Rigorous Coupled-wave Analysis Research Articles

Diffraction by gratings: from the C-method to the stochastic C-method

The Chandezon, or C-method, is an efficient and versatile numerical method for modeling diffraction problems involving smooth surface relief gratings. For some grating profiles, the C-method is limited when the height-to-period ratio exceeds a factor of three. This is due to the formation of ill-conditioned matrices for inversion. Here, the stochastic C-method (SCM) is introduced as a solution that leverages stochastic differential equations to overcome these numerical difficulties. The SCM is developed by altering the physical model of the grating profile function to include an additive Brownian noise component. The inclusion of noise dramatically expands the applicability of the C-method and enriches the physical model. Numerical experiments show that the SCM achieves a precision on the order of 10−5 for diffracted/transmitted amplitudes on sinusoidal profiles with height-to-period ratios as high as 72. These results are in agreement with those obtained using multi-precision and the rigorous coupled wave analysis (RCWA).

Localized Surface Plasmon Resonances of AgAl Alloy Nanoparticles Self-Organized by Annealing of Ag/Al Bilayer Films

Plasmonic applications have traditionally relied on noble metals such as gold (Au) and silver (Ag) for their excellent plasmonic performance in the visible and near-infrared spectrum. However, these metals are costly, scarce, and have limitations such as low stability (Ag) and interband transition losses, which restrict their spectral range. To address these issues, alternative plasmonic materials have been explored. One such material is aluminum (Al), which is inexpensive, abundant, and exhibits remarkable plasmonic properties in the UV region as well as wide tunability. Al is also compatible with complementary metal–oxide semiconductor (CMOS) fabrication processes and is very stable due to its ultrathin native oxide layer. Alloying different metals can combine their advantageous properties, resulting in enhanced tunable optical characteristics. This study investigates the LSPR properties of AgAl alloy nanoparticles grown after the annealing of precursor AgAl bilayer films. Interestingly, LSPRs were also observed in some cases for the as-deposited bilayers. The experimental results were complemented with simulations conducted via the rigorous coupled-wave analysis (RCWA) method. The investigated materials could be potentially useful for applications in energy harvesting or color printing.

Weakly guiding approximation of a three-dimensional waveguide model for extreme ultraviolet lithography simulation

A three-dimensional (3D) waveguide model is applied in extreme ultraviolet (EUV) lithography simulations. The 3D waveguide model is equivalent to rigorous coupled-wave analysis, but fewer field components are used to solve Maxwell’s equations. The 3D waveguide model uses two components of vector potential, A x and A y , corresponding to the two polarizations. The electric field of the A x polarization is approximately parallel to the x axis, and the electric field of the A y polarization is approximately parallel to the y axis. The 3D waveguide model solves a coupled vector wave equation for two polarizations. The refractive index of conventional EUV absorbers is close to that of vacuum. The weakly guiding approximation in optical fiber theory is applied to the 3D waveguide model. The coupled vector wave equations for the two polarizations are decoupled into two independent scalar wave equations. Maxwell’s equations are simplified to a set of scalar wave equations. The weakly guiding approximation reduces the computation time to solve the equations. The computation time required to solve the weakly guiding approximation is about 1/5 of the time to solve the original 3D waveguide model.

Ultrahigh reflectivity photonic crystal membranes with optimal geometry

Photonic crystal (PhC) structures with subwavelength periods are widely used for diffractive optics, including high reflectivity membranes with nanoscale thickness. Here, we report on a design procedure for 2D PhC silicon nitride membrane mirrors providing optimal crystal geometry using simulation results obtained with rigorous coupled-wave analysis. The Downhill Simplex algorithm, a robust numerical approach to finding local extrema of a function of multiple variables, is used to optimize the period and hole radius of PhCs with both hexagonal and square lattices, as the membrane thickness is varied. Following these design principles, nanofabricated PhC membranes made from silicon nitride have been used as input couplers for an optical cavity, resulting in a maximum cavity finesse of 33 000, corresponding to a reflectivity of 0.999 82. The role played by the spot size of the cavity mode on the PhC was investigated, demonstrating the existence of an optimal spot size that agrees well with predictions. We find that, compared to the square lattice, the hexagonal lattice exhibits a spectrally wider reflective range, less sensitivity to fabrication tolerances, and higher reflectivity for membranes thinner than 200 nm, which may be advantageous in cavity optomechanical experiments. Finally, we find that all of the cavities that we have constructed exhibit well-resolved polarization mode splitting, which we expect is due primarily to a small amount of anisotropic stress in the silicon nitride and PhC asymmetry arising during fabrication.

A comparative simulation study of light–matter coupling in 1D photonic crystals with 2D perovskite active layer

Light–matter interaction operating in the strong coupling regime offers wide prospects of applications going from nano-photonics to quantum communications. The most practical implementations are to embed the active matter into Fabry–Perot microcavities or photonic crystals. In this work we focus on the strong coupling of two-dimensional (2D) perovskite in 1D grating waveguide. We use rigorous coupled wave analysis to simulate electromagnetic wave confinement in the 1D waveguide. Various sets of waveguide geometrical parameters are examined to achieve the strong coupling regime for three different configurations of the active layer in the waveguide. To extract quantitatively the relevant physical parameters, such as strength of light–matter interaction, we develop a Hamiltonian formalism to reproduce results obtained by simulation. It is shown that the strongest interaction is to have the active layer inserted in the main slab of the waveguide. It is, however, still weaker by about 20% as compared to the use of Fabry–Perot microcavities.

Temperature-Dependent Localized Surface Plasmon Resonances of Noble Nanoparticles Covered with Polymers

Self-assembled gold and silver nanoparticles were fabricated in medium vacuum conditions on Corning glass substrates by means of DC magnetron sputtering. The samples were deposited either at 420 °C or 440 °C, or they were initially deposited at room temperature followed by post annealing. Subsequently, they were covered with three different polymers, namely Polystyrene-block-polybutadiene-blockpolystyrene (PS-b-PBD-b-PS), Polystyrene-co-methyl methacrylate (PS-co-PMMA) and Polystyreneblock-polyisoprene-block-polystyrene (PS-b-PI-b-PS), using spin coating. Localized surface plasmon resonances were recorded in the temperature range of −25 °C–100 °C. We show that the resonance position changes systematically as a function of temperature. Theoretical calculations carried out via the Rigorous Coupled Wave Analysis support the experimental results. Based on these findings, the investigated materials demonstrate potential as components for the development of temperature sensors.

Incoherent phenomena in anisotropic periodic structures: from modeling to experimental demonstration

The periodic structures are widely studied in numerous optical applications and there is a number of good tools for numerical modeling of such a structures (for example rigorous coupled-wave analysis, finite-difference time-domain, finite element method etc.). However, when it comes to the modeling of incoherent effects in many cases of practical interest, the current methods are not rigorous enough or depend on computationally demanding averaging of coherent response. In this paper, we present a novel approach to modeling of incoherent effects in structures with lateral periodicity based on scattering matrix formalism, as a way to describe optical response of a structure, and on application of incoherent wave summation in the form of infinite geometric series and generalized Mueller matrix calculus. This method can be combined with any of the existing coherent methods of modeling periodic structures and it offers significantly faster computational performance than partially coherent/incoherent methods based on averaging. It is compared with other methods for modeling of incoherent effects and also with experimental spectroscopic data. This method is then used to explain phenomena emerging from the complex interaction between diffraction grating and thick substrate.

Modeling full PCSELs and VCSELs using modified rigorous coupled-wave analysis

An integrated rigorous coupled-wave analysis (RCWA) algorithm is presented in this paper, which can simulate full vertical-cavity surface-emitting laser (VCSEL) and photonic crystal surface-emitting laser (PCSEL) structures. A classic RCWA can only analyze a structure when the light source is incident from the top, bottom, or both sides of the device. However, for VCSEL applications, the light source is generated in the middle and propagates in both directions. A bidirectional scattering matrix method and doubling algorithm are implemented in RCWA. The resonant wavelength and Q factor of a VCSEL can then be found in the output spectrum. The accuracy and execution speed are compared with those of the Lumerical finite-difference time-domain (FDTD) method for several VCSEL and PCSEL designs. The results show that the maximum discrepancy between RCWA and FDTD is less than 3 nm, and the difference in the far-field divergence angle is less than 0.5°. The speed of RCWA also outperforms FDTD simulation significantly.

End-to-end simulations of exoplanet detection with multiplexed volume Bragg gratings

Context. Direct imaging of exoplanets greatly enhances our ability to spectroscopically detect molecules in their atmospheres, but requires both excellent angular and spectral resolution. A highly promising approach uses multiplexed transmission gratings to achieve compressed sensing, but the published theoretical calculations are not realistic enough to determine whether the approach is feasible. Aims. We aim to determine the performance of recovering exoplanet signals of an instrument with multiplexed volume Bragg gratings in a full, detailed numerical simulation, specifically examining any effects caused by multiplexing several gratings. Methods. Our end-to-end simulation includes realistic stellar and planetary spectra, a closed-loop adaptive optics simulation, and rigorous coupled wave analysis to model the multiplexed gratings. We chose 20 passbands around expected methane absorption features that were optically combined using 20 gratings. Results. We find that exoplanet signals can be recovered down to contrasts of 10−6−10−7, without the addition of a coronagraph. Multiplexing a larger number of gratings improves the deepest recovered contrast, if photon noise is the dominant noise source. When residuals from our simple post-processing approach are the largest noise source, a larger number of multiplexed gratings decreases the performance, due to the stacking of the PSFs at different wavelengths. This is an artifact of our data reduction approach. Conclusions. We conclude that multiplexed Bragg gratings are a viable method to look for exoplanets with a compressed sensing approach, and additional gains may be made in post-processing.

Double-sided structure grating for dual-functional polarizing or non-polarizing splitter

In this paper, a novel bidirectional dual-functional metal-dielectric reflective grating with a double-sided structure is described, which can achieve transverse-electric (TE) polarization in the -1st order and transverse-magnetic (TM) polarization in the 0th order as a polarizing beam splitter in the upper part of the grating or TE and TM polarizations in the -1st order with polarization-independent property in the lower part of the grating. The proposed grating works at wavelength 1550 nm under Littrow mounting. Rigorous coupled-wave analysis (RCWA) is used to optimize grating parameters. The upper part can achieve diffraction efficiencies of 99.53% and 99.11% for TE polarization in the -1st order and TM polarization in the 0th order, respectively. In addition, the extinction ratios of -1st order and 0th order are 58.87 dB and 52.02 dB, respectively. The lower part can achieve diffraction efficiencies of 99.41% and 99.26% for TE and TM polarizations in the -1st order, respectively. Furthermore, wide incident wavelength and angular bandwidths can be obtained in both the upper and lower parts. Such a high-efficiency dual-functional grating has a wide range of applications in optical systems.

Rotating twisted templates for imprinting polarization gratings with a sub- to dozen-micron period.

In this Letter, we report and experimentally demonstrate what is to our knowledge a novel scheme for imprinting polarization gratings (PGs) with a pair of templates. Compared with the traditional method that a single template can only imprint PG with a single period, cascading two templates can control the period of imprinted PG at will. However, the low diffraction efficiency is inevitably caused by cascading two templates. Therefore, a rigorous coupled wave analysis (RCWA) is adopted to design a multi-twisted template to address this challenge. As a proof of concept, two multi-twisted templates with a period of 1.6 μm were fabricated, and PGs with a large period range from 0.4 to 48.6 μm were successfully imprinted. The proposed scheme is expected to enable rapid, robust, and high-quality mass production of beam steering, large-angle deflectors, and diffractive optical couplers.

Mitigating the Impact of Asymmetric Deformation on Advanced Metrology for Photolithography

Controlling overlay in lithography is crucial for improving the yield of integrated circuit manufacturing. The process disturbances can cause undesirable morphology changes of overlay targets (such as asymmetric grating), which can significantly impact the accuracy of overlay metrology. It is essential to decouple the overlay target asymmetry from the wafer deformation, ensuring that the overlay metrology is free from the influence of process-induced asymmetry (e.g., grating asymmetry and grating imbalance). Herein, we use an asymmetric grating as a model and show that using high-diffraction-order light can mitigate the impact of asymmetric grating through the rigorous coupled-wave analysis (RCWA) method. In addition, we demonstrate the diffraction efficiency as a function of the diffraction order, wavelength, and pitch, which has guiding significance for improving the measurement accuracy of diffraction-based overlay (DBO) metrology.

Simulation of gradient period polarization volume gratings for augmented reality displays

Augmented reality (AR) displays are gaining attention as next-generation intelligent display technologies. Diffractive waveguide technologies are progressively becoming the AR display industry's preferred option. Gradient period polarization volume holographic gratings (PVGs), which are considered to have the potential to expand the field of view (FOV) of waveguide display systems due to their wide bandwidth diffraction characteristics, have been proposed as coupling elements for diffraction waveguide systems in recent years. Here, what we believe to be a novel modeling method for gradient period PVGs is proposed by incorporating grating stacking and scattering analysis utilizing rigorous coupled-wave analysis (RCWA) theory. The diffraction efficiency and polarization response were extensively explored using this simulation model. In addition, a dual-layer full-color diffractive waveguide imaging simulation using proposed gradient period PVGs is accomplished in Zemax software using a self-compiled dynamic link library (DLL), achieving a 53° diagonal FOV at a 16:9 aspect ratio. This work furthers the development of PVGs by providing unique ideas for the field of view design of AR display.

Silver Nanoparticles’ Localized Surface Plasmon Resonances Emerged in Polymeric Environments: Theory and Experiment

Considering that the plasmonic properties of metallic nanoparticles (NPs) are strongly influenced by their dielectric environment, comprehension and manipulation of this interplay are crucial for the design and optimization of functional plasmonic systems. In this study, the plasmonic behavior of silver nanoparticles encapsulated in diverse copolymer dielectric environments was investigated, focusing on the analysis of the emerging localized surface plasmon resonances (LSPRs) through both experimental and theoretical approaches. Specifically, two series of nanostructured silver ultrathin films were deposited via magnetron sputtering on heated Corning Glass substrates at 330 °C and 420 °C, respectively, resulting in the formation of self-assembled NPs of various sizes and distributions. Subsequently, three different polymeric layers were spin-coated on top of the silver NPs. Optical and structural characterization were carried out by means of UV–Vis spectroscopy and atomic force microscopy, respectively. Rigorous Coupled Wave Analysis (RCWA) was employed to study the LSPRs theoretically. The polymeric environment consistently induced a red shift as well as various alterations in the LSPR amplitude, suggesting the potential tunability of the system.

Influence of Photonics and Plasmonics Effect on Ultrathin Amorphous Silicon Thin Film Solar Cell

Abstract Our recent surge in silicon derived materials has major demand in thin film photovoltaic (PV) modules and enhancing the significant numbers. The rigorous coupled wave analysis method is a simple and fast method also known as developed to determine the light absorption and cell efficiency. The optics of thin film solar cells (amorphous silicon) garnering crucial role in the photovoltaic market. In this work, an ultrathin thin film amorphous silicon solar cell PV performance investigated by periodically textured surfaces at the nanoscale level. This analysis of periodic textured substrate was deriving optimal surface textures. The nanogratings lead to light scattering mechanism with the higher order diffraction angle and enhanced the light absorption of the incidence spectrum. The influence of grating period and height, the collection of the charge carriers increased at various wavelengths from ultraviolet, visible and infrared spectral regions are discussed with the assistance of photonic and plasmonic modes. Finally, nanoscale engineering mechanism the optimized thin film amorphous silicon solar cell yielded the highest current densities (22.6 and 23.8 mA/cm2) in both polarization modes.

Process and optical modeling of black silicon

Black silicon is relevant for the photovoltaic industry when searching for low-reflectance, low-defect front surface, which is the goal of this work. We have fabricated samples using reactive ion etching (RIE) plus chemical etching for the smoothing, characterized them, and built modeling tools capable of reproducing the resulting geometric features, based on the process parameters. Reflectance is simulated using a proprietary rigorous coupled wave analysis (RCWA)-based tool, and compared with the experimental results. A good matching was achieved using a simple unit cell, and a better agreement when using a 0.5 square microns sample. Finally, an optimum trade-off between low reflectance and low thickness has been achieved.

Augmented reality display with high eyebox uniformity over the full field of view based on a random mask grating

Ensuring uniform illuminance in waveguide-based augmented reality (AR) display devices is crucial for providing an immersive and comfortable visual experience. However, there is a lack of a straightforward and efficient design method available to achieve illuminance uniformity over the full field of view. To address this issue, we propose a novel design that utilizes random mask gratings (RMGs) as the folding grating and the out-coupling grating. Unlike traditional approaches that modify the grating structure, we control the diffraction efficiency distribution by adjusting the filling factor of the mask while keeping the grating structure unchanged in one RMG. The grating structures are designed and optimized based on rigorous coupled wave analysis and particle swarm optimization. The feasibility of our method is verified by the simulation results in Lighttools. In the FOV range of 20°×15°, the eyebox uniformities of all fields are greater than 0.78, which can provide a good visual experience for users.

Eigendecomposition-free inverse design of meta-optics devices

The inverse design of meta-optics has received much attention in recent years. In this paper, we propose a GPU-friendly inverse design framework based on improved eigendecomposition-free rigorous diffraction interface theory, which offers up to 16.2 × speedup over the traditional inverse design based on rigorous coupled-wave analysis. We further improve the framework’s flexibility by introducing a hybrid parameterization combining neural-implicit and traditional shape optimization. We demonstrate the effectiveness of our framework through intricate tasks, including the inverse design of reconfigurable free-form meta-atoms.

Angular uniformity improvement of diffractive waveguide display based on region geometry optimization.

Augmented reality (AR) near-eye displays have significantly progressed due to advances in nanostructure fabrication. However, for diffractive waveguide AR displays requiring exit pupil expansion, the angular uniformity of each exit pupil position still needs to improve. In this paper, an angular uniformity improvement method based on region geometry optimization is proposed. This optimization method essentially introduces the interaction number of the light with the grating as one of the variables to manipulate the energy distribution. This distribution is obtained by the rigorous coupled wave analysis (RCWA) method and ray tracing process and is further optimized by a multi-objective genetic algorithm. A model is built, and the feasibility of the proposed method is verified. The diffractive waveguide system has a 10m m×10m m exit pupil size at the eye relief of 25mm and a field of view (FOV) of 21∘×12∘. After the optimization, the overall optical efficiency of the central field and the angular uniformity at the center exit pupil position increased from 0.9% and 66% to 3.1% and 80%, respectively.

Near-unity nonreciprocal thermal radiation in biaxial van der Waals material-Weyl semimetal heterostructures

Polaritons in natural van der Waals (vdW) materials have drawn significant attention as the unique optical properties and have opened doors for various potential applications. α-phase molybdenum trioxide (α-MoO3) has been revealed as a semiconducting anisotropic vdW material that holds great promise for manipulating orthogonal in-plane phonon polariton (PhP) modes in infrared (IR). Here, we consider an attenuated total reflection (ATR) geometry consisting of an α-MoO3 layer and a Weyl semimetal (WSM) interlayer sitting on a silver (Ag) substrate. The incorperated coupling prism is selected as KRS5 (thallium bromide-iodide) that can provide a larger wavevector. The spectral responses of the proposal have been investigated using electromagnetic (EM) simulations that utilize the anisotropic rigorous coupled-wave analysis (aRCWA). Our results demonstrate that the α-MoO3 thin layer can boost the constrast between the emissivity and absorptivity in a broad range of wavelength as compared to an uncoated device. The nonreciprocity at resonance wavelength can reach near-unity for both transverse electric (TE) and transverse magnetic (TM) wave incidence. The proposed α-MoO3/WSM heterostructure suggests great potentials for thermal radiation management with different demands.