Evolutionary Multi-Objective Optimization for Multi-Resonant Photonic Nanostructures Peter R. Wiecha1,∗ , Arnaud Arbouet1 , Christian Girard1 , Aurélie Lecestre2 , Guilhem Larrieu2 and Vincent Paillard1 Abstract— The tailoring of optical properties of photonic nanostructures is usually based on a reference design. The target optical behavior is obtained by variations of an initial geometry. This approach however can be of limited versatility, in particular if complex optical properties are desired. In order to design double-resonant photonic nano-particles, we attack the problem in the inverse way: We mathematically define an optical response and optimize multiple of such objective functions concurrently, using an evolutionary multi-objective optimization algorithm coupled to full-field electro-dynamical simulations. We demonstrate that this approach is extremely versatile, that it allows the consideration of technological limitations and that it yields a correct prediction of the optical response of nano-structures fabricated by state-of-theart electron beam lithography. We demonstrate the technique on multi-resonant photonic nanoparticles made from silicon, which belong to the emerging class of high refractive index dielectric nanostructures with applications such as field-enhanced spectroscopies.
I. I NTRODUCTION Plasmonic nanostructures, usually made from noble metals like gold, provide original optical properties thanks to the occurrence of size- and shape-dependent localized plasmon resonances [1]. Such nanoparticles share many characteristics with radio-frequency antennas, yet operate at optical wavelengths. This is why they are often called “optical antennas”. Functionalities like directional scattering [2], polarization conversion [3], localized heat generation [4] or nonlinear effects like second harmonic generation [5] are accessible. However, due to the large imaginary part of the dielectric permittivity, metals usually suffer from significant losses at optical frequencies. High-index dielectric nanostructures have thus attracted increasing interest as low-loss alternatives to plasmonic particles, since the imaginary part of their permittivity is almost zero [6]. Like in plasmonics, optical resonances can be tuned over and beyond the visible spectral range, which allows to provoke similar phenomena from nanoparticles of highindex dielectrics like silicon [7]. A noteworthy advantage of dielectrics apart from very low losses is the possibility to obtain purely magnetic resonances with strong enhancements of the magnetic near-field [8]. Compatibility with silicon technology finally is a further plus compared to plasmonics. The tailoring of optical properties via the shape of a particle is one of the great prospects in nano-photonics. Usually, a certain reference layout is chosen initially – often by intuitive 1 CEMES-CNRS,
Université de Toulouse, CNRS, UPS, Toulouse, France CNRS, INP, Toulouse, France
2 LAAS-CNRS, Université de Toulouse, ∗
[email protected]
considerations – which is then varied in order to maximize a target optical characteristic. However, this approach is limited by the initial choice of geometry. Furthermore, with finite means only small parameter spaces can be explored systematically. We therefore tackle the problem from the rear by defining a target optical property as function of several parameters describing a nanoparticle geometry. Hence, we transform the search of nanostructures into an optimization problem. II. E VOLUTIONARY O PTIMIZATION Unfortunately, such optimization problems lead to nonanalytic functions of huge parameter-spaces, impossible to be solved by classical maximization algorithms like variants of Newton’s method. Evolutionary optimization algorithms, mimicking nature’s selection process of the survival of the fittest, are a possible approach to complex problems. Such techniques have been employed for example in the design of the cross sectional shape of infinitely long wires to tune their far-field resonance [9] or in the design of plasmonic planar structures [10] and 3-dimensional nanoparticles [11], featuring strong near-field enhancements. Evolutionary algorithms are based on a population of individuals, each of them corresponding to one parameterset describing in our case the geometry of a nano-structure. In a subsequent step of evaluation and selection, the fittest candidates are chosen for reproduction. Then the parameters are mixed (crossover) and randomly “mutated”, comparable to DNA in nature, eventually leading to a new generation of individuals. This cycle of reproduction is repeated several times, which iteratively improves the best candidate to the problem. III. M ULTI -O BJECTIVE O PTIMIZATION Targeting simultaneously several objective functions complicates the design of photonic nanostructures. Evolutionary multi-objective optimization (EMO) is therefore a very promising approach to problems in nano-optics. In a multiobjective optimization, a single optimum cannot be defined. Instead, a whole set of “non-dominated” solutions exists. Each of these solutions (forming the so-called “Pareto-front”) is optimum in the sense that an improvement in one objective necessarily results in a worsening of at least one other target value. IV. D ESIGN OF M ULTI -R ESONANT NANO -S CATTERERS We couple an EMO algorithm to a full-field electrodynamical (ED) solver using the Green dyadic method
random Si scaerers
evolutionary optimization
polarization dependent scaering wavelength
selection 5µm evaluation
reproduction
500nm Fig. 1. Evolutionary multi-objective optimization on silicon nano-scatterers. Randomly initialized populations of particle geometries are evolved numerically to design multi-resonant nano-structures (top-view of selected structures shown in the small boxes). The outcome of the optimization is then used as a template for nano-fabrication by electron-beam lithography on SOI substrate (lithographic mask, SEM and darkfield microscopy images on the right).
(GDM). In this frequency-domain technique, the volume of the nano-particle is discretized, solving an optical LippmannSchwinger equation [12]. A great advantage of the GDM is that only the nano-structure itself needs to be discretized which usually leads to a very quick convergence. As for EMO, the speed of the ED simulation is crucial, since a huge number of simulations needs to be performed. We apply the EMO-scheme on silicon nano-scatterers, searching for possible alternatives to color-rendering plasmonic particles i.e. for diffraction limited printing [1]. As optimization target we use the scattering efficiency, defined as the ratio of scattering cross section with respect to the geometrical cross section (the “footprint”) of the structure. We then define two objectives: An optical resonance is desired at a first target wavelength for light polarized along the X-direction (λx ). Simultaneously, for a polarization along Y , a second resonance wavelength (λy ) is target of the optimization. Randomly initialized populations of particle geometries are evolved by the evolutionary algorithm until finally the solution with most similar scattering efficiencies for λx and λy is selected from the Pareto-front. These double-resonant nano-structures subsequently serve as templates for a photo-lithographic mask, used to fabricate samples by state-of-the-art electron-beam lithography and reactive ion etching on commercial silicon-on-insulator substrates (SOI, Si-layer thickness of ≈ 100 nm). Fig. 1 shows a scheme of the optimization process, SEM images of some optimized nano-particles as well as polarization filtered darkfield microscopy images of a green/blue color-switching double-arrow, composed of EMO designed particles. We could verify experimentally the predictions of the evolutionary optimizations for a vast number of different optimization targets. In conclusion, EMO has a tremendous potential for manifold applications in nano-photonics. Promising other optimization targets include – among others – light-harvesting structures for photovoltaics or thermo-plasmonics and structures for enhanced non-linear effects like harmonics generation or four-wave mixing.
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