Geometrical Optics Theory Research Articles

Revisiting the ‘magic condition’ on the basis of the Takagi–Taupin theory

A numerical framework based on the integral solution of the Takagi–Taupin equations has been developed for cylindrically bent Laue crystals. On the basis of this framework, diffraction geometries that satisfy the `magic condition' have been studied from the perspective of dynamical theory. The numerical findings indicate that, in certain diffraction geometries, the focusing behaviour of cylindrically bent Laue crystals will be notably influenced by dynamical effects and the foci of different energies will not converge as predicted by the `magic condition', which is derived from geometric optics theory. These dynamical effects are further explained through a direct numerical analysis of the influence function.

A mathematical model to generate a light reflector that produces a uniform irradiance distribution from a point light source

In this work, we present the development of a mathematical model profile for a light reflector that generates a uniform irradiance distribution from a point light source on a flat region at a fixed distance. The mathematical model is based on the theory of geometric optics, particularly the concepts of reflection, ray tracing, point light sources, spherical caps, and elliptical curvatures. Numerical simulations illustrate that the mathematical model is adequate because its geometry produces a uniform irradiance from a point source. This model allows manufacturers reflectors smaller than parabolic reflectors, which are currently used to generate this type of uniform light irradiance distribution, allowing these devices to be used in smaller optoelectronic systems.

X-ray fluorescence detection system for trace elements in oil based on graphite curved crystal technology

Curved crystal technology is a key technology in the field of X-ray fluorescence trace detection. To detect trace sulfur, chlorine, and silicon in oil, the key parameters of Johann-type graphite curved crystal design were calculated based on geometrical optics theory and mathematical modeling, and a curved crystal optical system was designed. Subsequently, the hardware, firmware, software, and mechanical structure were designed according to specific requirements. Following the development of the relevant equipment, laboratory tests were performed on standard petroleum samples. Experimental results show that technology based on bent crystals can significantly reduce the detection limit of the instrument, and the detection limit of sulfur, chlorine, and silicon can reach below 1 ppm. In accordance with the experimental results, possible interference factors were analyzed, and suitable solutions were proposed. The performance of the proposed system was thus directly verified, showcasing its applicability in detecting trace elements.

Design method for lens desensitization adapted to refractive index error

The design of optical systems not only considers the imaging performance but also the manufacturing difficulty and feasibility of the system. In practice, errors in the manufacturing process of glass materials and deviations in glass material parameters introduced in complex environments can both lead to degradation in the imaging quality of optical systems. Optical systems that are sensitive to glass material errors face increased manufacturing difficulty and reduced stability. This paper, based on geometrical optics theory, establishes an evaluation function for refractive index error sensitivity and analyzes its relationship with optical parameters and glass materials. It proposes a design method to reduce the refractive index sensitivity of optical systems. Through simulation verification and analysis using examples, the validity of the desensitization design method is confirmed.

Degradation mechanism of measuring performance of optical particle counter under temperature-pressure coupling effect

To investigate the degradation mechanism of measuring the performance of an optical particle counter (OPC) under temperature-pressure coupling, this study first establishes a theoretical calculation model of gas refractive index and then elucidates the comprehensive influence mechanism of temperature and pressure on gas optical properties. Furthermore, the experimental measurement technique and measuring device for gas refractive index are built. By comparing the theoretical and experimental results in the temperature range of 48 °C–560 °C and the pressure range of 0.9–4.6 MPa, the difference between the two errors is just 0.05%, indicating the accuracy of the theoretical model of the refractive index of gas. Secondly, a dynamic model of optical measurement volume (OMV) under high-temperature and high-pressure conditions was established using geometrical optics theory, and the impacts of gas temperature and pressure variations on OPC measurement performance were investigated. The gas temperature (100 °C–1000 °C) and pressure (1–4.6 MPa) are shown to have opposing effects on the OMV, with gas pressure being more relevant. Finally, in order to eliminate the effect of gas refractive index change on the optical measurement performance of OPC, a parallel light model is proposed to solve the problem of the degradation of OPC measurement performance under temperature and pressure coupling conditions.

Design and Study of a Two-Dimensional (2D) All-Optical Spatial Mapping Module.

Sequentially timed all-optical mapping photography is one of the main emerging ultra-fast detection technologies that can be widely applicable to ultra-fast detection at the picosecond level in fields such as materials and life sciences. We propose a new optical structure for an all-optical spatial mapping module that can control the optical field of two-dimensional imaging while improving spectral resolution and detector sensor utilization. The model of optical parameters based on geometrical optics theory for the given structure has been established, and the theoretical analysis of the inter-frame energy crosstalk caused by incident beam spot width, chromatic aberration, and main errors of the periscope array has been conducted. The optical design of the two-dimensional (2D) all-optical spatial mapping module was finally completed using ZEMAX OpticStudio 2018 software. The results show that our optical module can realize targets of 16 frames and 1.25 nm spectral resolution.

Optical Simulation Design of a Short Lens Length with a Curved Image Plane and Relative Illumination Analysis.

This study proposes a three-lens design with a short lens length and explores the curved imaging plane and performs a relative illumination analysis. There are two ways to reduce the lens length: shortening the back focal and lens group lengths. We derived the relevant parameter relationships of three lenses using the first-order geometric optics theory. The optical lens length can be controlled within 2 mm. The shorter the lens length, the larger the angle of the chief ray in the image space, resulting in an increase in the field curvature and astigmatism. Third-order Seidel aberrations can be effectively reduced by a curved image plane. We also derived the equations for relative illuminance, solid angle, surface transmittance, and internal transmittance for the short three-lens design. The optical lens design uses a curved image plane to shorten the distance from the off-axis beam image space to the image plane and reduce the incident angle of the chief ray on the image plane. The formula and design results verified by Code V software (version 11.2) show that both the solid angle and relative contrast of the lens can be increased. For the proposed three-lens design with a short lens length, the semi-field angle is 32°, F/# is 2.7, the effective focal length is 1.984 mm, the image plane area is 2.16 mm × 1.22 mm, and the curvature radius of the concave image plane is 3.726 mm. Moroever, the MTF (100 lp/mm) is larger than 52%, the lateral color aberration is less than 2.12 μm, the optical distortion is less than 2.00%, and the relative illumination is greater than 68%.

An analytical urban temperature model with building heterogeneity using geometric optical theory

The enhancement of the living conditions in big cities since the end of the last century is closely related to changes in the thermal environment and besides in urban microclimate, particularly for metropolitan areas. In this context, a knowledge of the spatial and temporal variability of urban heat island (UHI) became an increasing matter of concern, which can be measured from land surface temperature (LST). Actually, LST can be derived from thermal infrared (TIR) remote sensing observations to ensure the necessary spatial and time frequency coverage. But a full exploitation of satellite TIR data cannot be achieved without accounting for the strong anisotropy of urban landscape. Hitherto, a poor investigation was focused on the modeling and the analysis of the directional anisotropies of LSTs considering the complexity of urban building surfaces (e.g., heterogeneity of building morphology and temperature distribution) whereas it is fundamental to establish reliable critical indicators derived from energy balance. Herein, we propose an analytical model to simulate the angular signatures of urban temperatures, in which the geometric optical theory is considered to model the direct radiances of the main components (i.e., sunlit and shaded street, roof and wall). The built model assumes a random distribution of low/middle-rise and high-rise buildings, which depicts realistically the heterogeneity of urban architectural distribution. We evaluated the proposed model using both measured datasets from airborne and satellite sensors and a simulated dataset from a 3D ray-tracing model so-called discrete anisotropic radiative transfer (DART). Results indicate that 1) the proposed model is effective for simulating directional anisotropies of LSTs, with a root mean square error (RMSE) lower than 0.90 °C and R2 > 0.49 for comparison with measured datasets; and 2) the directional anisotropies of LSTs are significantly affected by variations in building height, with values possibly exceeding 1.5 °C. The proposed model can be perceived as a useful tool to analyze the contribution of each component and to assess the impact of urban structure. Furthermore, it can serve to improve urban radiation budget estimations in mixed pixels.

Effect of the focusing system on measurements in gravitational wave detection telescope

Telescopes primarily transmit and receive laser beams over long distances as part of a gravitational wave interferometric measurement system. Due to factors such as optical design, fabrication, and alignment, the wavefront at the exit pupil of the telescope inevitably experiences distortion, resulting in wavefront aberrations that couple with pointing jitter to generate tilt-to-length (TTL) coupling noise. During the process of gravitational wave detection, the large distance between the primary and secondary mirrors and temperature fluctuations in space can cause significant axial misalignment between them. This results in a substantial displacement of the primary-secondary mirror system’s primary focus along the axial direction, further degrading the wavefront at the exit pupil of the telescope. The TTL coupling noise caused in this scenario will affect the detection of gravitational waves, thus requiring the adjustment of the position of the three-four mirror system through the focusing system to minimize TTL coupling noise. In this paper, the model for TTL coupling noise was established using the first 36 orders of Zernike polynomials. The misalignment model of the primary-secondary mirror system was derived using geometric optics theory. The study investigates the influence of the telescope focusing system before and after focusing on the wavefront aberrations and TTL coupling noise at the exit pupil of the telescope. The analysis indicates that with a misalignment of 7.56 μm in the axial distance between the primary and secondary mirrors, the addition of a focusing system reduces the wavefront error at the exit pupil of the telescope from 0.0328 λ to 0.0046 λ. Furthermore, the maximum coupling noise between wavefront distortion and pointing jitter is reduced from 4 pm Hz−1/2 to 0.4 pm Hz−1/2. This provides valuable insights for the design of gravitational wave detection telescopes and the study of focusing systems.

Research on Propagation Characteristics Based on Channel Measurements and Simulations in a Typical Open Indoor Environment

At present, it is difficult to obtain the indoor propagation loss quickly and accurately by directly using measurements in the millimeter wave band. To solve this problem, in this paper, a ray tracing method suitable for indoor scenes based on geometric optics theory, the uniform theory of diffraction and image theory is presented; the space-alternation generalized expectation-maximization (SAGE) algorithm is used to analyze the measured data and the multipath information of the wireless channel is analyzed; three deep learning models are used to predict the path loss at different receiving distances based on 1600 sets of path loss data. The results show that the comparison between the ray tracing and experimental results shows a good agreement. Moreover, the root-mean-square error (RMSE) and mean absolute error (MAE) of the long short-term memory (LSTM) network are the smallest, and the LSTM has a better fitting effect on the propagation loss sequences predicted at more distant locations when compared with the recurrent neural network (RNN) and gate recurrent unit (GRU) methods, which can better reflect the propagation trend. This provides theoretical support for the layout of base stations and network optimization in typical open indoor environments.

The impact of a droplet's light refraction on the size enlargement of its internal objects

Liquid droplets are objects of study in many industries due to their wide range of applications, including in agriculture and energy. When studying a droplet with a deformed object inside it, the length or radius of the object inside it will not be directly measured by photographing of the side view due to the refraction effect. In order to solve this problem, this paper analyzes the magnification of the object inside the droplet by geometric optics theory, derives the formula of magnification, and verifies it using practical experiments and simulations to illustrate the validity of the formula. It is also found that the amplification effect decreases as the internal object increases. The conclusions of this paper can correct the size amplification effect of the object inside the droplet due to refraction, which is a guideline for experimental and computational analysis in the field of droplet-related research.

Single-Pixel Imaging for Partially Occluded Objects

In some complex scene, such as military reconnaissance and other fields, the target object may be partially occluded by some opaque obstacles, which increase the difficulty to obtain the complete object image. In traditional optical imaging system, the lenslet array or the reference beam and special algorithms are usually needed to solve this problem, which increase the complexity of the system. As a computational optical imaging technology, the problem of single-pixel imaging (SPI) for an occluded object has not been researched yet as we know. Here we propose an SPI system to recover the occluded object image. Firstly, the images of object and occlusion from different perspectives are obtained by detecting the light field at different positions. Secondly, the limited conditions of the system parameters are derived by using geometric optics theory to ensure complete imaging. Thirdly, the feasibility of the scheme and the correctness of the limited conditions are verified by experiments. The results in this article can promote the application of SPI technology in many complex practices.

Analysis of 5G channel loss and its influencing factors in substation based on ray tracing algorithm

Abstract5G (fifth generation) signals have small coverage and poor stability, and 5G channels are prone to loss in substations densely populated with metal equipment. Due to the particularity of the spatial layout in the substation, traditional physical or statistical channel models cannot accurately calculate the 5G channel loss in the substation. Therefore, this paper introduces the basic idea of ray tracing algorithm for studying ray propagation. Based on the geometric optics theory of ray propagation, the effective channel path of 5G signal in substation is found. Through the electromagnetic reflection theory and consistent diffraction theory, the energy loss caused by signal reflection and diffraction is calculated. Combined with the relationship between the free space loss of 5G signal on the transceiver path and the change of electric field, the solution method of 5G channel loss in substation is deduced. According to this method, (the influence of signal receiver height, 5G antenna height and its spatial position in horizontal and vertical directions on 5G channel loss of substation is studied in turn. By optimizing the layout of communication facilities according to the influence law, the reliability of 5G channel in substation can be realized).

Desensitization design method of a freeform optical system based on local curve control.

In this Letter, an error sensitivity evaluation function of freeform optical systems is proposed, and a desensitization design method is established. This method adopts the idea of micro-elements and, based on geometric optics theory, studies the relationship between the local curve of the freeform surface and the change of the wavefront error (ΔWE) when the optical system is disturbed by the position error, and realizes the desensitization design of the optical system. By simply changing the evaluation function, the method can be applied to the desensitization design of any optical system with any surface (spherical, aspheric, and freeform surface).

Design of a measurement system for measuring geometric errors of a workpiece table during two-axis synchronous motion

With the increasing customer demand for high-precision workpieces, the accuracy performance of CNC machine tools has become an increasingly important hot topic for almost all machine tool manufacturers. For fine surface finish workpieces, the performance of multi-axis synchronous motion has a significant impact on lathe processing. Accordingly, it is necessary to calibrate and identify the geometric errors of the workpiece table during multiple-axis synchronous motion. Therefore, this study proposes a novel noncontact optical measurement system for measuring the geometric errors of the workpiece table while the two axes are moving simultaneously. In addition, the horizontal CNC lathe machine iTC-2000LM, produced by Tongtai machine & tool Co., Ltd., was taken as the research vehicle. The proposed measurement system comprises an optics module composed of several reflectors, lenses, and cubic beam splitters; a sensing module composed of multiple two-dimensional position-sensitive detectors (PSDs); and a laser. Based on the principle of geometric optics theory and the optical triangulation method, complete mathematical models that describe the geometric errors of the workpiece table during two-axis synchronous motion were established using the skew-ray tracing method. Additionally, the essential features of the proposed measurement system were verified using the optical design software Zemax. Finally, a laboratory prototype system was constructed to experimentally verify the proposed measurement system. As mentioned above, the main contribution of this study is to design a novel and highly efficient measurement system with established measurement mathematical models and experimental verifications.

Optical element to generate zero-order quasi-Bessel beam with "focal length".

An optical element has been invented to generate a zero-order quasi-Bessel beam with a certain distance to the element, which does not exist in the zero-order quasi-Bessel beam by using a traditional axicon. The cross section of designed element is an isosceles triangle whose equal sides are circumscribed by two semi-ellipses. Using a well-developed three-dimensional (3D)-printing technique, we have fabricated a series of elements working at terahertz (THz) frequency. Both simulated and experimental results clearly show that there is a certain distance between the generated quasi-Bessel beam and this element. A physical analysis based on geometric optics theory is performed to explain the obtained results. Because it is a refractive transmitted optical element, we propose that it can be also realized at another frequency band if the relevant processing techniques are available.

Characterization of NDSI Variation: Implications for Snow Cover Mapping

The normalized difference snow index (NDSI) plays an important role in mapping snow cover with spaceborne visible and shortwave-infrared imagery. The NDSI variation depends on illuminating-viewing geometry and snow physical properties, including equivalent grain size (EGS), snow depth (SD) and impurity concentration, as well as fractional snow cover (FSC) within a mixed pixel; however, it is still not fully understood. To quantifiably characterize the pattern of snow NDSI variation, we use a light scattering model of snow to calculate bidirectional reflectance and consequent NDSI values for a wide range of illuminating-viewing geometries, SD, and EGS values. In the model designated bicontinuous snow model using Geometric Optics theory and Radiative Transfer Equation (bicontinuous-GO/RTE), snowpack is represented by a bicontinuous microstructure, and bidirectional reflectance is simulated based on geometric optics and vector radiative transfer equation. The discrete ordinates radiative transfer (DISORT) algorithm is used to simulate the soot concentration effect on snow NDSI. A soil spectral reflectance model (SOILSPECT) is utilized with the assumption of the linear spectral mixture of snow and soil to quantify the effect of FSC on NDSI. As for discontinuous forests, an analytical hybrid geometric-optical and radiative transfer (GORT) model in conjunction with the bicontinuous-GO/RTE model and the PROpriétésSPECTrales (PROSPECT) model is used to examine the effect of canopy cover, which is related to the maximal FSC viewable to satellites. Modeling results indicate that: 1) snow NDSI is comparably low at off-nadir viewing angles, and this effect is exacerbated by the decline in solar elevation but limited by large EGS; 2) snow NDSI increases with EGS yet becomes saturated at EGS of 500 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> ; 3) the effect of SD that is as low as 1 cm on NDSI is rarely noticeable; 4) the concentration of internally mixed soot up to 1 ppm has little reducing effect on snow NDSI (<0.05); 5) the angular effects on the NDSI thresholding method to identify binary snow cover for soil background are nonsignificantly pronounced in the cross plane (CP); 6) NDSI increases linearly with FSC for soil background at nadir as well as in the CP; 7) the SNOw-MAPping algorithm (SNOMAP) method could omit snow cover at large zenith angles for solar zenith angles <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$&gt; 20^{\circ }$ </tex-math></inline-formula> in forests; and 8) forests complicate the nonlinear relationship between NDSI and canopy cover with fully snow-covered ground beneath canopies. These findings imply important uncertainty sources of binary and FSC mapping with NDSI.

Effects of leaf age and canopy structure on gross ecosystem production in a subtropical evergreen Chinese fir forest

Despite the advances in technologies to derive the maximum rate of Rubisco carboxylation (Vcmax) seasonality at large-scale, factors controlling the temporal dynamics of Vcmax is largely unknown without extensive field measurements at stand scale. In addition, state-of-the-art process-based terrestrial ecosystem models had not accounted the complex canopy structure for estimating radiation interception by vegetation. Both of which could lead to uncertainties in gross ecosystem production (GEP) estimations. Here, we examined the respective and combined effects of leaf age and canopy structure on GEP by integrating the Farquhar photosynthesis model with a two-leaf (sunlit and shaded) canopy radiation interception model based on the Geometric Optical and Radiative Transfer (GORT) theory. We observed that the Vcmax of new leaves was approximately 34.4% higher than that of mature leaves. Considering the seasonal dynamics of Vcmax caused by new leaf expansion, the average bias between the modeled GEP and that derived from the eddy covariance (EC) in the growing season at 8-day temporal scale was reduced from -8% to -1%, with slightly higher correlation (from 0.9 to 0.92) and reduction in root mean squared error (from 6.0 to 3.9 g C m−2 8d−1). Most importantly, the total effect of new leaf expansion and canopy gaps during the growing season was +322 g C m−2 yr−1 and -114 g C m−2 yr−1, which was approximately 22.5% and 8.1% of the total GEP, respectively. Thus, it is important to consider both leaf age and canopy structure when developing a robust terrestrial ecosystem model. We highlighted that Vcmax seasonality and realistic canopy radiation interception simulation are critical for accurate estimation of canopy GEP for evergreen forests.

End-to-end complex lens design with differentiate ray tracing

Imaging systems have long been designed in separated steps: experience-driven optical design followed by sophisticated image processing. Although recent advances in computational imaging aim to bridge the gap in an end-to-end fashion, the image formation models used in these approaches have been quite simplistic, built either on simple wave optics models such as Fourier transform, or on similar paraxial models. Such models only support the optimization of a single lens surface, which limits the achievable image quality. To overcome these challenges, we propose a general end-to-end complex lens design framework enabled by a differentiable ray tracing image formation model. Specifically, our model relies on the differentiable ray tracing rendering engine to render optical images in the full field by taking into account all on/off-axis aberrations governed by the theory of geometric optics. Our design pipeline can jointly optimize the lens module and the image reconstruction network for a specific imaging task. We demonstrate the effectiveness of the proposed method on two typical applications, including large field-of-view imaging and extended depth-of-field imaging. Both simulation and experimental results show superior image quality compared with conventional lens designs. Our framework offers a competitive alternative for the design of modern imaging systems.

Optical nano-imaging via microsphere compound lenses working in non-contact mode.

Microsphere lens for nano-imaging has been widely studied because of its superior resolving power, real-time imaging characteristic, and wide applicability on diverse samples. However, the further development of the microsphere microscope has been restricted by its limited magnification and small field-of-view. In this paper, the microsphere compound lenses (MCL) which allow enlarged magnification and field-of-view simultaneously in non-contact imaging mode have been demonstrated. A theoretical model involving wave-optics effects is established to guide the design of MCL for different magnifications and imaging configurations, which is more precise compared with common geometric optics theory. Experimentally, using MCL to image the specimen with a tunable magnification from 2.8× to 10.3× is realized. Due to the enlarged magnification, a high-resolution target with 137 nm line width can be resolved by a 10× objective. Besides, the field-of-view of MCL is larger than that of a single microsphere and can be further increased through scanning working manner, which has been demonstrated by imaging a sample with ∼76 nm minimum feature size in a large area. Prospectively, the well-designed MCL will become irreplaceable components to improve the imaging performances of microsphere microscope just like the compound lens in the conventional macroscopic imaging system.