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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 16 — Jul. 30, 2012
  • pp: 17591–17599
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Are scaling laws of sub-optical wavelength electric field confinement in arrays of metal nanoparticles related to plasmonics or to geometry?

M. Essone Mezeme and C. Brosseau  »View Author Affiliations


Optics Express, Vol. 20, Issue 16, pp. 17591-17599 (2012)
http://dx.doi.org/10.1364/OE.20.017591


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Abstract

In this work, we describe finite element simulations of the plasmonic resonance (PLR) properties of a self-similar chain of plasmonic nanostructures. Using a broad range of conditions, we find strong numerical evidence that the electric field confinement behaves as ( Ξ/λ ) PLR EF E -γ , where EFE is the electric field enhancement, Ξ is the linear size of the focusing length, and λ is the wavelength of the resonant excitation. We find that the exponent γ is close to 1, i.e. significantly lower than the 1.5 found for two-dimensional nanodisks. This scaling law provides support for the hypothesis of a universal regime in which the sub-optical wavelength electric field confinement is controlled by the Euclidean dimensionality and is independent of nanoparticle size, metal nature, or embedding medium permittivity.

© 2012 OSA

Scaling laws are ubiquitous in many natural and engineering systems and have been a rich source of physics insight for over 30 years [1

1. P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, 1979).

,2

2. P. Meakin, Fractals, Scaling, and Growth Far from Equilibrium (Cambridge University Press, 1998).

]. An especial challenge exists when the equations of the underlying physical system are unknown and one must then rely on numerical simulation or data to make predictions. On the other hand, precise control over the energy confinement has proven pivotal for the study of plasmon resonance (PLR) properties in nanostructures. Recent advances in the development of nanoplasmonics and light harvesting [3

3. A. Maier, Plasmonics: Fundamental and Applications (Springer, 2007).

] suggest that the spatial confinement of plasmon excitations at metal-dielectric interfaces or in hybrid (metallodielectric) nanostructures can be applied to enable entirely new structures with versatile applications ranging from highly sensitive sensors to photovoltaics. These can typically be shaped on subwavelength scales. Particular interest [4

4. A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010). [CrossRef] [PubMed]

21

21. COMSOL Multiphysics User’s Guide, version 3.4; Comsol Inc. (2006).

] has been focused on linear chains of N resonantly coupled plasmonic nanostructures because these systems are very sensitive to an applied electric field, giving rise to extremely high electric fields (hot spots), i.e. nanosphere cascade nanolenses yielding electric field enhancement in the nanogap between two nanoparticles which can exceed the excitation field by a factor of 103 at the smallest particle.

Presently, efforts are underway to understand the relationship between the electric field enhancement (EFE), the linear size of the focusing length (Ξ), and the geometry of the plasmonic (metal) phase in nanoplasmonic systems. Not long ago it was hypothesized that the locality of plasmon dispersion in a self-similar chain of magnetoplasmonic core-shell two-dimensional (2D, equivalently, circular infinite cylindrical) nanostructures embedded in a host matrix might be governed by a scaling law [6

6. K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91(22), 227402 (2003). [CrossRef] [PubMed]

]. This is (Ξ/λ)PLREFE-1.5, where λ denotes the free space wavelength of the resonant excitation. Fundamental questions remain unanswered about such a law, (i) whether self similar chains of three-dimensional (3D) particles can be characterized within a simple universal scaling behavior, (ii) whether it is valid for any particle geometry, or dependent on physical characteristics of the chain, and (iii) whether these observations are consistent with the PLR characteristics. We stress that a complete understanding of the scope of universality of this law still evades our grasp, e.g. for chains of N>5 nanostructures the applicability of scaling law is debatable because nonlocal (quantum confinement) effects lead to significant plasmon broadening in metal nanoparticles of diameter smaller than 10 nm. We note that DNA has been used to design plasmonic nanostructures, such as plasmonic molecules, polymers and crystals [14

14. V. M. Shalaev, “Electromagnetic properties of small-particle composites,” Phys. Rep. 272(2-3), 61–137 (1996). [CrossRef]

]. While numerous investigations have shown that EFE can reach 103 (hot spots) in nanosystems [19

19. F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112(46), 17983–17987 (2008). [CrossRef]

], there are only a few calculations on predicting Ξ in 2D [8

8. M. Essone Mezeme, S. Lasquellec, and C. Brosseau, “Subwavelength control of electromagnetic field confinement in self-similar chains of magnetoplasmonic core-shell nanostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026612 (2011). [CrossRef] [PubMed]

] and 3D [4

4. A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010). [CrossRef] [PubMed]

7

7. Z. Li, Z. Yang, and H. Xu, “Comment on “Self-similar chain of metal nanospheres as an efficient nanolens”,” Phys. Rev. Lett. 97(7), 079701, discussion 079702 (2006). [CrossRef] [PubMed]

] systems.

Consider the schematic of the typical array of N metallic nanospheres in Fig. 1
Fig. 1 The geometry of the problem is shown schematically. The coordinate system used in the calculations is indicated. The external field is applied to the system in the x direction. The position of the hot spot corresponding to the maximum field enhancement is indicated by the dot near the smallest particle. The numerical parameters for calculations were: R1 = 185 nm and L = 1226 nm. This two-phase system consists of a self-similar chain of plasmonic nanospheres (phase 2) embedded in a surrounding medium (phase 1). The (i + 1)th sphere has outer radius Ri+1=kRi. The spherical inclusions have permittivity ε2=ε2'jε2" and the host’s permittivity reads ε1=ε1' with ε1'=1.77 in the THz range of frequencies [22].
. The dielectric properties of the embedding medium can be assimilated to water. We assume ideally smooth interfaces between rigid phases. The spheres are obtained using scaled-down copies of an initial geometry. A recursive algorithm, i.e. Ri+1=kRi and di,i+1=Ri+1 can be developed to generate any occurrence of such self-similarity for a given iteration i. Here, Ri denotes the radius of the i-th iteration and i parameterizes the iteration process (as illustrated in Fig. 1). In all simulations the radius corresponding to the first iteration is held fixed at R1 = 185 nm. The choice of this parameter and the number of iterations should be consistent with the overall volume L3 of the cubic cell. Clearly, the process cannot be carried up to a high number of iterations. In this paper, we were able to perform calculations up to four iterations for the geometric parameter ranges: k = 0.30-0.35 and =0.30.6. We will assume throughout that the time dependence of the electric field excitation, assumed to be directed along the x direction, is proportional to exp(2πjct/λ), where c is the speed of light in vacuum. At long wavelength the physics of the system turns out to allow a further simplification. It must be borne in mind that the validity of this long-wavelength behavior is rooted in the fact that all length scales must be much smaller than λ. To ensure that this constraint was satisfied, we use L = 1226 nm. We employ a continuum modelling approach built upon constitutive equations which can capture the material behavior on experimentally relevant scales. That is, when the local electrical response in terms of a position dependent permittivity. The consistency and validation of this procedure was verified by agreement (not shown) of our calculations with those obtained from Kramers-Kronig causality relationships [22

22. M. Essone Mezeme, S. Lasquellec, and C. Brosseau, “Long-wavelength electromagnetic propagation in magnetoplasmonic core-shell nanostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(5), 057602 (2010). [CrossRef] [PubMed]

].

Gold and silver are chosen as model shell materials and were modelled using the Drude model, i.e. ε2(ω)=ε2'ωp2/ω(ωjωc). For Au, plasma frequency ωp/2π=2228 THz, collision frequency ωc/2π=6.0 THz [16

16. J. Borneman, K.-P. Chen, A. Kildishev, and V. Shalaev, “Simplified model for periodic nanoantennae: linear model and inverse design,” Opt. Express 17(14), 11607–11617 (2009). [CrossRef] [PubMed]

], and ε'=7. For Ag, plasma frequency ωp/2π=2149 THz, collision frequency ωc/2π=12.2 THz [31

31. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–20 (1983). [CrossRef] [PubMed]

33

33. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

], and ε'=2.48. Notice that the penetration depth of electromagnetic waves at optical frequencies is about 20 nm for Au. While nonlocality turns out to be a generic feature of small nanoobjects, our calculations show that if we only consider the conventional Drude’s form EFE can be significantly modified (Fig. 2(d)
Fig. 2 (a) Universal scaling of the relative focusing length, Ξ/λ, with respect to the PLR wavelength of the excitation, as a function of EFE for the array of nanoparticles shown in Fig. 1 at various model parameters. The figure is plotted on a log-log scale and the slope of the solid line is −1. (a) =0.6 fixed. Symbols are (open squares) k = 0.30, (open circles) k = 0.31, (open triangles) k = 0.32, (solid squares) k = 0.33, (solid diamonds) k = 0.34, (solid triangles) k = 0.35. The metal phase is assumed to be Au. (b) k = 0.33. Symbols are: (open diamonds) =0.3, (solid circles) =0.4, (open triangles) =0.5, (solid squares) =0.6. The metal phase is assumed to be Au. (c) l = 0.6 and k = 0.33. Symbols are: (solid squares) metal phase is Au and surrounding medium is water; (solid circles) metal phase is Au and surrounding medium is air. (d) =0.6 and k = 0.33. Symbols are: (solid squares) metal phase is Au and no FSC is considered, (solid circles) metal phase is Ag and no FSC is considered, (solid triangles) metal phase is Au and FSC is considered, (solid diamonds) Fe3O4-Au CS nanoparticles (t = 0.2) and no FSC is considered.
) compared to the case including the finite-size correction (FSC) leading to an enhanced rate of electron scattering. The FSC was included by changing ωc in Drude’s model of permittivity by ωc+AvF/R, where vF is the Fermi velocity for bulk Au, R is the radius of the particle, and A is a constant of order unity [9

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

,17

17. J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B 76(24), 245403 (2007). [CrossRef]

,19

19. F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112(46), 17983–17987 (2008). [CrossRef]

,21

21. COMSOL Multiphysics User’s Guide, version 3.4; Comsol Inc. (2006).

]. This is consistent with earlier studies [19

19. F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112(46), 17983–17987 (2008). [CrossRef]

,20

20. G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. 109(12), 124306 (2011). [CrossRef]

,30

30. S. Foteinopoulou, J. P. Vigneron, and C. Vandenbem, “Optical near-field excitations on plasmonic nanoparticle-based structures,” Opt. Express 15(7), 4253–4267 (2007). [CrossRef] [PubMed]

,34

34. E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120(1), 357–366 (2004). [CrossRef] [PubMed]

38

38. S. E. Sburlan, L. A. Blanco, and M. Nieto-Vesperinas, “Plasmon excitation in sets of nanoscale cylinders and spheres,” Phys. Rev. B 73(3), 035403 (2006). [CrossRef]

]. We later return to discuss this point for the effective permittivity. An immediate consequence of the Drude’s model when nonlocal surface effects are neglected is that in the near-field region EFE is scale invariant.

Our main results are summarized in the (Ξ/λ)PLRvs EFE diagram shown in Fig. 2. The main feature of the present study is that this diagram displays a scaling law (Ξ/λ)PLREFE-γwith an exponent γ1. This relationship is found to be independent of the values of k and investigated, the metal’s composition and the dielectric medium that surrounds the metal nanoobject, suggesting that it could be universal. With regard to the actual value of γ = 1, it is interesting to observe that it is significantly lower than the 1.5 found for nanodisks. The panels in Fig. 2 present the simulations for a range of model parameters k and , two kinds of noble metals, and two kinds of embedding medium (water and air in order to compare with Refs [4

4. A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010). [CrossRef] [PubMed]

7

7. Z. Li, Z. Yang, and H. Xu, “Comment on “Self-similar chain of metal nanospheres as an efficient nanolens”,” Phys. Rev. Lett. 97(7), 079701, discussion 079702 (2006). [CrossRef] [PubMed]

].). In all cases, we see from Fig. 2 that there is only a weak sensitivity of the scaling behavior upon the various sets of parameters. Remarkably, we find that the ratio of the γ values found between the 2D [8

8. M. Essone Mezeme, S. Lasquellec, and C. Brosseau, “Subwavelength control of electromagnetic field confinement in self-similar chains of magnetoplasmonic core-shell nanostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026612 (2011). [CrossRef] [PubMed]

] and the current 3D cases is to the inverse ratio of their Euclidean dimensions. With a further increase of N we find that EFE is much larger (Fig. 3
Fig. 3 A comparison of EFE with the estimates of the cascade amplification coefficient gN and g¯N, suggested by different authors [47,45], for the various cases of metal phase/embedding medium considered in Fig. 2(a), 2(b), 2(c). Squares (resp. crosses) correspond to gN (resp. g¯N). The solid line corresponds to EFE=gN or g¯N.
) than gN=Q(RN/R1)lnQ/|lnk|, where Qε2'/ε2" denotes the quality factor of the surface plasmon resonance and ε2 is the permittivity of the metal at the surface-plasmon resonance frequency, as was suggested by Stockman et al. [4

4. A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10(7), 2574–2579 (2010). [CrossRef] [PubMed]

,5

5. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87(16), 167401 (2001). [CrossRef] [PubMed]

]. It is also interesting to note that Dai and associates [21

21. COMSOL Multiphysics User’s Guide, version 3.4; Comsol Inc. (2006).

] had hinted at the possibility that cascade amplification produces a local field enhanced by a factor of g¯N=QN at the Nth particle. Figure 3 compares also the EFE values achieved as the structure geometry and metal nature are varied and Dai et al.’s estimate.

Figure 3 shows that EFE continues to grow for powers far above the Dai et al. estimate. A similar analysis reveals that γ1 persists when core-metal shell particles are considered at a specific of the metal layer’s thickness (ei=tRi with t = 0.2) instead of full particles (Fig. 2). The simulations for Fe3O4-Au CS nanospheres were procedurally similar from the 2D ones reported in [8

8. M. Essone Mezeme, S. Lasquellec, and C. Brosseau, “Subwavelength control of electromagnetic field confinement in self-similar chains of magnetoplasmonic core-shell nanostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(2), 026612 (2011). [CrossRef] [PubMed]

]. Initially, it has been expected that the value of γ could be accounted for through the fractal dimension of these arrays, i.e. df=ln2/|lnk|, but this hope was dashed when it became clear our simulations that probe the (Ξ/λ)PLRvs EFE scaling behavior are incapable of distinguishing between types of nanoparticle, metal, and embedding medium but are typically controlled by the Euclidean dimensionality. It is therefore tempting to suggest that it is a universal and robust property of self-similar chains of plasmonic nanoparticles and, most likely, which comes from the resonant excitation of a damped plasmon mode.

A couple of remarks on our results are in order. Interestingly, we find that they show qualitatively good consistency with recently reported identical universal scaling behavior of plasmon coupling in metal nanoshells and that in metal nanoparticles [39

39. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A plasmon ruler equation,” Nano Lett. 7(7), 2080–2088 (2007). [CrossRef]

41

41. P. K. Jain and M. A. El-Sayed, “Surface plasmon coupling and its universal size scaling in metal nanostructures of complex geometry: elongated particle pairs and nanosphere trimers,” J. Phys. Chem. C 111, 17451–17454 (2007). [CrossRef]

]. As another point of comparison, it is important to note that finite-difference time-domain computations demonstrated a nanolens effect which can convert a diffraction limited Gaussian beam into a sub-wavelength focus as small as λ/10 for self-similar Ag nanosphere array embedded in glass [42

42. J. Kottmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express 8(12), 655–663 (2001). [CrossRef] [PubMed]

44

44. S. V. Boriskina and B. M. Reinhard, “Molding the flow of light on the nanoscale: from vortex nanogears to phase-operated plasmonic machinery,” Nanoscale 4(1), 76–90 (2011). [CrossRef] [PubMed]

]. Our finding is consistent with recently published works that have highlighted the importance of carefully considering the issue of meshing the computational domain when calculating EFE and ε [6

6. K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91(22), 227402 (2003). [CrossRef] [PubMed]

,7

7. Z. Li, Z. Yang, and H. Xu, “Comment on “Self-similar chain of metal nanospheres as an efficient nanolens”,” Phys. Rev. Lett. 97(7), 079701, discussion 079702 (2006). [CrossRef] [PubMed]

,38

38. S. E. Sburlan, L. A. Blanco, and M. Nieto-Vesperinas, “Plasmon excitation in sets of nanoscale cylinders and spheres,” Phys. Rev. B 73(3), 035403 (2006). [CrossRef]

,45

45. J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, “Electrodynamic effects in plasmonic nanolenses,” Phys. Rev. B 77(11), 115419 (2008). [CrossRef]

]. From a practical point of view, the scaling law is very useful, because all the quantities involved can be measured experimentally and do not rely on microscopic details. Only very recently, promising measurements have been reported of the optical-field enhancement from well controlled plasmonic arrays [23

23. R. Elghanian, J. J. Storhoff, R. C. Mucic, R. L. Letsinger, and C. A. Mirkin, “Selective colorimetric detection of polynucleotides based on the distance-dependent optical properties of gold nanoparticles,” Science 277(5329), 1078–1081 (1997). [CrossRef] [PubMed]

28

28. B. Ding, Z. Deng, H. Yan, S. Cabrini, R. N. Zuckermann, and J. Bokor, “Gold nanoparticle self-similar chain structure organized by DNA origami,” J. Am. Chem. Soc. 132(10), 3248–3249 (2010). [CrossRef] [PubMed]

,45

45. J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, “Electrodynamic effects in plasmonic nanolenses,” Phys. Rev. B 77(11), 115419 (2008). [CrossRef]

49

49. V. G. Kravets, G. Zoriniants, C. P. Burrows, F. Schedin, C. Casiraghi, P. Klar, A. K. Geim, W. L. Barnes, and A. N. Grigorenko, “Cascaded optical field enhancement in composite plasmonic nanostructures,” Phys. Rev. Lett. 105(24), 246806 (2010). [CrossRef] [PubMed]

].

We end with a brief discussion of the effective permittivity for these self-similar chains of nanospheres which, to the best of our knowledge, was not considered in the majority of past works.

Figure 4
Fig. 4 A comparison of the imaginary parts of the effective permittivity for self-similar chains of Au nanospheres embedded in water with (green line) or without (black line) FSC. =0.6 and k = 0.33. The asterisk indicates the PLR spectral position corresponding to the maximum field enhancement. (a) first iteration (b) second iteration, (c) third iteration, and (d) fourth iteration.
shows the imaginary parts of the effective permittivity which consist of many resonant peaks across a wide spectral region and are characterized by an intricate interplay between them (a full discussion of the permittivity spectra is beyond the scope of this study and will be the object of a future work). With respect to the maximum field enhancement, one can see a systematic redshift of the PLR (marked by the asterisk in Fig. 4) when the iteration number increases. The electrostatic resonance spectrum of nanoscale particles can be complex due to their multiband nature. This issue has been recently addressed in a particularly pointed fashion by Mayergoyz and associates [50

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

]. However, an analytical framework of general applicability for the collective PLR behavior of an arbitrary (non-translation invariant) plasmonic array of nanostructures is lacking. Although our calculations are performed without the nonlocality correction of ε (black line in Fig. 4), there is no significant change in the ε" results when the FSC is taken into account (green line in Fig. 4) with the exception of the 4th iteration for which the low-frequency modes vanish (Fig. 4(d)). To put the result shown in Fig. 2 into perspective, we provide also the ε" data for Fe3O4-Au CS nanosphere arrays (Fig. 5
Fig. 5 Same as in Fig. 4 for self-similar chains of Fe3O4-Au CS nanoparticles embedded in water without FSC (blue line). =0.6, k = 0.33, and t = 0.2. The value of ε" for self-similar chains of Au nanospheres embedded in water without (black line) FSC is shown for comparison.
).

Interestingly, the data in Fig. 5 show that the PLR characteristics (blue line in Fig. 5) corresponding to the maximum field enhancement show a different spectral profile than the corresponding case of full Au nanoparticles. Although huge, tunable responses to electric perturbations are possible in this system, i.e. by varying t and the CS phases, experimental difficulties to fabricate self-similar plasmonic arrays will prevent perfect tuning [19

19. F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112(46), 17983–17987 (2008). [CrossRef]

,20

20. G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. 109(12), 124306 (2011). [CrossRef]

,23

23. R. Elghanian, J. J. Storhoff, R. C. Mucic, R. L. Letsinger, and C. A. Mirkin, “Selective colorimetric detection of polynucleotides based on the distance-dependent optical properties of gold nanoparticles,” Science 277(5329), 1078–1081 (1997). [CrossRef] [PubMed]

28

28. B. Ding, Z. Deng, H. Yan, S. Cabrini, R. N. Zuckermann, and J. Bokor, “Gold nanoparticle self-similar chain structure organized by DNA origami,” J. Am. Chem. Soc. 132(10), 3248–3249 (2010). [CrossRef] [PubMed]

,49

49. V. G. Kravets, G. Zoriniants, C. P. Burrows, F. Schedin, C. Casiraghi, P. Klar, A. K. Geim, W. L. Barnes, and A. N. Grigorenko, “Cascaded optical field enhancement in composite plasmonic nanostructures,” Phys. Rev. Lett. 105(24), 246806 (2010). [CrossRef] [PubMed]

].

Acknowledgments

We acknowledge financial support from the Ph.D. funding programme (grant programme 211-B2-9/ARED) of the Conseil Régional de Bretagne. Lab-STICC is UMR CNRS 6285.

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J. Borneman, K.-P. Chen, A. Kildishev, and V. Shalaev, “Simplified model for periodic nanoantennae: linear model and inverse design,” Opt. Express 17(14), 11607–11617 (2009). [CrossRef] [PubMed]

17.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B 76(24), 245403 (2007). [CrossRef]

18.

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23(17), 1331–1333 (1998). [CrossRef] [PubMed]

19.

F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112(46), 17983–17987 (2008). [CrossRef]

20.

G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. 109(12), 124306 (2011). [CrossRef]

21.

COMSOL Multiphysics User’s Guide, version 3.4; Comsol Inc. (2006).

22.

M. Essone Mezeme, S. Lasquellec, and C. Brosseau, “Long-wavelength electromagnetic propagation in magnetoplasmonic core-shell nanostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(5), 057602 (2010). [CrossRef] [PubMed]

23.

R. Elghanian, J. J. Storhoff, R. C. Mucic, R. L. Letsinger, and C. A. Mirkin, “Selective colorimetric detection of polynucleotides based on the distance-dependent optical properties of gold nanoparticles,” Science 277(5329), 1078–1081 (1997). [CrossRef] [PubMed]

24.

S. Bidault, F. J. García de Abajo, and A. Polman, “Plasmon-based nanolenses assembled on a well-defined DNA template,” J. Am. Chem. Soc. 130(9), 2750–2751 (2008). [CrossRef] [PubMed]

25.

S. J. Tan, M. J. Campolongo, D. Luo, and W. Cheng, “Building plasmonic nanostructures with DNA,” Nat. Nanotechnol. 6(5), 268–276 (2011). [CrossRef] [PubMed]

26.

E. Hutter and J. H. Fendler, “Exploitation of localized surface plasmon resonance,” Adv. Mater. (Deerfield Beach Fla.) 16(19), 1685–1706 (2004). [CrossRef]

27.

X. Huang, S. Neretina, and M. A. El-Sayed, “Gold nanorods: From synthesis and properties to biological and biomedical applications,” Adv. Mater. (Deerfield Beach Fla.) 21(48), 4880–4910 (2009). [CrossRef]

28.

B. Ding, Z. Deng, H. Yan, S. Cabrini, R. N. Zuckermann, and J. Bokor, “Gold nanoparticle self-similar chain structure organized by DNA origami,” J. Am. Chem. Soc. 132(10), 3248–3249 (2010). [CrossRef] [PubMed]

29.

V. Poponin and A. Ignatov, “Local field enhancement in star-like sets of plasmon nanoparticles,” J. Korean Phys. Soc. 47, S222–S228 (2005).

30.

S. Foteinopoulou, J. P. Vigneron, and C. Vandenbem, “Optical near-field excitations on plasmonic nanoparticle-based structures,” Opt. Express 15(7), 4253–4267 (2007). [CrossRef] [PubMed]

31.

M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–20 (1983). [CrossRef] [PubMed]

32.

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

33.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

34.

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120(1), 357–366 (2004). [CrossRef] [PubMed]

35.

J. P. Kottmann and O. J. F. Martin, “Retardation-induced plasmon resonances in coupled nanoparticles,” Opt. Lett. 26(14), 1096–1098 (2001). [CrossRef] [PubMed]

36.

H. Xu, “Multilayered metal core-shell nanostructures for inducing a large and tunable local optical field,” Phys. Rev. B 72(7), 073405 (2005). [CrossRef]

37.

C. L. Nehl, N. K. Grady, G. P. Goodrich, F. Tam, N. J. Halas, and J. H. Hafner, “Scattering spectra of single gold nanoshells,” Nano Lett. 4(12), 2355–2359 (2004). [CrossRef]

38.

S. E. Sburlan, L. A. Blanco, and M. Nieto-Vesperinas, “Plasmon excitation in sets of nanoscale cylinders and spheres,” Phys. Rev. B 73(3), 035403 (2006). [CrossRef]

39.

P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A plasmon ruler equation,” Nano Lett. 7(7), 2080–2088 (2007). [CrossRef]

40.

P. K. Jain and M. A. El-Sayed, “Universal scaling of plasmon coupling in metal nanostructures: Extension from particle pairs to nanoshells,” Nano Lett. 7(9), 2854–2858 (2007). [CrossRef] [PubMed]

41.

P. K. Jain and M. A. El-Sayed, “Surface plasmon coupling and its universal size scaling in metal nanostructures of complex geometry: elongated particle pairs and nanosphere trimers,” J. Phys. Chem. C 111, 17451–17454 (2007). [CrossRef]

42.

J. Kottmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express 8(12), 655–663 (2001). [CrossRef] [PubMed]

43.

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120(1), 357–366 (2004). [CrossRef] [PubMed]

44.

S. V. Boriskina and B. M. Reinhard, “Molding the flow of light on the nanoscale: from vortex nanogears to phase-operated plasmonic machinery,” Nanoscale 4(1), 76–90 (2011). [CrossRef] [PubMed]

45.

J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, “Electrodynamic effects in plasmonic nanolenses,” Phys. Rev. B 77(11), 115419 (2008). [CrossRef]

46.

G. Das, F. De Angelis, M. L. Coluccio, F. Mecarini, and E. Di Fabrizio, “Spectroscopy nanofabrication and biophotonics,” Proc. SPIE 7205, 720508, 720508-10 (2009). [CrossRef]

47.

F. Le, D. W. Brandl, Y. A. Urzhumov, H. Wang, J. Kundu, N. J. Halas, J. Aizpurua, and P. Nordlander, “Metallic nanoparticle arrays: A common substrate for both surface-enhanced Raman scattering and surface-enhanced infrared absorption,” ACS Nano 2(4), 707–718 (2008). [CrossRef] [PubMed]

48.

J. Kneipp, X. Li, M. Sherwood, U. Panne, H. Kneipp, M. I. Stockman, and K. Kneipp, “Gold nanolenses generated by laser ablation-efficient enhancing structure for surface enhanced Raman scattering analytics and sensing,” Anal. Chem. 80(11), 4247–4251 (2008). [CrossRef] [PubMed]

49.

V. G. Kravets, G. Zoriniants, C. P. Burrows, F. Schedin, C. Casiraghi, P. Klar, A. K. Geim, W. L. Barnes, and A. N. Grigorenko, “Cascaded optical field enhancement in composite plasmonic nanostructures,” Phys. Rev. Lett. 105(24), 246806 (2010). [CrossRef] [PubMed]

50.

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

51.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007). [CrossRef]

52.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008). [CrossRef]

53.

V. Castel and C. Brosseau, “Electron magnetic resonance study of transition-metal magnetic nanoclusters embedded in metal-oxides,” Phys. Rev. B 77(13), 134424 (2008). [CrossRef]

54.

V. Castel and C. Brosseau, “Magnetic field dependence of the effective permittivity in BaTiO3/Ni nanocomposites observed via microwave spectroscopy,” Appl. Phys. Lett. 92(23), 233110 (2008). [CrossRef]

55.

B. M. Ross and L. P. Lee, “Plasmon tuning and local field enhancement maximization of the nanocrescent,” Nanotechnology 19(27), 275201 (2008). [CrossRef] [PubMed]

56.

K. Li, L. Clime, B. Cui, and T. Veres, “Surface enhanced Raman scattering on long-range ordered noble-metal nanocrescent arrays,” Nanotechnology 19(14), 145305 (2008). [CrossRef] [PubMed]

57.

H. Rochholz, N. Bocchio, and M. Kreiter, “Tuning resonances on crescent-shaped noble-metal nanoparticles,” New J. Phys. 9(3), 53–70 (2007). [CrossRef]

58.

J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, “Fabrication of crescent-shaped optical antennas,” Adv. Mater. (Deerfield Beach Fla.) 17(17), 2131–2134 (2005). [CrossRef]

59.

J. Kim, G. Liu, Y. Lu, and L. Lee, “Intra-particle plasmonic coupling of tip and cavity resonance modes in metallic apertured nanocavities,” Opt. Express 13(21), 8332–8338 (2005). [CrossRef] [PubMed]

60.

L. Yang, X. Luo, and M. Hong, “Self-similar chain of nanocrescents as a surface-enhanced Raman scattering substrate,” J. Comput. Theor. Nanosci. 7(8), 1364–1367 (2010). [CrossRef]

61.

Y. Luo, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband light harvesting nanostructures robust to edge bluntness,” Phys. Rev. Lett. 108(2), 023901 (2012). [CrossRef] [PubMed]

OCIS Codes
(160.1245) Materials : Artificially engineered materials
(260.2065) Physical optics : Effective medium theory
(160.4236) Materials : Nanomaterials
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(260.2710) Physical optics : Inhomogeneous optical media

ToC Category:
Metamaterials

History
Original Manuscript: May 25, 2012
Revised Manuscript: July 12, 2012
Manuscript Accepted: July 15, 2012
Published: July 18, 2012

Citation
M. Essone Mezeme and C. Brosseau, "Are scaling laws of sub-optical wavelength electric field confinement in arrays of metal nanoparticles related to plasmonics or to geometry?," Opt. Express 20, 17591-17599 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-17591


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References

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  5. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett.87(16), 167401 (2001). [CrossRef] [PubMed]
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  8. M. Essone Mezeme, S. Lasquellec, and C. Brosseau, “Subwavelength control of electromagnetic field confinement in self-similar chains of magnetoplasmonic core-shell nanostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(2), 026612 (2011). [CrossRef] [PubMed]
  9. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science311(5758), 189–193 (2006). [CrossRef] [PubMed]
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  12. 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]
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  15. Properties of Nanostructured Random Media, V. M. Shalaev, ed. (Springer, 2002).
  16. J. Borneman, K.-P. Chen, A. Kildishev, and V. Shalaev, “Simplified model for periodic nanoantennae: linear model and inverse design,” Opt. Express17(14), 11607–11617 (2009). [CrossRef] [PubMed]
  17. J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76(24), 245403 (2007). [CrossRef]
  18. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett.23(17), 1331–1333 (1998). [CrossRef] [PubMed]
  19. F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112(46), 17983–17987 (2008). [CrossRef]
  20. G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys.109(12), 124306 (2011). [CrossRef]
  21. COMSOL Multiphysics User’s Guide, version 3.4; Comsol Inc. (2006).
  22. M. Essone Mezeme, S. Lasquellec, and C. Brosseau, “Long-wavelength electromagnetic propagation in magnetoplasmonic core-shell nanostructures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.81(5), 057602 (2010). [CrossRef] [PubMed]
  23. R. Elghanian, J. J. Storhoff, R. C. Mucic, R. L. Letsinger, and C. A. Mirkin, “Selective colorimetric detection of polynucleotides based on the distance-dependent optical properties of gold nanoparticles,” Science277(5329), 1078–1081 (1997). [CrossRef] [PubMed]
  24. S. Bidault, F. J. García de Abajo, and A. Polman, “Plasmon-based nanolenses assembled on a well-defined DNA template,” J. Am. Chem. Soc.130(9), 2750–2751 (2008). [CrossRef] [PubMed]
  25. S. J. Tan, M. J. Campolongo, D. Luo, and W. Cheng, “Building plasmonic nanostructures with DNA,” Nat. Nanotechnol.6(5), 268–276 (2011). [CrossRef] [PubMed]
  26. E. Hutter and J. H. Fendler, “Exploitation of localized surface plasmon resonance,” Adv. Mater. (Deerfield Beach Fla.)16(19), 1685–1706 (2004). [CrossRef]
  27. X. Huang, S. Neretina, and M. A. El-Sayed, “Gold nanorods: From synthesis and properties to biological and biomedical applications,” Adv. Mater. (Deerfield Beach Fla.)21(48), 4880–4910 (2009). [CrossRef]
  28. B. Ding, Z. Deng, H. Yan, S. Cabrini, R. N. Zuckermann, and J. Bokor, “Gold nanoparticle self-similar chain structure organized by DNA origami,” J. Am. Chem. Soc.132(10), 3248–3249 (2010). [CrossRef] [PubMed]
  29. V. Poponin and A. Ignatov, “Local field enhancement in star-like sets of plasmon nanoparticles,” J. Korean Phys. Soc.47, S222–S228 (2005).
  30. S. Foteinopoulou, J. P. Vigneron, and C. Vandenbem, “Optical near-field excitations on plasmonic nanoparticle-based structures,” Opt. Express15(7), 4253–4267 (2007). [CrossRef] [PubMed]
  31. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt.22(7), 1099–20 (1983). [CrossRef] [PubMed]
  32. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  33. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).
  34. E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys.120(1), 357–366 (2004). [CrossRef] [PubMed]
  35. J. P. Kottmann and O. J. F. Martin, “Retardation-induced plasmon resonances in coupled nanoparticles,” Opt. Lett.26(14), 1096–1098 (2001). [CrossRef] [PubMed]
  36. H. Xu, “Multilayered metal core-shell nanostructures for inducing a large and tunable local optical field,” Phys. Rev. B72(7), 073405 (2005). [CrossRef]
  37. C. L. Nehl, N. K. Grady, G. P. Goodrich, F. Tam, N. J. Halas, and J. H. Hafner, “Scattering spectra of single gold nanoshells,” Nano Lett.4(12), 2355–2359 (2004). [CrossRef]
  38. S. E. Sburlan, L. A. Blanco, and M. Nieto-Vesperinas, “Plasmon excitation in sets of nanoscale cylinders and spheres,” Phys. Rev. B73(3), 035403 (2006). [CrossRef]
  39. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A plasmon ruler equation,” Nano Lett.7(7), 2080–2088 (2007). [CrossRef]
  40. P. K. Jain and M. A. El-Sayed, “Universal scaling of plasmon coupling in metal nanostructures: Extension from particle pairs to nanoshells,” Nano Lett.7(9), 2854–2858 (2007). [CrossRef] [PubMed]
  41. P. K. Jain and M. A. El-Sayed, “Surface plasmon coupling and its universal size scaling in metal nanostructures of complex geometry: elongated particle pairs and nanosphere trimers,” J. Phys. Chem. C111, 17451–17454 (2007). [CrossRef]
  42. J. Kottmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express8(12), 655–663 (2001). [CrossRef] [PubMed]
  43. E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys.120(1), 357–366 (2004). [CrossRef] [PubMed]
  44. S. V. Boriskina and B. M. Reinhard, “Molding the flow of light on the nanoscale: from vortex nanogears to phase-operated plasmonic machinery,” Nanoscale4(1), 76–90 (2011). [CrossRef] [PubMed]
  45. J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, “Electrodynamic effects in plasmonic nanolenses,” Phys. Rev. B77(11), 115419 (2008). [CrossRef]
  46. G. Das, F. De Angelis, M. L. Coluccio, F. Mecarini, and E. Di Fabrizio, “Spectroscopy nanofabrication and biophotonics,” Proc. SPIE7205, 720508, 720508-10 (2009). [CrossRef]
  47. F. Le, D. W. Brandl, Y. A. Urzhumov, H. Wang, J. Kundu, N. J. Halas, J. Aizpurua, and P. Nordlander, “Metallic nanoparticle arrays: A common substrate for both surface-enhanced Raman scattering and surface-enhanced infrared absorption,” ACS Nano2(4), 707–718 (2008). [CrossRef] [PubMed]
  48. J. Kneipp, X. Li, M. Sherwood, U. Panne, H. Kneipp, M. I. Stockman, and K. Kneipp, “Gold nanolenses generated by laser ablation-efficient enhancing structure for surface enhanced Raman scattering analytics and sensing,” Anal. Chem.80(11), 4247–4251 (2008). [CrossRef] [PubMed]
  49. V. G. Kravets, G. Zoriniants, C. P. Burrows, F. Schedin, C. Casiraghi, P. Klar, A. K. Geim, W. L. Barnes, and A. N. Grigorenko, “Cascaded optical field enhancement in composite plasmonic nanostructures,” Phys. Rev. Lett.105(24), 246806 (2010). [CrossRef] [PubMed]
  50. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B72(15), 155412 (2005). [CrossRef]
  51. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys.3(7), 477–480 (2007). [CrossRef]
  52. A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics2(6), 365–370 (2008). [CrossRef]
  53. V. Castel and C. Brosseau, “Electron magnetic resonance study of transition-metal magnetic nanoclusters embedded in metal-oxides,” Phys. Rev. B77(13), 134424 (2008). [CrossRef]
  54. V. Castel and C. Brosseau, “Magnetic field dependence of the effective permittivity in BaTiO3/Ni nanocomposites observed via microwave spectroscopy,” Appl. Phys. Lett.92(23), 233110 (2008). [CrossRef]
  55. B. M. Ross and L. P. Lee, “Plasmon tuning and local field enhancement maximization of the nanocrescent,” Nanotechnology19(27), 275201 (2008). [CrossRef] [PubMed]
  56. K. Li, L. Clime, B. Cui, and T. Veres, “Surface enhanced Raman scattering on long-range ordered noble-metal nanocrescent arrays,” Nanotechnology19(14), 145305 (2008). [CrossRef] [PubMed]
  57. H. Rochholz, N. Bocchio, and M. Kreiter, “Tuning resonances on crescent-shaped noble-metal nanoparticles,” New J. Phys.9(3), 53–70 (2007). [CrossRef]
  58. J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, “Fabrication of crescent-shaped optical antennas,” Adv. Mater. (Deerfield Beach Fla.)17(17), 2131–2134 (2005). [CrossRef]
  59. J. Kim, G. Liu, Y. Lu, and L. Lee, “Intra-particle plasmonic coupling of tip and cavity resonance modes in metallic apertured nanocavities,” Opt. Express13(21), 8332–8338 (2005). [CrossRef] [PubMed]
  60. L. Yang, X. Luo, and M. Hong, “Self-similar chain of nanocrescents as a surface-enhanced Raman scattering substrate,” J. Comput. Theor. Nanosci.7(8), 1364–1367 (2010). [CrossRef]
  61. Y. Luo, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband light harvesting nanostructures robust to edge bluntness,” Phys. Rev. Lett.108(2), 023901 (2012). [CrossRef] [PubMed]

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