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We present a general framework for inverse design of nanopatterned surfaces that maximize spatially averaged surface-enhanced Raman (SERS) spectra from molecules distributed randomly throughout a material or fluid, building upon a recently proposed trace formulation for optimizing incoherent emission. This leads to radically different designs than optimizing SERS emission at a single known location, as we illustrate using several 2D design problems addressing effects of hot-spot density, angular selectivity, and nonlinear damage. We obtain optimized structures that perform about 4 × better than coating with optimized spheres or bowtie structures and about 20 × better when the nonlinear damage effects are included.

Spatially incoherent light sources, such as spontaneously emitting atoms, naively require Maxwell's equations to be solved many times to obtain the total emission, which becomes computationally intractable in conjunction with large-scale optimization (inverse design). We present a trace formulation of incoherent emission that can be efficiently combined with inverse design, even for topology optimization over thousands of design degrees of freedom. Our formulation includes previous reciprocity-based approaches, limited to a few output channels (e.g. normal emission), as special cases, but generalizes to a continuum of emission directions by exploiting the low-rank structure of emission problems. We present several examples of incoherent-emission topology optimization, including tailoring the geometry of fluorescent particles, a periodically emitting surface, and a structure emitting into a waveguide mode, as well as discussing future applications to problems such as Raman sensing and cathodoluminescence., Comment: 29pages, 3 figures

Optical metasurfaces have been heralded as the platform to integrate multiple functionalities in a compact form-factor, potentially replacing bulky components. A central stepping stone towards realizing this promise is the demonstration of multifunctionality under several constraints (e.g. at multiple incident wavelengths and/or angles) in a single device -- an achievement being hampered by design limitations inherent to single-layer planar geometries. Here, we propose a general framework for the inverse design of volumetric 3D metaoptics via topology optimization, showing that even few-wavelength thick devices can achieve high-efficiency multifunctionality. We embody our framework in multiple closely-spaced patterned layers of a low-index polymer. We experimentally demonstrate our approach with an inverse-designed 3d-printed light concentrator working at five different non-paraxial angles of incidence. Our framework paves the way towards realizing multifunctional ultra-compact 3D nanophotonic devices., 16 pages, 4 figures

Surface-enhanced Raman spectroscopy is a powerful and versatile sensing method with a detection limit down to the single molecule level. In this article, we demonstrate how topology optimization (TopOpt) can be used for designing surface enhanced Raman scattering (SERS) substrates adhering to realistic fabrication constraints. As an example, we experimentally demonstrated a SERS enhancement factor of 5*10e4 for the 604 cm-1 Raman line of rhodamine 6G using metal nanostructures with a critical dimension of 20 nm. We then show that, by relaxing the fabrication constraints, TopOpt may be used to design SERS substrates with orders of magnitude larger enhancement factor. The results validate topology optimization as an effective method for engineering nanostructures with optimal performance and fabrication tolerance., 12 pages

Topology optimization methods for inverse design of nano-photonic systems have recently become extremely popular and are presented in various forms and under various names. Approaches comprise gradient and non-gradient based algorithms combined with more or less systematic ways to improve convergence, discreteness of solutions and satisfaction of manufacturing constraints. We here provide a tutorial for the systematic and efficient design of nano-photonic structures by Topology Optimization (TopOpt). The implementation is based on the advanced and systematic approaches developed in TopOpt for structural optimization during the last three decades. The tutorial presents a step-by-step guide for deriving the continuous constrained optimization problem forming the foundation of the Topology Optimization method, using a cylindrical metalens design problem as an example. It demonstrates the effect and necessity of applying a number of auxiliary tools in the design process in order to ensure good numerical modelling practice and to achieve physically realisable designs. Application examples also include an optical demultiplexer., Comment: 8 figures, 19 pages

Upconversion of sunlight with energy below the band gap of a solar cell is a promising technique for enhancing the cell efficiency, simply by utilizing a larger part of the solar spectrum. The present topical review addresses this concept and discusses the material properties needed for an efficient upconversion process with focus on both silicon and organic solar cells. To design efficient upconverters, insight into topics such as quantum-optics, nano-optics, numerical modeling, optimization, material fabrication, and material characterization is paramount, and the necessary concepts are introduced throughout the review. Upconversion modeling is done using rate equations, while optical modeling is done by solving Maxwell's equations using the finite element method. Topology optimization is introduced and used to generate geometries of gold nanoparticles capable of greatly enhancing the upconversion yield. Fabrication and experimental characterization methods are discussed. Some recent results are presented and finally the possibility of designing upconverting materials capable of increasing the short-circuit current in a solar cell is discussed.

The preliminary study reported here investigates if unit-cell inclusion-symmetries may be broken in time-reversalinvariant topological insulator designs, while maintaining the desired global behaviour of pseudo-spin-dependent edge state based bi-directional, back-scattering robust, energy propagation. By allowing symmetries to be broken additional geometrical design freedom is attained, which may turn out to enable an improvement of various performance measures such as bandwidth and field confinement. The particular study considers a time-reversal-invariant acoustic topological insulator design, designed using a modified version of our recently proposed topology optimization based method for designing photonic and acoustic topological insulators.1 This method relies on a carefully constructed model system combined with the application of density based topology optimization to design two carefully interfaced crystal phases to maximize the flow of energy through the system. Through simple modifications of the method, we demonstrate that it is possible to design structures with different symmetry conditions from those that have previously been investigated using the method.

Designing photonic topological insulators is highly non-trivial because it requires inversion of band symmetries around the band gap, which was so far done using intuition combined with meticulous trial and error. Here we take a completely different approach: we consider the design of photonic topological insulators as an inverse design problem and use topology optimization to maximize the transmission through an edge mode with a sharp bend. Two design domains composed of two different, but initially identical, C$_\text{6v}$-symmetric unit cells define the geometrical design problem. Remarkably, the optimization results in a photonic topological insulator reminiscent of the shrink-and-grow approach to quantum-spin-Hall photonic topological insulators but with notable differences in the topology of the crystal as well as qualitatively different band structures and with significantly improved performance as gauged by the band-gap sizes, which are at least 50 \% larger than previous designs. Furthermore, we find a directional beta factor exceeding 99 \%, and very low losses for sharp bends. Our approach allows for the introduction of fabrication limitations by design and opens an avenue towards designing PTIs with hitherto unexplored symmetry constraints., Comment: 7 pages, 5 figures

A new method for dose regularization in optimization-based proximity-effect-correction is proposed. In contrast to the commonly adopted approach of adding penalty terms to the objective function, a modified scheme is demonstrated where dose regularization is achieved via filtering and projection techniques. The resulting dose patterns are simple and two-toned, and can thus readily be applied in production. Furthermore, existing extensions developed in the context of topology optimization that build on top of the filtering framework, such as robust optimization and strict length scale control, can be adopted directly. The validity of the scheme is assessed in experiments, where the resolvable feature size of the considered 30 kV electron-beam lithography system is decreased from around 100 nm to a few tens of nm. A Python implementation of the scheme is made freely available.

Small manufacturing-tolerant photonic crystal cavities are systematically designed using topology optimization to enhance the ratio between the quality factor and mode volume, Q/V. For relaxed manufacturing tolerance, a cavity with a bow-tie shape is obtained which confines light beyond the diffraction limit into a deep-subwavelength volume. Imposition of a small manufacturing tolerance still results in efficient designs, however, with diffraction-limited confinement. Inspired by numerical results, an elliptic ring grating cavity concept is extracted via geometric fitting. Numerical evaluations demonstrate that for small sizes, topology-optimized cavities enhance the Q/V-ratio by up to two orders of magnitude relative to standard L1 cavities and more than one order of magnitude relative to shape-optimized L1 cavities. An increase in cavity size can enhance the Q/V-ratio by an increase in the Q-factor without a significant increase in V. Comparison between optimized and reference cavities illustrates that significant reduction of V requires big topological changes in the cavity.

This paper describes a systematic design study of periodic gold-nanostrip arrays placed on a thin film aimed at enhancing the electric field inside the film when irradiated by light. Based on the study, a "selection rule" is proposed, which provides optimization-based design methods with an a priori choice between field-enhancement dominated by coupling to guided modes, by plasmonic near-field enhancement or by a mix hereof. An appropriate choice of wavelength and grating period is shown to selectively suppress or include waveguiding effects for the optimized designs. The validity of the selection rule is demonstrated through a numerical topology optimization study in which gold nanostrips are optimized for electric-field enhancement in an erbium-doped TiO2 thin film, targeting increased spectral upconversion in the erbium ions. The obtained designs exhibit waveguide excitation within the predicted intervals and, for light polarized perpendicularly to the strips, plasmonic response outside.

Spectral conversion of the sunlight has been proposed as a method for enhancing the efficiency of photovoltaicvdevices, which are limited in current production by the mismatch between the solar spectrum and the wavelength range for efficient carrier generation. For example, the photo current can be increased by conversion of two low-energy photons (below the band gap of the absorber) to one higher-energy photon (i.e. upconversion). In this paper, we will review our ongoing activities aimed at enhancing such spectral-conversion processes by employing appropriately designed plasmonic nanoparticles. The nanoparticles serve as light-concentrating elements in order to enhance the non-linear upconversion process. From the theoretical side, we approach the optimization of nanoparticles by finite-element modelling of the plasmonic near fields in combination with topological optimization of the particle geometries. Experimentally, the nanostructures are formed by electronbeam lithography on thin films of Er3+-containing transparent materials, foremost TiO2 made by radio-frequency magnetron sputtering, and layers of chemically synthesized NaYF4 nanoparticles. The properties of theupconverter are measured using a variety of optical methods, including time-resolved luminescence spectroscopy on erbium transitions and spectrally resolved upconversion-yield measurements at ∼1500-nm-light excitation.The calculated near-field enhancements are validated using a technique of near-field-enhanced ablation by tunable, ultrashort laser pulses.

This paper proposes a topology optimization based approach for designing meta materials exhibiting a desired negative refraction with high transmission at a given angle of incidence and frequency. The approach considers a finite slab of meta material consisting of axis-symmetric designable unit cells subjected to an exterior field. The unit cell is designed to achieve the desired properties based on tailoring the response of the meta material slab under the exterior field. The approach is directly applicable to physical problems modeled by the Helmholtz equation, such as acoustic, elastic and electromagnetic wave problems. Acoustic meta materials with unit cell size on the order of half the wave length are considered as examples. Optimized designs are presented and their performance under varying frequency and angle of incidence is investigated.

This Letter reports on the experimental validation of a two-dimensional acoustic hyperbolic metamaterial slab optimized to exhibit negative refractive behavior. The slab was designed using a topology optimization based systematic design method allowing for tailoring the refractive behavior. The experimental results confirm the predicted refractive capability as well as the predicted transmission at an interface. The study simultaneously provides an estimate of the attenuation inside the slab stemming from the boundary layer effects—insight which can be utilized in the further design of the metamaterial slabs. The capability of tailoring the refractive behavior opens possibilities for different applications. For instance, a slab exhibiting zero refraction across a wide angular range is capable of funneling acoustic energy through it, while a material exhibiting the negative refractive behavior across a wide angular range provides lensing and collimating capabilities.