Hyperbolic Mirror Research Articles

Mirror-centered representation of a focusing hyperbolic mirror for X-ray beamlines.

Conic sections are commonly used in reflective X-ray optics. Hyperbolic mirrors can focus a converging light source and are frequently paired with elliptical or parabolic mirrors in Wolter type configurations. This paper derives the closed-form expression for a mirror-centered hyperbolic shape, with zero-slope at the origin. Combined with the slope and curvature, such an expression facilitates metrology, manufacturing and mirror-bending calculations. Previous works consider ellipses, parabolas, magnifying hyperbolas or employ lengthy approximations. Here, the exact shape function is given in terms of the mirror incidence angle and the source and image distances.

Analysis of a small-scale modified beam-down solar concentrator system for low temperature applications

A small-scale beam-down solar concentrator (BDSC) without a receiver concentrator (RC) was studied for the first time for its applicability to low-temperature heating systems. A comprehensive theoretical model was developed that combines solar ray tracing with existing BDSC models to include solar angle variations for different days in a year at any location. The model allows the estimation of image magnification based on the heliostat area and the resultant uniform solar enhancement area on the RR. A test setup with a footprint area of 2.25 m2 and a tower height of 1.5 m, consisting of 11 heliostats, a single hyperbolic central mirror, and a RR, was built and tested to validate the model. It was found that cosine efficiency is an essential factor that dictates the system's performance. Increasing the number of mirrors from 11 to 41 showed the number of suns on the RR will increase from 1.45 to 4.5. A 22% reduction in total insolation on the RR was observed at solar noon due to the shadow of the central mirror, suggesting a prominent effect of shadow in small-scale BDSC systems. The system analyzed could achieve temperatures up to 200 °C, making it suitable for low-temperature applications.

Hyperbolic caustics of light rays reflected by hyperbolic front mirrors

Hyperbolic caustics of light rays reflected by hyperbolic front mirrors

Off-axis representation of hyperbolic mirror shapes for X-ray beamlines.

Mirror-centered, closed-form expressions for hyperbolic surfaces used in X-ray beamlines have been derived. Hyperbolic mirrors create a virtual focus or source point and can be used to lengthen or shorten the effective focal distance of a compound optical system. The derivations here express off-axis segments of a hyperbolic surface in terms of the real and virtual focal distances and the incident glancing angle at the center of the mirror. Conventional mathematical expressions of hyperbolic shapes describe the surfaces in Cartesian or polar coordinates centered on an axis of symmetry, necessitating cumbersome rotation and translation to mirror-centered coordinates. The representation presented here, with zero slope and the origin at the central point, is most convenient for modeling, metrology, aberration correction, and general surface analysis of off-axis configurations. The direct derivation avoids the need for nested coordinate transforms. A series expansion provides a helpful approximation; the coefficients of the implicit equation are also provided.

Wide field-of-view x-ray imaging optical system using grazing-incidence mirrors.

A field-curvature-corrected imaging optical system for x-ray microscopy using only grazing-incidence mirrors is proposed. It combines a Wolter type I (WO1) mirror pair, which forms a real image, with field curvature correction (FCC) optics-a convex hyperbolic mirror pair-that form a virtual image; compensation of the field curvatures realizes a wide field-of-view (FOV) and high magnification. Ray-tracing and wave-optics simulations verified the efficacy of the design, for which a FOV width was 111µm-4.7 times larger than that for the uncorrected WO1 design. The addition of FCC optics also produced a 2.3-fold increase in magnification.

Интерференционный метод контроля выпуклых гиперболических поверхностей с использованием дифракционного оптического элемента

Interference control methods are making it possible to evaluate with high accuracy errors in the shape of the optical part surface profile. The interference pattern processing allows obtaining a map of the surface deviations at each of its points with an accuracy of half the wavelength. An interference method is proposed for testing the convex hyperbolic surfaces, which could be introduced to control mirrors with large aperture angles in the imaginary geometric focus. The proposed auto-collimation control scheme consists of a helium-neon laser with the wavelength of 632.8 nm, a meniscus lens and a planar axisymmetric diffractive optical element to correct the meniscus spherical aberration. Numerical method is presented for calculating the optical diffraction element using a phase profile on the example of the secondary hyperbolic mirror of the Millimetron space telescope. The developed scheme was simulated in the Zemax OpticStudio program. The approximation error of the calculated phase profile was evaluated depending on the number of phase coefficients. The Fizeau interferometer optical system is proposed to implement the developed method. The influence of errors in installing a controlled mirror on the interference pattern for axial displacement, transverse displacement and tilt was determined. The residual wave aberration was evaluated in the control system

Optical efficiency and performance optimization of a two-stage secondary reflection hyperbolic solar concentrator using machine learning

This paper focuses on the geometric analysis of parabolic trough solar collectors (PTCs) with components of different sizes. The concentration ratio of the receiver tube, intercept factor, and optical efficiency are the three main features for PTCs geometric optimization. This study proposes a python-based algorithm for calculating the optical efficiency of PTCs based on a secondary hyperbolic mirror. This algorithm is suitable for writing programs, inputting AutoCAD data into software for simulation of different incident angles and focal lengths of the secondary hyperbolic mirror concentrator, and processing simulation data. The error between the simulated and theoretical optical efficiency results is less than 3.13%. To provide an accurate optical efficiency computing method for geometric designing, energy analyses, and optimizing solar concentration collectors, this paper presents an available method of predicting optical efficiency for solar concentrating design based on machine learning is presented. The optical efficiency was fitted using machine learning approaches numerically with a coefficient of determination (R2 = 0.97874). The optical efficiency fitting formulas for PTCs are generated by analyzing the secondary reflection hyperbolic data of samples computed through a variable reduction technique, while the R2 values are obtained from linear regression. Then the final optical efficiency fitting formula is applied to adequately characterize optical efficiency under different geometrical configurations, incidence angles, and tracking models for the optical and geometrical optimization of PTCs. Data from tests involving the secondary hyperbolic mirror concentrators are used to validate the ray-tracing model and compute optical efficiency based on a secondary reflection hyperbolic mirror.

Calibration Method of the Monocular Omnidirectional Stereo Camera

Compact and low-cost devices are needed for autonomous driving to image and measure distances to objects 360-degree around. We have been developing an omnidirectional stereo camera exploiting two hyperbolic mirrors and a single set of a lens and sensor, which makes this camera compact and cost efficient. We establish a new calibration method for this camera considering higher-order radial distortion, detailed tangential distortion, an image sensor tilt, and a lens-mirror offset. Our method reduces the calibration error by 6.0 and 4.3 times for the upper- and lower-view images, respectively. The random error of the distance measurement is 4.9% and the systematic error is 5.7% up to objects 14 meters apart, which is improved almost nine times compared to the conventional method. The remaining distance errors is due to a degraded optical resolution of the prototype, which we plan to make further improvements as future work.

Image Sensory Experience of Artistic Design Based on the Role of Omnidirectional Vision Sensors

As a sensor with a wide field of view, the panoramic vision sensor is efficient and convenient in perceiving the characteristic information of the surrounding environment and plays an important role in the experience of artistic design of images. The transformation of visual and other sensory experiences in art design is to integrate sound, image, texture, taste, and smell with each other through reasonable rules, to create more excellent crossborder art design works. To improve the sensory experience that art design works bring to the audience, the combination of vision and other sensory experiences can maximize the advantages of multiple information dissemination methods and combine the omnidirectional visual sensor with the sensory experience of art design images. In the method part, this article introduces the omnidirectional vision sensor, art design image, and sensory experience modes and content and introduces the hyperbolic concave mirror theory and the Micusik perspective projection imaging model. In the experimental part, the experimental environment, experimental objects, and experimental procedures of this article are introduced. In the analysis part, this article analyzes the six aspects of image database dependency test, performance, comparison of different distortion types, false detection rate and missing detection rate, algorithm time‐consuming comparison, sensory experience analysis, and feature point screening. Among the feelings of the art design image, for the first image, 87.21% of the audience’s feelings are happy, indicating that the main idea of this image can bring joy to people. In the second image, the audience’s feelings are mostly sad. For the third image, more than half of the audience’s feelings are melancholy. For the fourth image, 69.34% of the audience’s inner feelings are calm. It explains that the difference in the content of art design images can bring different sensory experiences to people.

Gas Pipeline Inspection Using Autonomous Robots With Omni-Directional Cameras

Maintenance of pipelines is a major concern for oil and gas companies and millions of dollars are spent on preventive inspections every year. The type of equipment used in carrying out inspections is expensive and very sophisticated requiring very specialized manpower. Here, a novel approach to pipeline inspection is presented using a small mobile robot and a catadioptric omni-directional vision system. The system captures panoramic videos using conical and/or hyperbolic mirrors, which are transmitted to a monitoring station to examine the inner surface of gas pipelines. The videos are prepped, unwrapped, and un-warped to obtain a realistic flat image of the whole inner surface of the pipeline. The Simulated Annealing optimization method is implemented to optimize the system parameters using different mirror shapes and unwrapping and un-warping techniques. Simulations and experimental studies using a mobile robot equipped with a CCD camera and a conical mirror are presented here to demonstrate the viability of this inspection technique. The developed robotic system can be used to visually pre-scan pipelines before using much more expensive tools such as magnetic flux leakage (MFL) or ultrasound (UT) based inspection systems. These latter tools cost tens of thousands of dollars per hour and will be used only after suspicious areas are visually detected. This will reduce the inspection time and cost significantly.

Cassegrain-based concentrator with tailored mirrors.

In this study, we design a Cassegrain-based concentrator with tailored mirrors. The proposed concentrator comprises a primary optical element (POE) and a secondary optical element (SOE). The POE is a parabolic concave mirror and the SOE is a hyperbolic convex mirror. In order to achieve uniform irradiance distribution without a homogenizer, the POE is tailored and tilted to generate a uniform distribution by overlapping the energy well. The Cassegrain-based concentrator with tailored mirrors can achieve a geometric concentration ratio of 1236×, a concentration ratio of 1034×, an optical efficiency of 83.66%, an acceptance angle of ±0.38∘, a uniformity of 7.87, and an aspect ratio of 0.254.

Creating a large aspheric primary mirror using spherical segments

The use of aspheric mirrors is a common practice to design astronomical telescopes with a few optical elements. In the most preferred optical design Ritchey Chretien (RC), both primary and secondary mirrors are hyperboloid. Nowadays large telescopes are being built using small mirror segments, however, making aspheric off-axis mirror segments is still a challenge. We have conducted a study in which, we explored the possibility to mimic an aspheric hyperbolic primary mirror by making use of smaller spherical mirror segments. Three different methods have been used to form a large segmented aspheric primary of nearly 12m aperture. In the first method, fixed ROC(radius of curvature) spherical mirror segments are reconfigured by a piston, tip, and tilt (PTT). In the other two methods, in addition to PTT, ROC of the segments are also varied. We further attempted to reduce the telescope wave-front error by varying the segment size and the F ratio of the primary. We found out that none of these three methods provided acceptable image quality unless we incorporate the warping harness in the segment support. The use of the warping harness emulated by Zernike coefficient correction, remarkably reduced the wave-front error and delivered a decent image quality over a large field of view. In this paper, we present the results of our study on designing an RC type optics for a 12m class optical-NIR(Near Infrared) telescope using spherical mirror segments.

Reflections in a moving mirror

Euclid's law of reflection, r = i, rests on three assumptions: Fermat's principle of least time, constancy of speed of light, and that the mirror is stationary. For a moving mirror, the corresponding law is known as the relativistic reflection law; however, it was first inferred prior to relativity. While modern derivations rely on Lorentz transformations or Maxwell's equations, the original derivation was an elegant application of geometric optics. In this paper, we develop a new geometric approach which shows that the relativistic reflection from a plane mirror is equivalent to the ordinary reflection from a certain hyperbolic mirror, and that the incident and reflected rays are the focal rays of the hyperbola. We demonstrate that incident concentric spherical waves reflect as non-concentric spherical waves, and thereby derive the Doppler effect formula. We conclude with a class project for students to derive a generalization of Snell's law of refraction for a particular type of a moving interface.

Design and analysis for annular focus antenna with off-axial parabolic and slanting hyperbolic rotating surfaces configuration.

We propose a novel annular focus antenna with an off-axial parabolic primary mirror and slanting hyperbolic secondary mirror. Ray tracing based on three-dimensional vector reflection theory is utilized to design and analyze the structural parameters of the antenna. In ideal conditions, there is no geometric loss in theory and the transmitting efficiency increases by 24.54% compared with the conventional Cassegrain optical antenna. The transmitting efficiency is also calculated in the conditions of off-axial and on-axial defocus. The simulated results demonstrate that this design can increase transmission efficiency of the optical antenna effectively.

Ex situ metrology and data analysis for optimization of beamline performance of aspherical pre-shaped x-ray mirrors at the advanced light source.

Super high quality aspherical x-ray mirrors with a residual slope error of ∼100 nrad (root-mean-square) and a height error of ∼1-2 nm (peak-to-valley), and even lower, are now available from a number of the most advanced vendors utilizing deterministic polishing techniques. The mirror specification for the fabrication is based on the simulations of the desired performance of the mirror in the beamline optical system and is normally given with the acceptable level of deviation of the mirror figure and finish from the desired ideal shape. For example, in the case of aspherical x-ray mirrors designed for the Advanced Light Source (ALS) QERLIN beamline, the ideal shape is defined with the beamline application (conjugate) parameters and their tolerances. In this paper, we first discuss an original procedure and dedicated software developed at the ALS X-Ray Optics Laboratory (XROL) for optimization of beamline performance of pre-shaped hyperbolic and elliptical mirrors. The optimization is based on results of ex situ surface slope metrology and consists in minimization of the mirror shape error by determining the conjugate parameters of the best-fit ideal shape within the specified tolerances. We describe novel optical metrology instrumentation, measuring techniques, and analytical methods used at the XROL for acquisition of surface slope data and optimization of the optic's beamline performance. The high efficacy of the developed experimental methods and data analysis procedures is demonstrated in results of measurements with and performance optimization of hyperbolic and elliptical cylinder mirrors designed and fabricated for the ALS QERLIN beamline.

Production of High Frequency Elliptic and Hyperbolic Optic Mirrors

An article provides an overview of production and corresponding issues of one of the most important component of a satellite dish - a high frequency optic mirror that polarizes signals caught by a parabola into a sensor connected to computing (so-called receiver).The article describes an issue of production of high frequency elliptic and hyperbolic optic mirrors forming one of the most important functional components of parabolic satellites designed for ALMA, the biggest and the highest located international astronomic radio telescope in the world. It is situated at a high plateau Chajnantor in Chile, South America.Individual parts of the article are outlined in a sense of a flow of a production technological process including both theoretical and practical analyses of the given issue. A content of the article leads especially to a description and explanation of causes of individual problems during production of elliptic and hyperbolic optic mirrors.

Three-dimensional panoramic sensor based on a variable-frequency fringe pattern.

This paper aims to develop a 3D panoramic sensor with a catadioptric imaging system. First, the fringe pattern is designed in a polar coordinate system to match the dimensions of a hyperbolic mirror. Second, a variable-frequency fringe pattern is proposed to obtain the constant-frequency reconstruction in real space. Experiments are conducted to verify system accuracy and feasibility, and the experimental results show the effectiveness of the proposed 3D panoramic sensor.

Spectral Skyline Separation: Extended Landmark Databases and Panoramic Imaging.

Evidence from behavioral experiments suggests that insects use the skyline as a cue for visual navigation. However, changes of lighting conditions, over hours, days or possibly seasons, significantly affect the appearance of the sky and ground objects. One possible solution to this problem is to extract the “skyline” by an illumination-invariant classification of the environment into two classes, ground objects and sky. In a previous study (Insect models of illumination-invariant skyline extraction from UV (ultraviolet) and green channels), we examined the idea of using two different color channels available for many insects (UV and green) to perform this segmentation. We found out that for suburban scenes in temperate zones, where the skyline is dominated by trees and artificial objects like houses, a “local” UV segmentation with adaptive thresholds applied to individual images leads to the most reliable classification. Furthermore, a “global” segmentation with fixed thresholds (trained on an image dataset recorded over several days) using UV-only information is only slightly worse compared to using both the UV and green channel. In this study, we address three issues: First, to enhance the limited range of environments covered by the dataset collected in the previous study, we gathered additional data samples of skylines consisting of minerals (stones, sand, earth) as ground objects. We could show that also for mineral-rich environments, UV-only segmentation achieves a quality comparable to multi-spectral (UV and green) segmentation. Second, we collected a wide variety of ground objects to examine their spectral characteristics under different lighting conditions. On the one hand, we found that the special case of diffusely-illuminated minerals increases the difficulty to reliably separate ground objects from the sky. On the other hand, the spectral characteristics of this collection of ground objects covers well with the data collected in the skyline databases, increasing, due to the increased variety of ground objects, the validity of our findings for novel environments. Third, we collected omnidirectional images, as often used for visual navigation tasks, of skylines using an UV-reflective hyperbolic mirror. We could show that “local” separation techniques can be adapted to the use of panoramic images by splitting the image into segments and finding individual thresholds for each segment. Contrarily, this is not possible for ‘global’ separation techniques.

Experimental research of alignment error correction by aspheric mirror based on the function of imaging quality

Image definition will be influenced by alignment errors of mirrors in an optical system consisting of hyperbolic, parabolic or ellipse mirrors. The major factors of alignment errors are gravity, wind loads and heat exchange for some optical systems like ground-based telescopes, while vibration and temperature gradient for systems like space telescopes. Larger telescopes are more sensitive to these error sources, which becomes the concerns of researchers. So the alignment errors of mirrors must be corrected in time to keep systems working in best condition. In order to solve the problem, many methods are proposed based on the detection of wave-front errors using wave-front sensors like Hartmann-Shack. However, wave-front sensors may not be used or cause optical systems to be more complicated. For example, multi-fields must be tested when telescope is working. On the one hand, if a wave-front sensor is used, it must be moved around imaging plane, on the other hand, if more wave-front sensors are used, system must be more complicated. So a new method is discussed for alignment error correction by evaluating the quality of spot diagrams based on the using of stochastic parallel gradient descent (SPGD) algorithm. The method considers the performance metric like spot diagram radius as a function of control parameters and then uses the SPGD optimization algorithm to improve the performance metric. The control parameters include positions of mirrors. The iteration process must be used in the right way to control position parameters. If it is not considered, a problem may come up that positions of spot diagrams may be influenced by the iteration. Furthermore, spot diagrams will probably disappear from detectors. Then the radii of spot diagrams are not correct. So a better way is put forward by the combination of de-center and tilt of mirrors. The way ensures that the position error produced by de-center and tilt are compensated for. A formula is provided in this paper to give the relationship between them. Based on the analysis, an optical system is designed to verify the conclusion. The SPGD algorithm is achieved by computer programming and the position of the mirror is controlled by a hexapod. Firstly, the problem is verified that the spot diagram will disappear from the detector with a normal iteration process. Then the new way is implemented. In the iteration process, the spot diagram is always in the center of the detectors. In order to prove the feasibility of the method, three different alignment errors are tested and all of them each give an Airy disk finally. The experiment can provide reference for engineering practice.

Conversion of the electrostatic energy analyzer of charged particles into double spectrograph regime in energy and angle

The possibility to convert the combined electrostatic energy analyzer of charged particles into double spectrograph regime in energy and angle has been studied. The analyzer consists of the hyperbolic and cylindrical mirrors. The correlation, expressing condition of the first-order focal line straightening of the particles beams of various energies, has been obtained. The possibility of simultaneous registration of the finite energy interval of charged particle distribution by means of a position-sensitive detector has been substantiated.