Zernike Coefficients Research Articles

Deep learning-assisted wavefront correction with sparse data for holographic tomography

In this paper, a novel approach using deep learning-assisted wavefront correction in beam rotation holographic tomography to acquire three-dimensional images of native biological cell samples is described. With digitally recorded holograms, the wavefront aberration is contained in the reconstructed phases. However, there are large computation costs for compensating the phase aberration during the reconstruction. With the aid of a deep convolution network, we present an effective algorithm on the reconstructed phases with sparse data for active wavefront correction. To accomplish this, we developed a Res-Unet scheme to segment the cell region from its background aberration and a deep regression network for the representation of the aberration on Zernike orthonormal basis. Moreover, a sparse data fitting algorithm was used to predict the Zernike coefficients of whole scanning angles from the collected sparse data. As a result, the proposed algorithm is capable of accurately correcting the background aberration and much faster than the original plain algorithm.

Effect of Accommodation on Peripheral Higher Order Aberrations

Knowledge of the effect of accommodation on image quality of peripheral retina is crucial for better understanding of the visual system, but only a few studies have been carried out in this area. This study was designed to evaluate the effect of accommodation on higher order aberrations from third to sixth Zernike polynomials in central and peripheral retina up to 23° off-axis. We used a Hartmann–Shack aberrometer to measure Zernike coefficients with both accommodated and non-accommodated eyes of 15 healthy subjects. Each Zernike coefficient, total higher order aberrations, spherical aberrations and astigmatism were compared between accommodated and non-accommodated status. Additionally, aberrations in the central retina were compared with the peripheral retina. Accommodation induced significant changes in the Zernike coefficients of vertical pentafoil C5−5 and secondary vertical tetrafoil C6−4 in central retina, secondary vertical astigmatism C4−2 on 23° of temporal retina, secondary vertical tetrafoil C6−4 and tertiary vertical astigmatism C6−2 on 10° of nasal retina, secondary vertical trefoil C5−3 and secondary vertical tetrafoil C6−4 on 23° of nasal retina, and horizontal tetrafoil C44, and secondary horizontal tetrafoil C64 on 23° of inferior retina (p < 0.05). Total higher order aberration was lower in each retinal area examined with accommodation, but it was statistically significant only on 23° temporal retina and 11.5° and 23° of superior retina (p < 0.05). Spherical aberration decreased with accommodation on 23° temporal retina (p = 0.036). Astigmatism was similar in non-accommodated and accommodated eyes. Overall, accommodation affected higher order aberration (HOA) asymmetrically in different peripheral retinal areas.

Сравнение результатов коррекции миопии с помощью visx customvue wavefront-guided lasik и стандартным lasik

Purpose. To compare visual and clinical outcomes of wavefront-guided VISX CustomVue (VISX, Santa Clara, CA) LASIK and conventional LASIK. Methods. This study retrospectively evaluated 130 eyes of 65 patients, including 40 eyes of 20 patients in the CustomVue group and 90 eyes of 45 patients in conventional LASIK group. Uncorrected distance visual acuity (UDVA), corrected distance visual acuity, and refractive error were measured. Higher-order aberration (HOA) were measured by WaveScan system using a Hartmann-Shack sensor as their root mean square values (RMS HO), and individual Zernike coefficients of 3rd to 4th order before and 1 year after the surgery and compared using Mann-Whitney test or Student's t-test. Results. There was no significant difference in spherical equivalent between the groups. CV eyes showed superiority in number of eyes seeing 20/20 or better. RMS HO was less in CV group than that induced by conventional LASIK (0.37 ± 0.15 µm and 0.62 ± 0.26 µm, respectively). Preoperative mean RMS HO more than 0.3 µm was associated with a decrease in postoperative HOA in CV group. CV system induced significantly less spherical aberration (0.149 ± 0.157 µm and 0.364 ± 0.284 µm, respectively, р < 0,002). Conclusions. CustomVue seems to have better uncorrected visual acuity outcomes. It generally increases HOA, but statistically significant by less, compared with conventional LASIK. The most benefits have patients with RMS HO higher than 03 µm. Key words: CustomVue, conventional LASIK, wavefront-guided LASIK.

Prediction of manifest refraction using machine learning ensemble models on wavefront aberrometry data

PurposeTo assess the performance of machine learning (ML) ensemble models for predicting patient subjective refraction (SR) using demographic factors, wavefront aberrometry data, and measurement quality related metrics taken with a low-cost portable autorefractor. MethodsFour ensemble models were evaluated for predicting individual power vectors (M, J0, and J45) corresponding to the eyeglass prescription of each patient. Those models were random forest regressor (RF), gradient boosting regressor (GB), extreme gradient boosting regressor (XGB), and a custom assembly model (ASB) that averages the first three models. Algorithms were trained on a dataset of 1244 samples and the predictive power was evaluated with 518 unseen samples. Variables used for the prediction were age, gender, Zernike coefficients up to 5th order, and pupil related metrics provided by the autorefractor. Agreement with SR was measured using Bland-Altman analysis, overall prediction error, and percentage of agreement between the ML predictions and subjective refractions for different thresholds (0.25 D, 0.5 D). ResultsAll models considerably outperformed the predictions from the autorefractor, while ASB obtained the best results. The accuracy of the predictions for each individual power vector component was substantially improved resulting in a ± 0.63 D, ±0.14D, and ±0.08 D reduction in the 95% limits of agreement of the error distribution for M, J0, and J45, respectively. The wavefront-aberrometry related variables had the biggest impact on the prediction, while demographic and measurement quality-related features showed a heterogeneous but consistent predictive value. ConclusionsThese results suggest that ML is effective for improving precision in predicting patient's SR from objective measurements taken with a low-cost portable device.

Aberration analysis and compensate method of fully connected neural network in deep ultraviolet lithography

Thermal aberration of a projection lens is caused by lens heating and leads to degradation of image quality. Two tasks have been undertaken in this article: first, analysis of the impact of single aberration on imaging quality, and second, use of machine learning to calculate the combination of Zernike aberration coefficients to reduce the influence of aberration on the process window. The impact of aberration on deep ultraviolet (DUV) lithography imaging has been analyzed separately for 16 Zernike terms. The influence of single Zernike coefficient on depth of focus, mask error enhancement factor, and image log slope is comprehensively analyzed by source mask optimization. However, more than one Zernike term is present in the aberration of a real DUV optics. A fully connected neural network (FCNN) model based on machine learning is widely used in the screening of multivariate experiments and can solve the problem of the large number of input variables and the random values of the input data. Its network structure is simple to debug, with fast screening speed and high precision. To make reasonable budget for aberration, an FCNN model is used to find some combinations of Zernike coefficients for different lithographic performance indicators. This implies that the FCNN model has the ability to select a better Zernike combination while each individual Zernike term value falls in the range of −30 to 30 mλ, which plays a guiding role in DUV optics design.

Measurement of the magnification and distortion of high-precision imaging system using dual-fiber point diffraction interferometer

Magnification and distortion are two important parameters for high-precision imaging systems. Point diffraction interferometers (PDIs) can measure the magnification, distortion, and wavefront aberration of imaging systems with high precision. However, determining the precise pinhole alignment of the classical PDI is difficult. A new method for measurement of the magnification and distortion based on a dual-fiber point diffraction interferometer (DFPDI) is proposed. The end faces of two fibers are placed on the object plane of the optics under test and imaged to the image plane. The distance between the image points in the x and y directions are proportional to the Z2 and Z3 Zernike coefficients of the wavefront measurement result, respectively. The measurements of the image placement shift and precise alignment of the point diffraction pinhole are realized rapidly with high accuracy. The feasibility of the method is verified experimentally. The wavefront aberration, magnification, and distortion of a 5 × reduction lens with numerical aperture (NA) of 0.3 is measured jointly. The measurement uncertainties (3σ) of the magnification in the x and y directions and distortion are 756 ppm, 793 ppm, and 0.233 μm, respectively. Error analysis shows that the position error of the object- and image-plane stages is the main error source. An improved measurement scheme with a pinhole–pinhole pairs array in the object plane and a pinhole–window pairs array in the image plane is proposed. The influence of the position errors of the stages is eliminated with optimized measurement procedure. The DFPDI’s measurement repeatability (3σ) of the Z2 and Z3 coefficients is 0.65 and 0.33 nm, respectively, corresponding measurement uncertainties (3σ) of the magnification (in the x and y directions) and distortion can reach 1.88 ppm, 1.69 ppm, and 0.812 nm, respectively.

Spatiotemporal statistics of the turbulent piston-removed phase and Zernike coefficients for two distinct beams.

In the context of adaptive optics for astronomy, one can rely on the statistics of the turbulent phase to assess a part of the system's performance. Temporal statistics with one source and spatial statistics with two sources are well known and widely used for classical adaptive optics systems. A more general framework, including both spatial and temporal statistics, can be useful for analysis of the existing systems and to support the design of future ones. In this paper, we propose an expression of the temporal cross power spectral densities of turbulent phases in two distinct beams, which is from two different sources to two different apertures. We consider the phase either as it is, without a piston, or as its decomposition on Zernike modes. The general formulas allow coverage of a wide variety of configurations, from single-aperture to interferometric telescopes equipped with adaptive optics, with the possibility to consider apertures of different sizes and/or sources at a finite distance. The presented approach should lead to similar results with respect to existing methods in the Fourier domain, but it is focused on temporal frequencies rather than spatial ones, which might be convenient for some aspects such as control optimization. To illustrate this framework with a simple application, we demonstrate that the wavefront residual due to the anisoplanatism error in a single-conjugated adaptive optics system is overestimated when it is computed from covariances without taking into account the temporal filtering of the adaptive optics loop. We also show this overestimation in the case of a small-baseline interferometer, for which the two beams are significantly correlated.

Effect of Central Corneal Thickness on Corneal Higher Order Aberrations After Cataract Surgery.

To investigate the changes of corneal higher order aberrations (HOAs) before and after phacoemulsification in patients with different central corneal thickness (CCT). In this retrospective non-randomized, non- comparative cases series, 51 eyes of 51 patients who underwent phacoemulsification were enrolled. The corneal total HOAs and Zernike coefficients (3rd and 4th order) over the 3- and 4-mm zones were measured by the iTrace aberrometer (Tracey Technologies). Participants were divided into three groups (528 µm or less, 529 to 550 µm, and greater than 550 µm) depending on the CCT tested by the LenStar LS900 (Haag Streit AG). The corneal aberrations between groups were compared with analysis of variance, the variation of corneal aberrations before and after phacoemulsification was analyzed by the paired-samples t test, and generalized linear models were used to compare postoperative aberrations between groups. Significant differences were found in oblique trefoil at 3 mm and vertical trefoil, oblique trefoil, and HOAs at 4 mm in the 528 µm or less group; vertical trefoil, oblique trefoil, oblique quadrafoil, and oblique secondary astigmatism at 3 mm and oblique trefoil and oblique secondary astigmatism at 4 mm in the 529 to 550 µm group; and vertical quadrafoil at 3 mm in the greater than 550 µm group. After corrected deviation of preoperative aberrations, there were significant differences in postoperative vertical coma and oblique secondary astigmatism at 4 mm when compared to the 528 µm or less and greater than 550 µm groups. The lower the CCT, the greater the change in corneal aberrations in patients with cataract after phacoemulsification. Surgeons should be aware of the potential for worse visual quality related to surgically induced aberrations in patients with thin corneas. [J Refract Surg. 2021;37(12):842-847.].

THE ACCURACY OF ZERNIKE RECONSTRUCTION OF OCULAR ABERRATIONS WITH MEASUREMENT NOISE

The ocular aberrations (wavefront) can be used to describe the vision defects, and wavefront sensing devices (aberrometers) are used to measure the ocular aberrations (OAs). The Zernike polynomials (ZPs) have been successfully used in wavefront reconstruction algorithms of aberrometers for years. As the aberrometer measurements consist with noise, the measurements observed using aberrometers are not always accurate. It is therefore necessary to take multiple measurements from the patient in each sitting. However, taking multiple measurements also causes to induce the variations on measurements, and this variability of measurements is significant, and leads errors in clinical practice. Therefore, the variability should be taken into account during wavefront reconstruction. Consequently, unlike prior work, the accuracy of Zernike representation for the ocular aberrations with noise (due to the system noise and the variability of measurements) is studied. In the study, the noisy data is fitted using ZPs. Data sets are created using extracted data from arbitrary OAs, that is, synthetic data sets are used. Normally distributed very small random numbers are added into the data sets in order to create measurement noise. The magnitude of added noisy has been changed to range the signal-to-noise ratio (SNR) from 55 dB to 20 dB. The corresponding Zernike coefficients for each noised wavefront are computed, and visual acuity is used to examine the deviation of the reconstructed wavefront from the data set. The study concludes that the Zernike polynomials are not good enough to represent the wavefront with noisy level less than certain value of SNR which is greater than the reference SNR values of aberrometers, and note that the aberrometer SNR is always between 20-30 dB.

Extended aberration analysis in symmetry-free optical systems - part I: method of calculation.

The imaging performance evaluation of symmetry-free optical systems is of great importance during the correction process of optical design. However, due to the complexity and limitations of the available tools, the higher-order aberrations in the system cannot be well analyzed and are hard to control. In this paper, the theoretical background and the mathematical approach of a quantitative analysis method for surface-decomposed transverse aberration in the symmetry-free systems are introduced. With the mixed ray-tracing calculation in both real and paraxial cases, the implementations of full-order intrinsic/induced aberration, as well as the surface-additive Zernike coefficient fitting method are demonstrated. The applications of this method help assess the correction performance considering the relatively critical surfaces in an arbitrary off-axis system. The reliability and the accuracy of the method will be evaluated in part II with a test system. And as an illustration of the practical usage of the method for optical design, the corresponding applications on a group of lithographic systems will also be demonstrated.

Misalignment algorithm of a wide-field survey telescope based on third-order quadratic nodal aberration theory

Off-axis three-mirror anastigmatic (TMA) telescopes with designed decenters and tilts, have many misalignment degrees of freedom and strong coupling between each misalignment degree of freedom. Therefore, it is difficult to establish misalignment equations only using A222 and A131 in nodal aberration theory (NAT). In addition, for off-axis TMA optical systems with designed decenters and tilts, the robustness of the existing fifth-order NAT misalignment calculation algorithm based on high-order Zernike coefficients and boresight errors decreases, so it is difficult to realize its engineering application. To solve the issue of insufficient practicality of the existing misalignment algorithm based on fifth-order NAT, a third-order NAT calculation algorithm based on quadratic aberration field decenter vectors is derived and established. Two concepts of inherent aberration field decenter vector and misalignment aberration field decenter vector are proposed. Taking an off-axis TMA optical system with 6-m focal length as the research object, simulations, and alignment verification experiments were carried out. Compared with the existing fifth-order NAT misalignment algorithm, the results show that when measurement noise is not considered, the two methods can both obtain convergent calculation results, and the average RMS wavefront errors (WFE) of the optical system are both corrected to be below 0.0574 waves. When different levels of measurement noise are introduced, the robustness of the fifth-order NAT misalignment algorithm decreases, and there are even cases where the optical system completely fails to be corrected. However, the algorithm based on quadratic aberration field decenter vectors shows better robustness. Under different levels of measurement noise, this algorithm could correct the average RMS WFE of the optical system to around 0.0574 waves.

Near Vision Tasks and Optical Quality of the Eye.

PurposeTo study the effect of near-vision reading task on optical quality of the eye when performed on a computer monitor and on printed paper, and to identify which of the two results in greater changes.MethodsTwo groups of subjects performed a 30-min reading task in two different conditions: on a computer monitor and on printed paper. Ocular, corneal, and internal wavefront aberrations (Zernike coefficients up to 6 order), root-mean-square of low- and high-order aberrations, spherical equivalent, vectoral components of ocular astigmatism (J45 and J0), and the compensation factor between internal and corneal aberrations were measured before and after the tasks. Their changes were analyzed in each group and between groups.ResultsStatistically significant changes in wavefront aberrations and in root mean square of low- and high-order aberrations were observed in both groups which was significantly greater when the task was performed on printed paper. Partial loss of compensation mechanism and variation in spherical equivalent in a negative direction occurred after both reading tasks; however, it was statistically significant only with printed paper reading task. The vectoral components of ocular astigmatism did not show statistically significant changes in either groups.ConclusionNear-vision reading tasks can change the optical quality of the eye, especially when the task is performed on printed paper.

Passive Athermal Optical Design Method Considering Thermal-Induced Surface Deformation

Temperature variation not only results in changes in refractive index, radius, thickness, and air space, but also leads to surface deformation due to the mismatch in thermal expansion coefficients between glass and mechanical materials. However, existing passive athermal optical design methods cannot optimize thermal-induced surface deformation, and optimization methods usually focus on structural optimization or thermal control rather than optical optimization. Here, we investigate the deterioration in image quality caused by thermal-induced surface deformation and propose a passive athermal optical design method to reduce deterioration. To this end, MATLAB was utilized to jointly call finite element analysis (FEA) software (COMSOL) and optical design software (Code V) to realize the data exchange of an optical–mechanical–thermal integrated analysis for iterative optical optimization. This process makes automatic iterative optimization possible by transforming parametric FEA results into Zernike coefficients in each iteration of optimization. The theoretical and design examples indicate that our method can effectively reduce the degradation in image quality with surface deformation. Our method provides an optical optimization approach for optical designers to work on a passive athermal optical design by considering thermal-induced surface deformation.

Finite-element-method study of tilt mirrors without PZTs using different clamping methods under laser irradiation

Tilt mirrors play a fundamental role in correcting aberrations in adaptive optics systems. To analyze the influence of clamping methods and materials on temperature rising, deformations and high-order aberrations of tilt mirrors without PZTs during the evaluation of laser irradiation of multilayer coatings, a finite-element model of tilt mirror with a diameter of 100 mm and thicknesses of 12.5 mm was constructed. The deformation and high-order aberrations were simulated after applying a laser to the mirrors. The appearance of peak-to-valley and high-order aberrations became more significant as the tilt mirror was supported on the edge at 3 points 120° apart. Using 3 points at the back and 3 points on the side 120° apart with polytetrafluoroethylene (PTFE) protection clamping lessened the effect, and the PV was only 0.0828 µm. The corresponding 65th-order Zernike coefficients were lower, so the clamping approach is ideal for fixing the tilt mirror for evaluating the multilayer coatings. Also, the temperature rising and thermal deformation of tilt mirrors made from various materials under laser irradiation were calculated. The results show that the tilt mirror made of ultra-low expansion microcrystalline glass had the highest temperature rising and the lowest thermal deformation. The temperature rising and PV was 10.56 ℃ and 0.004 µm, respectively. Beryllium and silicon carbide materials are suitable for increasing the resonance frequency and reducing the weight of tilt mirrors. These results are also useful for designing the clamping methods of large size folding mirror, telescope fast steering mirror and small scanner mirror, etc.

Zernike Coefficient Prediction Technique for Interference Based on Generation Adversarial Network

In the paper, we propose a novel prediction technique to predict Zernike coefficients from interference fringes based on Generative Adversarial Network (GAN). In general, the task of GAN is image-to-image translation, but we design GAN for image-to-number translation. In the GAN model, the Generator’s input is the interference fringe image, and its output is a mosaic image. Moreover, each piece of the mosaic image links to the number of Zernike coefficients. Root Mean Square Error (RMSE) is our criterion for quantifying the ground truth and prediction coefficients. After training the GAN model, we use two different methods: the formula (ideal images) and optics simulation (simulated images) to estimate the GAN model. As a result, the RMSE is about 0.0182 ± 0.0035λ with the ideal image case and the RMSE is about 0.101 ± 0.0263λ with the simulated image case. Since the outcome in the simulated image case is poor, we use the transfer learning method to improve the RMSE to about 0.0586 ± 0.0035λ. The prediction technique applies not only to the ideal case but also to the actual interferometer. In addition, the novel prediction technique makes predicting Zernike coefficients more accurate than our previous research.

Higher order aberrations and retinal image quality during short-term accommodation in children

Changes in higher order aberrations (HOA’s) and retinal image quality during accommodation have not previously been examined in children. This study measured ocular HOA’s in ninety non-myopic, school-aged children during short-term accommodation tasks at 0, 3, 6, and 9 D demands presented via a Badal optometer mounted to a Hartmann-Shack wavefront aberrometer (COAS-HD, Wavefront Sciences). Eighty-four participants who exhibited active accommodation were included in the analyses. An eighth order Zernike polynomial was fit across a 2.3 mm, 4 mm, and natural pupil diameter to evaluate changes in refractive power vectors (M, J180, and J45), accommodation errors (lags and leads), HOA root mean square (RMS) variables, individual Zernike coefficients, and the visual Strehl ratio based on the optical transfer function (VSOTF). All HOA RMS variables changed significantly with accommodation, with the greatest change observed for the 9 D demand. Of the individual Zernike coefficients, primary (C40) and secondary spherical aberration (C60) exhibited the greatest magnitude of change, becoming negative and positive with increasing accommodation, respectively. The VSOTF changed significantly with greater accommodation for both the 4 mm and natural pupil size, becoming significantly worse for the 9 D demand. HOA’s increase and retinal image quality decreases significantly during higher levels of accommodation in children, similar to adults. These findings provide a greater understanding of the optical properties of children’s eyes and insights into possible mechanisms for the association between accommodation, near work, and refractive error development.

Unified analytical method for Zernike coefficient transformation of scaled, rotated, and translated pupils based on Shack's vector multiplication.

Zernike polynomials play an essential role in characterizing and analyzing wavefront aberrations. Transformation of weighted coefficients for Zernike modes is required when pupil scaling, rotation, and/or translation exist. Here, a novel method based on Shack's vector multiplication is first proposed to derive the transformation relation. The derived modes resulting from pupil scaling, rotation, and/or translation for each individual mode are easily indicated via this method; thus, the effect of each kind of pupil change could be studied qualitatively and quantitatively. Its remarkable computational efficiency against the direct integral is demonstrated by simulation. The method introduced here provides a generalized methodology to analyze the relationship between weighted coefficients for different description basis sets.

Optimization of Virtual Shack-Hartmann Wavefront Sensing.

Virtual Shack–Hartmann wavefront sensing (vSHWS) can flexibly adjust parameters to meet different requirements without changing the system, and it is a promising means for aberration measurement. However, how to optimize its parameters to achieve the best performance is rarely discussed. In this work, the data processing procedure and methods of vSHWS were demonstrated by using a set of normal human ocular aberrations as an example. The shapes (round and square) of a virtual lenslet, the zero-padding of the sub-aperture electric field, sub-aperture number, as well as the sequences (before and after diffraction calculation), algorithms, and interval of data interpolation, were analyzed to find the optimal configuration. The effect of the above optimizations on its anti-noise performance was also studied. The Zernike coefficient errors and the root mean square of the wavefront error between the reconstructed and preset wavefronts were used for performance evaluation. The performance of the optimized vSHWS could be significantly improved compared to that of a non-optimized one, which was also verified with 20 sets of clinical human ocular aberrations. This work makes the vSHWS’s implementation clearer, and the optimization methods and the obtained results are of great significance for its applications.

Agreement between Scheimpflug Camera and the Swept-source Optical Coherence Tomography Measurements in Keratometry and Higher-order Aberrations.

PurposeTo evaluate the compatibility of corneal curvature and astigmatism, and higher-order aberrations (HOAs) measured by the Scheimpflug camera Pentacam HR and the swept-source optical coherence tomography ANTERION.MethodsThis prospective study included normal subjects with no ophthalmic history. Steep keratometry (K), flat K, astigmatism and its axis of the anterior and posterior surfaces, total corneal power, and HOAs using the two instruments were compared. To compare the mean values of the measurements, a paired t-test was used. Bland-Altman analysis was applied to assess the agreement between the two devices.ResultsFifty-three eyes of 53 subjects were evaluated. There were statistically significant differences for steep K, astigmatism, and vector J0, J45 in the anterior surface and total corneal power between the two devices (p < 0.05). There were also significant differences in the most of the keratometric values of the posterior corneal surface (p < 0.05) except J0 (p = 0.410). Both devices showed strong positive correlations in steep K, flat K, astigmatism (r > 0.81, p < 0.001) with wide ranges of a 95% limit of agreement. Vectoral components were significantly correlated (r > 0.78, p < 0.001) with narrow 95% limit of agreement, except J45 of the posterior surface (r = 0.39, p = 0.004). In the corneal HOAs, there were statistically significant differences in the vertical coma, horizontal trefoil, spherical aberration, and root mean square of each fifth- and sixth-order Zernike coefficient (p = 0.043, p = 0.041, p < 0.001, p < 0.001, and p < 0.001, respectively). Other HOAs showed moderate to strong positive correlations (r > 0.37, p < 0.05). Most HOAs, except for the horizontal trefoil, showed clinically acceptable agreements. The total root mean square of HOAs was not significantly different between the two devices (p = 0.122).ConclusionsMost of the keratometric values cannot be used interchangeably. However, the vectoral component of astigmatism showed clinically good agreement. Several HOAs have statistically significant differences; however, almost all HOAs showed acceptable agreements, except for the horizontal trefoil.

Wavefront sensing method for the sparse aperture optical system

A method based on the Zernike coefficients transformation is proposed for the wavefront sensing of a sparse aperture system. We deduce a matrix that allows the Zernike transformation between the sub-apertures’ wavefronts and the global counterparts in a sparse aperture system. With the matrix and the given global aberrations, the aberrations of the sub-apertures can be calculated through the transformation. The simulation results indicate that the increase of filling factor of the sparse aperture system could lead to the increase of orders of the Zernike polynomials, which describe the sub-aperture’s wavefront. The root mean square value of the detected Zernike coefficients of the sub-aperture is determined by its azimuth angle when the filling factor is fixed.