Risley Prisms Research Articles

Super-Resolution and Wide-Field-of-View Imaging Based on Large-Angle Deflection with Risley Prisms.

A novel single camera combined with Risley prisms is proposed to achieve a super-resolution (SR) imaging and field-of-view extension (FOV) imaging method. We develop a mathematical model to consider the imaging aberrations caused by large-angle beam deflection and propose an SR reconstruction scheme that uses a beam backtracking method for image correction combined with a sub-pixel shift alignment technique. For the FOV extension, we provide a new scheme for the scanning position path of the Risley prisms and the number of image acquisitions, which improves the acquisition efficiency and reduces the complexity of image stitching. Simulation results show that the method can increase the image resolution to the diffraction limit of the optical system for imaging systems where the resolution is limited by the pixel size. Experimental results and analytical verification yield that the resolution of the image can be improved by a factor of 2.5, and the FOV extended by a factor of 3 at a reconstruction factor of 5. The FOV extension is in general agreement with the simulation results. Risley prisms can provide a more general, low-cost, and efficient method for SR reconstruction, FOV expansion, central concave imaging, and various scanning imaging.

Symmetries of Scan Patterns of Laser Scanners with Rotational Risley Prisms

Laser scanners with rotational Risley prisms produce scan patterns that can be classified as a type of rhodonea curve, because of their symmetry. The present study builds upon the novel, graphical method that we have developed to simulate and analyze such patterns using a commercially available mechanical design program, CATIA V5R20 (Dassault Systems, Paris, France). This graphical method has both the advantage of simplicity (regarding analytical methods) and of generating exact scan patterns (in contrast to approximate methods). The aim of this work is to utilize this method to study symmetry properties of scan patterns produced by a pair of rotational Risley prisms. A multi-parameter analysis is performed, considering the characteristic parameters of the approached scanner: refractive indexes and angles of the prisms, as well as their rotational velocities. Furthermore, Marshall’s parameters are considered: ratios of the rotational velocities and of the prism angles. Because the symmetries are identical for scan patterns generated by all the four possible configurations of a pair of prisms (and only their dimensions are different), the shape of these patterns is studied only for one of these configurations. Therefore, without losing generality, the symmetries of the patterns produced by such scanners are determined, as well as some of their characteristic dimensions. Rules-of-thumb were obtained to be able to generate patterns that are appropriate for a specific application. Moreover, we demonstrated that the existing symmetries simplify the process of obtaining the scan patterns by identifying repetitive parts (defined as structures of symmetry) of a pattern.

Correction of Pointing Deviation of Risley Prisms Based on Levenberg-Marquardt Algorithm

Correction of Pointing Deviation of Risley Prisms Based on Levenberg-Marquardt Algorithm

Motion Deblurring Technology Research for Risley Prism Scanning Imaging System Based on Dynamic Virtual Camera

Motion Deblurring Technology Research for Risley Prism Scanning Imaging System Based on Dynamic Virtual Camera

Multivariable Decoupling and Nonlinear Correction Method for Image-Based Closed-Loop Tracking of the Risley Prisms System.

Image-based closed-loop tracking (IBCLT) is an important part of the process of target tracking. The Risley prism system has a unique advantage in improving the target tracking ability because of its compact and lightweight structure. Compared with traditional target tracking equipment, the Risley prism system has two difficulties in the process of IBCLT. First, the Risley prism is a complex coupling system of double input and double output. Second, the Risley prism itself is a nonlinear system. These problems lead to decrease in dynamic response and inconsistent target tracking capabilities. Thus, this paper proposes a method to implement multivariable decoupling and reduce the nonlinear effect. First, the boresight error of IBCLT is decoupled to the azimuth and elevation directions by the rotation matrix error-decoupling (RMED) method. Second, the gains of IBCLT in azimuth and elevation directions are independent variables that comes from two functions of the target elevation angle. The experimental results show that the IBCLT error deviation of different static targets in the field of view is within 0.025 arcsec, which is 70% lower compared with the fixed gain method. Furthermore, the steady-state error deviation of moving targets is controlled within 2.5 arcsec. These experimental results prove the feasibility and effectiveness of the proposed method.

Analytical inverse solution of a rotational achromatic Risley prism based on Fourier series fitting

High-precision analytical inverse solution is a significant issue faced by the rotational achromatic Risley prism system. The current inverse solution method is based on numerical iteration, which is time-consuming. To solve this problem, we propose an analytical inverse solution method for a rotational achromatic Risley prism system based on Fourier series fitting. In the proposed method, the expression of the difference between the rotation angle and elevation angle of beam deflection is inversely fitted using the forward solution model. The analytical inverse solution is achieved using a two-step method based on this fitted expression. Simulation results of the analytical inverse solution show that the maximum azimuthal error was less than 2.23e−8 arcsec and the maximum elevation error was less than 1.33e−4 arcsec. Experimental results prove that with 1 arcsec pointing accuracy, the time consumption of the proposed method is only 0.108 ms, whereas that of the previous numerical-iteration-based method is 1.480 ms.

Research on achromatic Risley prism in normal tracing angle measurement system

The normal tracing angle measurement system based on Risley prisms can eliminate the influence of aberrations other than the dispersion of Risley prisms on the angle measurement accuracy. Although the traditional dispersion optimization method based on the extreme point can make the beam directions of different wavelengths more consistent after passing through the Risley prisms, the light intensity distribution that deviates from the extreme point wavelength causes the spot centroid to deviate from the target position, thus introducing the angle measurement error. In this paper, we propose a highly symmetry-glued achromatic Risley prisms optimization method for the needs of a normal tracing angle measurement system. Our method not only realizes the beam direction as closely as possible after the deflection of the Risley prisms but also compensates for the centroid error caused by the beam deviating from the central wavelength. ZEMAX simulation shows that, in the normal tracing angle measurement system, within the angle measurement range of ±700 ″ and the light source wavelength range of 600 to 700 nm, the angle measurement error introduced by the dispersion of the optimized Risley prisms is not >0.01 ″ .

Geometric Method: A Novel, Fast and Accurate Solution for the Inverse Problem in Risley Prisms

Today, mechanical tracking systems are becoming increasingly compact, enabling a new range of civil and military applications. These include aerial laser scanning, for which Risley prisms are used. In Risley systems, the so-called inverse problem, which focuses on obtaining the angles of the prisms for a given target coordinate, has not yet been solved mathematically. As a consequence, approximate approaches have been used, but the solutions obtained have significant errors and a lack of precision. To improve accuracy, iterative methods, which are computationally intensive, have also been implemented. In this paper, an analytical process which we call the geometric method is presented, and we verified that this strategy highly improves accuracy and computational speed. Using this method in an iterative process gives accuracies of up to 1 pm in only three iterations. This high accuracy would allow the geometric method to be applied in fields such as lithography, stereolithography, or 3D printing.

A Dual-Layer Full-Phase Frequency-Selective Surface for Continuous 2-D Beam Scanning

Based on the generalized frequency selective surface (FSS) theory, we prove that the dual-layer dual-polarization transmission FSS can achieve 360° phase coverage at a limited loss. As a verification, a dual-layer dual-polarization full-phase unit cell with vias is proposed, which can achieve a phase range of more than 360° under 2 dB transmission loss. In addition, we innovatively apply the proposed dual-layer full-phase FSS to the design of a continuous 2-D beam scanning antenna based on the Risley prism (RP). This scheme effectively reduces the number of dielectric substrate layers required by the phase shifting surface (PSS), reduces the rotating scanning inertia, and effectively improves the aperture efficiency. The measured results show that the maximum scanning range of the 2-D beam scanning antenna can reach ±46° in the elevation plane and 360° in the azimuth plane. When operating at the center operating frequency of 21 GHz, the maximum gain of the antenna is 35.75 dBi with an aperture efficiency of 44.1%.

A Simplified Configuration of Beam-Steerable Risley Prism Antennas: Principles and Validation

Beam-steerable Risley prism antennas (RPAs) typically require at least three separated components, including a stationary feed source and two independently rotatable gradient-phase-shift surfaces. In this letter, we propose a simplified two-component RPA configuration that employs a gradient-phase feed source and a gradient-phase transmitarray. The proposed configuration is validated by a novel circularly polarized (CP) RPA in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> -band through simulations. It consists of an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{8}\times \text{8}$</tex-math></inline-formula> gradient-phase CP patch array feed and a 10 cm diameter CP transmitarray. The total height of this system is 14.6 mm, which is less than one wavelength at 19 GHz. A left-hand CP beam scan from broadside to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{55}^\circ$</tex-math></inline-formula> is demonstrated. The simulated RPA achieved a broadside directivity of 22.3 dBi at 19 GHz (42.7% aperture efficiency) and a 3 dB directivity bandwidth of 9.2%.

Three-dimensional computational ghost imaging using a dynamic virtual projection unit generated by Risley prisms.

Computational ghost imaging (CGI) using stereo vision is able to achieve three-dimensional (3D) imaging by using multiple projection units or multiple bucket detectors which are separated spatially. We present a compact 3D CGI system that consists of Risley prisms, a stationary projection unit and a bucket detector. By rotating double prisms to various angles, speckle patterns appear to be projected by a dynamic virtual projection unit at different positions and multi-view ghost images are obtained for 3D imaging. In the process of reconstruction, a convolutional neural network (CNN) for super-resolution (SR) is adopted to enhance the angular resolution of reconstructed images. Moreover, an optimized 3D CNN is implemented for disparity estimation and 3D reconstruction. The experimental results validate the effectiveness of the method and indicate that the compact system with flexibility has potential in applications such as navigation and detection.

Validation of a set of compact and low-cost Stokes lenses aimed at refractive astigmatism measurement for optometric practice

We describe the construction and characterization of three Stokes lenses that can be used in optometric practice to measure different ranges of refractive astigmatism using subjective routines. The proposed set of Stokes lenses is assembled from Risley prism mounts coming from a Nidek phoropter in which the optical prisms are replaced by three different sets of cylindrical lenses ( ± 0.75, ± 1.25, and ± 1.75 D) coming from a trial lens case to cover a different astigmatism range (1.5, 2.5, and 3.5 D in modulus, respectively). We present the experimental characterization of the three Stokes lenses with the power vector notation and use an automatic lensmeter (Topcon CL-300) for dioptric power measurement. The measured dioptric power profiles are in concordance with the theoretical predictions, and statistical testing reveals no differences between measured and theoretical values for the three lenses, neither in spherical nor in cylindrical components (p ≥ 0.05 for all cases). Bland–Altman plots show very good agreement between measured and theoretical values with limits of agreement below 0.12 D for both M and J components for the three lenses. These Stokes lenses can be easily constructed from regular Risley prism mounts, so they can be used in almost any phoropter model for refractive error compensation.

Risley prisms as a conformal transformation device: II. Derivatives of the absolute functions and an attributive analysis of the control singularities.

Risley prisms have quite a few advantages over their competitive mirror scanning systems, a highly commendable one being the conformal transformation of the angular motions of the two prisms into a 2D pattern in the polar frame. However, in such a transformation, knotty problems relating to the so-called control singularities unpreventably occur when the two prisms are either in parallel with their apexes aligned or in opposition to each other. The aim of Part II of this study is an extension of the theories and their applications presented in Part I of this study [J. Opt. Soc. Am. A39, 1540 (2022)JOAOD60740-323210.1364/JOSAA.463724] to a parametric analysis of the singularities and their describing functions for a better understanding of how the scanning geometry affects the attributes of the singularities and how we can reduce their unfavorable influence by properly selecting the parameters of this two-prism system.

Risley prisms as a conformal transformation device. I. Complex and graphic analyses of mapping images conformality.

Mapping conformality and the singularities in the control law of the Risley prisms for target tracking are two internally related topics well suited for an investigation in one study. This is because an intrinsic ability of the Risley prisms is to produce 2D scan patterns in a polar frame by means of a combination of two 1D motions of the two prisms when their orientations are specified by two independent variables expressible as the x and y coordinates of a single point in a rectangular frame. In transformation of points between the rectangular and the polar frames, a singularity occurs inevitably at the pole where r=0 in the polar frame. For a better understanding of the two topics and their relationship, this study presents an in-depth analysis, starting from the first-order approximation, and then the far-field condition is considered and finally is the rigorous theory developed for the Risley-prism-based beam steering system under the basis of a consideration of the effect of the thickness of the Risley prisms on the accuracy of beam steering. The rigorous theory provides correct results of mapping conformality in both the near and the far zones of the beam steering field.

A Low-Profile, Risley-Prism-Based, Beam-Steerable Antenna Employing a Single Flat Prism

We present a low-profile, mechanically beam-steerable antenna that implements the Risley prism beam-steering concept using only a single flat prism. The proposed antenna achieves 2-D beam steering and consists of two parts: a holographic leaky-wave antenna and a phase-shifting surface (PSS) that acts as a flat prism. The lower holographic leaky-wave antenna acts as the feed for the prism and provides a phase-shifting gradient over its output aperture. When paired with the flat prism, the system can provide 2-D beam steering by mechanically rotating the two layers against each other. Unlike conventional Risley-prism-based beam-steerable antennas, the proposed approach results in a very low-profile antenna and does not need spatial illumination of a two-prism system with a separate feed antenna. The flat prism is implemented using a PSS consisting of low-pass, hexagonal-shaped, spatial phase shifters. A prototype antenna with an overall thickness of only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.1\lambda _{0}$ </tex-math></inline-formula> at 10 GHz was designed, fabricated, and experimentally characterized. Measurement results agree well with theoretical predictions and both show wide-angle beam scanning ranging from 0° to 57° in the elevation plane and 0° to 360° in the azimuth plane. The proposed antenna system has an equivalent aperture diameter of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$7.1\lambda _{0}$ </tex-math></inline-formula> and a measured peak gain of 22.3 dBc at 10 GHz corresponding to an aperture efficiency of 34.5%. The proposed concept is expected to be useful in designing low-profile, beam-steerable antennas in which the scanning speed can be traded off to reduce system complexity.

Time-of-flight measurement-based three-dimensional profiler system employing a lightweight Fresnel-type Risley prism scanner

A time-of-flight measurement-based three-dimensional (3D) profiler system employing a lightweight scanning system is demonstrated. To reduce the weight of the scanning system, and thereby achieve faster scanning speeds, two Fresnel prism sheets were employed as the scanning optics and installed to work as a pair of Risley prisms. Each Fresnel prism sheet has a diameter of 102 mm and mass of 15 g, which is about 12 times lighter than ordinary bulky prism. By scanning the laser beam with the developed scanning system, a 3D point cloud image of a target object located 8 m away could be successfully obtained. The image distortion was removable by correcting six geometrical parameters of the scanner using a simple optimization algorithm. It was confirmed by the experiment that once the distortion has been corrected, it is valid for other scanning speeds (and trajectories), enabling 3D profile measurements that do not require postprocessing of measured data. Measurement results for a standard target composed of square extrusions were in good agreement with the reference values, with deviations of <1 mm.

Multiview three-dimensional imaging using a Risley-prism-based spatially adaptive virtual camera field.

We present a three-dimensional (3D) imaging system that incorporates a stationary camera and Risley prisms. By planning prism rotation to generate a spatially adaptive virtual camera field, the system allows multiple virtual cameras in the field to capture any object from different perspectives for 3D reconstruction. An automatic virtual camera calibration method based on perspective projection and geometric optics is developed to enable virtual camera field construction and characterization. Moreover, a 3D computational reconstruction framework is proposed for multiview information fusion using the virtual camera field. This framework combines nonlinear distortion correction with epipolar geometry computation to efficiently perform image rectification and stereo matching, which can further facilitate 3D object reconstruction through multiview triangulation. The experiments on synthetic and real data validate the feasibility and flexibility of our 3D imaging technique.

Virtual image array generated by Risley prisms for three-dimensional imaging

We present a compact three-dimensional (3D) imaging system based on a virtual image array, which employs a single camera and Risley prisms. The original multi-perspective images are captured by the fixed camera with a dynamically deflected boresight. By applying planar homographies to original images, we obtain light field samples from a planar virtual image array. Moreover, a depth estimation framework on the epipolar-plane image (EPI) is adopted to generate 3D profiles of scenes. In the experiment, two-dimensional (2D) projections from a 5 × 5 virtual image array are obtained to exhibit the 3D imaging capability. The system has the potential to achieve stereovision in numerous applications such as industrial vision and visual navigation.

Robust Risley prism control based on disturbance observer for system line-of-sight stabilization.

In this paper, a robust control based on disturbance observer is proposed to improve the tracking accuracy of the Risley prism system (RPS). Applying the flexible thin-wall ring mechanism in the RPS causes a series of tracking and pointing challenges. Disturbances such as friction, shaft deformation, and model perturbation significantly deteriorate the tracking and pointing accuracy of the RPS. Two different observer-based control methods are proposed to guarantee the tracking precision of the RPS. Moreover, the disturbance observation and compensation (DOC) performance of the proposed methods is analyzed and compared. Finally, simulation and experiment results indicate that the proposed control methods, especially the DOC-expanded state observer control mode, obtain the best performance for disturbance rejection in the RPS.

Rotation matrix error-decoupling methods for Risley prism closed-loop tracking

In a rotational double-prism photoelectric tracking system based on image feedback, the rotation angle of the prisms must be adjusted to correct the tracking error. This study proposes a rotation matrix error-decoupling method that removes the strong coupling between the tracking errors in the x and y directions and the prism rotation in the rotational double-prism tracking device. Based on the tracking error after decoupling, the rotations of the same and reverse directions are simultaneously applied to both prisms to achieve closed-loop tracking. The high precision and robustness of the method were confirmed in simulation and experimental tracking evaluations.