Zernike Polynomial Fitting Research Articles

Optical–Mechanical Integration Analysis and Validation of LiDAR Integrated Systems with a Small Field of View and High Repetition Frequency

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.

Numerical investigation of aero-optical effects around the turret based on delayed detached eddy simulation and unsteady Reynolds averaged Navier-Stokes

Delayed Detached Eddy Simulation (DDES) and Unsteady Reynolds Averaged Navier-Stokes (URANS), based on the two-equation Shear Stress Transport (SST) model, are implemented to investigate the flow features and the aero-optical distortions around the turret. The Mach number is Ma = 0.4 and the Reynolds number is Re = 1.43 × 106. Instantaneous and time-averaged flow fields are presented to compare the ability of DDES and URANS in predicting the flow features. The instantaneous results show that DDES can resolve the abundant flow structures and more disordered density distributions than URANS. The time-averaged pressure coefficient and the density distribution of both methods are generally similar, but the time-averaged turbulent kinetic energy of URANS is far higher than that of DDES. The time-averaged pressure coefficient of DDES is closer to experimental data. In the windward view, typical surface flow features of DDES and URANS are similar. In the leeward view, there are remarkable differences of typical flow features between DDES and URANS. At the six angles of elevation, 60°, 76°, 90°, 103°, 120°, and 132°, the spatial-temporal wavefront distortions are calculated and discussed with the geometric ray-tracing method and the Zernike polynomial fitting, respectively. In spatial distribution, the wavefront distortions of DDES and URANS are slightly different from the experimental data. At the angles of 60°, 76°, 90°, and 103°, the tendencies of wavefront distortion of DDES at different tracing distances are the same with that of URANS, which is due to the same ability of two methods to resolve the density distributions in the attached flow region. However, the results of DDES agree well with the experimental results at the angles of 120° and 132°, which is bigger than the results of URANS. For temporal characteristics, the frequencies of wavefront distortions of DDES are obviously higher than that of URANS. The amplitudes of wavefront distortions by DDES are about 3 to 5 times higher than that by URANS. At the cases of two different FLHs at Ma = 0.4, the flow structures are totally similar, and the tendencies of wavefront distortion with θ are also similar. At the cases of three Mach number, the compression has a big influence on the wavefront distortion.

Correcting retrace and system imaging errors to achieve nanometer accuracy in full aperture, single-shot Fizeau interferometry.

X-ray mirrors with single-digit nanometer height errors are required to preserve the quality of ultra-intense photon beams produced at synchrotron or free electron laser sources. To fabricate suitable X-ray mirrors, accurate metrology data is needed for deterministic polishing machines. Fizeau phase-shifting interferometers are optimized to achieve accurate results under nulled conditions. However, for curved or aspheric mirrors, a limited choice of reference optic often necessitates measurement under non-nulled conditions, which can introduce retrace error. Using experimental measurements of a multi-tilted calibration mirror, we have developed an empirical model of Fizeau retrace error, based on Zernike polynomial fitting. We demonstrate that the model is in good agreement with measurements of ultra-high quality, weakly-curved X-ray mirrors with sags of only a few tens of microns. Removing the predicted retrace error improves the measurement accuracy for full aperture, single shot, Fizeau interferometry to < 2 nm RMS.

Compensation machining of freeform based on fast tool servo system.

With the advantages of large field of view, low cost and simple structure, the freeform optical system has extensive requirements in space exploration and other fields. However, the current machining methods for freeform are difficult to meet the requirements of optical use. Based on a developed fast servo tool (FTS) device, this paper proposes an error compensation turning method for freeforms. Firstly, the Zernike polynomial fitting method is used to reconstruct the freeform surface shape error obtained from off-line measurement, and the offset compensation is used to correct the tool path. Then, the compensation processing physical system is built to simulate the off-line compensation processing of the workpiece to verify the feasibility of compensation processing. Finally, the turning compensation processing of convex freeform aluminum mirror is carried out, and the surface accuracy of the workpiece meets the requirements of visible band. The research results have important practical significance for realizing the fast response machining of free-form surface mirror.

Separate the primary aberration for the deformation error of optical flat in 300-mm-aperture vertical Fizeau interferometer

As a key component in a phase-shifting interferometer, the optical flat is often affected by supporting methods and generates deformation. In this paper, we propose a method to separate the deformation error through rotation measurement and Zernike polynomial fitting. We studied the principles, simulations, and experiments of this method. First, a support model is established in Comsol, and the deformation of the optical flat is analyzed. Then, the numerical simulation of the separation method for the deformation is performed, and the corresponding experiment is carried out on a Φ300 mm vertical Fizeau interferometer to verify the correctness of the method. The results show that numerical calculation and experimental measurement results are in good agreement, the difference between the PV value and the RMS value are only 0.004λ and 0.002λ, respectively. Finally, the error term that affected the calculation accuracy was analyzed. This proves that the proposed method can effectively separate the deformation introduced by the support.

Modular Kirigami Arrays for Distributed Actuation Systems in Adaptive Optics

The demands on adaptive optics are increasing to achieve high performance and meet the requirements for accuracy and reliability with these systems. At present, common technologies for actuator arrays are highly developed and designed in complex arrangements leading to bulky devices. In this work, we propose an approach for designing modular actuation arrays by means of kirigami metasheets realigned in parallel and leveled in plane. The study focuses on the numerical analysis of different cut patterns to couple lift-off motions within a thin, scalable sheet. A design of a modular actuation array is explained and illustrated in the context of a deformable mirror. The optimal lift-off positions within the array are calculated for this example by a proposed positioning algorithm, which makes use of common methods in adaptive optics such as Zernike polynomial fitting. The algorithm is validated by several simulations calculating the membrane deflection, and the overall concept is discussed.

Robust phase unwrapping algorithm based on Zernike polynomial fitting and Swin-Transformer network

Phase unwrapping plays an important role in optical phase measurements. In particular, phase unwrapping under heavy noise conditions remains an open issue. In this paper, a deep learning-based method is proposed to conduct the phase unwrapping task by combining Zernike polynomial fitting and a Swin-Transformer network. In this proposed method, phase unwrapping is regarded as a regression problem, and the Swin-Transformer network is used to map the relationship between the wrapped phase data and the Zernike polynomial coefficients. Because of the self-attention mechanism of the transformer network, the fitting coefficients can be estimated accurately even under extremely harsh noise conditions. Simulation and experimental results are presented to demonstrate the outperformance of the proposed method over the other two polynomial fitting-based methods. This is a promising phase unwrapping method in optical metrology, especially in electronic speckle pattern interferometry.

High-Precision Calibration of Phase-Only Spatial Light Modulators

In the fields of optics and photonics, phase-only spatial light modulators (SLMs) play an increasingly important role in wave-front engineering. However, the SLMs are subject to wavefront distortion arising from the imperfection in the birefringence effect and physical structure of modulators. This paper presents a simple self-interference phase calibration method applicable to liquid-crystal SLM. We build an interferometric imaging system based on the Pancharatnam phase-shifting to measure the phase distribution of a light beam coming from SLMs. Two types of phase modulation errors of SLMs can be characterized in the measurement process: the erroneous gamma curve and shape aberration. The former belongs to dynamic phase distortion and is measured through a four-step Pancharatnam phase-shifting interference, which allows a one-shot recording of interference pattern via a polarization camera; the latter represents static phase distortion and is extracted from the interference between light waves coming from different regions of the SLM panel by using Zernike polynomial fitting. Our method has the advantages of high-precision pixel-wise phase correction and robustness against environmental disturbance and thus can facilitate the applications of SLM in optical field manipulation.

Robust phase unwrapping algorithm for interferometric applications based on Zernike polynomial fitting and Wrapped Kalman Filter

We propose a noise robust phase unwrapping algorithm based on the state-space model of Zernike polynomials fitting and the Wrapped Kalman Filter. In our method, the phase map is modeled as the linear combination of Zernike polynomials and the phase unwrapping problem reduces to the estimation of the fitting coefficients. To accurately calculate those coefficients under a noisy condition, the Wrapped Kalman Filter with a weighting strategy based on the second difference of the wrapped phase map is developed. As a result, our proposed method can successfully recover an absolute phase from a wrapped phase map corrupted by heavy noise. The results of synthetic data and experimental data show the outperformance of the proposed methods against other three representative algorithms.

Performance of Zernike polynomials in reconstructing raw-elevation data captured by Pentacam HR, Medmont E300 and Eye Surface Profiler

PurposeTo investigate the capability of Zernike polynomials fitting to reconstruct corneal surfaces as measured by Pentacam HR tomographer, Medmont E300 Placido-disc and Eye Surface Profiler (ESP). MethodsThe study utilised a collection of clinical data of 527 participants. Pentacam HR raw elevation data of 660 eyes (430 healthy and 230 keratoconic) were fitted to Zernike polynomials of order 2 to 20. Same analyses were carried out on 158 eyes scanned by Medmont E300 Placido-disc and 236 eyes were scanned by ESP for comparison purposes. The Zernike polynomial ​fitting was carried out using a random 80% of each individual eye surface's data up to a corneal radius of 5 mm and the root means squared fitting error (RMS) was calculated for the unused 20% portion of the surface data. The process was carried out for the anterior and posterior surfaces of the corneal measurements of the Pentacam HR and the anterior surfaces only with the ESP and the Medmont E300 measurements. ResultsStatistical significances in reduction of RMS were noticed up to order 14 among healthy participants (p < 0.0001 for right eyes, p = 0.0051 for left eyes) and up to order 12 (p < 0.0001 for right eyes, p = 0.0002 for left eyes) in anterior surfaces measured by the Pentacam. Among keratoconic eyes, statical significance was noticed up to order 12 in both eyes (p < 0.0001 for right eyes, p = 0.0003 for left eyes). The Pentacam posterior corneal data, both right and left, healthy and keratotic eyes recorded significance (p < 0.0001) in reduction of RMS up to order 10 with same RMS values of 0.0003 mm with zero standard deviation. RMS of fitting Zernike polynomials to Medmont data up to order 20 showed a consistent reduction in RMS with the increase of the fitting order with no rise at high fitting orders. Minimum RMS = 0.0047 ± 0.0021 mm, 0.0046 ± 0.0019 mm for right and left eyes respectively were recorded at order 20 and were more than 15 times the minimum RMS of the Pentacam. RMS of fitting Zernike polynomials to ESP data also showed a consistent reduction in RMS with the increase of the fitting order with no sign of any rise at high fitting orders. Similar to the Medmont, minimum RMS of 0.0005 ± 0.0003 mm, 0.0006 ± 0.0003 mm was recorded at order 20 for right and left eyes respectively and was 2 times the minimum RMS of the Pentacam for right eyes and 1.7 times the minimum RMS of the Pentacam for left eyes. ConclusionsOrders 12 and 10 Zernike polynomials almost perfectly matched the raw-elevation data collected from Pentacam for anterior and posterior surfaces, respectively for either healthy or keratoconic corneas. The Zernike fitting could not perfectly match the data collected from Medmont E300 and ESP.

Absolute measurement method for a spherical surface by random off-axis rotation based on Zernike polynomials.

An off-axis rotation absolute measurement method by regarding random shift-rotation movement as off-axis rotation is proposed. The off-axis rotation center is calculated by matching feature points on the measured surface. Then the surface profile is calculated relying on Zernike polynomial fitting. Multiple rotations are carried out to reduce symmetrical and environment errors. N(N-1)/2 results can be collected after N times rotation. Compared to traditional methods, this method can obtain accurate results with high efficiency and does not need high precision positioning.

Quantitative phase imaging in digital holographic microscopy based on image inpainting using a two-stage generative adversarial network

Based on the hologram inpainting via a two-stage Generative Adversarial Network (GAN), we present a precise phase aberration compensation method in digital holographic microscopy (DHM). In the proposed methodology, the interference fringes of the sample area in the hologram are firstly removed by the background segmentation via edge detection and morphological image processing. The vacancy area is then inpainted with the fringes generated by a deep learning algorithm. The image inpainting finally results in a sample-free reference hologram containing the total aberration of the system. The phase aberrations could be deleted by subtracting the unwrapped phase of the sample-free hologram from our inpainting network results, in no need of any complex spectrum centering procedure, prior knowledge of the system, or manual intervention. With a full and proper training of the two-stage GAN, our approach can robustly realize a distinct phase mapping, which overcomes the drawbacks of multiple iterations, noise interference or limited field of view in the recent methods using self-extension, Zernike polynomials fitting (ZPF) or geometrical transformations. The validity of the proposed procedure is confirmed by measuring the surface of preprocessed silicon wafer with a Michelson interferometer digital holographic inspection platform. The results of our experiment indicate the viability and accuracy of the presented method. Additionally, this work can pave the way for the evaluation of new applications of GAN in DHM.

Phase-aberration compensation via deep learning in digital holographic microscopy

Digital holographic microscopy (DHM), a quantitative phase-imaging technology, has been widely used in various applications. Phase-aberration compensation in off-axis DHM is vital to reconstruct topographical images with high precision, especially for microstructures with a small background or a dense phase distribution. We propose a numerical method based on deep learning combined with DHM. First, a convolutional neural network (CNN) recognizes and segments the sample and the background area of the hologram. Zernike polynomial fitting is then executed on the extracted background area. Finally, the whole process of phase-aberration compensation is automatically performed. To obtain a robust and accurate deep-learning model for hologram segmentation, we collected many holograms corresponding to several samples that had different morphological characteristics. The experimental results confirm that the trained CNN can accurately segment the sample from the background area of the hologram, and that this method is applicable and effective in off-axis DHM.

Stitching interferometric characterization of geometric errors of hemispherical shells

Geometric errors including the surface error and thickness variation of a hemispherical shell are determinant of the system performance, e.g. the frequency splitting of a hemispherical resonator gyroscope. Therefore, complete characterization at 0.1 µm level of accuracy is required which is difficult with current roundness measuring instruments. This paper presents an interferometric method for axially eccentric hemispherical shells. The outer surface and inner surface are measured individually by simply translating the shell along its anchor axis. The two measurements are registered after correction of pixel misalignment and mapping distortion through ray tracing. Thickness variation is then calculated from their difference. A multi-axis platform is designed to scan multiple subapertures covering the hemispherical shell. All subapertures are stitched together with misalignment corrected, according to the minimized inconsistency in the overlapping region of neighboring subapertures. The full aperture map is thus obtained for both the surface error and thickness variation. Polar harmonic representation is suggested by applying annular Zernike polynomial fitting. The method is experimentally demonstrated and validated with roundness cross test.

Comparison of processing speed of typical wavefront reconstruction methods for lateral shearing interferometry.

Lateral shearing interferometry is widely applied in wavefront sensing, optical components testing, and defect inspection. The procedure of reconstructing the wavefront is the most specific difference between lateral shearing interferometry and other classical methods such as the Fizeau and Twyman interferometers. The speed and accuracy are two main features to evaluate the performance of one wavefront reconstruction method. In this work, optimized procedures for three typical wavefront reconstruction methods-the iterative FFT wavefront reconstruction method (FFT method), the partial differential least-squares method (LSQ method), and the difference Zernike polynomial fitting method (DZF method)-are designed. The calculation speeds of the three wavefront reconstruction methods are evaluated with different GPUs and CPUs. According to the test results, the DZF method is the fastest method both in the GPUs and CPUs. The shortest processing times of the DZF, FFT, and LSQ methods are 100, 449, and 494 ms, respectively, with the wavefront size of 1024×1024pixels. The calculation speeds of the FFT method and the LSQ method are similar in the CPUs, and the FFT method is faster in the GPUs. The relationship between the consumed time and the wavefront size is an exponential function in the CPUs and a power function in the GPUs. Generally speaking, GPUs' processing speeds are faster than CPUs'. But CPUs can be faster than GPUs when the test wavefront sizes are smaller than 64×64pixels. Besides, the differences between the consumed times of different CPUs are relatively smaller than those of the GPUs.

Method for absolute measurement of flat surface.

Interferometry is a relative measurement method for optical surface testing, and thus its testing accuracy depends on the accuracy of the reference surface. Absolute measurement is one of the most effective methods to improve the testing accuracy in interferometry. We present an efficient absolute measurement method based on Zernike polynomial fitting algorithms. With our proposed method, the profiles of both the test surface and the reference surface can be calculated simultaneously. We further carried out simulation analysis to scientifically evaluate the test accuracy of the proposed method. Finally, we conducted actual experiments to demonstrate the feasibility and practicability of our method.

Simulation and experimental study of laser-induced thermal deformation of spectral beam combination grating.

The multilayer dielectric (MLD) grating is a critical device for combining multiple laser beams into a single beam in a spectral beam combining (SBC) system. We established a theoretical thermal deformation model of the laser-irradiated MLD grating. Thermal deformation on the surface of the grating is simulated according to a series of parameters including the laser irradiation time, laser power density, and substrate size. To verify the model, we exposed a 960 l/mm, 50×50×1.5 mm3 grating to a laser power density of 3.61 kW/cm2 and observed the temperature change. We used a Twyman-Green interferometer to measure the interference fringes on the grating surface. Based on the Fourier-transform method and a Zernike polynomial fitting method, the real-time grating surface profile is reconstructed. The results show that substrate thickness increase or area decrease can reduce thermal deformation, the average decreases are 18.3% and 19.9%, respectively. The discussion and analysis of the grating thermal deformation are potentially valuable for designing grating to decrease the thermal deformation and improve the combined beam quality of a SBC system.

Phase aberration compensation of digital holographic microscopy with curve fitting preprocessing and automatic background segmentation for microstructure testing

In the digital holographic microscopy, the phase aberration is a key problem limiting the quantitative phase measurement, especially for the microstructure whose phase distribution is dense with few and discontinuous background region. A numerical phase aberration compensation method with global curve fitting preprocessing and automatic extraction of the background region for microstructure testing is proposed. The global curve fitting is implemented to compensate most of the tilt, quadratic and partial high order aberrations and make the profile of the measured object significantly distinguishable. Then the background region is automatically extracted by directly segmenting the converted grayscale phase image. At last, the Zernike polynomial fitting is performed on the extracted background region, and the obtained coefficients are used to calculate the phase aberration of the whole phase image. Simulation and experimental results demonstrated that the proposed method is able to accurately extract background region and compensate the phase aberration for the complicated dense microstructure.

Probe misalignment calibration in fiber point-diffraction interferometer.

The measuring probe integrated with multiple fiber point-diffraction sources can be applied to measure both the three-dimensional coordinates and highly accurate point-diffraction wavefront. The probe determines the achievable measurement accuracy of fiber point-diffraction interferometer (PDI), in which the fiber exit end plane is required to be parallel with the detector plane. The probe misalignment due to fabrication error could introduce significant measurement error. A high-precision method is proposed to calibrate the probe misalignment in fiber PDI, including the central positioning based on phase difference and tilt adjustment based on Zernike polynomials fitting. Both numerical simulation and experiments have been carried out to demonstrate the feasibility and accuracy of the proposed probe misalignment calibration method. The proposed method provides a feasible way to address the processing uncertainty on measuring probe in fiber PDI, and enables high-precision geometry alignment and misalignment calibration in the interferometric testing systems with case of no imaging lens.

Positions of Ocular Geometrical and Visual Axes in Brazilian, Chinese and Italian Populations

ABSTRACTPurpose: To identify the relative positions of geometrical and visual axes of the eye and present a method to locate the visual center when the geometrical axis is taken as a reference.Methods: Topography elevation data was collected using a Pentacam HR ® topographer from 2040 normal eyes of 1020 healthy participants in Brazil, China and Italy. A three-dimensional, rotation algorithm, a first-order Zernike polynomial fit and a nonlinear least-squares error function was used within an optimization function to locate the geometrical axis and the visual center of each eye.Results: The right eyes of participants were significantly more tilted than left eyes throughout the topography scanning process (p < 0.001). The visual centers were always located in the nasal-superior quadrant, although the visual centers of fellow eyes were not symmetrically located. Mean distances between the visual center and the geometrical center in right eyes were 0.8 ± 0.29 mm, 0.56 ± 0.18 mm and 0.91 ± 0.34 mm among Brazilian, Chinese and Italian participants, respectively, and located at angular positions of 38.7 ± 24.5°, 23.0 ± 29.8° and 23.1 ± 28.1° from the nasal side. However, in left eyes, mean distances were 0.76 ± 0.33 mm, 0.45 ± 0.12 mm and 0.75 ± 0.33 mm at polar angles from the nasal side of 59.3 ± 29.0°, 50.6 ± 44.5° and 61.8 ± 34.1°, respectively.Conclusions: Fellow eyes do not perform similarly during the fixation process, with right eyes tending to tilt more than left eyes, and the visual centers of the fellow eyes positioned differently relative to the geometrical centers.