.89(25), 253104 (2006). [CrossRef]
  41. F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101(15), 157403 (2008). [CrossRef] [PubMed]
  42. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: evolution of sub- and super-radiant modes,” Nano Lett.11(3), 1020–1024 (2011). [CrossRef] [PubMed]

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