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Abstract
Plasmonic groove structures, which are widely known for their absorbent properties of light, are numerically investigated and optimized. Genetic algorithms have been successfully used to aid in the design of two-dimensional high efficiency wide-angle plasmonic groove absorbers for visible wavelengths. We demonstrate that the genetic algorithm is a powerful and flexible evolutionary optimization tool, able to handle high challenging design tasks by optimizing several complex problems currently of high interest to the optics and photonics community. The novel proposed periodic groove structure exhibits absorption above 90% for ultra-broadband wavelengths ranging from 300 to 700 nm. The resonant modes induce localized zero wavevector plasmon polaritons in the metallic material, which favors absorption and may also enhance non-linear optical processes.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The geometric parameters that define the spatial distribution of materials in the metamaterials devices greatly influence the electromagnetic behavior of such structures. Finding the ideal combination of parameters that promotes the desired electromagnetic response becomes an arduous task as such geometric parameters are quite numerous and can assume infinite values. Obtaining the response of a given structure by numerical simulation can take from tens of seconds to a few minutes. However, the number of possible structures is enormous, consequently, the search for the ideal combination to determine the optimal geometry and material configuration is a strenuous task, which requires immense computational time and yielding little fruitfulness if performed manually by a designer. For the task of searching for an ideal structure, it is therefore imperative to use an automated search tool. The genetic algorithms method is widely used and finds several applications in nanooptics [1–4]. In this work, we use genetic algorithms to aid in the design of plasmonic absorbers that find potential applications in refractive index sensing [5–7], solar cells and photodetectors [8–11]
The search methods based on evolutionary algorithms are classically used in problems where the search space is multidimensional, like the problem here considered, a multiparameter search for a structure with an ideal electromagnetic response. Genetic algorithms (GA) are a class of iterative search method inspired by sexual reproduction [12–14]. The GA operates from a first random generation of solutions, which are tested in the objective function. In this work we propose a broadband absorber for applications in the visible electromagnetic spectrum with polarization independent and wide angle response. In order to obtain the maximum absorption we use GA to optimize all geometric parameters of the structure. From this initial population, subsequent generations are created from combinations based on evolutionary techniques common in biology (selection, crossover, mutation, and elitism). The selection consists in choosing the individuals of one generation that will serve as the basis for the subsequent generation. Selection must ensure the diversity of the population as well as increase the chances of evolution through reproduction. The crossover promotes the blending between the characteristics of a generation, the parents, to create new individuals, the offspring. These offspring have parents' genes randomly mixed. An algorithm that implements only selection and crossover quickly converges to a single solution of the entire population. In such case, the diversity between individuals disappears gradually and after some iterations, the population becomes completely homogeneous as the search converges to a local minimum without a proper solution. This problem is solved by using mutation, which consists of random changes in random genes of individuals, thus ensuring greater population variety and the scanning a larger area of the solution space, while avoiding premature convergence of the population. Elitism ensures the maintenance of the best individuals within the population by escalating a give generation with their parents.
1.1 Structures based on metasurfaces optimized by GA
The iterative search based on genetic algorithm was implemented for the problem of designing a two-dimensional high efficiency wide-angle plasmonic groove absorbers for visible wavelengths. The methodology for the automated search of new structures involves the interaction between two computational packages, COMSOL and MATLAB. The implementation of the GA is performed in MATLAB while the COMSOL is used to model and enable the evaluation of the absorption of each structure.
The structure is modeled in COMSOL without the definition of values for the geometric parameters. The materials are also not defined in the generic template. The boundary conditions, field excitation and the main parameters of the mesh are defined in this template. MATLAB edits the COMSOL model file, adds the due missing parameters and COMSOL is then executed in batch mode to calculate the absorption coefficients for the desired frequency range. These results are imported back to MATLAB to feedback the algorithm.
The genetic algorithm’s loop begins with the generation of individuals with random genes, which are the geometric parameters and the materials. The absorbance profiles of each individual are loaded to MATLAB and evaluated in a fitness function, which seeks to maximize absorption. The stopping criterion is then evaluated, which consists of a maximum number of generations or the stabilization of the evolution curve.
1.2 GA Methodology
The selection method used is the fitness proportional selection, also known as the roulette method. The method increases the chances of reproduction of an individual in proportion to their adaptation to the environment. The fitter an individual is, the greater the chances of it being selected to be used in reproduction. The block diagram shown in Fig. 1
describes the genetic algorithm’s operational sequenceThe diversity of the individuals is guaranteed by the mutation stage, which consists of random increments occurring at randomly assigned values of genes given by a uniform probability distribution. The chance of mutation, or likelihood of the mutation occurring in a given gene, is an adjustable parameter that determines the degree of diversity of a generation. An important parameter of this stage is amplitude of the mutation, represented by a percentage of the available range of the gene in question, respecting the constraints of the problem. The amplitude of the mutation is a percentage factor that determines the impact of this stage on the gene, if it actually happens. The amplitude of the mutation increases every time the search begins to converge to a plateau in order to increase the diversity and the search area of the algorithm.
The new automatically generated structures obey the constraints of the problem and do not repeat themselves over the generations. Repeating the simulation of a particular individual is a waste of computational time and should be avoided.
In order to avoid wasting computational time with the accomplishment of diverse simulations of structures very similar to each other, the parameter similarity (S) was defined.
Newly created specimens are compared to the previous simulated structures through the similarity parameter. If the result of the similarity of this new structure to any other already evaluated ones is larger than a chosen Similarity Limit (SL), the structure is discarded and the creation process will restart. The dices are rolled once again, a new structure is created and the process of evaluating the similarity is repeated until the resulting structure is sufficiently different from the previous ones.1.3 Modeling of the structure
The proposed structure, shown in Fig. 2
, which is based on the infrared broadband absorber [15] was modeled using seven genes (w1, h1, w2, h2, P, dielectric and metal) that determine the phenotypic expression of slit widths and heights and the materials. Because of the periodicity along the x direction, only one unitary cell with P width needs to be analyzed by imposing periodic boundary conditions and since there is no variation of the structure along the z direction, the problem can be further reduced to a bidimensional one, where the cross section (xy plane) is considered. The range of possible values of the geometrical parameters in the genes are shown in Table 1
Table 1. - Defined Limits for the Values of the Genes
The objective function used is a form of the root-mean-square deviation, which measures the aggregated differences between values of a desired electromagnetic behavior, i.e. a given absorbance spectrum, and the values actually obtained by simulation at each wavelength. This approach has also been used in optical coating design [16].The objective function used to determine the quality of each structure is the average of the individual contributions (at each wavelength) of the Euclidean distances between the ideal behavior and the simulated one, as shown in:
where is the vector containing the wavelength absorption curve, and are the optimal response and response obtained for a given index wavelength respectively and n is the quantity of simulated wavelengths. The function tends to unity when the response curve obtained is equal to the ideal curve and to zero when the response curve obtained is the complement the ideal curve. The genetic algorithm was programmed to maximize the objective function.1.4 Validation of simulation and search methodology for a multiband absorption structure
In order to validate the proposed methodology, a known structure was used [15]. The structure allows the selective absorption of two distinct bands in the infrared band, with peaks at 1530nm and 2940nm. The structure was simulated in this work using FEM and the obtained results were compatible. The absorption spectra of the structure proposed in the literature and same structure simulated in this work are shown in Fig. 3
Fig. 3 Infrared absorbance spectra with peaks at 1530nm and 2940nm of the structure proposed in the literature [15], in dashed blue, and from the structure simulated in this work in continuous red. The metal used was silver and the parameters were P = 1100nm, w1 = 200nm, h1 = 300nm, w2 = 50nm and h2 = 350.
The GA was configured to perform the automatic search for the already known structure, using the absorbance profile of the previously simulated structure as a target. The convergence has been observed for fitness F = 92.63% after 73 generations, although the search was configured to run 150 generations. The values of the geometric parameters obtained do not correspond to those presented in the literature, so the optimized structure is different. The spatial distribution of the magnetic field and electric field, the fitness evolution, spatial distribution of the materials used and the absorption spectrum can be visualized in Fig. 4
Fig. 4 (a) Absorbance spectrum of (c) best structure with peaks at 1530nm and 2940nm, obtained by genetic algorithm search (b) with fitness evolution for 150 generations. The obtained absorption spectra is displayed in blue and the target curve is displayed in red. The distribution profiles (d) of the magnetic field and (e) of the electric field at wavelength of 1580nm indicates the presence of resonance in the structure’s cavities.
The high absorption is possible because conventional surface plasmons have a wave vector larger than that of light in a vacuum, which makes it necessary for their excitation to use a coupling medium that increases the momentum to enable the transfer of energy from the photon to the SPP. This medium is commonly a prism, or a region with periodic grids. The state of plasmon polaritons, however, may exist at the interface between a metal and a dielectric of a periodically arranged structure, such as a Bragg lattice or a photonic crystal that uses metal in its composition. This state of plasmon polaritons may have a wave vector with a size close to zero, as a consequence, it is possible to optically excite this state directly from the air [17,18].
2. Visible range absorbers
The structure proposed in the literature presented in the previous section operates in the infrared band, however, the methodology we used allows searching for absorbance profiles in other frequency bands. In order to test the evolution of the genetic algorithm in other bands, additional results were obtained for two different absorbance profiles in the visible range: maximum absorption over the visible band and a “low-pass” filter.
The structures were simulated for both TE polarization and TM polarization. The stop criterion adopted for the searches was the number of generation equal to 150. Multiple searches performed indicated that with this number of generations there is the convergence of the search to a plateau, so that the execution of the algorithm does not lead to better results.
2.1 Structure with absorption throughout the visible range
The first structure had targeted the maximization of the absorbance over the visible band. Thin visible range absorbers previously studied have demonstrated simulated absorption of 90% using silver and silica [19], while the best structure we found was able to absorb above 93% using nickel. The spatial distribution of the TE and TM magnetic fields as well as the absorbance spectrum and GA evolution graph are shown in Fig. 5
Fig. 5 Absorbance spectrum of (c) best structure for maximum absorbance over the visible band obtained by genetic algorithm search (b) with fitness evolution evaluated for 150 generations. The simulated absorption coefficient is displayed in blue and the objective absorption is displayed in red. The magnetic field distribution profiles for (d) TM mode and for (e) TE mode indicate the presence of resonance in the structure’s cavities.
The best structure obtained through the optimization process for ultra-broadband absorption has a grooved profile and demonstrated absorption improvement (AI), defined by , from 50% up to 170% when compared to a slab of the same material (Fig. 6
Fig. 6 Absorbance comparison between grooved (orange triangles) and slabbed structures (blue dots). The increased absorbance factor is shown in circled blue line. The optimized grooved structure has demonstrated absorbance improvement from 50% to 170% when compared to a slab of the same material.
The proposed structure demonstrated excellent wide-angle response, as seen in Fig. 7
Fig. 7 Absorbance spectrum as a function of the incident angles on the grooved optimized structure. The material slab for normal incidence is also shown as reference.
2.2 Structure with absorption type of “low-pass”
By targeting the genetic algorithm to a specific absorbance profile, in this case a low-pass filter for the range of 500 to 800nm with cutoff wavelength set to 600nm, we were able to obtain fitness of approximately 83%. Similar absorbance profile is also achievable by using a metamaterial composed by arrayed microscopic manipulated gold nanoparticles and a layer of silica followed by a layer of gold [20]. The structure found should be easier to fabricate, although the 30nm walls (w1 – P) can be quite challenging. The results for this search are shown in Fig. 8
Fig. 8 (a) Absorbance spectrum of (c) best structure for low pass filtering with 600nm as cutoff wavelength obtained by genetic algorithm search (b) with fitness evaluated for 150 generations. The simulated absorbance is displayed in blue and the target absorption is displayed in red. The magnetic field distribution profiles at the peak absorbance of 98% for (d) TM mode and for (e) TE mode indicate the presence of resonance in the structure’s cavities.
3. Conclusions
We proposed an automated approach by using GA combined with MATLAB and COMSOL for the optimization of two-dimensional absorbers based on nanogratings. We demonstrate that genetic algorithm is a powerful and flexible evolutionary optimizer tool, able of handling high complexity absorption properties design tasks by solving several complex problems of high current interest to the optics and photonics community. The optimization algorithm presented fast convergence and successfully arrives to structures with wide-angle absorbance of up to 99% and incident angles of up to 60°. High absorption is obtained with the proposed absorber because the state of plasmon polaritons has a wave vector with a size close to zero allowing optical excitation directly from the air, causing standing surface waves, which maximizes absorption in the metallic-dielectric interface. The designed structures have great potential in applications requiring insensitive angle for maximum absorbance, such as energy harvesting and sensing.
Funding
Conselho Nacional de Desenvolvimento Científico e Tecnológico (305762/2015-0) and Fundação de Amparo à Pesquisa do Estado da Bahia (APP0079/2016)
Acknowledgments
The authors would like to acknowledge CAPES, CNPq, UFBA, FAPESB, and IFBA.
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