Geometrical Optics Research Articles

A tactile perception method with flexible grating structural color

Abstract Affordable high-resolution cameras and state-of-the-art computer vision techniques have led to the emergence of various vision-based tactile sensors. However, current vision-based tactile sensors mainly depend on geometric optics or marker tracking for tactile assessments, resulting in limited performance. To solve this dilemma, we introduce optical interference patterns as the visual representation of tactile information for flexible tactile sensors. We propose a novel tactile perception method and its corresponding sensor combining structural colors from flexible blazed gratings with deep learning. The richer structural colors and finer data processing foster the tactile estimation performance. The proposed sensor has an overall normal force magnitude accuracy of 6 mN, a planar resolution of 79 μm, and a contact-depth resolution of 25 μm. This work presents a promising tactile method that combines wave optics, soft materials, and machine learning. It demonstrates well in tactile measurement, and can be expanded into multiple sensing fields.

Improved calculation of single-scattering properties of frozen droplets and frozen-droplet aggregates observed in deep convective clouds

Abstract. During multiple field campaigns, small quasi-spherical ice crystals, commonly referred to as frozen droplets (FDs), and their aggregates (frozen-droplet aggregates, FDAs) have been identified as the predominant habits in the upper regions of deep convective clouds (DCCs) and their associated anvils. These findings highlight the significance of FDs and FDAs for understanding the microphysics and radiative properties of DCCs. Despite the prevalence of FDs and FDAs at the tops of DCCs where they directly contribute to cloud radiative effect, the detailed single-scattering properties (e.g., scattering-phase function P11 and asymmetry parameter g) of FDs and FDAs remain highly uncertain. This uncertainty is mainly due to insufficient in situ measurements and the resolution of cloud probes, which hinder the development of idealized shape models for FDs and FDAs. In this study, two shape models, the Gaussian random sphere (GS) and droxtal (DX), are proposed as possible representations for the shapes of FDs and FDAs measured in situ. A total of 120 individual models of GSs and 129 models of DXs were generated by varying their shapes. Furthermore, by attaching these individual models in both a homogeneous and heterogeneous manner, three different types and a total of 404 models of FDAs were created: (1) aggregates of GSs; (2) aggregates of DXs; and (3) combinations of GSs and DXs, which are called habit mixtures (HMs). The P11 and g values of the developed models were calculated using a geometric optics method at a wavelength of 0.80 µm and then compared with those obtained using a polar nephelometer (PN) during the CIRCLE-2 field campaign to assess the models. Both individual-component ice crystals (i.e., either GS or DX) and homogeneous-component aggregates (i.e., either aggregates of GSs or aggregates of DXs) showed substantial differences compared with the PN measurements, whereas the P11 of the HMs was found to most accurately match the P11 measured in situ, reducing the differences to +0.87 %, +0.88 %, and −5.37 % in the forward-, lateral-, and backward-scattering regions, respectively. The g value of the HMs was found to be 0.80, which falls within the range of the PN measurement (0.78 ± 0.04). The root-mean-square error for the HM was minimized to a value of 0.0427. It was shown that the novel HMs developed in this study demonstrated better performance than in previous research where HMs were developed indirectly by weighting the calculated P11 of shape models to interpret in situ measurement. The results of this study suggest potential implications for enhancing the calculation of single-scattering properties of ice crystals in DCCs.

A simple model of a gravitational lens from geometric optics

We propose a simple geometric optics analog of a gravitational lens with a refractive index equal to one at large distances and scaling like n(r)2=1+C2/r2, where C is a constant. We obtain the equation for ray trajectories from Fermat's principle of least time and the Euler equation. Our model yields a very simple ray trajectory equation. The optical rays bending, reflecting, and looping around the lens are all described by a single trigonometric function in polar coordinates. Optical rays experiencing fatal attraction are described by a hyperbolic function. We use our model to illustrate the formation of Einstein rings and multiple images.

The Twisting of Radio Waves in a Randomly Inhomogeneous Plasma

Polarization of electromagnetic waves carries a large amount of information about their astrophysical emitters and the media they passed through, and hence is crucial in various aspects of astronomy. Here we demonstrate an important but long-overlooked depolarization mechanism in astrophysics: when the polarization vector of light travels along a nonplanar curve, it experiences an additional rotation, in particular for radio waves. The process leads to depolarization, which we call “geometric” depolarization (GDP). We give a concise theoretical analysis of the GDP effect on the transport of radio waves in a randomly inhomogeneous plasma under the geometrical optics approximation. In the case of isotropic scattering in the coronal plasma, we show that the GDP of the angle of arrival of the linearly polarized radio waves propagating through the turbulent plasma cannot be ignored. The GDP effect of linearly polarized radio waves can be generalized to astrophysical phenomena, such as fast radio bursts and stellar radio bursts, etc. Our findings may have a profound impact on the analysis of astrophysical depolarization phenomena.

System of N Thin Coaxial Lenses

In geometrical optics, in a system of two thin coaxial lenses, there are several standard formulas, including “𝟏𝑭=𝟏𝒇𝟏+𝟏𝒇𝟐−𝒅𝒇𝟏𝒇𝟐”. The purpose of this paper is to generalize these formulas to the case of a system of an arbitrary number of thin lenses. In particular, this paper proves that the focal length Fn of a system of n thin coaxial lenses is given by 𝟏𝑭𝒏=Σ{(−𝟏)𝒎Π[(Σ𝟏𝒇𝒓𝒔𝒂𝒔𝒓𝒔=𝒂𝒔−𝟏+𝟏 𝟎=𝒂𝟎<𝒂𝟏<⋯<𝑎𝒎<𝒂𝒎+𝟏=𝒏;𝒅𝒏=𝟏 )𝒅𝒂𝒔]𝒎+𝟏𝒔=𝟏}𝒏−𝟏𝒎=𝟎, where, fr is the focal length of the rth lens, and dr is the distance between the rth lens and (r+1)th lens. For a fixed value of m, all combinations of values of the a’s (satisfying the condition “0 = a0 < a1 < … < am < am+1 = n”) are taken in the inner sum.

Differentiable design of freeform diffractive optical elements for beam shaping by representing phase distribution using multi-level B-splines

The generation of a specific laser beam profile on the work surface is key to various laser beam shaping tasks, relying heavily on diffractive optical elements (DOEs). Most beam-shaping DOEs are designed using iterative Fourier transform algorithms (IFTAs), which generally have slow convergence and prone to stagnate at local minima. Moreover, the microreliefs generated by IFTAs tend to be irregular, complicating manufacturing and causing uncontrolled scattering of light. We propose a differentiable DOE design method that applies a phase-smoothness constraint using multi-level B-splines. A multi-scale gradient-descent optimization strategy, naturally linked with the multi-level B-splines, is employed to robustly determine the optimized phase distribution that is fully continuous. This, in turn, can lead to more regular DOE microreliefs, which can simplify the fabrication process and be less sensitive to changes in wavelength and working distance. Furthermore, our method can also design a fully continuous freeform lens, distinguished from most freeform lens design approaches by its foundation in physical optics rather than geometrical optics. Simulation and experimental results of several design tasks demonstrate the effectiveness of the proposed method.

Research on wireless channel model based on improved ray tracing algorithm that considers multiple diffuse scattering and employs pyramid-shaped ray tubes as the carrier

This paper extensively utilizes fine three dimensional environmental data obtained from laser point clouds. Based on theories such as geometrical optics and effective roughness theory, a deterministic wireless channel model is established, which integrates higher-order diffuse scattering. This model is referred to as the ray tracing fusion with higher-order diffuse scattering model. To expedite the collision calculation between rays and the scene, this paper introduces a combined approach of voxelization and signed distance field, resulting in a remarkable 16-fold improvement in computational speed. Moreover, aiming to balance accuracy and efficiency, the paper systematically analyzes the optimization computation parameters of the model. Finally, the proposed model is validated using measurement data in the frequency range of 1 GHz to 6 GHz in mountainous terrain. The results indicate that the predicted outcomes of the proposed model have an accuracy within 6 dB compared to the measurement results, and are superior to ITU-R P.1546, which is an international standard recommended by the International Telecommunication Union for modeling electromagnetic wave propagation in undulating terrain. This provides necessary technical support for network planning and optimization.

Lensing Point-spread Function of Coherent Astrophysical Sources and Nontrivial Wave Effects

Most research on astrophysical lensing has been conducted using the geometric optics framework, where there exists a clear concept of lensing images. However, wave optics effects can be important for coherent sources, e.g., pulsars, fast radio bursts, and gravitational waves observed at long wavelengths. There, the concept of lensing images needs an extension. We introduce the concept of the “lensing point-spread function” (LPSF), the smoothed flux density distribution of a coherent point source after being lensed, as a generalization of the lensing image concept at finite frequencies. The frequency-dependent LPSF captures the gradual change of the flux density distribution of the source from discrete geometric images at high frequencies to a smooth distribution at low frequencies. It complements other generalizations of lensing images, notably the imaginary images and the Lefschetz thimbles. Being a footprint of a lensing system, the LPSF is useful for theoretical studies of lensing. Using the LPSF, we identify a frequency range with nontrivial wave effects, where both geometric optics and perturbative wave optics fail, and determine this range to be ∣κ∣−1 ≲ ν ≲ 10, with κ and ν being the dimensionless lens amplitude and the reduced observing frequency, respectively. Observation of LPSFs with nontrivial wave effects requires either very close-by lenses or very large observing wavelengths. The potential possibilities are the lensing of gravitational waves, the plasma lensing of Milky Way pulsars, and lensing by the solar gravitational lens.

Longitudinal resolution of three-dimensional integral imaging in the presence of noise

The two-point source longitudinal resolution of three-dimensional integral imaging depends on several factors including the number of sensors, sensor pixel size, pitch between sensors, and the lens point spread function. We assume the two-point sources to be resolved if their point spread functions can be resolved in any one of the sensors. Previous studies of integral imaging longitudinal resolution either rely on geometrical optics formulation or assume the point spread function to be of sub-pixel size, thus neglecting the effect of the lens. These studies also assume both point sources to be in focus in captured elemental images. More importantly, the previous analysis does not consider the effect of noise. In this manuscript, we use the Gaussian process-based two-point source resolution criterion to overcome these limitations. We compute the circle of confusion to model the out-of-focus blurring effect. The Gaussian process-based two-point source resolution criterion allows us to study the effect of noise on the longitudinal resolution. In the absence of noise, we also present a simple analytical expression for longitudinal resolution which approximately matches the Gaussian process-based formulation. Also, we investigate the dependence of the longitudinal resolution on the parallax of the integral imaging system. We present optical experiments to validate our results. The experiments demonstrate agreement with our Gaussian process-based two-point source resolution criteria.

Problem of Surface Waves on Water in Higher School Laboratory Workshop

Traditionally, in lecture courses on General Physics, the example of waves propagating on the water surface is used for the qualitative description of wave phenomena. At the same time, in corresponding laboratory courses, there are almost no quantitative tasks on the same topic, the purpose of which would be to measure the surface wave characteristics. The reason for this difference must be associated with the complexity of the physical mechanism of generation and propagation of waves on the water surface. In this work, a laboratory problem has been developed to measure the group velocity of circular waves traveling on the water surface. It uses a standard laboratory water tank equipped with a surface wave generator. The proposed experiment offers an opportunity for quantitative measurement of surface wave characteristics, a dispute of only qualitative observations of wave phenomena commonly found in current curricula, due to the introduction of two improvements. First, to produce the water waves’ rich in-contrast representations on a screen, the optimal generator amplitude is specially set for each experiment; and second, to measure the group wavelength, the device magnification is preliminary calculated based on the laws of geometric optics. The tests carried out have shown that the relative accuracy of these measurements is quite acceptable: within a few percent. The introduction of the problem of surface waves on water into the General Physics laboratory workshop for engineering specialties of technical universities will contribute to improving the quality of mastering the subject by students.

Applications of the Stone–Weierstrass theorem in the Calderón problem

We give examples on the use of the Stone–Weierstrass theorem in inverse problems. We show uniqueness in the linearized Calderón problem on holomorphically separable Kähler manifolds and in the Calderón problem for nonlinear equations on conformally transversally anisotropic manifolds. We also study the holomorphic separability condition in terms of plurisubharmonic functions. The Stone–Weierstrass theorem allows us to generalize and simplify earlier results. It also makes it possible to circumvent the use of complex geometrical optics solutions and inversion of explicit transforms in certain cases.

Effects of multiple reflections on nonlinear absorption measurements.

Accurate measurement of nonlinear absorption coefficients by transmission-based techniques, such as Z-scan and its modifications, remains challenging due to high measurement errors. In this Letter, we claim that one of the reasons for that could lie in ignoring multiple reflections of light in plane-parallel samples in the presence of multi-photon excitation. Closed-form formulations for transmittance and reflectance coefficients, considering both linear and multiphoton absorption, are obtained in the geometrical optics approximation. The contributions of multiple reflections to the total transmittance are calculated for several II-VI materials reported in the literature. The results show a nonlinear dependence on incident intensity, ranging from 15.7% down to 7.9% for CdS, from 10.3% down to 5.8% for CdSe, and from 18.1% down to 9.1% for ZnSe. The dependence of the contribution of multiple reflections on the sample thickness is shown to exhibit a near exponential decay.

Research on the NI-MLA Method for Enhancing the Spot Position Detection Accuracy of Quadrant Detectors Under Atmospheric Turbulence

The distorted spots induced by atmospheric turbulence significantly degrade the spot position detection accuracy of the quadrant detector (QD). In this paper, we utilize angular measurement and homogenization characteristics of non-imaging microlens array (NI-MLA) systems, effectively reducing the distortion degree of the spots received on the QD target surface, thereby significantly enhancing the spot detection accuracy of the QD. First, based on the principles of geometric optics and Fourier optics, it is proved that the NI-MLA system possesses the angular measurement characteristic (AMC) within the paraxial region while deriving and verifying the focal length of the system. Then, the QD computation curve characteristics of the system under non-turbulence are explored. This study further elucidates the mathematical principle of the NI-MLA system for mitigating the spot position detection random error of QD (SPDRE-QD) and discusses in depth the relationship between the NI-MLA system’s capability to mitigate the SPDRE-QD and the system’s parameters under various turbulence intensities. Finally, it is experimentally verified that the root-mean-square error (RMSE) of the QD computation values using the NI-MLA system are reduced by a significant improvement of at least 2.44 times and up to 17.36 times compared with that of the conventional optical system of QD (COS-QD) under turbulence conditions ranging from weak to strong.

Model of time-distance curve of electromagnetic waves diffracted on a local feature in the georadar study of permafrost zone rock layers

In GPR (georadar) studies, one of the most popular procedures for determining electromagnetic waves propagation velocity in a rock mass is the selection of theoretical hyperbolic time-distance curves and subsequent comparison with the time-distance curve obtained from a GPR measurement. This procedure is based on the model of homogeneous medium, but nowadays the subject of GPR study is often inhomogeneous media, such as horizontally layered media characteristic of loose permafrost zone sediments. The paper presents the findings of studying the formation of hyperbolic time-distance curves of georadar impulses in a horizontally layered medium without taking into account the dispersion and absorption of electromagnetic waves. On the basis of geometrical optics laws, formulas were derived to calculate the shape of the hyperbolic lineup of georadar impulses reflected from a local feature in a multilayer frozen rock mass. On the example of a permafrost zone rock mass containing a layer of unfrozen rocks, the effect of the thicknesses of rock layers and their relative dielectric permittivity on the apparent dielectric permittivity resulting from the calculation of the theoretical hyperbolic time-distance curve was shown. The conditions under which it is impossible to determine the presence of a layer of unfrozen rocks from a hyperbolic time-distance curve are also presented. The established regularities were tested on synthetic georadar radargrams calculated in the gprMax software program. The findings of the theoretical studies were confirmed by the comparison with the results of the analysis of the georadar measurements computer simulation data in the gprMax system (the relative error was less than 0.5%).

Device response principles and the impact on energy resolution of epitaxial quantum dot scintillators with monolithic photodetector integration

Epitaxial quantum dot (QD) scintillator crystals with picosecond-scale timing and high light yield have been created for medical imaging, high energy physics and national security applications. Monolithic photodetector (PD) integration enables the sensing of photons generated within the waveguiding crystal and allows a wide range of scintillator-photodetector coupling geometries. Until recently, these doubly novel devices have suffered from complex, high variance responses to monoenergetic sources which significantly reduces their precision and accuracy. The principles governing the overall device response have now been discerned and embodied by an expression derived within a geometrical optics framework which considers optical properties, surface roughness and photodetector coupling geometry. Response variation due to these factors was sufficiently reduced to obtain material-related energy resolution values of 2.4% with alpha particles. These findings place energy resolution alongside luminescence timescale, photon yield, and radiation hardness as outstanding properties of these engineered materials.

Gravitational spinoptics in a curved space-time

In this paper we discuss propagation of the weak high-frequency gravitational waves in a curved spacetime background. We develop a so-called spinoptics approximation which takes into account interaction of the spin of the field with the curvature of the background metric. This is achieved by modifying the standard geometric optics approximation by including the helicity sensitive terms of the order 1/ω in the eikonal equation. The novelty of the approach developed in this paper is that instead of study of the high-frequency expansion of the equations for the gravitational field perturbations we construct the effective action for the gravitational spinoptics. The gravitational spinoptics equations derived by variation of the effective action correctly reproduce the earlier obtained results. However, the proposed effective action approach is technically more simple and transparent. It allows one to reduce the study of the high-frequency gravitational waves to study classical dynamics of massless particles with internal discrete degree of freedom (helicity). The formalism is covariant and it can be applied for arbitrary vacuum space-time background.

Stream Theory

Purpose: The main objective of this research was trying to solve the misunderstood nature of light by introducing and studying fluid optic and its influence on physical science. Methodology: The research adopted observation and a study of already published online and school library books on optics and physical science. Observation is always the best approach to quote a natural occurrence and behavior, the researcher observed the behaviour of thin light by isolating themself for a long time in a dark room having roof holes. Findings: The research findings revealed that; light is not basically a wave, a particle or a photon. Light is sensibly a fluid. Fluid optics has no assumptions and weaknesses as other earlier optical theories as fluid optics clearly brings the channels of light (The streams) and its close link, agreement and influence on physical science as physical nature is brought into focus. Unique contribution to Theory, Practice and Policy: Stream theory completely solves the mystery of light and brings proper understanding of optics now fluid optics. Adoption of the theory and its concepts completely exposes physical science into focus with ease in understanding the natural phenomenon related to optics as the theory avoids assumptions and connects geometrical optics and physical optics to one singular optical study (fluid optics) as a step to singularity achievement in physics. (It is that and it happened that way).

Effect of Bi2O3 in Bi2O3–SrO–ZnO–B2O3 glass: Structural, mechanical, optical, and gamma ray shielding modifications

The effect of Bi2O3 in Bi2O3–SrO–ZnO–B2O3 glasses has been evaluated in this study. The density of the glasses increased from 5.127 g/cm³ to 6.214 g/cm³ as Bi2O3 concentration increased from 35 % to 50 %. The rise in molar mass and molar volumes points towards the opening of the network. The structural modifications increased the elastic moduli from 30.110 to 30.944 GPa for Young modulus; 17.653–17.860 GPa for bulk modulus; 13.242–13.662 GPa for shear modulus and 35.309–36.077 GPa for longitudinal modulus, respectively. The FTIR spectra indicate the conversion of BO4 to BO3 with increasing Bi2O3 concentration which accounts for the increase of the non-bridgingoxygens (NBOs). The band gap energy decreases from 3.165 to 3.020 eV with increasing Bi2O3 content. The transmission factors (TF) and other relevant parameters were evaluated for four different thicknesses of the glasses (0.5, 1, 1.4 and 2 cm). The 0.5 cm thick glass retained the highest TF when compared to other thicknesses. The TF for 0.5 cm thick glass is about 70 %, and drops to about 20 % for 2 cm thickness, which confirms that at 2 cm thickness, the glass materials shield better than 0.5, 1 and 1.4 cm. Furthermore, the addition of Bi2O3 causes a decrease in TF, so the glass with a high Bi2O3 content shields gamma radiation better than those with a low Bi2O3 content. The half value layer (HVL) values reduce as the Bi2O3 content is increased at all examined energies, confirming that the glass sample with highest Bi2O3 content has the superior shielding capability compared to the remaining examine glasses.

Simple, innovative and attractive approach for geometrical optics, chromatic aberration and colour combination experiments

Simple, innovative and attractive approach for geometrical optics, chromatic aberration and colour combination experiments

Development of Collaborative Online Learning Model Based on Case Method in Optics Courses to Train Creative and Communication Skills

Learning models really need to be developed to meet the demands of the world of work and keep up with existing technological developments, especially in the field of physics. This study aims to produce a collaborative online learning model based on the case method in optics courses. This research uses the Research and Development (R&D) method with the ADDIE model. The ADDIE model has five steps that need to be carried out in a structured manner because they are systematic, namely analysis, design, development, implementation and evaluation. The number of participants in this research consisted of 12 physics education students from Uhamka Jakarta and 13 physics education students from Unismuh Makassar. The number of lecturers involved was 5 people, with details of 3 from Unismuh Makassar lecturers and 2 from Uhamka Jakarta lecturers, all participants, both lecturers and students, carried out activities in the collaborative LMS that had been developed previously. The based on the process of implementing a collaborative online learning program. The analysis stage is the first step which aims to match the learning outcomes with the type or form of the digital module in the Optics course chosen. The design stage is to organize and design learning activities and videos. The development stage is the stage of building digital modules by assembling them into SPADA Unismuh Makassar. The number of students in the high improvement category in creative skills is 50% and in written communication skills is 60%. This shows that the use of collaborative online learning design in optics lecture activities related to geometric optics material has moderate effectiveness in improving creative thinking and communicative skills.