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The Zernike polynomial method is extensively used for atmospheric phase screen generation but is limited by insufficient high-frequency components. Calculating higher-order terms introduces challenges in computational efficiency and numerical instability when using the direct method. This paper analyzes these issues and proposes replacing the direct method with recursive radial formulas. We evaluate four recursive algorithms (Barmak’s, q-recursive, Prata’s and Kintner’s) for their performance in phase screen generation, focusing on computational speed and numerical stability. Our results demonstrate that recursive methods achieve a 10–20-times improvement in computational efficiency and maintain numerical stability even for high-order expansions. The main novelty of this study lies in the comprehensive comparison and validation of these recursive strategies for high-accuracy atmospheric phase screen simulation.

Integrated systems are facing complex and changing environments with the wide application of atmospheric LiDAR in civil, aerospace, and military fields. Traditional analysis methods employ optical software to evaluate the optical performance of integrated systems, and cannot comprehensively consider the influence of optical and mechanical coupling on the optical performance of the integrated system, resulting in the unsatisfactory accuracy of the analysis results. Optical–mechanical integration technology provides a promising solution to this problem. A small-field-of-view LiDAR system with high repetition frequency, low energy, and single-photon detection technology was taken as an example in this study, and the Zernike polynomial fitting algorithm was programmed to enable transmission between optical and mechanical data. Optical–mechanical integration technology was employed to obtain the optical parameters of the integrated system under a gravity load in the process of designing the optical–mechanical structure of the integrated system. The experimental validation results revealed that the optical–mechanical integration analysis of the divergence angle of the transmission unit resulted in an error of 2.586%. The focal length of the telescope increased by 89 μm, its field of view was 244 μrad, and the error of the detector target surface spot was 4.196%. The continuous day/night detection results showed that the system could accurately detect the temporal and spatial variations in clouds and aerosols. The inverted optical depths were experimentally compared with those obtained using a solar photometer. The average optical depth was 0.314, as detected using LiDAR, and 0.329, as detected by the sun photometer, with an average detection error of 4.559%. Therefore, optical–mechanical integration analysis can effectively improve the stability of the structure of highly integrated and complex optical systems.

The TDICCD aerial camera was developed to study the relationship between the structure and optical system. Based on the camera outputs, integrated analysis and experimental methods were proposed. The proposed method was then used to both study and verify the influence of thermal disturbance on the optical performance and optimal aerial camera design. The nodal displacement of the optical surface under thermal disturbance was calculated via the finite element method. The resulting data were fitted to Zernike polynomial coefficients using the Zernike polynomial. Additionally, a method of calculating rigid body displacement was also proposed to determine the effects of rigid optical system displacement. The method calculates the RMS and PV parameters by fitting the surface distortion data. The fitted Zernike polynomial coefficients were input to ZEMAX software to obtain the optical system response. The influence of thermal disturbance on the optical performance of the aerial camera was analyzed. The analysis results have shown that the low-temperature conditions have a more prominent impact on the optical performance of aerial cameras. The radial and axial lens steady-state temperature range was 2.06 °C in conduction temperature of − 40 °C. At the same time, the aerial camera surface was frosted at − 40 °C to carry out the low-temperature experiment, which verified the results obtained for a large temperature difference environment. Finally, results were verified experimentally.

Nowadays, progressive addition lenses are widely applied to correct people’s vision disorders, but the design and machining of progressive addition lenses are still difficult. Generally, a progressive addition lens has a freeform front surface and a spherical back surface. In this article, the design of the front surface was simplified as a minimization problem of a functional, and the solution was obtained directly using a variational-difference method. After solving this problem, the description of the surface was discussed, and an evaluation method was proposed to analyze the fitting accuracy of Zernike polynomial method and B-spline interpolation method with different degrees. As a result, the progressive addition lens surface was constructed by the method with smaller fitting error. Moreover, a new generation algorithm of entrance parameters for tool path generation was put forward, which can reduce the interpolation error. The experimental results indicate that the design method was feasible and the machined surface quality was satisfactory using the proposed algorithm.

We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-[...] Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-[...] Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give [...]-analogues of all these extensions. In each case in addition to generating functions and three term recursions we provide raising and lowering operators and show that the polynomials are eigenfunctions of second-order partial differential or [...]-difference operators. [ProQuest: [...] denotes formulae omitted.]

This work discovers two hidden cases of blockwise recurrence in Zernike computations. Based on these findings, a new computation scheme for Zernike polynomials is proposed. It uses one Pascal triangle for all internal factors, thus avoiding calculations of factorials, cos/sin, inverse matrix, etc., and meets the requirements of computational accuracy, high speed, low memory footprint, and flexibility.

In light of fine learning ability in the existing uncertainties, a sage revised reiterative even Zernike polynomials neural network (SRREZPNN) control with modified fish school search (MFSS) method is proposed to control the six-phase squirrel cage copper rotor induction motor (SSCCRIM) impelled continuously variable transmission assembled system for obtaining the brilliant control performance. This control construction can carry out the SRREZPNN control with the cozy learning law, and the indemnified control with an assessed law. In accordance with the Lyapunov stability theorem, the cozy learning law in the revised reiterative even Zernike polynomials neural network (RREZPNN) control can be extracted, and the assessed law of the indemnified control can be elicited. Besides, the MFSS can find two optimal values to adjust two learning rates with raising convergence. In comparison, experimental results are compared to some control systems and are expressed to confirm that the proposed control system can realize fine control performance.

When structured light is propagated through the atmosphere, turbulence results in modal scattering and distortions. An extensively studied example is that of light carrying orbital angular momentum (OAM), where the atmosphere is treated as a phase distortion and numerical tools extract the resulting modal cross-talk. This approach focuses on the light itself, perturbed by the atmosphere, yet does not easily lend itself to physical insights, and fails to ask a pertinent question: where did the OAM that the beam gained or lost come from? Here, we address this by forgoing the beam and instead calculating the OAM of the atmosphere itself. With this intuitive model we are able to draw general conclusions on the impact of atmospheric turbulence on OAM beams, which we confirm experimentally. Our work alters the perspective on this problem, opening new insights into the physics of OAM in turbulence, and is easily extended to other structured light fields through arbitrary aberrations.

Retrieving the pupil phase of a beam path is a central problem for optical systems across scales, from telescopes, where the phase information allows for aberration correction, to the imaging of near-transparent biological samples in phase contrast microscopy. Current phase retrieval schemes rely on complex digital algorithms that process data acquired from precise wavefront sensors, reconstructing the optical phase information at great expense of computational resources. Here, we present a compact optical-electronic module based on multi-layered diffractive neural networks printed on imaging sensors, capable of directly retrieving Zernike-based pupil phase distributions from an incident point spread function. We demonstrate this concept numerically and experimentally, showing the direct pupil phase retrieval of superpositions of the first 14 Zernike polynomials. The integrability of the diffractive elements with CMOS sensors shows the potential for the direct extraction of the pupil phase information from a detector module without additional digital post-processing. Retrieving the pupil phase of a optical beam path is a central problem for imaging systems across scales. The authors use Diffractive Neural Networks to directly extract pupil phase information with a single, compact optoelectronic device.

The new developments in Cryo-EM Single Particle Analysis are helping us to understand how the macromolecular structure and function meet to drive biological processes. By capturing many states at the particle level, it is possible to address how macromolecules explore different conformations, information that is classically extracted through 3D classification. However, the limitations of classical approaches prevent us from fully understanding the complete conformational landscape due to the reduced number of discrete states accurately reconstructed. To characterize the whole structural spectrum of a macromolecule, we propose an extension of our Zernike3D approach, able to extract per-image continuous flexibility information directly from a particle dataset. Also, our method can be seamlessly applied to images, maps or atomic models, opening integrative possibilities. Furthermore, we introduce the ZART reconstruction algorithm, which considers the Zernike3D deformation fields to revert particle conformational changes during the reconstruction process, thus minimizing the blurring induced by molecular motions.