Wavefront Error Research Articles

Combining EFC with spatial LDFC for high-contrast imaging on Subaru/SCExAO

Context.Exoplanet direct imaging is a key science goal of current ground-based telescopes as well as of future ground-based extremely large telescopes and space-based telescopes. Several high-contrast imaging (HCI) systems for direct exoplanet imaging have been developed and are implemented on current telescopes. Despite recent developments in HCI systems, the contrast they deliver is limited by non-common path aberrations (NCPAs) and residual wavefront errors of the adaptive optics (AO) system. To overcome this limitation and reach higher contrast, HCI systems need focal plane wavefront-sensing and control (FPWFS&C) techniques.Aims.We propose a method that provides both deep contrast and a 100% duty cycle by combining two complementary FPWFS&C methods: electric field conjugation (EFC), and spatial linear dark field control (LDFC). The ultimate goal of this work is to generate the high contrast zone, which is called the dark hole, in the focal plane by using EFC and to maintain the contrast within the high-contrast zone by using spatial LDFC without interrupting science observations. We describe the practical implementation, quantify the linearity range over which LDFC can operate, and derive its photon-noise-limited dynamical performance.Methods.We implemented EFC+LDFC on the Subaru Coronagraphic Extreme Adaptive Optics (SCExAO) instrument using its internal light source (off-sky). We first deployed the implicit EFC (iEFC) algorithm to generate the dark hole with a classical Lyot coronagraph (CLC) with a 114 mas diameter focal-plane mask at 1550 nm wavelength. This iEFC algorithm was deployed with pair-wise probes. Using iEFC with pair-wise probes, we directly measured the response matrix of the deformable mirror (DM) modes and built the control matrix by inverting the response matrix. After the calibration process, we generated the dark hole by closing the iEFC loop. When the dark hole was generated, we implemented spatial LDFC to restore and maintain the contrast of the dark hole. In the tests shown here, we introduced static and quasi-static speckles, and then we operated spatial LDFC in closed loop to verify its performance. We used numerical simulations to derive linearity range and photon-noise-limited dynamical performance.Results.Using iEFC, we generated the dark hole with a ~2×10−7contrast in a narrow-band filter (λ= 1550 ± 25 nm). We reached a contrast floor limited by the camera noise. Comparison between pre- and post-iEFC images shows that with iEFC in closed-loop operation, an improvement in contrast of a factor ≈ 100–500× was reached across the dark hole. In the spatial LDFC experiments, we were able to nearly fully remove the speckles generated by the DM perturbation and maintain the contrast of the dark hole.Conclusions.This work presents the first laboratory demonstration of combining two FPWFS&C methods, iEFC and spatial LDFC. Linear range and photon-noise-limited sensitivity are provided to derive close-loop performance for on-sky systems. Our results provide a promising approach for taking advantages of both high contrast and a 100% science duty cycle for HCI systems.

Myopia Control Dose Delivered to Treated Eyes by a Dual-focus Myopia-control Contact Lens.

This study examined the optical impact of a DF contact lens during near viewing in a sample of habitual DF lens wearing children. Seventeen myopic children aged 14 to 18 years who had completed 3 or 6 years of treatment with a DF contact lens (MiSight 1 Day; CooperVision, Inc., San Ramon, CA) were recruited and fit bilaterally with the DF and a single-vision (Proclear 1 Day; CooperVision, Inc.) contact lens. Right eye wavefronts were measured using a pyramidal aberrometer (Osiris; CSO, Florence, Italy) while children accommodated binocularly to high-contrast letter stimuli at five target vergences. Wavefront error data were used to compute pupil maps of refractive state. During near viewing, children wearing single-vision lenses accommodated on average to achieve approximate focus in the pupil center but, because of combined accommodative lag and negative spherical aberration, experienced up to 2.00 D of hyperopic defocus in the pupil margins. With DF lenses, children accommodated similarly achieving approximate focus in the pupil center. When viewing three near distances (0.48, 0.31, and 0.23 m), the added +2.00 D within the DF lens treatment optics shifted the mean defocus from +0.75 to -1.00 D. The DF lens reduced the percentage of hyperopic defocus (≥+0.75 D) in the retinal image from 52 to 25% over these target distances, leading to an increase in myopic defocus (≤-0.50 D) from 17 to 42%. The DF contact lens did not alter the accommodative behavior of children. The treatment optics introduced myopic defocus and decreased the amount of hyperopically defocused light in the retinal image.

Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging

Context. Wavefront sensing and control (WFSC) will play a key role in improving the stability of future large segmented space telescopes while relaxing the thermo-mechanical constraints on the observatory structure. Coupled with a coronagraph to reject the light of an observed bright star, WFSC enables the generation and stabilisation of a dark hole (DH) in the star image to perform planet observations. Aims. While WFSC traditionally relies on a single wavefront sensor (WFS) input to measure wavefront errors, the next generation of instruments will require several WFSs to address aberrations with different sets of spatial and temporal frequency contents. The multiple measurements produced in such a way will then have to be combined and converted to commands for deformable mirrors to modify the wavefront subsequently. Methods. We asynchronously operate a loop controlling the high-order modes digging a DH and a control loop that uses the rejected light by a Lyot coronagraph with a Zernike wavefront sensor to stabilize the low-order aberrations. Using the HiCAT testbed with a segmented telescope aperture, we implement concurrent operations and quantify the expected cross-talk between the two controllers. We then present experiments that alternate high-order and low-order control loops to identify and estimate their respective contributions. Results. We show an efficient combination of the high-order and low-order control loops, keeping a DH contrast better than 5 × 10−8 over a 30 min experiment and stability improvement by a factor of 1.5. In particular, we show a contrast gain of 1.5 at separations close to the DH inner working angle, thanks to the low-order controller contribution. Conclusions. Concurrently digging a DH and using the light rejected by a Lyot coronagraph to stabilize the wavefront is a promising path towards exoplanet imaging and spectroscopy with future large space observatories.

Changes in Wavefront Error of the Eye for Different Accommodation Targets under the Application of Phenylephrine Hydrochloride

Pharmacological dilation of the eye to have a larger pupil diameter may allow a better understanding of the wavefront error changes with accommodation. This work aimed to investigate whether dilation of the pupil with Phenylephrine Hydrochloride (PHCl) application changes the accommodative response and the Zernike coefficient magnitude with accommodative demand when computed to a common pupil size. Sixteen right eyes of healthy young subjects were measured with the commercial Hartmann–Shack aberrometer IRX3 (Imagine Eyes, Orsay, France) 30 min after two drops of 1.0% PHCl were applied. The eye wavefronts for accommodative demands from 0 to 5 D were measured in natural conditions and after pupil dilatation. Statistically significant differences between both conditions were found for the Zernike coefficients C3−1, C31, C40 and C60. Without the effect of PHCl, higher values were found for all higher-order Zernike coefficients (HOA). With increased accommodative response, an increase of C3−1 and a decrease of C31 was observed and the C40 becomes more negative; the change from positive to negative is shown in the accommodative demand of 1.5 D. Conversely, C60 increases with increasing accommodative demand. To conclude, the results demonstrated that the mydriatic effect of PHCl causes changes in the magnitude of HOA when accommodation is stimulated. The trends observed in the different Zernike coefficients were the same reported in previous studies.

Structural aberration coefficients for freeform imaging systems

Freeform imaging systems are a subject of current interest. To aid in the design of such systems, we introduce structural aberration coefficients for plane symmetric imaging with XY-polynomial freeform surfaces that may also be tilted. Further, we extend the coefficients to systems with an arbitrary number of components and specifically provide analytical equations for two-mirror systems. Distinguishing between system configuration and structural parameters enhances clarity, which we highlight by deriving new conditions for minimum root-mean-square (RMS) spot size and minimum RMS wavefront error in aplanatic two-mirror telescopes.

Generalization of learned Fourier-based phase-diversity wavefront sensing.

Proper initialization of the nonlinear optimization is important to avoid local minima in phase diversity wavefront sensing (PDWS). An effective neural network based on low-frequency coefficients in the Fourier domain has proved effective to determine a better estimate of the unknown aberrations. However, the network relies significantly on the training settings, such as imaging object and optical system parameters, resulting in a weak generalization ability. Here we propose a generalized Fourier-based PDWS method by combining an object-independent network with a system-independent image processing procedure. We demonstrate that a network trained with a specific setting can be applied to any image regardless of the actual settings. Experimental results show that a network trained with one setting can be applied to images with four other settings. For 1000 aberrations with RMS wavefront errors bounded within [0.2 λ, 0.4 λ], the mean RMS residual errors are 0.032 λ, 0.039 λ, 0.035 λ, and 0.037 λ, respectively, and 98.9% of the RMS residual errors are less than 0.05 λ.

Direct wavefront sensing with a plenoptic sensor based on deep learning.

Traditional plenoptic wavefront sensors (PWS) suffer from the obvious step change of the slope response which leads to the poor performance of phase retrieval. In this paper, a neural network model combining the transformer architecture with the U-Net model is utilized to restore wavefront directly from the plenoptic image of PWS. The simulation results show that the averaged root mean square error (RMSE) of residual wavefront is less than 1/14λ (Marechal criterion), proving the proposed method successfully breaks through the non-linear problem existed in PWS wavefront sensing. In addition, our model performs better than the recently developed deep learning models and traditional modal approach. Furthermore, the robustness of our model to turbulence strength and signal level is also tested, proving the good generalizability of our model. To the best of our knowledge, it is the first time to perform direct wavefront detection with a deep-learning-based method in PWS-based applications and achieve the state-of-the-art performance.

Using a deep learning algorithm in image-based wavefront sensing: determining the optimum number of Zernike terms

The turbulent atmosphere usually degrades the quality of images taken on Earth. Random variations of the refractive index of light cause distortion of wavefronts propagating to ground-based telescopes. Compensating these distortions is usually accomplished by adaptive optics (AO) approaches. The control unit of AO adjusts the phase corrector, such as deformable mirrors, based on the incoming turbulent wavefront. This can be done by different algorithms. Usually, these algorithms encounter real-time wavefront compensation challenges. Although many studies have been conducted to overcome these issues, we have proposed a method, based on the convolutional neural network (CNN) as a branch of deep learning (DL) for sensor-less AO. To this objective, thousands of wavefronts, their Zernike coefficients, and corresponding intensity patterns in diverse conditions of turbulence are generated and fed into the CNN to predict the wavefront of new intensity patterns. The predictions are done for considering the different number of Zernike terms, and the optimum number is achieved by comparing wavefront errors.

Accounting for Polychromatic Light in Virtual Shack-Hartmann Wavefront Sensing

Virtual Shack-Hartmann wavefront sensing (vSHWS) has some significant advantages and is promising for aberration measurement in the field of biomedical optical imaging. The illumination sources used in vSHWS are almost broadband, but are treated as monochromatic sources (only using center wavelength) in current data processing, which may cause errors. This work proposed a data processing method to take into account the multiple wavelengths of the broadband spectrum, named multiple-wavelength centroid-weighting method. Its feasibility was demonstrated through a series of simulations. A wavefront generated with a set of statistical human ocular aberrations was used as the target wavefront to evaluate the performance of the proposed and current methods. The results showed that their performance was very close when used for the symmetrical, but the wavefront error of the proposed method was much smaller than that of the current method when used for the asymmetrical spectrum, especially for the broader spectrum. These results were also validated by using 20 sets of clinical human ocular aberrations including normal and diseased eyes. The proposed method and the obtained conclusions have important implications for the application of vSHWS.

The analysis of the far-field phase and the tilt-to-length error contribution in space-based laser interferometry

The arm length of the space-based interferometer for gravitational wave detection is 108–109 , and picometer precision is required. The wavefront error in the far-field coupled with the pointing jitter is a major noise source, which raises higher requirements of the wavefront quality of the telescope. By extending the analytical solutions of the far-field phase to 21 Zernike aberrations, this paper demonstrates that the far-field phase could be regarded as the sum of the effects of the individual aberrations and a phase plane with its slope related to the wavefront RMS. For individual aberrations, only low-order aberrations have a significant effect on the far-field phase, but high-order aberrations will contribute to the coupling terms which is shown as a phase slope. So there is a definite corresponding relationship between the aberrations and the far-field phase, and it can be easily illustrated graphically. Based on the analytical solutions of the far-field phase, we proposed to consider the tilt-to-length (TTL) error together with the far-field phase. The TTL error is also one of the main noise sources in space-based laser interferometry, which presents big difficulties in the optical system. We found that TTL error could be added to the far-field phase as a piston term during pointing, which then help to bring the stationary point closer to the z-axis and reduce the phase noise detected at the stationary point. By considering the TTL together with the far-field phase, the requirement of the telescope could be λ/25 wavefront RMS (λ = 1064 ). And there will be a large tolerance range for the TTL error, so it is only necessary to evaluate the TTL error and limit it to a suitable range, not to eliminate it.

Alignment with a combination of deflectometry and the sine condition test.

We propose a new alignment method that is based on relative measurements with an on-axis test setup composed of a pixelated camera and a monitor. By combining deflectometry and the sine condition test, the new method eliminates the necessity of moving a test instrument to multiple field points but still estimates the state of alignment by measuring both the off-axis and on-axis performances of the system. Additionally, it can be a very cost-effective option for certain projects as a monitor, and a camera may be substituted for the return optic and the interferometer required in a conventional interferometric method. We explain the concept of the new alignment method using a meter-class Ritchey-Chrétien telescope. Additionally, we present a new metric, the Metric for Misalignment Indicators (MMI), which represents the transmitted wavefront error caused by misalignment in the system. Then we demonstrate the validity of the concept using simulations where a poorly aligned telescope is the starting point to show that this method has a larger dynamic range compared to the interferometric method. Even considering realistic levels of noise, the new alignment method works successfully, as there is an improvement of two orders of magnitude in the final MMI after three iterations of alignment. The MMI of perturbed telescope models is about 10µm but, after alignment, the MMI converges to one-tenth of a micrometer.

Evaluation method for the design results of space gravitational-wave telescopes

Unlike traditional imaging telescopes, space gravitational-wave telescope has higher requirements for wavefront error and must meet the stability requirement of tilt-to-length (TTL). The tolerance analysis results of current telescopes can judge whether the wavefront error meets the requirements, but they cannot be used as the judgment result of the TTL noise coupling coefficient of the telescope. To analyze the influence of manufacturing errors of the telescope on the coupling coefficient of TTL noise, based on the error amount and the Zernike polynomial first-order sensitivity matrix, we established the sensitivity matrix of the mapping relationship between telescope manufacturing errors and coupling coefficients. For example, consider the design results of an off-axis four-mirror space gravitational-wave telescope. If the wavefront error was used as the judgment result to determine the tolerance distribution, the cumulative probability of the coupling coefficient meeting the requirements would be 66.7%. Furthermore, using the coupling coefficient as the control requirement and determining the tolerance allocation results according to the coupling coefficient sensitivity matrix, the cumulative probability increased to 93.8%. The necessity of using the coupling coefficient as an evaluation method for the design results of gravitational-wave telescopes was verified. This evaluation method can provide meaningful guidance for the design results of gravitational-wave telescopes.

Spherical Aberration of Cataractous Eyes and Its Relationship With Age, Ocular Biometry, and Various IOL Platforms.

To determine the median spherical aberration (SA) of the cataractous population, how it relates to biometry, and the theoretical effect of different intraocular lens (IOL) platforms. A retrospective chart review of patients undergoing cataract surgery evaluation with a high quality Pentacam (Oculus Optikgeräte GmbH) were included. Age, gender, Q-value, mean total SA, higher order aberration root mean square wavefront error, and equivalent keratometry were collected from the Holladay report and axial length and anterior chamber depth (ACD) from the IOLMaster 700 (Carl Zeiss Meditec AG). Data from 1,725 eyes of 999 patients were collected. SA had a median of 0.37 µm (95% confidence interval: 0.36 to 0.38. Age (r = .136, P < .001), Q-factor (r = .743, P < .001), and higher order aberration root mean square wavefront error (r = .307, P < .001) were positively correlated with SA. Average equivalent keratometry (r = -.310, P < .001) was negatively correlated with SA. Axial length (r = -0.037, P = .120) and ACD (r = .004, P = .856) had no association with SA. Up to 1,499 (86.9%) theoretically had SA moved closer to zero with IOLs that had negative SA. Up to 102 (5.9%) had SA theoretically worsened. SA is not normally distributed, suggesting that there may be no "average" SA that IOLs should aim to correct. Patients might benefit from tailoring IOL choice to individual SA. Without access to SA data, eyes with steeper average keratometry or younger patients may have less SA, which could influence IOL choice. [J Refract Surg. 2023;39(2):89-94.].

Suppression of coupling between optical aberration and tilt-to-length noise in a space-based gravitational wave telescope.

Coupling between exiting wavefront error of space gravitational wave telescopes and tilt-to-length (TTL) noise affects the measurement accuracy. Using the LISA Pathfinder signal, we analyzed cancellation and superposition of TTL coupling noise under various optical aberrations. We proposed proportion requirements of any two aberrations amplitude when noise was cancelled and an aberration amplitude control requirement when noise was superposed. Taking them as the aberration control requirements of gravitational wave telescope optical system, the exiting wavefront error requirements was reduced while suppressing the TTL coupling noise. A 40× optical telescope system with detection aperture φ=200 mm was designed. The exiting wavefront error was relaxed from 0.02 λ to 0.0496 λ. The maximum coupling coefficient value did not exceed 6.9448 pm/µrad within a pointing jitter angle of ±300 µrad. The proposed approach should be useful in future telescope design.

Influence and Correction of Wavefront Primary Aberrations for Radio Telescopes using Aberration Theory

Dual-reflector antennas are widely used in astronomical observations and satellite communication. Structural deformations of the reflectors for radio telescopes are inevitable in outside working conditions due to exterior environment loads, which will cause distortion in the surface of the primary reflector and displacement of the subreflector, then lead to gain degradation and misalignment. In this paper, the influence and correction of misalignment in a dual-reflector antenna have been studied. From the perspective of wavefront aberration, a method is proposed to correct the wavefront primary aberration by adjusting the subreflector position. The characteristics of wavefront errors caused by structural deformation of the reflector have been analyzed, and relationships between the position motions of the subreflector and the Seidel wavefront aberrations are derived. The adjustment quantities of the subreflector are also derived. The results show the appropriate positional change of the subreflector in the lateral and axial directions can effectively correct the effects of the tilt and defocus in the primary aberrations caused by antenna structural deformations.

Optical filter wavefront distortion: out-of-band to in-band predictions and the effect of the illumination source bandwidth.

The wavefront distortion (WFD) of a surface with an optical filter coating is ideally measured at the operating wavelength (λ) and angle of incidence (θ) of the filter. However, this is not always possible, requiring that the filter be measured at an out-of-band wavelength and angle (typically λ=633n m and θ=0∘). Since the transmitted wavefront error (TWE) and reflected wavefront error (RWE) can depend on the measurement wavelength and angle, an out-of-band measurement may not give an accurate characterization of the WFD. In this paper, we will show how to predict the wavefront error (WFE) of an optical filter at the in-band wavelength and angle from a WFE measurement at an out-of-band wavelength and different angle. This method uses (i)the theoretical phase properties of the optical coating, (ii)the measured filter thickness uniformity, and (iii)the substrate's WFE dependence versus the angle of incidence. Reasonably good agreement was achieved between the RWE measured directly at λ=1050n m (θ=45∘) and the predicted RWE based on an RWE measurement at λ=660n m (θ=0∘). It is also shown through a series of TWE measurements using a light emitting diode (LED) and laser light sources that, if the TWE of a narrow bandpass filter (e.g.,an 11nm bandwidth centered at λ=1050n m) is measured with a broadband LED source, the WFD can be dominated by the chromatic aberration of the wavefront measuring system-hence, a light source that has a bandwidth narrower than the optical filter bandwidth should be used.

Real-time wavefront correction using diffractive optical networks.

Real-time wavefront correction is a challenging problem to present for conventional adaptive optics systems. Here, we present an all-optical system to realize real-time wavefront correction. Using deep learning, the system, which contains only multiple transmissive diffractive layers, is trained to realize high-quality imaging for unknown, random, distorted wavefronts. Once physically fabricated, this passive optical system is physically positioned between the imaging lens and the image plane to all-optically correct unknown, new wavefronts whose wavefront errors are within the training range. Simulated experiments showed that the system designed for the on-axis field of view increases the average imaging Strehl Ratio from 0.32 to 0.94, and the other system intended for multiple fields of view increases the resolvable probability of binary stars from 30.5% to 69.5%. Results suggested that DAOS performed well when performing wavefront correction at the speed of light. The solution of real-time wavefront correction can be applied to other wavelengths and has great application potential in astronomical observation, laser communication, and other fields.

Determination of phase change correction on gauge block measurement in two different interferometric measurement systems

Phase change correction (PCC) is an important correction value of the end effect in an optical interferometry system. Normally, this value is used to compensate for gauge block measurement by an optical interferometry system based on ISO 3650:1998. Two different interferometric measurement systems in terms of fringe fraction measurement were performed to determine the phase change correction by a five-stacking method. These results are used to determine the length measurement of gauge blocks in an optical interferometer technique and consequently, to evaluate the uncertainty of gauge blocks measurement. The preliminary results for steel gauge block are shown that the value of phase change correction in a phase shift gauge block interferometer (PSGBI) system and a standard uncertainty are 35.2 nm and 5.8 nm, respectively. In contrast, the values from an average slits gauge block interferometer (ASGBI) system and a standard uncertainty are 66.0 nm and 6.0 nm, respectively. We found that the phase correction from the PSGBI system is lower than ASGBI about 0.53 - 0.56 times because the different of wave front correction in two interferometric systems. However, the lengths of gauge blocks of all materials measured by the two systems were consistent as assessed by En number. According to the study, we can conclude that phase change correction is based on the characteristics of each GBI system, surface texture characteristic in term of wringing condition and material types of gauge block and optical plates such as the fringe fraction measurement technique, and wave front error compensation. Consequently, measurements that require a high accuracy should determine the phase change correction before each measurement due to this value is not interchangeable.

JWST NIRCam Defocused Imaging: Photometric Stability Performance and How It Can Sense Mirror Tilts

We use JWST NIRCam short-wavelength photometry to capture a transit lightcurve of the exoplanet HAT-P-14 b to assess performance as part of instrument commissioning. The short-wavelength precision is 152 ppm per 27 s integration as measured over the full time series compared to a theoretical limit of 107 ppm, after corrections to spatially correlated 1/f noise. Persistence effects from charge trapping are well fit by an exponential function with short characteristic timescales, settling on the order of 5–15 minutes. The short-wavelength defocused photometry is also uniquely well suited to measure the real-time wave-front error of JWST. Analysis of the images and reconstructed wave-front maps indicates that two different hexagonal primary mirror segments exhibited “tilt events,” where they changed orientation rapidly in less than ∼1.4 s. In some cases, the magnitude and timing of the flux jumps caused by tilt events can be accurately predicted with a telescope model. These tilt events can be sensed by simultaneous longer-wavelength NIRCam grism spectral images alone in the form of changes to the point-spread function, diagnosed from the full width at half maximum. They can also be sensed with the fine guidance sensor instrument from difference images. Tilt events possibly from sudden releases of stress in the backplane structure behind the mirrors were expected during the commissioning period because they were found in ground-based testing. Tilt events have shown signs of decreasing in frequency but have not disappeared completely. The detectors exhibit some minor (less than 1%) deviations from linear behavior in the first few groups of each integration, potentially impacting absolute fluxes and transit depths on bright targets, where only a handful of groups are possible. Overall, the noise is within 50% of the theoretical photon noise and read noise. This bodes well for high-precision measurements of transiting exoplanets and other time variable targets.

Toward a Phase Adaptive Stabilization System for Next-Generation Single-Dish Sub-mm Telescope Part—I: Multipath Phase Measurements Based on Ultrastable Microwave Signal Distribution

Large-aperture, single-dish, submillimeter-wave (sub-mm) telescopes are important to our understanding of the universe, but it is a long-term outstanding challenge to make the single-dish sub-mm telescopes larger while keeping a high accuracy of their surfaces. A bottleneck in the context of the size and the operating wavelengths of such telescopes is the deterioration of wavefront errors (phase shift) due to the deformation of reflective surfaces. Here proposed is a phase adaptive stabilization system (PASS), functioning as adaptive optics system of radio frenquency, an essential tool for overcoming the challenge, which is based on a state-of-the-art ultra-stable microwave signal distribution (UMSD) technology. The UMSD system with a phase drift of less than 11 fsec rms over 600 sec, is specially proposed for the PASS, which is an innovation. The results of outfield experiments have shown that the PASS is sufficiently accurate to measure the changes in the excess path length down to the level of 20 μm, which is the highest accuracy reported up to now. Our study marks a major step towards the phase adaptive stabilization system.