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 [1–10]. 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]. Recently, this geometry has been the subject of extensive experimental and theoretical works [12–22]. For example, the authors of reference [18] 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, 20] and Rodriguez-Oliveros and Sánchez-Gil [21,22] have employed an integral formalism to theoretically study the scattering and resonant properties of metallic nanostars and nanoflowers. Moreover, the authors of [22] have used a three-dimensional approach to study the thermal properties of such a nanostructure.

Notwithstanding the advances mentioned in the previous paragraph, the optimal design of a plasmonic nanostructure for a specific application had not received too much attention until recently. This can be attributed to the non-trivial relationship between the geometrical and material features of the nanostructure with the light it scatters. Also, the complexity of this particular type of inverse problem makes the use of classical optimization techniques such as Steepest Descent or Conjugate Gradient Methods [23] a not practical proposition, as for certain problems the computation of the gradient vector is not straightforward. Despite this fact, the approach described in reference [24] makes use of some variant of the Steepest Descent method to tune the spectral position of the resonance associated to a special kind of nanoparticle. In essence, the authors of that work introduce weak random perturbations on the surface of the nanoparticle, whose initial shape is arbitrarily established. The change on the shape will be accepted depending on whether the resonance shifts or not towards the desired spectral position. Despite its effectiveness, this iterative approach strongly depends on the initial guess and may become time consuming if it is far from a canonical geometry. An alternative way to solve the inverse problem is the one shown in references [25–32], where the authors of those works have employed the so known Direct Search Methods [33], which do not require the derivatives of the function to be optimized but only its values. The results reported in [32], for example, clearly show that using this kind of approach leads to optimal configurations that are not easily imaginable with traditional trial and error blind strategies. It seems thus well worth to explore further the possibilities and limitations of stochastic optimization in Plasmonics. In this contribution we take as starting point the evolutionary technique introduced in [30]. In a first stage, we further assess its performance through some examples involving the maximization of resonant properties of isolated metallic nanoparticles for different material and geometrical conditions. Then, we extend its applicability to more complex configurations as dimer nanoantennas.

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 ε_II(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 [34,35]. 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.

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The system in Fig. 1 is illuminated with a p-polarized wave of the form

With reference to Fig. 1, in this work we aim at maximizing some resonant or scattering property of the nanostructure as, for example, its Scattering Cross Section (SCS) at a given wavelength λ or its associated near-field intensity at a given position r₀. For this, we make use of the rigorous numerical method, based on Green’s Integral Theorem, described in Refs. [19, 36]. Then, the complex scattering amplitude is given by

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] and take Ampere’s Equation

As stated in [30], Eq. (3) can be maximized employing multidimensional optimization techniques through the reformulation of the original scattering problem as an optimization one, whose solution is given by the optimal geometry that generates the maximum Scattering Cross Section of a nanoparticle at an established wavelength. Thus, Eq. (3) can be written as

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] 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, 38].

Although the operational principles of Evolution Strategies can be found elsewhere [39], for the sake of clearness we will briefly recall them here. The first step prior to the beginning of the optimization process is the random generation of an ensemble of vectors p that will conform the initial population ${P_{\mu }^{〈g〉}|}_{g=0}$, where μ is the number of elements within the population and g is the associated iteration of the algorithm. A canonical evolutionary optimization algorithm is based on the application, over a defined number of iterations or generations, of two genetic operators with well defined roles: Recombination and Mutation. Whereas the former exploits the search space through the exchange of information between different elements of the population, the latter explores the search space through the introduction of random variations in the population. The application of these genetic operators over the initial population leads to the generation of a secondary population $P_{\lambda }^{〈g〉}$ of λ elements. It is at this stage of the evolutionary loop that the link between the physics of the problem studied and the optimization algorithm is established. That is, 𝒪(p|ω) is evaluated considering each element of the secondary population. Then, only those elements of $P_{\lambda }^{〈g〉}$ leading to regions of the search space where 𝒪(p|ω) is maximized/minimized will be retained, as part of the population $P_{\mu }^{〈g+1〉}$, for the next iteration of the evolutionary loop. The procedure is repeated until a defined termination criterion has been achieved. The respective sizes of the initial and the secondary populations, $P_{\mu }^{〈g〉}$ and $P_{\lambda }^{〈g〉}$ remain constant throughout the entire search process.

It is convenient to mention that the main difference between a Non-Elitist and an Elitist strategy is the selection scheme. The former choices the best elements only from the secondary population generated by means of the genetic operators, whereas the latter selects the best elements from an intermediate population generated from the union of the initial and secondary populations. As a consequence, the Non-Elitist strategy looses the information of a promising element of the initial population. The Elitist strategy, on the other hand, keeps the promising elements throughout the search process until they are replaced by better ones or the termination criteria has been reached.

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], given by the following parametric equations

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, 22, 30], or superellipsoids [41].

 

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

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The parameter r_int is not present in the original expression [40]. 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]

 

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).

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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 ε_II(ω) is obtained through the interpolation of the tabulated experimental data from reference [43]. 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 θ_inc = 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] 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]. 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], 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.

To define the research space in which the ES looks for the optimal solution, we set the upper and lower limits for the respective parameters in Gielis’ Superformula to 1 ≤ n₁ ≤ 10, 1 ≤ n₂ ≤ 10, 1 ≤ n₃ ≤ 10, and 20.0 nm ≤ r_int ≤ 80.0 nm. We followed [40] to establish the values of n₁, n₂ and n₃ and we considered them as real positive numbers. The limits of the parameter r_int serve to control the deepness of the resultant roughness on the geometry. Let us note that we did not consider the parameter m, related to the nanoparticle’s number of vertices, as an objective variable. This parameter is fixed prior to the beginning of the optimization loop and remains invariable throughout it. Also, to limit the maximum size of the nanoparticle’s cross section, we set the external radio of the imaginary circle that circumscribes it to rad_ext = 100 nm.

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 rad_ext = 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.

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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.

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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 [19,20]. 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.

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For completeness, we looked for the optimum employing an alternative population-based bio-inspired algorithm known as the Particle Swarm Optimization method (PSO) [44,45]. This heuristic technique imitates the collective and collaborative behavior of a population of individuals such as, for example, a flock of birds, a swarm of bees or a school of fishes during their exploration of a given extension of land/sea in search for food. When an individual finds a place potentially rich in eatables, it communicates its finding to the other individuals, which in turn make use of this global knowledge, together with their own search experience, to modify their respective velocities and positions to move towards the indicated area. This process is repeated several times until the entire population has converged to the portion of land richest in food.

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.

The respective scattering spectra of the stars in Fig. 5(a) are depicted in Fig. 5(b). The blueshift in the spectrum of the non-elitist star can be attributed to its small size, whereas the redshift in the spectrum of the PSO can be related to its size and the fact that the interstices of the PSO star are sharper than those of the Elitist Strategy.

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

The natural next step is to study the effect of the wavelength on the optimization process. For this, we arbitrarily set the wavelength to λ = 633 nm and to λ = 785 nm and then we repeated the previous numerical experiments keeping the illumination conditions, the research space and the number of peaks m = 4. The corresponding spectra and optimal geometries obtained with the (μ/ρ + λ) − ES are depicted in Fig. 7. For comparison, the dashed curve depicts the spectrum and geometry corresponding to λ = 532 nm. A clear peak (LSP) can be distinguished at the established wavelength. The convergence behavior for these numerical experiments presented, in all the twenty realizations, a dynamics similar to that shown in Fig. 4.

 

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.

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In order to obtain the LSP at red-shifted frequencies, not only shape but the size as well might play a key role. As a matter of fact, our stochastic approach indirectly explores both throughout its search in the parameter’s space. The resulting optimized geometries are essentially upscaled versions of each other, in such a way that the star-like shape that maximizes the SCS is preserved (their aspect ratios are similar), whereas the size increases to enforce the LSP red-shifts. Both, the shape and the size, are managed within the limits imposed upon the geometrical parameters. Furthermore, this behavior, together with the one depicted in Fig. 5, seems consistent with the experimental results reported by Berkovitch et al. [46]. The authors of this work explain the tunability of the resonance, in the infrared, as a consequence of the interplay between the concavity present in the cross-section of a square-like nanoparticle and the width of its basis.

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.

The dashed curve in Fig. 8 illustrates the poor performance of the ES when the wavelength is λ = 532 nm. On the other hand, when λ = 633 nm, the red solid curve shows a clear convergence towards the maximum. This situation can be better visualized in Fig. 9, where for consistency we have used the same line style as in Fig. 8. In Fig. 9(a) we depict the star-like structures obtained after optimization; the remarkable difference between the two geometries is a direct consequence of changing the material and its dependency on the wavelength. This fact can be appreciated also in Fig. 9(b), where the dashed and solid curves represent the respective scattering spectra of the geometries in Fig. 9(a). The vertical solid black lines are located at the wavelengths considered for the optimization process. There is a redshift of about 63 nm for the dashed spectrum and 7 nm for the solid one. We observed this behavior for each of the twenty realizations of the evolution strategy. In essence, the results shown in Figs. 8(a) and 8(b) were obtained considering a spectral position in which the absorption effects are important, specially for gold for which the onset of interband transitions is below λ = 600 nm. Thus, although the optimization method reaches the best optimized configuration within the parameter’s search space, in practice it is unable to yield a satisfactory LSP for Au NPs at such wavelength.

 

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

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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.

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To illustrate further the effect of the material on the topology of the fitness function, the SCS in the present example, let us consider the following special situation. Throughout our numerical experiments we have found that n₁ and rad_int are the most important parameters for the optimization process. It seems thus reasonable to assume the parameters n₂ and n₃ of the superformula as fixed. Also, for consistency we keep the number of peaks m = 4. The previous assumptions will allow us to compute the SCS as a function of only two parameters (n₁ and rad_int). The results obtained are the surfaces depicted in Fig. 10.

 

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.

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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

The next step in our study was to observe the influence of the roughness on the convergence towards the optimum. For this, we considered an structure of silver illuminated with a p-polarized plane wave at a wavelength λ = 532 nm. Also, for the optimization process, we kept the same initial conditions of our previous numerical experiments. We just incremented the number of tips of the nanoparticle. That is, we first set m = 5 to consider a non symmetrical structure, then we set m = 6. It is noteworthy to mention that such parameters could lead to pentagons, hexagons or similar. However, the geometries obtained after the optimization are, once again, the star-like geometries depicted in Fig. 11(a), whose respective scattering spectra are shown in 11(b). To facilitate the visualization, we have used the same line styles in both figures. Although not shown here, the results from twenty realizations seem to confirm our initial hypothesis concerning the elitism as the best strategy for the present problem.

 

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.

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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] 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].

 

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

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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 [34,35]. 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]. 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.

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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].

We next consider that the wavelength of interest is λ = 510 nm. This poses certain challenge to the optimization algorithm, since at such high frequencies, close to that of the LSP of a nanosphere, no dipole-like modes are expected to appear for dimer nanoantennas. Hence, we allow the algorithm to explore a larger parameters’ space: L ∈ [50, 250] and h ∈ [10, 50]. The results are shown in Fig. 14 for two different realizations; it should be emphasized that similar results are found for the other 18 realizations. This somehow indicates that there are, at least, two local maxima of the NF in the parameter search space. The NF and SCS spectra reveal two different configurations yielding NF maxima at lambda=510 nm: (i) a narrow higher-order, 3lambda/2 resonance for the dimer nanoantenna with thinner arms; (ii) a broad resonance with mixed character for the dimer nanoantenna with thicker arms. Enhancement factors |E|² ∼ 35, 14 are obtained, respectively, for the optimized dimer nanoantennas. Incidentally, such 3λ/2 resonance is much sharper than the first-order, dipole-like resonance [19], exhibiting indeed an asymmetric, Fano-like line shape [48].

 

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).

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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.