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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

The rates of optical processes, such as two-photon absorption and spontaneous photon emission, are strongly dependent on the environment in which they take place, easily varying by orders of magnitude between different settings. Using topology optimization, we design a set of compact wavelength-sized devices, to study the effect of optimizing geometries for enhancing processes that depend differently on the field in the device volume, characterized by different figures of merit. We find that significantly different field distributions lead to maximization of the different processes, and - by extension - that the optimal device geometry is highly dependent on the targeted process, with more than an order of magnitude performance difference between optimized devices. This demonstrates that a univeral measure of field confinement is meaningless when evaluting device performance, and stresses the importance of directly targeting the appropriate metric when designing photonic components for optimal performance.

Periodic wavelength-scale surface patterns have long been used in the context of lasing and spontaneous emission to enhance emission by light trapping (distributed Bragg resonances). Buried within these well-known devices, however, are theoretical mysteries that are still being unravelled. A periodic surface grating actually creates a continuum of resonant modes, so what determines which single mode (if any) lases? Technically, what determines the stability of a periodic lasing mode: is it only the finite size of a surface that allows single-mode lasing, or can it arise for arbitrarily large structures? More generally, if one continuously deforms an unpatterned surface to maximize light emission, how is the symmetry broken and what optimal structures arise? We address these questions by combining new computational techniques for modeling and large-scale optimization of incoherent emission and lasing with new analytical results arising from perturbation and stability theory.

Optical nanocavities confine and store light, which is essential to increase the interaction between photons and electrons in semiconductor devices, enabling, e.g., lasers and emerging quantum technologies. While temporal confinement has improved by orders of magnitude over the past decades, spatial confinement inside dielectrics was until recently believed to be bounded at the diffraction limit. The conception of dielectric bowtie cavities (DBCs) shows a path to photon confinement inside semiconductors with mode volumes bound only by the constraints of materials and nanofabrication, but theory was so far misguided by inconsistent definitions of the mode volume and experimental progress has been impeded by steep nanofabrication requirements. Here we demonstrate nanometer-scale photon confinement inside 8 nm silicon DBCs with an aspect ratio of 30, inversely designed by fabrication-constrained topology optimization. Our cavities are defined within a compact device footprint of 4 lambda^2 and exhibit mode volumes down to V = 3E-4 lambda^3 with wavelengths in the lambda = 1550 nm telecom band. This corresponds to field localization deep below the diffraction limit in a single hotspot inside the dielectric. A crucial insight underpinning our work is the identification of the critical role of lightning-rod effects at the surface. They invalidate the common definition of the mode volume, which is prone to gauge meretricious surface effects or numerical artefacts rather than robust confinement inside the dielectric. We use near-field optical measurements to corroborate the photon confinement to a single nanometer-scale hotspot. Our work enables new CMOS-compatible device concepts ranging from few- and single-photon nonlinearities over electronics-photonics integration to biosensing.

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

We demonstrate new axisymmetric inverse-design techniques that can solve problems radically different from traditional lenses, including \emph{reconfigurable} lenses (that shift a multi-frequency focal spot in response to refractive-index changes) and {\emph{widely separated}} multi-wavelength lenses ($\lambda = 1\,\mu$m and $10\,\mu$m). We also present experimental validation for an axisymmetric inverse-designed monochrome lens in the near-infrared fabricated via two-photon polymerization. Axisymmetry allows fullwave Maxwell solvers to be scaled up to structures hundreds or even thousands of wavelengths in diameter before requiring domain-decomposition approximations, while multilayer topology optimization with $\sim 10^5$ degrees of freedom can tackle challenging design problems even when restricted to axisymmetric structures., Comment: 13 pages, 6 figures

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.

We previously introduced a preconditioner that has proven effective for h p -FEM discretizations of various challenging elliptic and hyperbolic problems. The construction is inspired by standard nested dissection, and relies on the assumption that the Schur complements can be approximated, to high precision, by Hierarchically-Semi-Separable matrices. The preconditioner is built as an approximate L D M t factorization through a divide-and-conquer approach. This implies an enhanced flexibility which allows to handle unstructured geometric meshes, anisotropies, and discontinuities. We build on our previous numerical experiments and develop a preconditioner-update strategy that allows us to handle matrix sequences arising from problems with slowly-varying coefficients. We investigate the performance of the preconditioner along with the update strategy in context of topology optimization of an acoustic cavity.

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

This paper corrects an error in the software provided with J. Opt. Soc. Am. B 38, 510 (2021)JOBPDE0740-322410.1364/JOSAB.405955.

Upconversion of sub-band-gap photons constitutes a promising way for improving the efficiency of silicon-based solar cells beyond the Shockley-Queisser limit. 1500 to 980 nm upconversion by trivalent erbium ions is well-suited for this purpose, but the small absorption cross section hinders real-world applications. We employ tailored gold nanostructures to vastly improve the upconversion efficiency in erbium-doped TiO$_2$ thin films. The nanostructures are found using topology optimization and parameter optimization and fabricated by electron beam lithography. In qualitative agreement with a theoretical model, the samples show substantial electric-field enhancements inside the upconverting films for excitation at 1500 nm for both s- and p-polarization under a wide range of incidence angles and excitation intensities. An unprecedented upconversion enhancement of 913(51) is observed at an excitation intensity of 1.7 Wcm$^{-2}$. We derive a semi-empirical expression for the photonically enhanced upconversion efficiency, valid for all excitation intensities. This allows us to determine the upconversion properties needed to achieve significant improvements in real-world solar-cell devices through photonic-enhanced upconversion., Comment: 9 pages (main text), 4 figures, 1 supporting information of 15 pages

We show that topology optimization (TO) of metallic resonators can lead to $\sim 10^2\times$ improvement in surface-enhanced Raman scattering (SERS) efficiency compared to traditional resonant structures such as bowtie antennas. TO inverse design leads to surprising structures very different from conventional designs, which simultaneously optimize focusing of the incident wave and emission from the Raman dipole. We consider isolated metallic particles as well as more complicated configurations such as periodic surfaces or resonators coupled to dielectric waveguides, and the benefits of TO are even greater in the latter case. Our results are motivated by recent rigorous upper bounds to Raman scattering enhancement, and shed light on the extent to which these bounds are achievable., Comment: 19 pages

An acoustic topological insulator (TI) is synthesized using topology optimization, a free material inverse design method. The TI appears spontaneously from the optimization process without imposing requirements on the existence of pseudo spin-1/2 states at the TI interface edge, or the Chern number of the topological phases. The resulting TI is passive; consisting of acoustically hard members placed in an air background and has an operational bandwidth of $\approx$12.5\% showing high transmission. Further analysis demonstrates confinement of more than 99\% of the total field intensity in the TI within at most six lattice constants from the TI interface. The proposed design hereby outperforms a reference from recent literature regarding energy transmission, field confinement and operational bandwidth., 6 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.

Resonance and wave-propagation problems are known to be highly sensitive towards parameter variations. This paper discusses topology optimization formulations for creating designs that perform robustly under spatial variations for acoustic cavity problems. For several structural problems, robust topology optimization methods have already proven their worth. However, it is shown that direct application of such methods is not suitable for the acoustic problem under consideration. A new double filter approach is suggested which makes robust optimization for spatial variations possible. Its effect and limitations are discussed. In addition, a known explicit penalization approach is considered for comparison. For near-uniform spatial variations it is shown that highly robust designs can be obtained using the double filter approach. It is finally demonstrated that taking non-uniform variations into account further improves the robustness of the designs.

This paper presents the experimental validation of an acoustic cavity designed using topology optimization with the goal of minimizing the sound pressure locally for monochromatic excitation. The presented results show good agreement between simulations and measurements. The effect of damping, errors in the production of the cavity, and variations in operating frequency is discussed and the importance of taking these factors into account in the modeling process is highlighted.

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.

Measuring the Green’s function over the entire volume of a room would typically require an unfeasible number of measurements, due to requirements on spatial sampling. To alleviate the need for excessive measurements, sparse reconstruction methods can be employed, as they make it possible to reconstruct a seemingly undersampled signal. The present study proposes a method for acquiring experimentally the Green’s function in a room by measuring directly the mode shapes of the room, based on the conception that any mode can be expanded into a number of propagating waves. If the modes are described in the wavenumber domain (as a plane-wave expansion), sparse reconstruction methods can be employed, under the implicit assumption that each mode shape is represented as the superposition of a small number of plane waves. In addition, it is assumed that the medium is source-free and homogeneous. The methodology is examined numerically and verified experimentally, based on measurements in a lightly damped rectangular room.

This letter presents a topology optimization study of metal nanostructures optimized for electric-field enhancement in the infrared spectrum. Coupling of such nanostructures with suitable ions allows for an increased photon-upconversion yield, with one application being an increased solar-cell efficiency by exploiting the long-wavelength part of the solar spectrum. In this work, topology optimization is used to design a periodic array of two-dimensional gold nanostrips for electric-field enhancements in a thin film doped with upconverting erbium ions. The infrared absorption band of erbium is utilized by simultaneously optimizing for two polarizations, up to three wavelengths, and three incident angles. Geometric robustness towards manufacturing variations is implemented considering three different design realizations simultaneously in the optimization. The polarization-averaged field enhancement for each design is evaluated over an 80 nm wavelength range and a ±15-degree incident angle span. The highest polarization-averaged field enhancement is 42.2 varying by maximally 2% under ±5 nm near-uniform design perturbations at three different wavelengths (1480 nm, 1520 nm, and 1560 nm). The proposed method is generally applicable to many optical systems and is therefore not limited to enhancing photon upconversion.

Various scatterers such as rough surfaces or nanostructures are typically used to enhance the low efficiency of Raman spectroscopy (surface-enhanced Raman scattering). In this work, we find fundamental upper bounds on the Raman enhancement for arbitrary-shaped scatterers, depending only on its material constants and the separation distance from the molecule. According to our metric, silver is optimal in visible wavelengths while aluminum is better in the near-UV region. Our general analytical bound scales as the volume of the scatterer and the inverse sixth power of the distance to the active molecule. For periodic scatterers, a second bound with surface-area scaling is presented. Simple geometries such as spheres and bowties are shown to fall short of the bounds. However, using topology optimization based inverse design, we obtain surprising structures maximizing the Raman enhancement. These optimization results shed light to what extent our bounds are achievable.

We consider the design of individual and periodic arrangements of metal or semiconductor nanoparticles for localized electromagnetic field enhancement utilizing a topology optimization based numerical framework as the design tool. We aim at maximizing a function of the electromagnetic field amplitude in a region of space through the introduction of nanoparticles in and/or near the region.

We design, fabricate, and characterize an ultracompact (2) photonic nanocavity in silicon with a deep subwavelength mode volume, which enables strongly enhanced light-matter interaction for applications in nanolasers and nonlinear nanophotonics.