Genetic-algorithm-aided ultra-broadband perfect absorbers using plasmonic metamaterials

Complete absorption of electromagnetic waves is paramount in today's applications, ranging from photovoltaics to cross-talk prevention into sensitive devices. In this context, we use a genetic algorithm (GA) strategy to optimize absorption properties of periodic arrays of truncated square-based pyramids made of alternating stacks of metal/dielectric layers. We target ultra-broadband quasi-perfect absorption of normally incident electromagnetic radiations in the visible and near-infrared ranges (wavelength comprised between 420 and 1600 nm). We compare the results one can obtain by considering one, two or three stacks of either Ni, Ti, Al, Cr, Ag, Cu, Au or W for the metal, and poly(methyl methacrylate) (PMMA) for the dielectric. More than 10^17 configurations of geometrical parameters are explored and reduced to a few optimal ones. This extensive study shows that Ni/PMMA, Ti/PMMA, Cr/PMMA and W/PMMA provide high-quality solutions with an integrated absorptance higher than 99% over the considered wavelength range, when considering realistic implementation of these ultra-broadband perfect electromagnetic absorbers. Robustness of optimal solutions with respect to geometrical parameters is investigated and local absorption maps are provided. Moreover, we confirm that these optimal solutions maintain quasi-perfect broadband absorption properties over a broad angular range when changing the inclination of the incident radiation. The study also reveals that noble metals (Au, Ag, Cu) do not provide the highest performance for the present application.


Introduction
Since its theoretical introduction in 1860 by G. Kirchhoff, the black body concept played a seminal role in the history of quantum mechanics and modern physics. 1 It originally referred to an idealized physical body of infinitely small thickness, that completely absorbs all incident rays, and neither reflects nor transmits any. 2 The modern acceptance of the term does not include the infinitely small thickness anymore but still preserves the requirement to absorb all incident electromagnetic radiation regardless of the wavelength, the polarization or the angle of incidence of the incoming radiation. First experimental realizations of black bodies at the end of the 19th century/ dawn of the 20th century consisted of metallic boxes with its interior walls blackened by mixed chromium, nickel and cobalt oxides. [3][4][5] The quest for a Perfect Electromagnetic Absorber (PEA), i.e. the materialization of the idealized black body, continued over the years for example due to the recent need of efficient solar energy harnessing or camouflage solutions, the development of photothermal detectors or in order to prevent crosstalk in nanoscale opto-electronics and quantum technologies. 6,7 Among efficient solutions, carbon black and carbon nanotubes materials provide ultra-broadband PEA with 98-99 % absorption from UV to far infrared 8 while nickel-phosphorus alloy reach 96 % on the 5 − 9µm range. 9 At the dawn of the present century, metamaterials and plasmonic materials provided new opportunities in order to mold the flow of light at the nanoscale and drastically enhance light-matter interactions. [10][11][12][13] Negative refraction, cloaking, superlensing, near-zero refractive index, surface enhanced Raman scattering, high energy concentrations at metal-dielectric interfaces are some examples of current interest among those hot topics in photonics. [14][15][16] Nevertheless, losses due to metallic components are an important drawback limiting current applications. 17 This drawback was turned into an advantage by Landy et al. who designed the first metamaterial PEA by using metallic resonators operating at a single wavelength. 18 The metamaterial approach was successfully applied during the last decade in order to tame the black body radiation and provide efficient PEA over a broad range of frequencies. [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] Some proposed theoretical structures are nevertheless facing some technological difficulties due to the current limitations in resolution and complexity of the structures, especially in the visible range. The field of plasmonic metamaterials PEA is now mature enough to enter a novel phase of in silico modeling in order to develop more realistic PEA, [35][36][37] while preserving the ultra-broadband character of the PEA as well as its low angular dependency.
Here, we report on ultra-broadband metamaterial PEA using periodic stacks of square metal/dielectric layers arranged in a pyramidal way (see Fig. 1). We investigate among twenty-four possible PEA which are realistic from an experimental point of view and based on eight common metals arranged either in one, two or three stacks of metal/dielectric layers. The fact that the number of layers is limited to a few ones, while preserving the performances, is attractive in view of the tradeoff between fabrication complexity and broad absorption properties. It provides realistic perspectives for device fabrication with current technology. No less than 10 17 configurations are explored by varying different experimental parameters such as the lateral size of the metal/dielectric layers, the thickness of the dielectric layer or the lateral periodicity between different individual structures. The search for the optimal configuration, i.e. the PEA with the largest absoptance of electromagnetic (EM) radiations over the visible to mid-infrared (MIR) range, is realized using a Genetic Algorithm (GA) strategy. Optimal configurations are reported as well as their absorptance spectra.
Moreover, fields maps and angular dependency of the optimal PEA are discussed. The latter is primordial for several applications of black coatings such as cross-talk prevention in qubits. 6,7 or stray light reduction. 38,39 Design Energy conservation imposes that reflectance R, transmittance T and absorptance A are related through R(λ, θ) + T (λ, θ) + A(λ, θ) = 1 (1) with λ being the wavelength of the incident EM radiation and θ the angle of incidence.
Maximizing absorption requires suppressing both transmission and reflection at the same time. The PEA can be made opaque, i.e. no transmission, by adding a metallic layer over the substrate. In order to lower the reflection, the PEA needs to be impedance matched.
Surface corrugations in a tapered way provide this gradual transition between the refractive indices of both incidence and substrate media. This anti-reflection strategy is well-known both in Nature or in transparent coatings. 40,41 The previous approach leads to suppressed transmission (T = 0) and drastically reduced reflection (R → 0). To minimize reflection and fully absorb the incident radiation in order to obtain a PEA, the energy transported by the radiation has to be dissipated through the excitation of eigenmodes of the structure. This can be done for example by exciting localized surface plasmons (LSP) at the interfaces of metal/dielectric square resonators. [42][43][44][45] The wavelength associated with these plasmonic resonances is proportional to the lateral dimensions of the metallic layers, conferring the desired broadband character once metallic resonators of varied sizes are stacked. Following this approach, truncated square-based pyramids made of twenty stacks of Au/Ge layers lead to an integrated absorptance of 98% of normally incident radiations over a 0.2-5.8 µm wavelength range. 46,47 However, fabrication of such elaborated structures, with nanometer resolution and high number of layers is currently too demanding in terms of time and resources.
There is thus a need for simplified structures to reach ultra-broadband PEA that are tractable experimentally. Therefore, we consider here a simpler PEA consisting of one, two or three stacks of metal/dielectric layers (see Fig. 1). 35,36 We set the thickness of all metal layers within the pyramid stack to 15 nm, 46 which is smaller than the skin depth of the metals considered over the targetted radiation wavelength range, i.e. from 420 to 1600 nm here. It allows therefore a coupling between the surface plasmons at the two sides of each metallic layer. Moreover, we consider poly(methyl methacrylate) (PMMA) as dielectric. 35,36,48 PMMA can be easily deposited using spin-coating techniques and its thickness can be simply varied by changing PMMA/solvent relative concentration in solution, or spinning speed for example.
A vast range of metals are considered in the present study, namely Ni, Ti, Al, Cr, Ag, Cu, Au or W. [49][50][51][52] Those metals correspond to widely used materials in current nanophotonics applications, and are readily deposited to form few nm-thick films through physical deposition techniques. The thickness t i of each dielectric layer and the lateral dimension L i of each metal/dielectric stack are adjustable parameters in the present GA-aided approach. The lateral periodicity P of the system also remains an adjustable parameter. This relaxation of the PEA dimensions, as compared with previous studies 46 where thickness was the same for The role of these different parameters can be understood as follows. The lateral dimensions L i of the metallic resonators essentially control the plasmonic resonances of the system. The thicknesses t i of the dielectric layers control the vertical coupling between these resonators while the periodicity P finally controls the lateral coupling between the pyramids. 46 The detailed variation range of these parameters are provided in SI. Furthermore, the PEA stands on a flat 60 nm-thick gold layer that blocks the transmission of the considered incident radiations and reflects any EM radiation not absorbed in the pyramids. This gold layer stands on a 5-nm thick chromium (Cr) adhesion layer followed by a 1 µm-thick amorphous silicon (a-Si) layer. We consider finally a semi-infinite Si substrate (ε = 16). This choice of a silicon substrate has no impact on our results since no radiation will actually reach this region.

Optimization schemes
For a system made of N stacks of metal/dielectric layers, we consequently have a total of 2N +1 parameters to consider (i.e., L i , t i and P for i ∈ [1, N]). Different methods are used in optics to determine optimal parameter combinations. The systematic evaluation of a whole grid of possible parameter combinations is generally limited to two or three parameters.
Local optimization methods such as the (quasi-)Newton method or gradient descent are more efficient in terms of the required number of function evaluations. 53 They converge however generally to the first local optimum encountered. Global optimization methods are preferable in this respect since they provide a wider exploration of the parameter space, which can eventually lead to higher-quality solutions (ideally the best-possible solution).
Genetic Algorithms (GA), 54  The objective function to be maximized (also called fitness or figure of merit) will be the integrated absorptance for normally incident radiations with wavelengths comprised between 420 and 1600 nm. It is formally defined by η(%) = 100 ×´λ max λ min A(λ)dλ λmax−λ min , where A(λ) refers to the absorptance of normally incident radiations at the wavelength λ, λ min =420 nm and λ max =1600 nm. This is the quantity that the Genetic Algorithm seeks at maximizing by exploring the geometrical parameters of the structure considered. It relies on an homemade Rigorous Coupled Waves Analysis (RCWA) code that solves Maxwell's equations exactly for stratified periodic systems. 41,62 A plane wave (PW) expansion of the electric permittivity of the PEA is made, in which the number of Fourier components is the key parameter to insure numerical convergence.

Results and discussion
Once the GA started, one can follow generation after generation the fitness of the best individual in the population as well as the mean fitness in the population. The best fitness usually increases rapidly in the first generations as better solutions are rapidly detected.
Progress becomes typically slower after these first generations as the algorithm must either escape a local optimum to detect a radically different solution (exploration) or refine the best solution found so far to finalize the optimization (exploitation). A good optimization algorithm must actually find a sound balance between these two aspects. The mean fitness of the population follows the best fitness, to a degree that depends on the convergence of the population to the best individual, convergence measured by the genetic similarity s defined in SM.  plane waves). This extensive study reveals that Ni, W, Cr or Ti represent the best materials for this application. It is noticeable that integrated absorptance values above 86.1% are achieved for the 3 × 4 = 12 optimized structures that correspond to these four metals.
The best structures, which consist of N = 3 stacks of Ni, W, Cr or Ti/PMMA layers, have all an integrated absorptance above 99%. Those results are, to the best of our knowledge, the best PEA proposed in recent literature over such an extended wavelength range in the visible and NIR. The geometrical parameters and the figure of merit obtained for the four best metals selected (Ni, W, Cr and Ti) are given in Table 1. The results that correspond to the other four metals (Al, Cu, Au and Ag) are given in Table 2. They will be discarded for this PEA application, essentially because of a lack of robustness of these solutions within the parameter space, i.e. sensitivity to experimental deviations, in addition to lower integrated absorptance values. It should be noted that three out of those four less convincing are noble metals (Cu, Au, Ag) and are usually considered as low-loss metals for plasmonic applications. 17,63 Considering that the choice of noble metals (Au, Ag, Cu) impacts on fabrication costs, it is interesting to note that they do not provide the highest performance for the present application.
High-quality solutions must actually meet two requirements in order to be easily implemented in real devices: (i) to provide the highest possible integrated absorptance η, but also (ii) to be robust with respect to slight variations of the geometrical parameters (compatibility with fabrication tolerances). We ideally want to find a broad optimum rather than a sharp one. When running the GA to find optimal geometrical parameters, it is possible to interpolate the data collected by the algorithm and establish maps of the fitness in 2-D planes that cross the n-dimension point finally established by the GA. These maps reveal the robustness of the final solution (i.e., its stability with respect to slight variations of the geometrical parameters). An example is given in Fig. 5 Table 1: Geometrical parameters and figure of merit η for optimal structures made of N = 1, 2 or 3 stacks of Ni/PMMA, W/PMMA, Cr/PMMA or Ti/PMMA (from top to bottom). PW: number of plane waves used in the calculations of η.   Ti / PMMA nm, 10 nm and 91 nm respectively. The lateral dimension L 1 is the most sensitive parameter in this case. All solutions presented in Table 1 were checked for their robustness, based on the inspection of the fitness maps established by the GA. They appeared to correspond to broad optima suitable for practical applications. On the contrary, the solutions presented in Table 2 were discarded because they actually correspond to sharp optima.
A quality check of the reliability of the presented solutions is performed by increasing the plane wave number to 21 × 21 (see Table 1 Table 2: Geometrical parameters and figure of merit η for optimal structures made of N = 1, 2 or 3 stacks of Al/PMMA, Cu/PMMA, Au/PMMA or Ag/PMMA (from top to bottom). PW: number of plane waves used in the calculations of η.

Plasmonic absorber characteristics
In order to better describe the physics of those PEA, we will focus from now on only on the best structure identified in this work, i.e. the one made of N = 3 stacks of Ni/PMMA. It provides an integrated absorptance of 99.3%. As shown above, the solution is robust with respect to deviations of the geometrical parameters (broad optimum), as confirmed by 2-D maps of the fitness around the optimum and the comparison between the results obtained with 11×11 and 21×21 plane waves.
We can calculate the Poynting vector S = 1 2 E × H * , where E and H refer here to the complex-number representation of the electric and magnetic fields, in order to show the energy flow through the structure. Moreover, we can determine the local absorption inside the PEA. Based on the method developed by Brenner, 64 the absorbed power P a in a volume V can be estimated by with ε(r, ω) the local complex electric permittivity. By normalizing the absorbed power P a to the incident power P i , one obtains the local absorptance A loc (r, ω) = P a /P i .

Conclusion
Numerical investigations of truncated square-based pyramids made of a few number (one to three) of alternating stacks of metal/dielectric layers are carried out. We focused on realistic configurations consisting of maximum three metallic layers stacked above each other.
The GA strategy allowed to explore 10 17 geometrical configurations and selected the optimal ones. Ni/PMMA, W/PMMA, Cr/PMMA and Ti/PMMA are determined as the best configurations for realizing PEA over the visible and NIR range, with integrated absorption higher than 99 % once three layers are considered. Those PEA are robust against geometrical parameter deviations that might occur during experimental realization. Moreover, these optimal solutions maintain quasi-perfect broadband absorption properties over a broad angular range when changing the incidence angle of EM radiation. This study offers guidelines for a realistic design of PEA, that can readily be fabricated using currently available micro/nanofabrication techniques, using modest resources.

Acknowledgement
The authors would like to thank N. Reckinger and L. Henrard for stimulating discussions.

Range of variation of the parameters for the Genetic Algorithm
We consider that the lateral periodicity P of the system can take values between 50 and 500 nm (by steps of 1 nm). The lateral dimension L i of each stack of metal/dielectric layers can take values between 50 and 500 nm (by steps of 1 nm). The thickness t i of each dielectric can take values between 50 and 250 nm (by steps of 1 nm). The subscripts i=1, 2 and 3 refer respectively to the stack at the apex, in the middle or at the bottom of each nanopyramid (Fig. 1). In order to obtain pyramidal structures, we require that the final solution satisfies L 1 < L 2 < L 3 ≤ P − 40 nm, where 40 nm represents the minimal imposed safe distance between adjacent pyramids for insuring realistic fabrication. When optimizing structures made of three stacks of metal/dielectric layers, there are actually seven parameters to determine (P , L 1 , t 1 , L 2 , t 2 , L 3 and t 3 ), with a total of 13,936,405,106,594,025 possible parameter combinations to consider if the relaxed constraint L 1 < L 2 < L 3 ≤ P is actually enforced during the optimization. for details). In the current version of our GA, mutations can be "isotropic" (in this case, the mutation operator is applied n times on a given DNA). The probability p iso to apply isotropic mutations is set to 0.2 initially. This value is adapted according to the success of this operator.

Description of the Genetic Algorithm
In order to converge more rapidly to the final solution, we establish at each generation a quadratic approximation of the fitness in the close neighborhood of the best-so-far individual (this approximation is based on the data collected by the genetic algorithm). If the optimum of this approximation is within the specified boundaries, it replaces the last random individual scheduled for the next generation (see Appendix B of Ref. 35 for details).
The data collected by the algorithm is also used to establish 2-D maps of the fitness, by using dedicated interpolation techniques. This is useful for monitoring the progress of the algorithm and for assessing the quality of the final solution.
The fitness of all individuals scheduled for the next generation is finally computed in parallel. The new population is sorted from the best individual to the worst. If the best individual of the new generation is not as good as the best individual of the previous gen-1 In a binary one-point crossover, the first n cut bits of the DNA of the children come from one parent. The remaining n bits − n cut bits come from the other parent. The point n cut at which the parents' DNA is exchanged is chosen randomly in the interval [1, n bits − 1].
2 If x 1 and x 2 are the real variables represented by the two parents, the children obtained by a real crossover between these parents will represent a variable x = x 1 + (2 * rnd − 0.5) × ( x 2 − x 1 ), where rnd is a random number uniformly distributed in [0,1]. eration, the elite of that previous generation replaces an individual chosen at random in the new generation. We repeat these different steps from generation to generation until a termination criterion is met.
Quality check of the optimization results based on the plane wave number A final quality criterion is certainly the reliability of the presented results. In order to confirm the quality of our solutions, we increased the number of plane waves in the RCWA calculations to 21×21 (instead of 11×11 when running the GA). The results obtained are given in Tables 1 and 2. The comparison between 11×11 PW and 21×21 PW in Table 1 reveals that the solutions selected on the basis of high η values and high robustness are also stable with respect to this numerical test (only slight deviations between η 11×11PW and η 21×21PW ). On the contrary, the solutions in Table 2 that were discarded, essentially because of the high sensitivity of η with respect to the geometrical parameters, turn out to be significantly affected by this increase of the number of plane waves used in the RCWA calculations (large deviations between η 11×11PW and η 21×21PW ). It proves that the solutions given in Table 2 were rightly discarded (they fail this last reliability criterion). The fact that solutions that sit on sharp optima are also solutions that require a higher number of plane waves for an accurate calculation is actually consistent. This observation suggests a simple criterion for testing the robustness of solutions (stability with respect to deviations of their geometrical parameters): testing the stability with respect to the number of plane waves used for the calculation. This approach does not require the calculation of 2-D maps.
A single calculation based on an increased number of plane waves may be sufficient to get a clue !