Geometrical Optics Model Research Articles

Mathematical modelling and optimizing design of heliostat field

The heliostat field is a form of solar thermal power plant, The heliostat field is a crucial component of solar thermal power plants and establishing a heliostat field is of great significance for reducing carbon emissions. and is of great significance in reducing carbon emissions This paper focuses on the mathematical modeling and optimization design problem of the heliostat field. For three different design scenarios, firstly, based on the data provided by the China Undergraduate Mathematical Contest in Modeling 2023, a geometric optical efficiency model is proposed to determine the annual average optical efficiency and output thermal power under specific heliostat field design parameters. Secondly, aiming at the optimal design parameters of the solar furnace field under the condition of rated annual average output thermal power is 60 MW\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{MW}$$\\end{document}, Particle Swarm Optimization (PSO) combing with efficiency model is utilized to determine the optimal design parameters of the heliostat field. Finally, this paper uses Quasi-Monte Carlo ray tracing (QMCRT) model to perform simulations, exploring how to obtain the optimal design parameters of the heliostat field with the goals of maximizing the output thermal power and ensuring high efficiency. Furthermore, the influence of normal disturbances on the annual average annual average thermal output power is investigated.

Depth localization of subsurface defects by optical dark-field confocal microscopy

Abstract Subsurface defects (SSD) in optical components pose a significant challenge for enhancing the power density of high-energy laser devices. This study investigated the issue of systematic deviation between the measured and actual depths of SSD when employing optical dark-field confocal microscopy for three-dimensional measurements, which is attributed to refractive index disparities between the sample and the observation environment. This paper introduces geometric and diffraction optical models for correcting errors in the SSD depth, along with a calculation method for determining the correction coefficient. By comparing the experimental data and model simulations, a linear relationship between the measured and actual depths was identified with linearity errors below 2.5% and a minimum of 0.67%. The correction coefficients derived from the optical diffraction model are in good agreement with those obtained experimentally. These findings offer valuable insights for calculating SSD depth correction coefficients across various scenarios and requirements to ensure precise measurements.

Geometric-Optical Model of Digital Holographic Particle Recording System and Features of Its Application

The paper proposes an equivalent optical scheme of an in-line digital holographic system for particle recording and a mathematical model that establishes a one-to-one correspondence between the dimensional and spatial parameters of a digital holographic image of a particle and the imaged particle itself. The values of the model coefficients used to determine the real size and longitudinal coordinate of a particle according to its holographic image are found by calibration. The model was tested in field and laboratory conditions to calibrate a submersible digital holographic camera designed to study plankton in its habitat. It was shown that four calibration measurements are sufficient enough to determine the model coefficients, and the developed design of the submersible digital holographic camera makes it possible to perform these measurements during the recording of each hologram. In addition, this neither requires data on the refractive index of the medium with particles nor on the parameters of the optical elements of the scheme. The paper presents the results of marine experiments in the Kara Sea and the Laptev Sea, as well as in fresh water in laboratory conditions and in Lake Baikal. The error in measuring the particle size in seawater without the use of the model is 53.8%, while the error in determining their longitudinal coordinates is 79.3%. In fresh water, the same errors were 59% and 54.5%, respectively. The error in determining the position of a particle with the use of the designed mathematical model does not exceed 1.5%, and the error in determining the size is 4.8%. The model is sensitive to changes in the optical properties of the medium, so it is necessary to perform calibration in each water area, and one calibration is quite sufficient within the same water area. At the same time, the developed design of the submersible holographic camera allows, if necessary, calibration at each holographing of the medium volume with particles.

Design and Verification of Geometric Optical Approximation Model Based on Ray Tracing Technique and Electromagnetic Equivalence Principle

Design and Verification of Geometric Optical Approximation Model Based on Ray Tracing Technique and Electromagnetic Equivalence Principle

Constructing the large-scale collimating solar simulator with a light half-divergence angle

Large-scale collimating solar simulators (CSSs) with a light half divergence angle (HDA) of less than 1° have been developed. The small HDA ensures that the CSS can effectively simulate the optical behavior between natural sunlight and optical devices. However, the construction of large-scale CSSs still has a strict threshold because additional optical modules (AOMs), such as large-area collimating mirrors, are usually required to correct the light into collimated light. The manufacturing, installation, and adjustment of these precision optical devices pose significant challenges. An excellent scheme to avoid AOMs is to use the collimating radiation module (CRM), which can directly produce collimated light. Unfortunately, CRM can only produce light with an HDA of more than 3° at present, far larger than that of CSS with AOMs. Here, we report a CRM that can directly produce light with excellent collimation (an HDA<0.955°) and uniformity (>90 %). We accomplished this by analyzing the deviations between an idealized geometric optical model and actual CRMs and eliminating them with high-precision parts and a high-resolution adjustment method. We further used 24 CRMs to prove that the single-module collimating solar simulator (SMCSS, a radiation area of 2.55 m × 1.57 m) could be modularly constructed by them. Experimental investigations involving light-concentrating experiments on a parabolic trough collector demonstrated the superior collimation and simulation capabilities of the SMCSS. By eliminating the need for AOMs, the CRM and SMCSS significantly reduce system complexity and cost and lower the construction threshold for large-scale CSSs. It will benefit all experimental scenarios that need large-area collimated light and greatly promote the application of large-scale CSS in civilian solar research.

Simulating snow-covered forest bidirectional reflectance by extending hybrid geometric optical–radiative transfer model

Accurate simulations of the bidirectional reflectance distribution function (BRDF) of snow-covered forests are crucial for retrieving snow and canopy properties, which affects the analysis of net radiation, radiative forcing, and climate change. The anisotropic reflectance of snow in conifer forests is greatly influenced by the high reflectance of snow and the complex travel pathways of radiation in 3D canopy structures, which complicates BRDF simulations. Although bidirectional reflection models have been widely established for snow-free forests, few studies have attempted to simulate the BRDF in snow-covered forests. Therefore, we propose a new snow-covered forest bidirectional reflectance (SFBR) model by coupling geometric optical and radiative transfer models, which includes a snow anisotropic reflectance model (ART), needle leaf optical properties model (LIBERTY), canopy radiative transfer model (4SAIL), and geometric optical model (GOST2). The SFBR model mainly focuses on conifer forests with snow backgrounds; it can simulate anisotropic reflectance spectra from 400 nm to 2500 nm and is the only model to fully consider discontinuous canopy distribution, topographic effects, and snow anisotropy. The SFBR model was validated by simulating the reflectance over snow-covered forests, the results of which showed good consistency with simulations of 3D scene model and remote sensing observations. The proposed SFBR model has the potential to quantify radiation interactions between the snow background and forest overstory, evaluate the sensitivity of canopy reflectance to snow and forest properties, and retrieve snow-covered forest parameters from satellite-observed reflectance data.

Modeling of a multireflector terahertz imaging system using a geometrical optics model based on angular spectrum theory.

We propose a simulation method for a multireflector terahertz imaging system. The description and verification of the method are based on an existing active bifocal terahertz imaging system at 0.22 THz. Using the phase conversion factor and angular spectrum propagation, the computation of the incident and received fields requires only a simple matrix operation. The phase angle is used to calculate the ray tracking direction, and the total optical path is used to calculate the scattering field of defective foams. Compared with the measurements and simulations of aluminum disks and defective foams, the validity of the simulation method is confirmed in the field of view of 50 cm × 90 cm at 8 m. This work aims to develop better imaging systems by predicting their imaging behavior for different targets before manufacturing.

Mental Representations and Cognitive Schemata of Ninth Grade Students for the Refraction of Light

The current research study deals with students’ mental representations and cognitive schemata of light refraction. In the study, 213 ninth grade students participated who had taken basic Geometric Optics courses on refraction and Snell’s law. The students were given three tasks in which they were asked to predict and explain the phenomenon of refraction. The results showed that the vast majority of them articulated their responses based on representations that were not compatible with the Geometric Optics model. Quite interestingly, the Multiple Correspondence Analysis led to five distinct cognitive schemata resulting from a fixed combination of representations.

Shadow is related to roughness but MODIS BRDF should not be used to estimate lateral cover

A simple geometric-optical model is used to show that shadow in optical remote sensing imagery contains information about surface roughness. The use of MODIS albedo products for the purpose of estimating surface roughness so that these products may aid in dust emission modeling has been proposed in previous studies (Chappell et al. 2010; Chappell and Webb 2016; Chappell et al. 2018). This approach is evaluated through geometric-optical modeling and by analysis of existing MODIS data. Little support is found for the utility of this approach for dust emission modeling. However, because there is clearly information about surface roughness in optical satellite imagery, some suggestions for the proper use of optical data to estimate surface roughness are made.Chappell, A., Van Pelt, S., Zobeck, T., & Dong, Z. (2010). Estimating aerodynamic resistance of rough surfaces using angular reflectance. Remote Sensing of Environment, 114, 1462–1470.Chappell, A., & Webb, N.P. (2016). Using albedo to reform wind erosion modeling, mapping and monitoring. Aeolian Research, 23, 63–78.Chappell, A., Webb, N.P., Guerschman, J.P., Thomas, D.T., Mata, G., Handcock, R.N., Leys, J.F., & Butler, H.J. (2018). Improving ground cover monitoring for wind erosion assessment using MODIS BRDF parameters. Remote Sensing of Environment, 204, 756–768.

Effective Surface Roughness Impact in Polarimetric GNSS-R Soil Moisture Retrievals

Single-pass soil moisture retrieval has been a key objective of Global Navigation Satellite System-Reflectometry (GNSS-R) for the last decade. Achieving this goal will allow small satellites with GNSS-R payloads to perform such retrievals at high temporal resolutions. Properly modeling the soil surface roughness is key to providing high-quality soil moisture estimations. In the present work, the Physical Optics and Geometric Optics models of the Kirchhoff Approximation are implemented to the coherent and incoherent components of the reflectometry measurements collected by the SMAP radar receiver (SMAP-Reflectometry or SMAP-R). Two surface roughness products are retrieved and compared for a single-polarization approach, critical for single-polarization GNSS-R instruments that target soil moisture retrievals. Then, a polarization decoupling model is implemented for a dual-polarization retrieval approach, where the ratio between two orthogonal polarizations is evaluated to estimate soil moisture. Differences between linear and circular polarization ratios are evaluated using this decoupling parameter, and the theoretical soil moisture error with varying decoupling parameters is analyzed. Our results show a 1-sigma soil moisture error of 0.08 cm3/cm3 for the dual-polarization case for a fixed polarization decoupling value used for the whole Earth, and a 2-sigma error of 0.08 cm3/cm3 when the measured reflectivity and the VOD are used to estimate the polarization decoupling parameter.

A novel retrieval model for soil salinity from CYGNSS: Algorithm and test in the Yellow River Delta

Soil salinization, which occurs mainly in arid and coastal regions, hampers agricultural development, and threatens food security in these regions. Therefore, acquiring the spatial distribution of soil salinization with high accuracy is paramount. However, obstacles continue to hinder attempts to use satellite remote sensing. Cyclone Global Navigation Satellite System (CYGNSS) provides a new alternative opportunity for soil salinization retrieval. In this study, soil electrical conductivity (EC) was chosen as the proxy of soil salinity. A CYGNSS-based model for soil EC retrieval (CIG) was proposed, and has been tested in the Yellow River Delta, China. The CIG model consisted of two modules. First, we modified the geometrical optics model to derive the surface’s Fresnel reflectivity from the incoherent scattering signal of the land surface with vegetation attenuation, and horizontal and vertical surface roughness calibrations. Second, soil EC was retrieved based on the gradient boost regression tree (GBRT) algorithm with the inputs of the surface’s Fresnel reflectivity and other ancillary variables. The in-situ measurements were used to compare the CIG retrieved soil EC against the retrievals from other classical machine learning approaches and the coherent model using CYGNSS data, and the optical SVIs model. CYGNSS signal was found to be highly sensitive to in-situ soil EC, and the proposed CIG model outperformed other models, with R = 0.730, RMSE = 1.318 mS/cm, and MAE = 0.570 mS/cm. In addition, several issues related to the model were also discussed, including the interactions between the incidence angle of the CYGNSS signal and soil EC, and the advantages and limitations of CYGNSS. The result suggested that the proposed CIG model, along with new CYGNSS data, can provide a promising method for monitoring land salinization on a large scale.

Spatial scale applicability of 4-Scale geometrical optics model canopy reflectance simulation.

Accurately clarifying the applicable spatial scale of 4-Scale model is conducive to improving the accuracy of its application in canopy reflectance simulation of different vegetation types, and to further improving the inversion accuracy of leaf area index, canopy density, and other parameters. Two forest plots (one for broad-leaved forest and one for mixed forest) with each area of 100 m×100 m in Maoershan Experimental Forest Farm, Shangzhi, Heilongjiang, were divided into the spatial scales of 10, 20, 30, 40 and 50 m, respectively. The 4-Scale model was used to simulate forest canopy reflectance. Local mean method, the nearest neighbor method, bilinear interpolation method, and cubic convolution method were used to convert Sentinel-2 images with spatial resolution of 10 m to other scales, with the results being evaluated. The simulated canopy reflectance and remote sensing pixel reflectance were compared and analyzed. The spatial scale of mixed forest and broad-leaved forest suitable for high-precision inversion parameters of 4-Scale model was determined. The results showed that the 4-Scale model underestimated the pixel forest canopy reflectance as a whole. The canopy reflectance of mixed forest and broad-leaved forest had the worst simulation effect at the 20 m scale. Both the root mean square error (RMSE) and the mean absolute error from (MAE) of red and near-infrared band were large. When the scale was >20 m, the simulation effect became better. The applicability of the model was the best when the mixed forest was 40 m and the broad-leaved forest was 30 m. The mean and standard deviation of the reflectance difference between the simulated value and the remote sensing pixel were the minimum in the red and near near-infrared bands, with the minimum RMSE and MAE. The simulation results of mixed forest and broad-leaved forest at 10 m scale were not stable, the rule of mean and standard deviation was inconsistent, and the difference between RMSE and MAE was large under the same band.

The Causes and Effects of Preannealed Radiative Property Variations Across Full‐Hard Advanced High‐Strength Steel Coils

This study investigates the connection between surface topography and radiative property variations across full‐hard (unannealed and cold‐rolled) advanced high‐strength steel (AHSS) coils. Directional–hemispherical reflectivities of samples extracted from various locations on a full‐hard DP‐780 coil are measured using a Fourier transform infrared spectroscopy spectrometer at wavelengths between 0.5 and 20 μm. Surface topography analysis is performed using optical profilometry, optical microscopy, and scanning electron microscopy. Surface height profiles are generated using 3D depth mapping of optical images. The surface profiles are then used in a geometric optical approximation model, which correlates the radiative properties to surface topography variations. Significant differences in radiative properties and surface topography are observed along the length and width of the coil. These variations in radiative properties may explain the temperature excursions that cause nonuniformities in mechanical properties observed across AHSS coils produced on continuous galvanizing lines in the industry.

Optimization Design of Heliostat Field Based on Biomathematical Models

The optimal design of heliostat field in solar power generation device is studied Heliostat field generates high temperature by focusing the sunlight reflected from the mirror surface, thus pushing the turbine to generate electricity. In order to improve the efficiency of power generation, this paper optimizes the parameters of heliostat field, establishes a single-objective optimization model, and uses mixed strategy whale optimization algorithm to solve it. Through geometric optical analysis and ray tracing model, the optical efficiency of heliostat field and the annual average output thermal power per mirror area are calculated. Finally, through the discussion and analysis of the model, the optimization design method is evaluated

Modelling of the backside pattern transfer in magic mirror (Makyoh) imaging

A model of magic-mirror (Makyoh) imaging with the mirror backside relief as the input is described. The mechanical pattern transfer is modelled by a convolution approach while the optical imaging using a nonlinear geometrical optical model, more general than previous approaches. Characteristic features of the imaging is analysed and the linear approximation is examined. Diffraction effects are also treated. Characteristic features for the ancient mirror and applications in semiconductor wafer inspection are discussed.

Unveiling the Role of the Beam Shape in Photothermal Beam Deflection Measurements: A 1D and 2D Complex Geometrical Optics Model Approach

The preponderance of laser beam shapes cannot be ruled out during the implementation of an optical experiment nor during the formulation of its theoretical background. The present work elucidates the role of Gaussian and top-hat beam shapes in generating and analysing the photothermal beam deflection (PBD) signals. The complex geometrical optics models encompassing the perturbations in the phase and amplitude of the probe beam with one-dimensional (1D) and two-dimensional (2D) approaches is employed to curve fit the PBD signal and are compared. From the fitted curve, the thermal diffusivity and conductivity of the sample are calculated with the 1D and 2D models. A uniform intensity distribution over the sample, like a top-hat beam, is achieved using an optical lens system and verified using a beam profiler. When the phase and amplitude of the PBD signal are fitted at different positions of the lens, i.e., in focussed and defocussed conditions, it is observed that difference in the measured thermal characteristics is about 30% for the Gaussian pump beam profile, whereas it is only &lt;4% for top-hat beam. Even though the fitting accuracy and sum of residues estimated for the 2D model are better than 1D, the ease of computation with the 1D model employing top-hat excitation suggests the application of the top-hat profile in photothermal experiments.

Impact of a Turbulent Ocean Surface on Laser Beam Propagation.

The roughness of the ocean surface significantly impacts air-to-sea imaging, oceanographic monitoring, and optical communication. Most current and previous methods for addressing this roughness and its impact on optical propagation are either entirely statistical or theoretical, or are ‘mixed methods’ based on a combination of statistical models and parametric-based physical models. In this paper, we performed experiments in a 50-foot-wave tank on wind-generated waves, in which we varied the wind speed to measure how the surface waves affect the laser beam propagation and develop a geometrical optical model to measure and analyze the refraction angle and slope angle of the laser beam under various environmental conditions. The study results show that the laser beam deviations/distortions and laser beam footprint size are strongly related to wind speed and laser beam incidence angle.

Sun-induced chlorophyll fluorescence is more strongly related to photosynthesis with hemispherical than nadir measurements: Evidence from field observations and model simulations

Solar-induced chlorophyll fluorescence (SIF) has been shown to be a novel proxy for terrestrial gross primary production (GPP). A growing number of ground-based automatic SIF observation systems equipped with hemispherical-conical and bi-hemispherical observation configurations have been developed in synergy with EC flux measurements across different ecosystems. However, the difference in the canopy SIF observed by these two types of configurations has not been well studied, which poses challenges in evaluating their performance in tracking GPP. In this study, we investigated SIF from both hemispherical-conical and bi-hemispherical observation configurations for their ability to track GPP in a maize field during the 2020 growth season. We found that bi-hemispherical SIF observations (SIFHemis) showed higher correlations with GPP at both diurnal and seasonal scales, and the superiority of SIFHemis for GPP estimation was also supported by Soil-Canopy-Observation of Photosynthesis and the Energy balance (SCOPE) model simulations. In addition, we found that the SIFHemis-GPP model established at a satellite overpass time (e.g., 09:30) outperformed the corresponding SIFNadir-GPP model in estimating both the half-hourly and daily GPP. The underlying mechanism for the advantage of this SIFHemis-GPP relationship was elucidated by a simplified geometrical optical model, which showed that the diurnal patterns of the observed sunlit and shaded leaves for the SIFHemis were consistent with those of the canopy GPP. Our study recommends a bi-hemispherical configuration setup for its superiority in monitoring GPP dynamics.

Stand Canopy Closure Estimation in Planted Forests Using a Geometric-Optical Model Based on Remote Sensing

Canopy closure, which is the ratio of the vertical projection area of the crowns to the area of forest land, can indicate the growth and tending situation of a forest and is of great significance for forest management planning. In this study, a geometric-optical model (GOST model) was used to simulate the canopy gap fraction of a forest. Then, a canopy closure estimation method using the gap fraction was discussed. In this study, three typical planted forest farms (the Mengjiagang (MJG), Gaofeng (GF), and Wangyedian (WYD) forest farms) containing the most commonly planted tree species in the north and south regions of China were selected, and field measurements were executed. The results show that the gap fraction (Pvg-c) had a higher correlation with the average projected area of the tree crowns, and the relationship was an exponential function, with R2 and RMSE values of 0.5619 and 0.0723, respectively. Finally, the applicability and accuracy of this method were evaluated using line transects, and a fisheye camera measured the canopy closure. The accuracy of the canopy closure estimated by the Pvg-c was 86.69%. This research can provide a reference for canopy closure estimation using a geometric-optical model.

Determine Corneal Biomechanical Parameters by Finite Element Simulation and Parametric Analysis Based on ORA Measurements.

Purpose: The Ocular Response Analyzer (ORA) is one of the most commonly used devices to measure corneal biomechanics in vivo. Until now, the relationship between the output parameters and corneal typical biomechanical parameters was not clear. Hence, we defined the output parameters of ORA as ORA output parameters. This study aims to propose a method to determine corneal biomechanical parameters based on ORA measurements by finite element simulation and parametric analysis. Methods: Finite element analysis was used to simulate the mechanics process of ORA measurements with different intraocular pressure (IOP), corneal geometrical parameters and corneal biomechanical parameters. A simplified geometrical optics model was built to simulate the optical process of the measurements to extract ORA output parameters. After that, 70% of the simulated data was used to establish the quantitative relationship between corneal biomechanical parameters and ORA output parameters by parametric analysis and 30% of the simulated data was used to validate the established model. Besides, ten normal subjects were included to evaluate the normal range of corneal biomechanical parameters calculated from ORA. Results: The quantitative relationship between corneal biomechanical parameters and ORA output parameters is established by combining parametric analysis with finite element simulation. The elastic modulus (E) and relaxation limit (G ∞) of the ten normal subjects were 0.65 ± 0.07 MPa and 0.26 ± 0.15, respectively. Conclusions: A method was proposed to determine corneal biomechanical parameters based on the results of ORA measurements. The magnitude of the corneal biomechanical parameters calculated according to our method was reasonable.