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Particle-swarm optimization of broadband nanoplasmonic arrays Carlo Forestiere,1,2 Massimo Donelli,3 Gary F. Walsh,1 Edoardo Zeni,3 Giovanni Miano,2 and Luca Dal Negro1,4,* 1
Department of Electrical and Computer Engineering & Photonic Center, Boston University, 8 Saint Mary’s Street, Boston, Massachusetts 02215, USA 2 Department of Electrical Engineering, Università degli Studi di Napoli Federico II, via Claudio 21, Napoli, 80125, Italy 3 Department of Information and Communication Technology, University of Trento, via Sommarive 14, Trento, 38050, Italy 4 Division of Materials Science and Engineering, Boston University, 8 Saint Mary’s Street, Boston, Massachusetts 02215, USA *Corresponding author:
[email protected] Received September 17, 2009; revised December 1, 2009; accepted December 7, 2009; posted December 11, 2009 (Doc. ID 117386); published January 11, 2010 We used the particle swarm optimization algorithm, an evolutionary computational technique, to design metal nanoparticle arrays that produce broadband plasmonic field enhancement over the entire visible spectral range. The resulting structures turn out to be aperiodic and feature dense Fourier spectra with many closely packed particle clusters. We conclude that broadband field-enhancement effects in nanoplasmonics can be achieved by engineering aperiodic arrays with a large number of spatial frequencies that provide the necessary interplay between long-range diffractive interactions at multiple length scales and near-field quasi-static coupling within small nanoparticle clusters. © 2010 Optical Society of America OCIS codes: 240.6680, 240.6695, 050.6624, 290.4020.
Recently, there has been a growing interest in plasmonic structures featuring high field enhancement across a wide frequency range for applications such as surface-enhanced Raman scattering (SERS) [1], radiative rate enhancement [2], solar cells [3], and label-free optical biosensors [4]. Until now, this effort has largely focused on the study of periodic and deterministic aperiodic structures [4–11]. Here, we propose an alternative design strategy based on the application of the particle-swarming optimization (PSO) algorithm. This method, developed by Kennedy and Eberhart [12], is an evolutionary optimization tool inspired by the social behavior of groups of insects and animals such as swarms of bees, flocks of birds, and shoals of fish. In a PSO system, a swarm of W trial solutions, called particles, flies around the multidimensional solution space trying to improve the particles’ position, based on knowledge of its own previous best performance and that of the entire swarm. PSO is widely used to optimize rf antenna array properties [13] and has been applied in photonics to design diffraction-grating filters [14]. The growing interest in PSO is due to its ability to avoid local maxima, which is the main disadvantage of the deterministic method [13], and to overcome stagnation problems, which occurs in genetic algorithms when there is a lack of genetic diversity [15]. In addition, PSO can be more easily implemented than other evolutionary algorithms, since it requires the tuning of few parameters [13]. In this Letter, a binary version of the PSO algorithm, BPSO [16], has been used to design an array of plasmonic nanospheres in order to achieve broadband field enhancement spanning the 400– 900 nm spectral range. Similar optimization techniques have 0146-9592/10/020133-3/$15.00
been widely used to design broadband antennas for rf applications (see, e.g., [17] and references therein). We first describe the BPSO, and then we use it with a recently developed coupled dipole approximation (CDA) code [18] to thin a square array of silver nanospheres. We find that the optimized structure is aperiodic and features dense Fourier transforms and high nanoparticle densities. In BPSO, a swarm consists of a set w = 1 , … . , W of N ⫻ N binary matrices (particles), whose bits 共m = 1 . . . N2兲 control the presence or the absence of a nanosphere in the array. A position and a velocity vector 共w , vw兲 are associated to each particle of the swarm. In particular, vw models the capacity of the particle to fly, at iteration k, from a given position in the solution space kw to another k+1w. The positions and velocities of the W particles in the swarm are initialized by a random number generator. How well a trial solution solves the problem is determined by the evaluation of a suitable cost parameter, called the fitness function (FF). At each iteration, the position and velocity vectors of each particle are updated by the action of a force attracting them toward both their own previous best positions pb and that of the entire swarm gb, i.e., w w w pb w gb vk,m = vk−1,m + C1共k,m − k,m 兲1 + C2共k,m − k,m 兲2 ,
共1兲 where  is the inertia factor, C1 and C2 are constants called cognition and social acceleration, and 1 and 2 are two randomly generated positive numbers between 0 and 1. The particle’s position is then updated considering the following relation: © 2010 Optical Society of America
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OPTICS LETTERS / Vol. 35, No. 2 / January 15, 2010
w k,m =
冦
w 1 if k,m ⬍
0
冧
1
w 1 + exp共− vk,m 兲 ,
otherwise
共2兲
and the velocity clipping boundary condition [13],
冦
Vmax
w, if vk,m ⬎ Vmax
冧
w w = − Vmax if vk,m ⬍ − Vmax . vk,m w vk,m
otherwise
共3兲
If the maximum number of iterations is reached or if the fitness of the best particle is within a predefined tolerance, the optimization process is stopped and the solution has been reached. Using the BPSO, we thinned a 55⫻ 55 element square array of 100 nm diameter silver nanospheres with 25 nm edge-to-edge separation, modeled with realistic dispersion data [19]. The number of particles in the swarm 共W兲 was 5, the inertia weight was held constant at 0.4, and the cognitive and social rates were both 2.0. It is worth noting that the considered population, composed of only five agents completely randomly generated, is very low considering the number of unknowns. The size of population was forced so low to keep the computational burden to a sustainable level. We aimed to find an array geometry that achieves high field enhancement spanning the entire visible spectrum when the structure is illuminated by a circularly polarized plane wave at normal incidence with unitary field intensity. Therefore a multiobjective fitness function was defined as a uniformly weighted sum of the maximum field enhancement in the array plane, as calculated by the CDA code, at 50 equispaced frequencies between 400 and 900 nm, namely, 50
FF =
兺 FEMAX共M兲.
共4兲
M=1
We have found that after only ten PSO iterations the FF has nearly reached a saturation value and that the best FF value is ultimately achieved after 60 steps. The optimization process was eventually stopped after 100 iterations.
It was found that, in the optimized silver nanoparticle array shown in Fig. 1(a), 1506 of the 3025 allowed positions (49%) are occupied and are arranged in a seemingly random distribution, as confirmed by the magnitude of its Fourier transform, reported in Fig. 1(b). As was recently shown [11], the degree of structural complexity of plasmonic arrays can be quantified by the spectral flatness (SF), which is defined as the ratio of the geometric mean to the arithmetic mean of its power spectrum [11]. An SF value of 1 indicates a completely flat spectrum, whereas a value of 0 indicates a perfectly band-limited signal. The SF of the optimized array was found to be 0.81, whereas a completely filled array would be 0.0004. It is worth noting that similar SF values can also be achieved by deterministic aperiodic arrays with absolutely continuous Fourier spectra [6–11]. Figure 2(a) shows the maximum fieldenhancement spectra of the optimized array (triangles) compared with the periodic array, in which all allowed positions are filled by a particle, and with the single particle. The maximum field enhancement across the entire interval 400– 900 nm is 35.9, occurring at 500 nm, while the spectral bandwidth, defined as the width of the frequency range, in nanometers, for which the field enhancement is above 37% 共1 / e兲 of its maximum value, is 180 nm. Analogous to the behavior of finitesize periodic arrays of small clusters, the large field enhancements observed here are due to the interplay of long-range photonic interactions with strong nearfield plasmonic coupling. Owing to the large number of spatial frequencies in the optimized aperiodic array, local nanoparticles clusters are inhomogeneously distributed and photonic gratings modes can be excited at many different wavelengths, giving rise to enhancement effects over a broad spectral range. Additionally, since plasmon waves couple strongly only in the near-field regime at very short distances, closely packed clusters are also needed in order to achieve high field enhancement [11]. Our optimized silver array has a particle density of 32 m−2, which can be compared to the aperiodic arrays reported in [11]. In Fig. 2(b) we also show the CDFE [18]. This function describes the fraction of the total area of the array covered by plasmonic enhanced fields with values
Fig. 1. Geometry of (a) the optimized silver nanoparticle array and (b) its Fourier transform magnitude (log scale). In (b) a = 125 nm is the minimum center to center distance, and the Fourier transform magnitude has been normalized to its maximum value. The central peak and the cross in the middle of the Fourier space result from the square symmetry of the finite-size array.
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Fig. 2. (Color online) (a) Maximum field-enhancement spectrum of the optimized array (triangles) of silver nanoparticles, illuminated, at normal incidence, by a circularly polarized plane wave of unitary intensity, compared with the performances of periodic array (circles), in which all allowed positions are filled by a particle, and with the single particle (squares). (b) Cumulative distribution function of field enhancement (CDFE) (logarithmic scale) versus wavelength (x axis) and field enhancement (y axis).
greater then a fixed threshold specified in the vertical axis. Although the CDFE does not give any information about the size of hot spots, it gives a quantitative measure of how the energy is spatially distributed, at each wavelength, over the surface of the array. The high field-enhancement localization of disordered arrays [20,21] can be also qualitatively understood in light of the uncertainly principle for optical waves [22]. According to this general principle, the ultimate limit to the localization of any wave in a given direction ⌬x is uniquely dictated by the spread (uncertainty) in the corresponding wavevector components ⌬kx, according to the well-known relation ⌬x 艌 ␣ / ⌬kx, where ␣ is a constant. It follows that if the Fourier spectrum of a plasmonic array is almost flat (large value of spectral flatness), as in the case of the BPSO optimized arrays, a large number of spatial frequencies (wavenumbers) are available to match in-plane scattering processes. This gives rise to efficient multiple scattering in the array plane, resulting in higher field enhancement and stronger wave localization. In summary, we have applied a BPSO algorithm to optimize plasmonic nanoparticle arrays for broadband field enhancement in the visible spectral range. We found that a large number of spatial frequencies is needed in order to provide the necessary interplay between long-range diffractive interactions at multiple length scales and near-field quasi-static coupling within small nanoparticle clusters. This approach may be utilized to design plasmonic structures for applications such as SERS [8], broadband radiative rate enhancement, solar cells [3], and label-free optical biosensors [4]. This work was partially supported by the Defense Advanced Research Projects Agency (DARPA)– Defense Sciences Office Chemical Communication project (ARM168), the U.S. Army through the Natick Soldier Center (W911NF-07-D-001), and the SMART Scholarship Program and the Air Force program Deterministic Aperiodic Structures for On-Chip Nanophotonic and Nanoplasmonic Device Applications under the award FA-9550-10-1-0019.
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