SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Jungtaek Kim
Pittsburgh, PA 15261, USA
jungtaek.kim@pitt.edu
Oliver Hinder
ohinder@pitt.edu
Mingxuan Li
mil152@pitt.edu
Paul W. Leu
pleu@pitt.edu
Nanophotonic structures have versatile applications including solar cells, anti-reflective coatings, electromagnetic interference shielding, optical filters, and light emitting diodes. To design and understand these nanophotonic structures, electromagnetic fields over time and calculate optical properties. In this work, we introduce frameworks and benchmarks to evaluate nanophotonic structures in the context of parametric structure design problems. The benchmarks are instrumental in assessing the performance of optimization algorithms and identifying an optimal structure based on target optical properties. Moreover, we explore the impact of varying grid sizes in electrodynamic simulations, shedding light on how evaluation fidelity can be strategically leveraged in enhancing structure designs.
Nanophotonic structures play a crucial role for a wide range of real-world applications such as solar cells, anti-reflective coatings, electromagnetic interference shielding, optical filters, and light emitting diodes. In particular, several studies have demonstrated that nanophotonic structures can enhance target performance for many applications [70, 71, 73, 21, 19, 39]. Electrodynamic simulations, which are based on Maxwell's equations, provide accurate predictions of the optical and electromagnetic properties of these structures [18]. Leveraging the utility of these simulations, we can combine them with optimization procedures to design and discover new and improved structures.
In this paper, we introduce datasets and benchmarks for nanophotonic structures, focusing on parametric design simulations in relation to optical properties. These benchmarks allow us to analyze the performance of optimization algorithms such as derivative-free algorithms [53] and Bayesian optimization [16], when it comes to identifying optimal photonic structures based on a target optical property. Our frameworks support two modes for fast prototyping of optimization methods: discretized search space mode and surrogate model mode. The discretized search space mode uses stored simulated results over grid query points; the surrogate model mode estimates any query points within a continuous search space based on training with our datasets. These modes allow users to quickly test their optimization algorithms without running expensive electrodynamic simulations. Our system facilitates the use of evaluation fidelity, which is controlled by a simulation
<sup>1</sup>Our implementation can be found at https://github.com/jungtaekkim/nanophotonic-structures.
resolution, and the handling of multiple objectives. Moreover, our frameworks can also provide electromagnetic fields as a function of position and time.
- Development of a generic simulation scheme and pipeline for nanophotonic structures in Python, based on the open-source software, Meep, and licensed under the MIT license;
- Creation of datasets of a myriad of nanophotonic structures for electromagnetic interference shielding, anti-reflection, and solar cells;
- Investigation into the effects of altering grid sizes in electrodynamic simulations, providing insights into tradeoffs between computational time and simulation accuracy;
- Introduction of benchmarks specifically designed for the optimization of parametric structures, facilitating the evaluation and comparison of different optimization algorithms.
In this section, we delve into the optical and electromagnetic properties of materials, explore nanophotonic structure designs, and review related literature.
2.1 Optical and Electromagnetic Properties of Materials
The optical and electromagnetic properties of materials can be determined by solving Maxwell's equations [18]. When studying materials featuring sizes smaller than wavelengths of interest, geometrical optics becomes unsuitable, and ray-tracing methods are inaccurate [7]. In such cases, it is crucial to use a simulation method that captures the wave-like nature of light to accurately address phenomena like interference and diffraction. Classical electromagnetic theory can predict reflection, absorption, and transmission spectra by solving Maxwell's equations:
$$
\nabla \cdot \boldsymbol {E} = \frac {\rho}{\epsilon_ {0}}, \quad \nabla \cdot \boldsymbol {B} = 0, \quad \nabla \times \boldsymbol {E} = - \frac {\partial \boldsymbol {B}}{\partial t}, \quad \nabla \times \boldsymbol {B} = \mu_ {0} \boldsymbol {J} + \mu_ {0} \epsilon_ {0} \frac {\partial \boldsymbol {B}}{\partial t}, \tag {1}
$$
where  $\pmb{E}$  and  $\pmb{B}$  are electric and magnetic fields,  $\rho$  and  $\pmb{J}$  are charge and current densities, and  $\epsilon_0$  and  $\mu_0$  are the permittivity and permeability of free space. The speed of light is defined as  $c = 1 / \sqrt{\mu_0\epsilon_0}$ .
As shown in Figure S1, light consists of two synchronized waves: electric fields and magnetic fields. These fields are characterized by an angular frequency  $\omega$ , which can also be expressed as a frequency  $f$ , energy  $E$ , or wavelength  $\lambda$ , with the relationships  $\omega = 2\pi f$ ,  $E = hf$ , and  $E = hc / \lambda$  where  $h$  is the Planck constant (i.e.,  $6.63 \times 10^{-34}$  J-sec) and  $c$  is the speed of light (i.e.,  $3 \times 10^{8}$  m/sec). The interaction of light with materials is characterized by either a complex refractive index,  $n(\omega) = n_r(\omega) + n_i(\omega)$ , or a complex permittivity,  $\epsilon(\omega) = \epsilon_r(\omega) + \epsilon_i(\omega)$ , both of which describe how light bends and is absorbed within that material. The relative permittivities of the materials used in this paper are detailed in Section E. An important aspect of light-material interactions is understanding the transmission, reflection, and absorption spectra from a specific light source as it interacts with a material at a given frequency, as depicted in Figure S2. As governed by the law of conservation of energy [18], the following equation is satisfied:
$$
R (\omega) + A (\omega) + T (\omega) = 1, \tag {2}
$$
where  $R(\omega), A(\omega)$ , and  $T(\omega)$  are the reflectance, absorbance, and transmittance of a material at a particular frequency  $\omega$ .
2.2 Designs of Nanophotonic Structures for Optoelectronic Applications
Numerous electrodynamic simulation techniques such as the transfer matrix method [6], finite element method [29], rigorous coupled-wave analysis [43], and finite-difference time-domain (FDTD) method [65] are important for nanophotonics research. FDTD, in particular, is widely used across various domains to explore how electromagnetic waves interact with different materials. Its applications span across photonic crystals, waveguides, plasmonics, and metamaterials. In our research, we harness the FDTD simulations for the design and optimization of optical devices, focusing on applications in solar cells, anti-reflection, and transparent electromagnetic interference shielding.
Solar Cells. Nanomaterials are revolutionizing solar cell technology, promising significantly enhanced efficiency and reduced costs. Certain nanostructures have the potential to surpass the Shockley-Queisser efficiency limit and capture light beyond the Yablonovitch or Lambertian limits [48, 59, 78, 9]. In this paper we investigate vertical nanowire arrays and nanosphere coatings, as they have proven abilities to enhance light trapping [32, 17, 72, 74, 19]. Our evaluation of solar cells is based on their ultimate efficiency, excluding transport and radiative losses, effectively focusing on optimizing solar absorption.
Anti-Reflection. Light traveling from air to glass partially reflects due to the disparity in index of refraction. Anti-reflection thin film coatings can be used to achieve perfect anti-reflection at a single wavelength and normal incidence angle. However, these structures are less effective for reflection across a range of wavelengths or incidence angles. This challenge has been tackled with moth eye [76, 63] and glasswing butterfly wing-inspired structures [60, 20], which contain sub-wavelength structures with an effective refractive index that gradually transits between those of the air and the glass. These structures achieve broad-spectrum and wide-angle anti-reflection, enhancing light emission from displays and improving light intake for photodetectors and solar cells.
Electromagnetic Interference Shielding. As the usage of electronic devices has grown, there is a growing demand for strategies to shield these devices from external electromagnetic waves and interference, which can lead to disruptions. Furthermore, transparent electromagnetic interference shielding has emerged as a significance field of interest in this context. However, a considerable challenge is balancing effective shielding performance and transparency [75, 40]. Notably, multi-layer films have been identified as a viable solution, successfully attaining high levels of both transparency and shielding effectiveness [42, 75]. Furthermore, the integration of nanocone structures into these films enhances their transparency, offering an innovative approach for this complex challenge [39].
2.3 Related Work
The field of molecular discovery has garnered extensive attention across numerous studies, leading to significant advancements [54]. One such contribution is the creation of a comprehensive dataset featuring a large array of organic molecules, complete with details on their geometric, energetic, electronic, and thermodynamic properties [52]. This dataset serves as a valuable resource for framing and addressing the challenges associated with molecular discovery. MoleculeNet, a tool developed using the DeepChem software, further extends this by consolidating various public molecular property datasets [77]. Moreover, benchmarks specifically designed for de novo molecular design have been introduced to aid in solving inverse molecular discovery problems [8].
The emergence of deep learning for nanophotonic device design, particularly in inverse design methods, has been discussed [62]. This era has also witnessed exploration into data-driven surrogate models for artificial electromagnetic materials, utilizing neural networks to capture the intricate behaviors of metamaterials, nanophotonics, and color filters [11]. A noteworthy study has conducted a comprehensive comparison of various graph neural networks, evaluating their effectiveness in learning the dynamics of simple physical systems [66]. Furthermore, a combination of Bayesian optimization and Bayesian neural networks has been effectively applied to tackle complex scientific challenges such as photonic crystal topology and quantum chemistry [34].
In the fields of machine learning and optimization, the community often leverages diverse benchmark functions. These include simple, cost-effective synthetic functions such as Branin, Rastrigin, and Ackley functions. The benchmarks extend beyond these elementary examples, encompassing a variety of scenarios for applications like black-box optimization [24, 68], hyperparameter optimization [14, 46, 57], and neural architecture search [80, 12, 36, 13]. Despite the diversity, a noticeable gap exists, as these benchmarks predominantly feature similar types of examples and problems, offering limited insight into the complex challenges prevalent in real-world scientific and engineering contexts.
3 Nanophotonic Structures and Their Design Problems
In this section we introduce the scheme of electrodynamic simulations, our nanophotonic structures, and their design problems. These include specifying possible material choices and ranges of sizes
for structure components. Moreover, we detail the consideration of simulation fidelity and multiple objectives. We also visualize how our objectives vary as we change parameters that define a structure.
3.1 Electrodynamic Simulations
We cover the scheme of electrodynamic simulations using the FDTD method - a widely adopted technique in time-domain simulations for nanophotonic applications. This method discretizes both spatial and temporal dimensions, enabling the approximation of derivatives at discrete points. Our simulations focus on exploring the optical behaviors of structures in two and three dimensions through the FDTD method [65].
For a light source, we employ a Gaussian-pulse wave, monitoring the resulting electric fields  $\pmb{E}(t)$  and magnetic fields  $\pmb{B}(t)$ . These fields are subsequently Fourier-transformed to yield their frequency-dependent counterparts  $\pmb{E}(\omega)$  and  $\pmb{B}(\omega)$ . A Yee grid, a computational mesh where the electric and magnetic field components are calculated, is used [79]. Our simulations are conducted using Meep, which is the open-source
FDTD simulation software licensed under the GNU General Public License [44].
Figure 1: Schematic of nanophotonic structure simulations. PMLs, light source, and reflection and transmission monitors are located in a simulation cell indicated by the dotted rectangle.
Figure 1 shows a schematic that illustrates the setup of our FDTD simulation, where the simulation domain is confined within the simulation cell indicated by the dotted rectangle. The size and type of this simulation cell are tailored to the specific nanophotonic structure investigated. We utilize periodic boundary conditions for the sides of the simulation cell and perfectly matched layers (PMLs) for the top and bottom of the cell. Periodic boundary conditions model semi-infinite arrays, while the PML boundary conditions ensure that fields radiate to infinity instead of reflecting [5].
Within the simulation cell, a light source is positioned near the top, while transmission and reflection monitors are situated near the bottom and below the source, respectively. The electromagnetic wave is chosen to be normally incident to the structure of interest. The monitors enable the calculation of flux from  $\mathrm{Re}\int E^{*}(\omega)\times B(\omega) / 2$  as a function of frequency, facilitating the derivation of reflectance  $R(\omega)$ , absorbance  $A(\omega)$ , and transmittance  $T(\omega)$ . To accurately measure reflectance, the total incident flux from an empty simulation cell is subtracted from the flux measured in the presence of the nanophotonic structure. The positioning of the light source and monitors is chosen to ensure that they are appropriately spaced from the top and bottom boundaries of the simulation cell.
For our simulations, we consider three types of spectral ranges: a single wavelength  $550\mathrm{nm}$  at the approximate center of the visible spectrum, the visible spectrum from  $380\mathrm{nm}$  to  $750\mathrm{nm}$  with standard illuminant D65 [27], and the AM1.5 global solar spectrum from  $280\mathrm{nm}$  to  $2500\mathrm{nm}$  [2].
3.2 Structures of Interest
In this section, we describe the specifics of the nanophotonic structures studied in this paper. The details of simulation cell sizes are presented in Section F. Table 1 provides the summary of the design space including parameters, materials, and parameters' ranges for each of these structures.
Three-Layer Film for Electromagnetic Interference Shielding. This film structure is aimed at mitigating electromagnetic interference, incorporating three distinct layers; see Figure 2a. The central layer is metallic, potentially consisting of silver (Ag), gold (Au), copper (Cu), or nickel (Ni). The adjacent layers can be selected from materials such as titanium dioxide  $(\mathrm{TiO}_2)$ , crystalline silicon (cSi), zinc oxide (ZnO), indium tin oxide (ITO), or aluminum-doped zinc oxide (AZO). We simplify our material selection by assuming the materials for the outer layers are identical. The three layers are characterized by their thicknesses:  $t_1$ ,  $t_2$ , and  $t_3$ . We use the electromagnetic wave of wavelength  $\lambda = 550$  nm, which approximately corresponds to green light.
(a) Three-layer film
(b) Anti-reflective nanocones
(c) Vertical nanowires
(d) Close-packed nanospheres
(e) Three-layer film with double-sided nanocones
(f) Combinatorial system with material blocks
Figure 2: Two- and three-dimensional nanophotonic structures. Different colors indicate different materials in each figure and a dotted region represents a simulation cell.
The objectives of this system include maximizing transmittance over  $t_1$ ,  $t_2$ , and  $t_3$  and tackling a multi-objective optimization problem for maximizing both electromagnetic interference shielding effectiveness and transmittance. The formula for shielding effectiveness is  $S = 20\log_{10}(1 + \eta_0t_2 / 2\rho)$ , where  $\eta_0 = 376.73\Omega$  is the free space impedance and  $\rho$  is the bulk metal resistivity.
Graded Index of Refraction Structures for Anti-Reflection. Here, we include a semi-infinite array of glass nanocones attached to a glass substrate, as illustrated in Figure 2b. The nanocones serve to gradually transition the index of refraction from air to glass, with fused silica as the glass of choice. This structure is defined by two parameters: the nanocones' height  $h$  and radius  $r$ . Assuming a close-packed square array, we determine the pitch of the nanocones is  $p = 2r$ . Our goal is to minimize solar reflection; see Section C for the details of solar reflection.
Vertical Nanowires for Solar Cells. We explore arrays of vertical nanowires composed of crystalline silicon (cSi), gallium arsenide (GaAs), or a perovskite structure of methylammonium lead iodide  $\mathrm{(CH_3NH_3PbI_3)}$ , as shown in Figure 2c. The structures are defined by the nanowire radius  $r$ , height  $h$ , and pitch  $p$ , with a fixed  $h = 200$  nm. The relationship  $2r \leq p$  has to be satisfied, which makes the diameter less than or equal to the pitch. We redefine this constraint using  $g = p - 2r$ , where  $g \geq 0$ . The nanowires are arranged in a square array, with optimization focused on maximizing solar absorption or ultimate efficiency; refer to Section D for further details.
Close-Packed Nanospheres for Solar Cells. This system features titanium dioxide  $(\mathrm{TiO}_2)$  nanospheres in a hexagonal array on top of a semiconductor thin film with possible materials including crystalline silicon (cSi), gallium arsenide (GaAs), or a perovskite structure of methylammonium lead iodide  $(\mathrm{CH}_3\mathrm{NH}_3\mathrm{PbI}_3)$ . The system, which is visualized in Figure 2d, is defined by the semiconductor layer thickness  $t$  and nanosphere radius  $r$ . Similar to the vertical nanowires, the optimization aims at maximizing solar absorption or ultimate efficiency; refer to Section D for details.
Table 1: Parameters, materials, and the search spaces of the parameters for nanophotonic structures. We assume that only one of AZO, cSi, ITO,  $\mathrm{TiO_2}$ , and ZnO should be used for one structural configuration in the three-layer film with or without double-sided nanocones.
Structure	Parameter	Materials	Lower Bound (nm)	Upper Bound (nm)
Three-layer film	t1	AZO, cSi, ITO, TiO2, ZnO	10	100
t2	Ag, Au, Cu, Ni	3	20
t3	AZO, cSi, ITO, TiO2, ZnO	10	100
Anti-reflective nanocones	r	Fused silica	5	150
h	1	300
Vertical nanowires	g	1	200
r	cSi, CH3NH3PbI3, GaAs	5	200
h	200	200
Close-packed nanospheres	t	cSi, CH3NH3PbI3, GaAs	100	400
r	TiO2	10	200
Three-layer film with double-sided nanocones	t1	AZO, cSi, ITO, TiO2, ZnO	10	50
t2	Ag, Au, Cu, Ni	3	20
t3	AZO, cSi, ITO, TiO2, ZnO	10	50
r1	AZO, cSi, ITO, TiO2, ZnO	20	50
h1	50	100
r2	AZO, cSi, ITO, TiO2, ZnO	20	50
h2	50	100
Combinatorial system with material blocks	-	Ag, Air, Au, AZO, cSi, CH3NH3PbI3, Cu, GaAs, ITO, Ni, TiO2, ZnO	-	-
Three-Layer Film with Nanocones for Electromagnetic Interference Shielding. Building upon the three-layer film concept, this structure incorporates double-sided nanocones for enhanced electromagnetic interference shielding; see Figure 2e. It comprises three layers and nanocones on both sides, where material options for the three-layer film are identical to the previously discussed three-layer film and nanocones are made of one of titanium dioxide  $(\mathrm{TiO}_2)$ , crystalline silicon (cSi), zinc oxide (ZnO), indium tin oxide (ITO), and aluminum-doped zinc oxide (AZO). Seven parameters define this structure: three layer thicknesses  $t_1$ ,  $t_2$ ,  $t_3$ , and two sets of cone heights  $h_1$ ,  $h_2$  and radii  $r_1$ ,  $r_2$ . Objective for the optimization of this structure can be either maximization of visible transmission or simultaneous maximization of visible transmission and shielding effectiveness, where light of the visible spectrum is injected; refer to the description of the aforementioned three-layer film.
**Combinatorial System with Material Blocks.** We introduce a novel and challenging problem: a combinatorial system composed of material blocks; see Figure 2f. The size of each block within the structure is  $(q, q)$  for two-dimensional systems or  $(q, q, q)$  for three-dimensional ones. Construction of this structure is accomplished by strategically selecting specific materials for all the blocks placed in a repeating unit. This unit consists of  $p/q$  blocks along the  $x$  and  $y$  directions, and  $t/q$  blocks along the  $z$  direction, where  $p$  denotes the repeating unit's pitch size and  $t$  represents the maximum thickness of the structure. Consequently, the repeating unit is comprised of either  $p/q \times t/q$  blocks for two-dimensional structures or  $p/q \times p/q \times t/q$  blocks for three-dimensional ones.
In this paper, we choose specific values:  $p = 200 \mathrm{~nm}$ ,  $t = 40 \mathrm{~nm}$ , and  $q = 10 \mathrm{~nm}$ . The range of materials considered for the material blocks includes silver (Ag), air, gold (Au), aluminum-doped zinc oxide (AZO), crystalline silicon (cSi), methylammonium lead iodide perovskite (CH₃NH₃PbI₃), copper (Cu), gallium arsenide (GaAs), indium tin oxide (ITO), nickel (Ni), titanium dioxide (TiO₂), and zinc oxide (ZnO). The AM1.5 solar spectrum is used for simulating this structure.
3.3 Simulation Fidelity and Multiple Objectives
As described above, the accuracy of simulation results are heavily influenced by the fidelity of the simulations. Simulations with a low fidelity level are faster but less accurate, while simulations with
(a) Low fidelity
(b) Medium fidelity
(c) High fidelity
Figure 3: Visualization of the transmittance of the three-layer film made of  $\mathrm{TiO_2 / Ag / TiO_2}$  for three different fidelity levels, where the second layer's thickness  $t_2$  is  $3\mathrm{nm}$ .
(b) Medium fidelity
(c) High fidelity
Figure 4: Visualization of the absorbance of the vertical nanowires made of cSi for three different fidelity levels, where the nanowire height  $h$  is  $200\mathrm{nm}$ .
a high fidelity level are more accurate but slower. There is an inherent tradeoff between speed and accuracy. Additionally, our frameworks can generate objective functions such as shielding effectiveness and transparency for three-layer films, both with and without nanocones. The introduction of multiple objectives adds a layer of complexity to the optimization problem. In summary, utilizing fidelity levels and multiple objectives enables us to expand our research, potentially exploring multi-fidelity optimization [15, 31, 4] and multi-objective optimization [37, 26, 3].
3.4 Visualization
For each type of nanophotonic structure featured in our frameworks, we visualize how specific optical properties change across different structural parameters, as shown in Figures 3, 4, and 6. It is important to note that the accuracy and smoothness of the optical property values are dependent on the fidelity level. Results from lower fidelity simulations tend to be less accurate and show more erratic fluctuations compared to results from their higher fidelity counterparts. This difference is evident in the comparisons provided in Figures 3 and 4. Additional visualization for more diverse examples is available in Section H, where the data
Figure 5: Examples of E-fields (left) and H-fields (right) out of plane for the combinatorial system with material blocks.
and examples provided in this paper pertain to two-dimensional nanophotonic structures.