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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 12 — Jun. 4, 2012
  • pp: 13146–13163
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Heuristic optimization for the design of plasmonic nanowires with specific resonant and scattering properties

D. Macías, P.-M. Adam, V. Ruíz-Cortés, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil  »View Author Affiliations


Optics Express, Vol. 20, Issue 12, pp. 13146-13163 (2012)
http://dx.doi.org/10.1364/OE.20.013146


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Abstract

In this contribution, we propose a computational tool for the synthesis of metallic nanowires with optimized optical properties, e.g. maximal scattering cross-section at a given wavelength. For this, we employ a rigorous numerical method, based on the solution of surface integral equations, along with a heuristic optimization technique that belongs to the population-based set known as Evolutionary Algorithms. Also, we make use of a general representation scheme to model, in a more realistic manner, the arbitrary geometry of the nanowires. The performance of this approach is evaluated through some examples involving various wavelengths, materials, and optimization strategies. The results of our numerical experiments show that this hybrid technique is a suitable and versatile tool straightforwardly extensible for the design of different configurations of interest in Plasmonics.

© 2012 OSA

1. Introduction

Over the past decades, an important amount of work has been devoted to study the scattering and resonant phenomena arising from the interaction between light and nanostructures with complex geometries [1

1. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003). [CrossRef]

10

10. E. A. Coronado, E. R. Encina, and F. D. Stefani, “Optical properties of metallic nanoparticles: manipulating light, heat and forces at the nanoscale,” Nanoscale 3, 4042–4059 (2011). [CrossRef] [PubMed]

]. Through various numerical, rigorous or approximative approaches, these works and references therein provide a better understanding of the linear or non-linear optical properties associated to the different morphologies of metallic nanoparticles. Among this set of possible shapes, star-like nanoparticles have shown a great potential for applications such as cancer hyperthermia therapy, thermal imaging, Surface-Enhanced Raman Scattering (SERS) or sensors design, to name but few examples [11

11. V. Giannini, A. Fernandez-Dominguez, Y. Sonnefraud, T. Roschuk, R. Fernandez-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6, 2498–2507 (2010). [CrossRef] [PubMed]

]. Recently, this geometry has been the subject of extensive experimental and theoretical works [12

12. C. G. Khoury and T. Vo-Dinh, “Gold nanostars for surface-enhanced raman scattering: synthesis, characterization and optimization,” J. Phys. Chem. C 112, 18849–18859 (2008).

22

22. R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express 20, 621–626 (2012). [CrossRef] [PubMed]

]. For example, the authors of reference [18

18. C. Hrelescu, T. K. Sau, A. L. Rogach, F. Jackel, G. Laurent, L. Douillard, and F. Charra, “Selective excitation of individual plasmonic hotspots at the tips of single gold nanostars,” Nano Lett. 11, 402–407 (2011). [CrossRef] [PubMed]

] have experimentally shown the possibility to selectively illuminate the tips of a nanostar through the controlled manipulation of the incident wavelength and the polarization. Furthermore, Giannini et al. [19

19. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A 24, 2822–2830 (2007). [CrossRef]

, 20

20. V. Giannini, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Surface plasmon resonances of metallic nanostars/nanoflowers for surface-enhanced Raman scattering,” Plasmonics 5, 99–104 (2010). [CrossRef]

] and Rodriguez-Oliveros and Sánchez-Gil [21

21. R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surface,” Opt. Express 19, 12208–12219 (2011). [CrossRef] [PubMed]

,22

22. R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express 20, 621–626 (2012). [CrossRef] [PubMed]

] have employed an integral formalism to theoretically study the scattering and resonant properties of metallic nanostars and nanoflowers. Moreover, the authors of [22

22. R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express 20, 621–626 (2012). [CrossRef] [PubMed]

] have used a three-dimensional approach to study the thermal properties of such a nanostructure.

The structure of this paper is as follows: Section 2 is devoted to the formulation of the problem and to introduce the rigorous numerical method to be employed throughout this work. The operational principles of the optimization technique used for the inverse problem are briefly described in Section 3, where we also illustrate and discuss its performance through some numerical experiments. Then, our approach is exploited in Sec. 4 and 5 to optimize two types of nanostructures with specific properties for given applications. The first are Ag and Au nanostars for SERS. The second nanostructures are dimer nanoantennas for enhanced fluorescence. Ultimately, in Section 6, we present a summary and give our final remarks.

2. Formulation of the problem

We consider the two-dimensional geometry shown in Fig. 1. The system is assumed invariant along the axis x2 and the profile of this structure is represented by the contour Γ(r), where r(x1, x3) is the position vector of a point that belongs to the profile. The particle is characterized by its frequency dependent dielectric constant εII(ω). The choice of a two-dimensional geometry is not restrictive; it corresponds to infinite (or very long) metal nanowires, and can be in turn applied to reproduce certain optical properties of metal nanoparticles [34

34. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007). [CrossRef] [PubMed]

,35

35. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express 15, 17736–17746 (2007). [CrossRef] [PubMed]

]. However, a three-dimensional geometry would increase considerably the complexity of the problem, as it must be treated in a vectorial way. At this stage it is out of the scope of this work.

Fig. 1 Geometry of the problem.

The system in Fig. 1 is illuminated with a p-polarized wave of the form
Ψ(r,t)=(0,ψ(r),0)exp{iωt},
(1)
where r = (x1, x3) and ψ(r) is the non null component of the magnetic field. In this geometry, only p-polarized light can excite localized surface plasmons.

In order to compute the total electric field in the neighborhood of the nanoparticle, in terms of φ(r′|ω) and χ(r′|ω), we follow [19

19. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A 24, 2822–2830 (2007). [CrossRef]

] and take Ampere’s Equation
×H(r)=iωcεIE(r)
(4)
as starting point. Also, we assume H(p)(r | ω) = (0, ψ(r | ω), 0) and, after some algebraic manipulations, we obtain
E1(p)(r|ω)=E1(p)(r|ω)incω4cdS{φ(r|ω)×[x3x3|rr|2(n(rr))H2(1)(nIωc|rr|)1nIωc|rr|H1(1)(nIωc|rr|)]χ(r|ω)x3x3nIωc|rr|H1(1)(nIωc|rr|)},
(5)
E2(p)(r|ω)=0,
(6)
and
E3(p)(r|ω)=E3(p)(r|ω)incω4cdS{φ(r|ω)×[x1x1|rr|2(n(rr))H2(1)(nIωc|rr|)dx3dx1nIωc|rr|H1(1)(nIωc|rr|)]+χ(r|ω)x1x1nIωc|rr|H1(1)(nIωc|rr|)},
(7)
where H1(1) and H2(1) are Hankel’s functions of first kind and orders 1 and 2, respectively. The components of the incident field in Eqs. (5) and (7) are also determined through Ampere’s Equation.

At this point, we consider convenient to note that our approach could be equally employed for the optimization of scattering or resonant features others than the SCS or the intensity of Electric-Field at a given position. The reason of this flexibility is that the operational principles of the optimization algorithm are independent from the underlying physics of the property to be optimized (fitness function). Then, to avoid a nonexistent limitation on the applicability of the method, we will refer to the fitness function as 𝒪(p|ω) and we will use a specific denotation whenever necessary for the clarity of the presentation.

3. Optimization of 𝒪(p|ω)

In [30

30. A. Tassadit, D. Macías, J. A. Sánchez-Gil, P. M. Adam, and R. Rodríguez-Oliveros, “Metal nanostars: stochastic optimization of resonant scattering properties,” Superlattice Microst. 49, 288–293 (2011). [CrossRef]

] we made use of a Non-Elitist Evolution Strategy ((μ/ρ, λ)-ES) to maximize Eq. (3). The preliminary results shown in that reference were encouraging; however, more numerical experiments were necessary to explore further the possibilities and limitations of that approach. To do this in an objective and systematic way, in this work we will also employ an Elitist Evolution Strategy ((μ/ρ + λ)-ES) as the one described in [37

37. D. Macías, A. Vial, and D. Barchiesi, “Application of evolution strategies for the solution of an inverse problem in near-field optics,” J. Opt. Soc. Am. A 21, 1465–1471 (2004). [CrossRef]

, 38

38. D. Macías, G. Olague, and E. R. Méndez, “Inverse scattering with far-field intensity data: random surfaces that belong to a well-defined statistical class,” Wave Random Complex 16, 545–560 (2006). [CrossRef]

].

3.1. Representation of the objective variables: Gielis’ Superformula

An important issue to take into account in any numerical optimization problem is the representation scheme. In the present context, we work with a discretized version of the geometry considered to solve the direct scattering problem. This step is crucial as an incorrect representation of the structure could lead to spurious effects in the computed magnitudes. To circumvent this situation, we employ a 2D version of Gielis’ Superformula [40

40. J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot. 90, 333–338 (2003). [CrossRef] [PubMed]

], given by the following parametric equations
x1=r(θ)cos(θ)andx3=r(θ)sin(θ),
(9)
where
r(θ)=rint[|cos(mθ4)a|n2+|sin(mθ4)b|n3]1n1.
(10)

The Eq. (9) offers a unified representation scheme that allows the generation of a wide variety of two-dimensional geometries through the variation of the parameters m, n1, n2, n3, a and b. This is illustrated in Fig. 2, where different profiles have been generated by means of Eq. (9). This reference gives an illustrative explanation of the relationship between the parameters m, n1, n2, n3, a and b and the associated shapes generated with them. Similar parametric formulas have been successfully exploited to solve direct scattering problems, based on 3D and 2D versions of Gielis’ superformula [21

21. R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surface,” Opt. Express 19, 12208–12219 (2011). [CrossRef] [PubMed]

, 22

22. R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express 20, 621–626 (2012). [CrossRef] [PubMed]

, 30

30. A. Tassadit, D. Macías, J. A. Sánchez-Gil, P. M. Adam, and R. Rodríguez-Oliveros, “Metal nanostars: stochastic optimization of resonant scattering properties,” Superlattice Microst. 49, 288–293 (2011). [CrossRef]

], or superellipsoids [41

41. T. Wriedt, “Using the T-Matrix method for light scattering computations by non-axisymmetric particles: superel-lipsoids and realistically shaped particles,” Part. Part. Syst. Charact 4, 256–268 (2002). [CrossRef]

].

Fig. 2 Geometries generated with Eq. (9). In all these cases we have assumed rint = 1.

The parameter rint is not present in the original expression [40

40. J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot. 90, 333–338 (2003). [CrossRef] [PubMed]

]. However, we have introduced it to guarantee the dimensional homogeneity of the resultant parametric equations needed in the present application. The advantages of using Gielis’ Superformula will become clear in the forthcoming sections, as it provides a suitable representation scheme for the numerical implementation of Green’s Surface Integral Theorem and the optimization process. Moreover, throughout this work we set a = b = 1 to avoid the generation of open geometries such as, for example, spirals.

3.2. Optimization technique: strategies and validation

At this stage some examples are convenient to illustrate the performance of the optimization technique proposed in the present work. Nevertheless, to facilitate the visualization of the operational principles the evolution strategies, we consider the benchmark function [42

42. J. G. Digalakis and K. G. Margaritis, “An experimental study of benchmarking functions for genetic algorithms,” Intern. J. Computer Math. 77, 481–506 (2002). [CrossRef]

]
f(xi)=i=12xi2for5.12xi5.12
(11)
where xi for i = 1,..., n are the components of the vector p of objective variables. The minimum of this function, represented with a blue diamond in Fig. 3, is located at (0,0). The convergence behavior of the Non-Elitist strategy is shown in the animation ( Media 1), where the red crosses are the elements of the initial population of each iteration of the evolutionary loop.

Fig. 3 Proof-of-concept test to illustrate of the convergence behavior of an evolution strategy. The contour lines correspond to the benchmark function given by Eq. (11) ( Media 1).

Within the context this contribution, we will consider two-dimensional isolated unsupported metallic nanostructures to keep the computational complexity into a manageable level. Their frequency-dependent dielectric constant εII(ω) is obtained through the interpolation of the tabulated experimental data from reference [43

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

]. The incidence medium is assumed to be air. Furthermore, the system in Fig. 1 is illuminated with a p-polarized plane wave at an angle of incidence θinc = 0°. Although not shown here for brevity reasons, we compared the results of our electromagnetic code with those obtained with Mie’s solution for the case of an infinite cylinder. Also, we performed a test of convergence similar to the one described in [19

19. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A 24, 2822–2830 (2007). [CrossRef]

] to determine the suitable number of sampling points to correctly describe the cross-section of the nanowire and, in this way, to avoid the appearance of numerical artifacts. In this respect, it is worth mentioning that despite the apparent sharpness in some of the shapes generated with Gielis’ Superformula, for example Figs. 2(a), 2(c), 2(d) or other geometries in forthcoming sections, the corners are rounded because of the angular sampling employed to generate the nanowire’s profile. Although advantageous for the numerical implementation of the electromagnetic code, this situation leads to a high density of sampling points in the corners or sharp edges of the geometry. A direct outcome of this could be a small argument of the Hankel’s functions required to solve the electromagnetic problem and their consequent divergence. However, we did not face this situation in any of the numerical experiments conducted throughout this work.

We begin our discussion with a comparison between the Elitist strategy ((μ/ρ + λ) − ES) to be used in this work and the Non-Elitist one ((μ/ρ, λ) − ES) used in [30

30. A. Tassadit, D. Macías, J. A. Sánchez-Gil, P. M. Adam, and R. Rodríguez-Oliveros, “Metal nanostars: stochastic optimization of resonant scattering properties,” Superlattice Microst. 49, 288–293 (2011). [CrossRef]

]. For the sake of consistency, we consider the same parameters as those used in that reference for the respective sizes of the initial and secondary populations, μ = 10 and λ = 100, and ρ = 2 for number of elements to be recombinated. Furthermore, we keep the maximum number of generations as the criterion to stop the evolutionary loop. As mentioned in [30

30. A. Tassadit, D. Macías, J. A. Sánchez-Gil, P. M. Adam, and R. Rodríguez-Oliveros, “Metal nanostars: stochastic optimization of resonant scattering properties,” Superlattice Microst. 49, 288–293 (2011). [CrossRef]

], extensive numerical experiments showed that there were no significant changes in the fitness function after g = 50 generations.

In the examples we present in this section, the optimization methods are tested for their relative success by searching for the optimal solution from twenty different initial states. That is, each realization of the algorithm started with a different initial population.

As a first example, let us look for the maximal Scattering Cross Section of a nanostructure made of silver. For this, we fix the wavelength to λ = 532 nm and the number of branches to m = 4, which means that the resulting geometry should resemble a structure with four vertices (see Fig. 2). The choice of this relatively simple structure is just to define the starting point for our forthcoming discussion.

Some typical results concerning the convergence behavior of the method just described are presented in Fig. 4. The blue and dark green solid curves correspond to the best realization of the (μ/ρ,λ) − ES and the (μ/ρ + λ) − ES, respectively. The oscillating behavior of the blue curve in Fig. 4 is typical of a non-elitist evolution strategy, which avoids the regions of attraction of local optima by allowing a temporary deterioration of the fitness value. The elitist strategy, on the other hand, presents a monotonic increase of the fitness value and has converged to what could be thought as the global optimum within the research space defined. In all the numerical experiments we conducted, the elitist evolution strategy outperformed the non-elitist strategy. We obtained similar results, not shown here for brevity, when we incremented the exterior radio to radext = 200 nm and also when we increased the number of generations to g = 500. This suggests the existence of a unique optimal star-like geometry, as those depicted in Fig. 5(a), that maximizes the SCS at the established wavelength, as shown in Fig. 5(b). In order to facilitate the visualization, we have employed the same line styles in Figs. 4 and 5.

Fig. 4 Convergence behavior of the (μ/ρ, λ) − ES, the (μ/ρ + λ) − ES and the PSO algorithms employed in this work.
Fig. 5 (a)Geometries and (b) Optimized Spectra obtained with the (μ/ρ, λ) − ES, the (μ/ρ +λ) − ES and the PSO algorithms. The solid black line shows the spectral position of the optimum.

In Fig. 6 we present the intensity maps, in logarithmic scale, corresponding to the geometry optimized at the wavelength of resonance and out of it. These results are consistent with those reported in references [19

19. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A 24, 2822–2830 (2007). [CrossRef]

,20

20. V. Giannini, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Surface plasmon resonances of metallic nanostars/nanoflowers for surface-enhanced Raman scattering,” Plasmonics 5, 99–104 (2010). [CrossRef]

]. However, in our case we observe the enhancement of the field at the tips of the star rather than in the interstices. The origin of this discrepancy can be explained in terms of the representation scheme. In this work we use Gielis’ superfomula to represent the geometry instead of Chebyshev’s polynomials, which are employed by the authors of those references. It is worth noting that Gielis’ Superformula allows the generation of geometries similar to those obtained with Chebyshev’s polynomials. This can be achieved just by setting the exponent n1 negative.

Fig. 6 Logarithmic near-field intensity maps associated to the geometry obtained with the (μ/ρ +λ) − ES corresponding to (a) the wavelength of resonance (λ = 532 nm) and (b) out of it (λ = 700) nm. The field intensity inside the nanostars is set to 1 for the sake of clarity.

In order to compare the ES and the PSO in an objective manner, we considered a swarm of size μ = 114 elements. Also, we fixed the number of iterations of the PSO to g = 50. Under these conditions, at least in what concerns the elitist strategy (μ/ρ + λ) − ES, both algorithms evaluate the fitness function the same number of times. Representative results of our numerical experiment are depicted with the solid red curves in Figs. 4 and 5. The convergence behavior in Fig. 4 shows that the dynamics of the (μ/ρ + λ) − ES and that of the PSO are very similar and their respective fitness values after 50 generations are quite close. This is also the case for the star-like geometries in Fig. 4(a), although the blue star associated to the non-elitist strategy is the smallest one.

Notwithstanding the slight differences in the convergence and the resulting geometries, the agreement between the results obtained with methods based on different operational principals (Elitist and non-Elitist ES, and PSO) provides us with confidence in our approach. In what follows, because of its satisfactory performance concerning both convergence and optimum objective function, we will employ the elitist ES.

4. Numerical results and discussion

4.1. Optimizing silver nanowires at different wavelengths

Fig. 7 (a)Resulting star-like geometries obtained after optimization, (b) SCS related to the geometries optimized with the (μ/ρ + λ) − ES considering different wavelengths. As in Fig. 6, the solid black vertical lines indicate the spectral position of the optimum.

4.2. Optimizing gold nanowires: Effect of the material

So far, we have considered nanostructures made of silver in all our numerical experiments. It seems thus interesting to study the influence that a different material could have on the performance of the optimization algorithm. For this, we repeat the numerical experiments under the same illumination conditions established in the beginning of our discussion. Also, in what concerns the optimization algorithm, we use the (μ/ρ + λ) Elitist strategy to look for the optimum from each of the twenty initial populations considered throughout this section. Moreover, we keep the number of generations as the criterion to stop them. For coherence in the presentation, we keep the parameter m = 4. Then, the only variant is that we use gold instead of silver as the material of the nanostructure to be optimized.

Fig. 8 Convergence behavior associated to the (μ/ρ + λ) − ES for a nanostar of gold. The wavelengths considered for the optimization are λ = 532 nm and λ = 633 nm. The nanostructure was illuminated with a p-polarized plane wave at normal incidence
Fig. 9 (a) Geometries and (b) Optimized Spectra obtained with the (μ/ρ +λ) − ES. The black vertical lines represent the wavelengths considered from the optimization process.

Fig. 10 SCS surface for : (a) λ = 532 nm and Ag, (b) λ = 532 nm and Au, (c) λ = 633 nm and Ag, (d) λ = 633 nm and Au. In all these cases the star-like geometry was illuminated with a p-polarized plane wave at normal incidence.

The surface shown in Fig. 10(a) corresponds to the SCS for a wavelength λ = 532 nm and a silver nanostar. These are the same illumination and material conditions we have considered so far to assess the performance of our stochastic optimization approach. The “rippled” appearance of the region where the maximum is located could possibly explain the convergence of the method to different sets of parameters that are quite close but not equal. Consequently, the respective scattering spectra of the star-like structures obtained present small shifts of few nanometers.

If instead of silver we consider gold and we keep the same wavelength and illumination conditions, we obtain the somehow expected but still surprising result shown in Fig. 10(b), where an evident modification of the topology has taken place just by changing the material. Moreover, the amplitude of the SCS has significantly decreased with respect to the value shown in Fig. 10(a). This explains why the evolution strategy converged, in all twenty realizations, to an asymmetrical star-like nanoparticle with a weak roughness.

If now we consider a wavelength at which the absorption is lower for both silver and gold, e.g. λ = 633 nm, their resulting SCS surfaces are shown in Figs. 10(c) and 10(d). Once again there is a significant modification of the topography, especially for gold. However, the “rippled” structure does not seem to affect the convergence of the evolutionary algorithm to symmetrical structures as those shown in Figs. 5(a), 7(a) and 9(a).

These results show how the choice of the material and the wavelength play a key role on the performance of the algorithm.

4.3. Optimizing nanowires with different symmetries

Fig. 11 (a)Geometries and (b) Optimized Spectra obtained with the (μ/ρ +λ) − ES for silver star-like geometries with five and six tips. As in previously, the solid black vertical line indicates the spectral position of the maximum.

An interesting situation from Fig. 11(b) is that, notwithstanding the asymmetry, both spectra are quite similar and we observed this same behavior even when we varied the angle of incidence. This fact has been previously pointed out by Giannini and Sánchez-Gil [19

19. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A 24, 2822–2830 (2007). [CrossRef]

] and can be explained in terms of the ratio between the nanoparticle’s size and the incident wavelength. That is, the results of numerical experiments showed that an increment in the size would result in different scattering spectra.

The intensity maps corresponding to the geometries in Fig. 11(a) are depicted in Fig. 12. As it was previously shown in Fig. 6, the enhancement of the field takes place in the tips and not in the interstices. Moreover, it suggests the possibility to selectively illuminate the tips for plamonic-based sensing applications, as suggested by the experimental evidence in presented reference [18

18. C. Hrelescu, T. K. Sau, A. L. Rogach, F. Jackel, G. Laurent, L. Douillard, and F. Charra, “Selective excitation of individual plasmonic hotspots at the tips of single gold nanostars,” Nano Lett. 11, 402–407 (2011). [CrossRef] [PubMed]

].

Fig. 12 Near-field intensity maps associated to the geometries shown in Fig. 11

5. Optimizing two-dimensional dimer nanoantennas

In this section, we apply the stochastic algorithm to a specific configuration of interest in nanostructure-enhanced fluorescence: gap nanoantennas [34

34. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007). [CrossRef] [PubMed]

,35

35. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express 15, 17736–17746 (2007). [CrossRef] [PubMed]

]. In particular, we aim at maximizing the near-field intensity at the center of the gap between a pair of coupled nanorods. Strictly speaking, we will do so for 2D rectangular nanorods for the sake of simplicity, which are equivalent to infinitely long, coupled rectangular nanowires. Indeed, it has been shown that some of the optical properties of such 2D dimers reproduced fairly well those of realistic dimer nanoantennas, specially concerning the longitudinal coupled modes [35

35. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express 15, 17736–17746 (2007). [CrossRef] [PubMed]

]. The starting configuration is the following. The dimer nanoantennas consists of two identical rectangles, but their dimensions (height h and length L) are allowed to vary within a certain range. Since the width W of the gap is critical for the field enhancement and is typically restricted by experimental (nanolithography) constraints, we will fix it at a reasonable value W = 20 nm; also, the dimer is assumed to be embedded in a surrounding medium with index of refraction n = 1.5, to roughly reproduce the impact of typical glass substrates on resonances. The dimer nanoantenna is illuminated with a monochromatic plane wave at a fixed frequency of interest, see Fig. 13(a); normal incidence and polarization parallel to the dimer axis is considered in order to ensure maximum coupling into the odd longitudinal mode.

Fig. 13 (a) Configuration of the dimer geometry to be optimized: the gap width is fixed at W = 20 nm. (b) NF (at the gap center) spectra (solid curve) corresponding to the dimer nanoantenna (L = 99 nm and h = 15 nm) optimized with the (μ/ρ + λ) − ES for maximum near-field intensity at the gap center at λ = 800 nm (normal incidence). The SCS is also shown (dashed curve) for comparison. Inset: Convergence behavior of the algorithm.

Two wavelengths are considered which provide illustrative examples in turn with slightly different phenomenology. First, we show in Fig. 13(b) the optimized dimer configuration of a typical realization and resulting Near-Field (NF) and SCS spectra at λ = 800 nm. Essentially, the optimization algorithm successfully retrieves the parameters L and h that yield a dipole-like, half-wavelength resonance, with a reasonably large NF intensity enhancement of |E|2 ∼ 60. Note that the maxima of the NF and SCS spectra are closed to the wavelength of interest, λ = 800 nm, but slightly shifted. Recall that this energy shift of the NF with respect to the far-field (SCS) has been explained as stemming from damping in Ref. [47

47. J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon systems,” Nano Lett. 11, 1280–1283 (2011). [CrossRef] [PubMed]

].

Fig. 14 NF at the gap center (solid curves) and SCS (dashed curves) spectra of two dimer nanoantennas optimized with the (μ/ρ +λ) − ES for maximum near-field intensity at the gap center at λ = 510 nm (normal incidence). The results shown correspond to two realizations of the algorithm: L = 99.7 nm and h = 15 nm (red curve), and L = 120 nm and h = 40 nm (blue curve).

6. Summary and concluding remarks

This contribution illustrates the great potential, and possible limitations, of Stochastic Optimization as alternative way to manipulate, in a systematic and controlled manner, the scattering and resonant properties of isolated and multiple metallic nanoparticles. Furthermore, the representation scheme employed to model the shape of the nanoparticles not only is a general tool easy to implement in a computing language, but it also offers the possibility to generate a wide variety of geometries. The modular structure of this computational tool should allow to extend its applicability to other kind of multi-physics or multi-scale problems involving three-dimensional geometries as periodic arrays of supported nanoparticles and forward solvers others than the integral formalism employed in this work. This approach should serve as the starting point for the synthesis and characterization of optimal nanostructures, with specific scattering or spectral features, prior to their fabrication.

Acknowledgments

DM acknowledges Prof. E. R. Méndez for fruitful discussions during the preparation of this work. JASG and RRO acknowledge support both from the Spain Ministerio de Economía y Competitividad through the Consolider-Ingenio project EMET ( CSD2008-00066) and NANOPLAS ( FIS2009-11264), and from the Comunidad de Madrid (grant MICROSERES P2009/TIC-1476).

References and links

1.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003). [CrossRef]

2.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137–141 (2003). [CrossRef]

3.

C. Rockstuhl, M. G. Salt, and H. P. Herzig, “Analyzing the scattering properties of coupled metallic nanoparticles,” J. Opt. Soc. Am. A 21, 1761–1768 (2004). [CrossRef]

4.

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B 72, 195429 (2005).

5.

C. Noguez, “Surface plasmons on metal nanoparticles: the influence of shape and physical environment,” J. Phys. Chem. C 111, 3806–3819 (2007). [CrossRef]

6.

V. Myroshnychenko, E. Carbó-Argibay, I. Pastoriza-Santos, J. Pérez-Juste, L. M. Liz-Marzán, and F. García de Abajo, “Modeling the optical response of highly faceted metal nanoparticles with a fully 3D boundary element method,” Adv. Mater. 20, 4288–4293 (2008). [CrossRef]

7.

W. Ding, R. Bachelot, R. Espiau de Lamaestre, D. Macias, A. L. Baudrion, and P. Royer, “Understanding near/far-field engineering of optical dimer antennas through geometry modification,” Opt. Express 17, 21228–21239 (2009). [CrossRef] [PubMed]

8.

U. Guler and R. Turan, “Effect of particle properties and light polarization on the plasmonic resonances in metallic nanoparticles,” Opt. Express 18, 17322–17338 (2010). [CrossRef] [PubMed]

9.

J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express 19, 23386–23399 (2011). [CrossRef] [PubMed]

10.

E. A. Coronado, E. R. Encina, and F. D. Stefani, “Optical properties of metallic nanoparticles: manipulating light, heat and forces at the nanoscale,” Nanoscale 3, 4042–4059 (2011). [CrossRef] [PubMed]

11.

V. Giannini, A. Fernandez-Dominguez, Y. Sonnefraud, T. Roschuk, R. Fernandez-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6, 2498–2507 (2010). [CrossRef] [PubMed]

12.

C. G. Khoury and T. Vo-Dinh, “Gold nanostars for surface-enhanced raman scattering: synthesis, characterization and optimization,” J. Phys. Chem. C 112, 18849–18859 (2008).

13.

P. Senthil Kumar, I. Pastoriza-Santos, B. Rodríguez-González, F. Javier García de Abajo, and L. M. Liz-Marzán, “High-yield synthesis and optical response of gold nanostars,” Nanotechnology 19, 015606 (2008). [CrossRef] [PubMed]

14.

C. Hrelescu, T. K. Sau, A. L. Rogach, F. Jäckel, and J. Feldmann, “Single gold nanostars enhance Raman scattering,” Appl. Phys. Lett. 94, 153113 (2009). [CrossRef]

15.

S. Barbosa, A. Agrawal, L. Rodríguez-Lorenzo, I. Pastoriza-Santos, R. A. Alvarez-Puebla, A. Kornowski, H. Weller, and L. M. Liz-Marzán, “Tuning size and sensing properties in colloidal gold nanostars,” Langmuir 26, 14943–14950 (2010). [CrossRef] [PubMed]

16.

L. Rodríguez-Lorenzo, R. A. Álvarez-Puebla, F. J. García de Abajo, and L. M. Liz-Marzán, “Surface-enhanced Raman scattering using star-shaped gold colloidal nanoparticles,” J. Phys. Chem. C 114, 7336–7340 (2010). [CrossRef]

17.

S. K. Dondapati, T. K. Sau, C. Hrelescu, T. A. Klar, F. D. Stefani, and J. Feldmann, “Label-free biosensing based on single gold nanostars as plasmonic transducers,” ACS Nano 4, 6318–6322 (2010). [CrossRef] [PubMed]

18.

C. Hrelescu, T. K. Sau, A. L. Rogach, F. Jackel, G. Laurent, L. Douillard, and F. Charra, “Selective excitation of individual plasmonic hotspots at the tips of single gold nanostars,” Nano Lett. 11, 402–407 (2011). [CrossRef] [PubMed]

19.

V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A 24, 2822–2830 (2007). [CrossRef]

20.

V. Giannini, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Surface plasmon resonances of metallic nanostars/nanoflowers for surface-enhanced Raman scattering,” Plasmonics 5, 99–104 (2010). [CrossRef]

21.

R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surface,” Opt. Express 19, 12208–12219 (2011). [CrossRef] [PubMed]

22.

R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express 20, 621–626 (2012). [CrossRef] [PubMed]

23.

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. 49, 409–436 (1952).

24.

P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nanoparticles,” Nano Lett. 11, 2329–2333 (2011). [CrossRef] [PubMed]

25.

D. Macías and A. Vial, “Optimal design of plasmonic nanostructures for plasmon-interference assisted lithography,” Appl. Phys. B 93, 159–163 (2008). [CrossRef]

26.

T. Grosges, D. Barchiesi, T. Toury, and G. Gréhan, “Design of nanostructures for imaging and biomedical applications by plasmonic optimization,” Opt. Lett. 33, 2812–2814 (2008). [CrossRef] [PubMed]

27.

I. Grigorenko, S. Haas, A. Balatsky, and A. F. J. Levi, “Optimal control of electromagnetic field using metallic nanoclusters,” New J. Phys. 10, 043017 (2008). [CrossRef]

28.

C. Forestiere, M. Donelli, G. F. Walsh, E. Zeni, G. Miano, and L. Dal Negro, “Particle-swarm optimization of broadband nanoplasmonic arrays,” Opt. Lett. 35, 133–135 (2010). [CrossRef] [PubMed]

29.

S. Kessentini, D. Barchiesi, T. Grosges, and M. Lamy de la Chapelle, “Selective and collaborative optimization methods for plasmonics: a comparison,” PIERS Online 7, 291–295 (2011).

30.

A. Tassadit, D. Macías, J. A. Sánchez-Gil, P. M. Adam, and R. Rodríguez-Oliveros, “Metal nanostars: stochastic optimization of resonant scattering properties,” Superlattice Microst. 49, 288–293 (2011). [CrossRef]

31.

M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Light concentration in the near-field of dielectric spheroidal particles with mesoscopic sizes,” Opt. Express 19, 2847–2858 (2011). [CrossRef]

32.

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012) [CrossRef] [PubMed]

33.

R. Wehrens and M. B. Lutgarde, “Classical and nonclassical optimization methods,” in Encyclopedia of Analytical Chemistry, R. A. Meyers, ed. (Wiley, 2000), pp. 9678–9689.

34.

O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007). [CrossRef] [PubMed]

35.

O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express 15, 17736–17746 (2007). [CrossRef] [PubMed]

36.

C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150–2161 (2003). [CrossRef]

37.

D. Macías, A. Vial, and D. Barchiesi, “Application of evolution strategies for the solution of an inverse problem in near-field optics,” J. Opt. Soc. Am. A 21, 1465–1471 (2004). [CrossRef]

38.

D. Macías, G. Olague, and E. R. Méndez, “Inverse scattering with far-field intensity data: random surfaces that belong to a well-defined statistical class,” Wave Random Complex 16, 545–560 (2006). [CrossRef]

39.

H. G. Beyer, The Theory of Evolution Strategies (Springer-Verlag, 2001).

40.

J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot. 90, 333–338 (2003). [CrossRef] [PubMed]

41.

T. Wriedt, “Using the T-Matrix method for light scattering computations by non-axisymmetric particles: superel-lipsoids and realistically shaped particles,” Part. Part. Syst. Charact 4, 256–268 (2002). [CrossRef]

42.

J. G. Digalakis and K. G. Margaritis, “An experimental study of benchmarking functions for genetic algorithms,” Intern. J. Computer Math. 77, 481–506 (2002). [CrossRef]

43.

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

44.

R. C. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proceedings of the Sixth International Symposium on Micro Machine and Human Science (IEEE,1995), pp. 39–43. [CrossRef]

45.

R. Poli, J. Kennedy, and T. Blackwell, “Particle Swarm Optimisation: an overview,” Swarm Intell. 1, 33–57 (2007). [CrossRef]

46.

N. Berkovitch, P. Ginzburg, and M. Orenstein, “Concave plasmonic particles: broad-band geometrical tunability in the near-infrared,” Nano Lett 10, 1405–1408 (2010). [CrossRef] [PubMed]

47.

J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon systems,” Nano Lett. 11, 1280–1283 (2011). [CrossRef] [PubMed]

48.

F. López-Tejeira, R. Paniagua-Domínguez, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Fano-like interference of plasmon resonances at a single rod-shaped nanoantenna,” New J. Phys. 14, 023035 (2012). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(290.3200) Scattering : Inverse scattering
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Optics at Surfaces

History
Original Manuscript: March 29, 2012
Revised Manuscript: April 27, 2012
Manuscript Accepted: April 30, 2012
Published: May 25, 2012

Citation
D. Macías, P.-M. Adam, V. Ruíz-Cortés, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, "Heuristic optimization for the design of plasmonic nanowires with specific resonant and scattering properties," Opt. Express 20, 13146-13163 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13146


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References

  1. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B107, 668–677 (2003). [CrossRef]
  2. W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun.220, 137–141 (2003). [CrossRef]
  3. C. Rockstuhl, M. G. Salt, and H. P. Herzig, “Analyzing the scattering properties of coupled metallic nanoparticles,” J. Opt. Soc. Am. A21, 1761–1768 (2004). [CrossRef]
  4. U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: A boundary integral method approach,” Phys. Rev. B72, 195429 (2005).
  5. C. Noguez, “Surface plasmons on metal nanoparticles: the influence of shape and physical environment,” J. Phys. Chem. C111, 3806–3819 (2007). [CrossRef]
  6. V. Myroshnychenko, E. Carbó-Argibay, I. Pastoriza-Santos, J. Pérez-Juste, L. M. Liz-Marzán, and F. García de Abajo, “Modeling the optical response of highly faceted metal nanoparticles with a fully 3D boundary element method,” Adv. Mater.20, 4288–4293 (2008). [CrossRef]
  7. W. Ding, R. Bachelot, R. Espiau de Lamaestre, D. Macias, A. L. Baudrion, and P. Royer, “Understanding near/far-field engineering of optical dimer antennas through geometry modification,” Opt. Express17, 21228–21239 (2009). [CrossRef] [PubMed]
  8. U. Guler and R. Turan, “Effect of particle properties and light polarization on the plasmonic resonances in metallic nanoparticles,” Opt. Express18, 17322–17338 (2010). [CrossRef] [PubMed]
  9. J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express19, 23386–23399 (2011). [CrossRef] [PubMed]
  10. E. A. Coronado, E. R. Encina, and F. D. Stefani, “Optical properties of metallic nanoparticles: manipulating light, heat and forces at the nanoscale,” Nanoscale3, 4042–4059 (2011). [CrossRef] [PubMed]
  11. V. Giannini, A. Fernandez-Dominguez, Y. Sonnefraud, T. Roschuk, R. Fernandez-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small6, 2498–2507 (2010). [CrossRef] [PubMed]
  12. C. G. Khoury and T. Vo-Dinh, “Gold nanostars for surface-enhanced raman scattering: synthesis, characterization and optimization,” J. Phys. Chem. C112, 18849–18859 (2008).
  13. P. Senthil Kumar, I. Pastoriza-Santos, B. Rodríguez-González, F. Javier García de Abajo, and L. M. Liz-Marzán, “High-yield synthesis and optical response of gold nanostars,” Nanotechnology19, 015606 (2008). [CrossRef] [PubMed]
  14. C. Hrelescu, T. K. Sau, A. L. Rogach, F. Jäckel, and J. Feldmann, “Single gold nanostars enhance Raman scattering,” Appl. Phys. Lett.94, 153113 (2009). [CrossRef]
  15. S. Barbosa, A. Agrawal, L. Rodríguez-Lorenzo, I. Pastoriza-Santos, R. A. Alvarez-Puebla, A. Kornowski, H. Weller, and L. M. Liz-Marzán, “Tuning size and sensing properties in colloidal gold nanostars,” Langmuir26, 14943–14950 (2010). [CrossRef] [PubMed]
  16. L. Rodríguez-Lorenzo, R. A. Álvarez-Puebla, F. J. García de Abajo, and L. M. Liz-Marzán, “Surface-enhanced Raman scattering using star-shaped gold colloidal nanoparticles,” J. Phys. Chem. C114, 7336–7340 (2010). [CrossRef]
  17. S. K. Dondapati, T. K. Sau, C. Hrelescu, T. A. Klar, F. D. Stefani, and J. Feldmann, “Label-free biosensing based on single gold nanostars as plasmonic transducers,” ACS Nano4, 6318–6322 (2010). [CrossRef] [PubMed]
  18. C. Hrelescu, T. K. Sau, A. L. Rogach, F. Jackel, G. Laurent, L. Douillard, and F. Charra, “Selective excitation of individual plasmonic hotspots at the tips of single gold nanostars,” Nano Lett.11, 402–407 (2011). [CrossRef] [PubMed]
  19. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A24, 2822–2830 (2007). [CrossRef]
  20. V. Giannini, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Surface plasmon resonances of metallic nanostars/nanoflowers for surface-enhanced Raman scattering,” Plasmonics5, 99–104 (2010). [CrossRef]
  21. R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surface,” Opt. Express19, 12208–12219 (2011). [CrossRef] [PubMed]
  22. R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express20, 621–626 (2012). [CrossRef] [PubMed]
  23. M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand.49, 409–436 (1952).
  24. P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nanoparticles,” Nano Lett.11, 2329–2333 (2011). [CrossRef] [PubMed]
  25. D. Macías and A. Vial, “Optimal design of plasmonic nanostructures for plasmon-interference assisted lithography,” Appl. Phys. B93, 159–163 (2008). [CrossRef]
  26. T. Grosges, D. Barchiesi, T. Toury, and G. Gréhan, “Design of nanostructures for imaging and biomedical applications by plasmonic optimization,” Opt. Lett.33, 2812–2814 (2008). [CrossRef] [PubMed]
  27. I. Grigorenko, S. Haas, A. Balatsky, and A. F. J. Levi, “Optimal control of electromagnetic field using metallic nanoclusters,” New J. Phys.10, 043017 (2008). [CrossRef]
  28. C. Forestiere, M. Donelli, G. F. Walsh, E. Zeni, G. Miano, and L. Dal Negro, “Particle-swarm optimization of broadband nanoplasmonic arrays,” Opt. Lett.35, 133–135 (2010). [CrossRef] [PubMed]
  29. S. Kessentini, D. Barchiesi, T. Grosges, and M. Lamy de la Chapelle, “Selective and collaborative optimization methods for plasmonics: a comparison,” PIERS Online7, 291–295 (2011).
  30. A. Tassadit, D. Macías, J. A. Sánchez-Gil, P. M. Adam, and R. Rodríguez-Oliveros, “Metal nanostars: stochastic optimization of resonant scattering properties,” Superlattice Microst.49, 288–293 (2011). [CrossRef]
  31. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Light concentration in the near-field of dielectric spheroidal particles with mesoscopic sizes,” Opt. Express19, 2847–2858 (2011). [CrossRef]
  32. C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett.12, 2037–2044 (2012) [CrossRef] [PubMed]
  33. R. Wehrens and M. B. Lutgarde, “Classical and nonclassical optimization methods,” in Encyclopedia of Analytical Chemistry, R. A. Meyers, ed. (Wiley, 2000), pp. 9678–9689.
  34. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett.7, 2871–2875 (2007). [CrossRef] [PubMed]
  35. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express15, 17736–17746 (2007). [CrossRef] [PubMed]
  36. C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B20, 2150–2161 (2003). [CrossRef]
  37. D. Macías, A. Vial, and D. Barchiesi, “Application of evolution strategies for the solution of an inverse problem in near-field optics,” J. Opt. Soc. Am. A21, 1465–1471 (2004). [CrossRef]
  38. D. Macías, G. Olague, and E. R. Méndez, “Inverse scattering with far-field intensity data: random surfaces that belong to a well-defined statistical class,” Wave Random Complex16, 545–560 (2006). [CrossRef]
  39. H. G. Beyer, The Theory of Evolution Strategies (Springer-Verlag, 2001).
  40. J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot.90, 333–338 (2003). [CrossRef] [PubMed]
  41. T. Wriedt, “Using the T-Matrix method for light scattering computations by non-axisymmetric particles: superel-lipsoids and realistically shaped particles,” Part. Part. Syst. Charact4, 256–268 (2002). [CrossRef]
  42. J. G. Digalakis and K. G. Margaritis, “An experimental study of benchmarking functions for genetic algorithms,” Intern. J. Computer Math.77, 481–506 (2002). [CrossRef]
  43. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
  44. R. C. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proceedings of the Sixth International Symposium on Micro Machine and Human Science (IEEE,1995), pp. 39–43. [CrossRef]
  45. R. Poli, J. Kennedy, and T. Blackwell, “Particle Swarm Optimisation: an overview,” Swarm Intell.1, 33–57 (2007). [CrossRef]
  46. N. Berkovitch, P. Ginzburg, and M. Orenstein, “Concave plasmonic particles: broad-band geometrical tunability in the near-infrared,” Nano Lett10, 1405–1408 (2010). [CrossRef] [PubMed]
  47. J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon systems,” Nano Lett.11, 1280–1283 (2011). [CrossRef] [PubMed]
  48. F. López-Tejeira, R. Paniagua-Domínguez, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Fano-like interference of plasmon resonances at a single rod-shaped nanoantenna,” New J. Phys.14, 023035 (2012). [CrossRef]

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