Wavefront Error Research Articles

Design Considerations for Wavefront Sensing with Self-referencing Interferometers in Adaptive Optics Systems

In this work, we show the design and implementation of wavefront sensing with a self-referencing interferometer (SRI). The SRI is developed to aid adaptive optics (AO) control, via deformable mirrors, in correcting wavefront error from atmospheric turbulence in free-space optical (laser-based) communication links. The SRI is used here given its potential to outperform more common wavefront sensors in functioning over weak through strong turbulence conditions. In this study, we identify and analyse the key parameters in the SRI's optical design and show guiding principles for its subsequent image processing.

Super-oscillation sub-diffraction focusing with emulated atmospheric turbulence

The super-oscillation (SO) phenomena successfully applied to super-resolution optical telescopes have yet to study the existence of atmospheric turbulence (AT). In this paper, we first experimentally investigate the sub-diffraction focusing of the SO light field in the atmospheric-like turbulence. The 137-element adaptive optics (AO) system is utilized to correct the dynamic wavefront aberration produced by the AT and obtain the AO closed-loop RMS of the residual wavefront error ∼λ/15. For a proofed telescope (clear aperture 12 mm and focal length 1000 mm) @λ=632.8 nm, the FWHM of the experimental SO spot under the modest AT (wavefront error RMS ∼0.57λ) is about 0.79 times of the Airy spot compared with the non-AT result of 0.76 times, while the strong AT (wavefront error RMS ∼1.35λ) erases the sub-diffraction focusing effect of the SO field and the average side-lobe intensity increases significantly. This study is promising for super-resolving telescope, synthetic aperture for visible imaging, etc.

Telescope Alignment Method Using a Modified Stochastic Parallel Gradient Descent Algorithm

To satisfy the demands of high image quality and resolutions, telescope alignment is indispensable. In this paper, a wavefront sensorless method based on a modified stochastic parallel gradient descent algorithm (SPGD) called the adaptive moment estimation SPGD (Adam SPGD) algorithm is proposed. Simulations are carried out using a four-mirror telescope, whose aperture is 6 m and fields of view are Φ2°. Three misalignments are shown as examples. Positions of the secondary mirror and third mirror are employed to compensate aberrations. The results show that merit functions and energy distributions of corrected images match with the designed ones. The mean RMS of residual wavefront errors is smaller than λ/14 (λ = 0.5 μm), indicating that the misalignments are well compensated. The results verify the effectiveness of our method.

Characterization of birefringence inhomogeneity of KAGRA sapphire mirrors from transmitted wavefront error measurements

Characterization of birefringence inhomogeneity of KAGRA sapphire mirrors from transmitted wavefront error measurements

Improving neural-network-based prediction models for misalignment in off-axis three-mirror anastigmat telescopes

The performance of a telescope system heavily relies on the precise alignment of the mirrors. The off-axis three-mirror anastigmat (TMA) telescope presents unique challenges due to its complex optical design. Each optical element within the off-axis TMA telescope is inherently introduced with theoretical eccentricity and tilt. Furthermore, the incorporation of freeform surfaces and other optical elements with intricate surface features typically leads to low initial alignment accuracy of the optical path. With this low initial alignment accuracy and the noise of the measurements, the prediction of the misalignment of the telescope is getting harder. A fully connected neural network architecture is proposed as a misalignment calculation method for an off-axis TMA telescope system with a freeform surface. Random training data is created by using optical design software. The sensitivity of the mirrors to the wavefront error is quantified and incorporated into the loss function of the neural network to improve prediction accuracy. Adding the noisy measurement samples to the training data creates a noise-immune neural network model. Simulation results show that our model can successfully predict misalignment of the mirrors with noisy measurement values.

Demonstration of a photonic lantern focal-plane wavefront sensor: measurement of atmospheric wavefront error modes and low wind effect in the nonlinear regime

Demonstration of a photonic lantern focal-plane wavefront sensor: measurement of atmospheric wavefront error modes and low wind effect in the nonlinear regime

Positioning of the test surface in a CGH null test by cat's eye interference

Interferometric null test of optical aspheres and freeforms is one of the most established methods, in spite of recognized limitations related to positioning of the test surface. Slight misalignment of the surface in the null test system can introduce remarkable wave aberrations. Current approaches for positioning the test surface are based on a series of fiducial marks or retroreflectors set at given spots around the test surface, which consequently suffer the problem of datum transformation. We propose a method for positioning the test surface in a null test by cat's eye interference without any fiducial marks or retroreflectors. A computer-generated hologram (CGH) is used as null optics fabricated with null test pattern, alignment pattern and positioning pattern. A part of test beam with relatively small f/number is diffracted through the positioning pattern and then focuses on certain spots of the test surface. The cat's eye reflection from the test surface returns to the interferometer and interferes with the reference beam. It is then possible to precisely position the test surface by measuring the wavefront error of the cat's eye interference. The design method for such a CGH positioning pattern is presented, following the physical constraint that the surface normal at the cay's eye reflection is right the angular bisector of the positioning beam. Analysis on the positioning performance is then presented to show the sensitivity to misalignment including defocus, lateral shift and tip-tilt, which at last is experimentally verified by measuring an even asphere with a CGH.

The Active Optics System on the Vera C. Rubin Observatory: Optimal Control of Degeneracy among the Large Number of Degrees of Freedom

The Vera C. Rubin Observatory is a unique facility for survey astronomy that will soon be commissioned and begin operations. Crucial to many of its scientific goals is the achievement of sustained high image quality, limited only by the seeing at the site. This will be maintained through an active optics system that controls optical element misalignments and corrects mirror figure error to minimize aberrations caused by both thermal and gravitational effects. However, the large number of adjustment degrees of freedom available on the Rubin Observatory introduces a range of degeneracies, including many that are induced by noise due to imperfect measurement of the wave-front errors. We present a structured methodology for identifying these degeneracies through an analysis of image noise level. We also present a novel scaling strategy based on truncated singular value decomposition that mitigates the degeneracy and optimally distributes the adjustment over the available degrees of freedom. Our approach ensures the attainment of optimal image quality, while avoiding excursions around the noise-induced subspace of degeneracies, marking a significant improvement over the previous techniques adopted for Rubin, which were based on an optimal integral controller. This new approach is likely to also yield significant benefits for all telescopes that incorporate large numbers of degrees of freedom of adjustment.

Corneal Epithelial Wavefront Error as a Novel Diagnostic Marker for Epithelial Basement Membrane Dystrophy.

Synopsis: Corneal epithelial wavefront error and epithelial thickness variance qualify as highly sensitive and specific biomarkers for epithelial basement membrane dystrophy (EBMD). The biomarkers show a normalization after treatment of EBMD with phototherapeutic keratectomy. Purpose: To gauge the diagnostic value of epithelial basement membrane dystrophy (EBMD), a novel spectral-domain optical coherence tomography (SD-OCT)-based imaging modality for simultaneous morphological (thickness profile) and refractive (optical wavefront) assessment of the corneal epithelial layer in one of the most common but often underdiagnosed corneal dystrophies. Methods: In this prospective observational study, a total of 32 eyes of 32 patients diagnosed with EBMD and 32 eyes of 32 healthy control subjects were examined with high-resolution anterior segment SD-OCT (MS-39; CSO, Florence, Italy). Various epithelial thickness and epithelial wavefront-derived terms were compared between groups and receiver operating characteristic (ROC) curves were computed to analyze the diagnostic capacity of the respective parameters. A total of 17 of 32 EBMD patients underwent treatment with phototherapeutic keratectomy (PTK) and were followed up for 3 months. Results: Epithelial thickness variance (60.4 ± 56.7 µm versus 7.6 ± 6.1 µm) and interquartile range (11.0 ± 6.9 versus 3.3 ± 1.9 µm) were markedly elevated in EBMD patients as compared with healthy controls (both with p < 0.001). Epithelial wavefront analysis showed a highly statistically significant excess in all examined aberration terms in EBMD patients (all with p < 0.001). Significantly greater areas under the curve (AUCs) were yielded by the epithelial wavefront-derived parameters (e.g., total epithelial wavefront error: AUC = 0.966; 95% confidence interval (CI) 0.932-1) than by the epithelial thickness-derived parameters (e.g., variance: AUC = 0.919; 95% CI 0.848-0.990). Conclusions: Corneal epithelial wavefront aberrometry proved valuable as an objective biomarker for EBMD, with high sensitivity and specificity. PTK resulted in a reduction of morphological and refractive epithelial irregularities in EBMD.

Optical design of a 5× zoom afocal telescope with deformable mirrors

All-reflective zoom afocal telescopes can be integrated with existing imaging systems as foreoptics, enabling flexible imaging with adjustable resolution. This paper proposes an approach to designing a 5× zoom afocal telescope using deformable mirrors. Based on matrix optics, the first-order parameters for a coaxial reflective 5× zoom afocal telescope with a fixed primary mirror and three deformable mirrors are determined. To prevent obscuration, the mirrors are tilted at their vertices, which introduces asymmetric aberrations. To correct these aberrations, XY polynomial surfaces are employed on the primary mirror and an additional flat fold mirror. The final design and image simulations indicate that the system achieves good image quality across all zoom positions, enabling 5× zoom imaging with a maximum root mean square wavefront error of less than 0.125λ (λ = 632.8 nm), thereby validating the efficacy of the proposed design methodology.

Effect of Small Angle Misalignments on Ocular Wavefront Zernike Coefficients

Purpose: To assess the possible impact of minor changes in fixation on wavefront measurements as a potential constraint in detecting subtle temporal variations in ocular wavefront error. Methods: Twelve healthy subjects with an average age of 36.3 ± 8.8 were instructed to put their heads in the aberrometer’s chin-rest and look at a fixation target that was embedded in the device. The fixation targets were readily observable to the participants without accommodation, thanks to the aberrometer’s Badal system. When each eye was staring at the target, its wavefront aberration was recorded three times and then averaged for further analysis. The averaged Zernike coefficients were rescaled to the smallest value of the maximum round pupil found among all eyes (4.41 mm), and this procedure was repeated for each target. Results: Alteration of the fixation targets caused changes to the Zernike coefficients of defocus (C(2,0)), vertical trefoil (C(3,–3)), vertical coma (C(3,–1)), horizontal coma (C(3,1)), oblique trefoil (C(3,3)), primary spherical aberration (C(4,0)), and secondary spherical aberration (C(6,0)), but the changes were not statistically significant. Nevertheless, an alteration in the target’s size and shape exhibited a significant correlation across all of the aforementioned coefficients in both eyes (p &lt; 0.05). The total RMS of aberrations and the RMS of the spherical-like aberrations were both lowest while choosing the larger Maltese cross, and the bigger E-letter minimized the RMS of HOA and comatic aberrations. Conclusion: The aberrometric changes occur as a consequence of altering the fixational gaze and are within the range of the changes found after performing a near-vision task, so they might potentially act as a confounding factor when attempting to identify such small variations in the ocular wavefront. Using a smaller E-letter (5 arcmin) as an internal fixation target resulted in the least standard deviation of measurements, fixational stability, and higher accuracy in ocular wavefront measurements.

High-precision, real-time wavefront sensing via sparse diffractive deep neural networks

As a key component of adaptive optics systems, wavefront sensing technology is an important way to effectively obtain aberrant phases in optical systems for high-capacity optical communications and high-quality imaging in relevant applications ranging from biological imaging to astronomical observation. To enhance the time efficiency of detection, the wavefront sensing with diffraction deep neural network (D2NN) directly calculates the wavefront information in the optical field. However, the compactness of the D2NN structure and the accuracy of wavefront prediction are important bottlenecks, restricting its practical application. Here, we design a multi-layer compact D2NN based on Bayesian optimization, called sparse D2NN (SD2NN), to achieve high-precision, real-time direct wavefront sensing. The experimental results demonstrated a reduction in the root-mean-square error (RMSE) of the SD2NN wavefront sensing of approximately 45.4%, along with a reduction in the axial length of approximately 82% in comparison to the unoptimized fully connected D2NN. This resulted in the attainment of a minimum layer distance of 8.77 mm. In addition, we additionally explored the effects of network depth and neuron size on the wavefront sensing performance of SD2NN and further summarized the general law of diffraction layer distance and neuron size. The proposed method will provide a reliable means of designing miniaturized integrated wavefront sensing chips.

Absolute testing of rotationally symmetric surfaces with computer-generated holograms

Extremely high accuracy is demanded for optics working at very short wavelength. Interferometric testing of optical aspheres or freeform surfaces requires null optics, typically computer-generated holograms (CGHs), to balance the wave aberrations. The measurement uncertainty is primarily limited by the accuracy of the test wavefront, which is predominantly influenced by the CGH and the interferometer optics. Absolute testing is fundamental to achieving accuracy much higher than that of the test wavefront through error separation. This paper presents a method for absolute testing of rotationally symmetric surfaces with CGH null optics. The basic assumption is that the off-axis hologram fabricated by raster scanning beam writing has negligible error of rotationally symmetric component due to pattern error of the CGH. Consequently, the wavefront error contributed by the CGH and the transmission flat can be completely separated from the absolute surface shape by combining the N-position method and the shift-rotation method. A theoretical model for absolute testing is proposed under the assumption. Experimental cross test is then presented to validate the method with sub-nanometer uncertainty. The assumption is further confirmed by characterizing the fabrication error of the hologram structures using a white light interferometer. Finally, the effect of noise, translation error, rotation error and eccentricity of rotation on the absolute testing is analyzed.

Suppression of vortex electromagnetic wave divergence by modulating the wave front dislocation line

Abstract When the beam is transmitted in the atmosphere, under the influence of atmospheric turbulence, the beam will occur energy attenuation, wavefront distortion and other phenomena, which will seriously affect the transmission and reception of information. Based on the PB phase principle enhances the focusing capability of the auto-focusing Airy Gaussian vortex beam by regulating the wavefront error curvature of the vortex beam, and then enhances the resistance of the beam against atmospheric turbulence. The results reveal the curvature of the wavefront error curve and the peak intensity of the light field of the beam can be changed by changing the beam polarization state parameters n and m in the focal plane. The auto-focusing Airy Gaussian vortex beam with the polarization state parameters m=1, n=4 increases the peak magnitude of the light field intensity in the focal plane of the auto-focusing Airy Gaussian vortex beam with the polarization state parameters m=0, n=0 to 1.99 times. Demonstrate that geometric phase tuning the auto-focusing Airy Gaussian vortex beams can enhance resistance to atmospheric turbulence. Furthermore, this paper extends this method to the field of millimeter wave. Under the transmission distance of 1km, the peak intensity of the initial vortex electromagnetic wave after the geometric phase adjustment is increased to 3.33 times, and the receiving radius of the target surface is reduced to 0.24 of the initial vortex electromagnetic wave. It shows that geometric phase-regulated vortex electromagnetic wave can suppress vortex electromagnetic wave divergence and have potential applications in wireless communication.

Research on Distortion Control in Off-Axis Three-Mirror Astronomical Telescope Systems

With off-axis reflection systems with specific distortion values serving as objectives or collimators, it is possible to compensate and correct for spectral line bending in spectroscopic instruments. However, there is limited research on the precise control of distortion, which poses particular challenges in large field-of-view optical systems. This paper presents a method for controlling distortion in off-axis reflection systems. Based on Seidel aberration theory and the relationship between distortion wavefront error and primary ray error, we construct objective functions with structural constraints and aberration constraints. The initial structure with specific distortion values is then solved using a differential evolution algorithm. The effectiveness and reliability of this method are verified through the design of an off-axis three-reflection system. The method provided in this study facilitates the design of remote sensing instruments.

Design and Analysis of Orthogonal Polarization Point Diffraction Pinhole Plate

The pinhole plate is a key component of the point diffraction interferometer (PDI). The reasonable improvement and simulation of this device would enhance the application of point diffraction interferometry technology during the measurement of wavefronts. The traditional point diffraction interferometry measurement method is easily disturbed by environmental noise, making it difficult to obtain high-precision dynamic measurements. This paper introduces a four-step phase-shift PDI that can be employed in a common optical path. By using the principle of the finite-difference time-domain method (FDTD), a simulation model of the orthogonal polarization point diffraction pinhole plate (OP-PDPP) structure is established. The results show that when Cr is used as the film material in the pinhole plate, the parameters include a film thickness of 150 nm, a pinhole diameter of 2 μm, a wire grid period of 150 nm, and a wire grid width of 100 nm; in addition, the comprehensive extinction ratio of the pinhole plate is the greatest and the diffraction wavefront error is the smallest. Finally, the constructed experimental system is used to test the wavefront of a flat sample with a 25.4 mm aperture, and the test results are compared with those of the ZYGO interferometer. The difference in the peak-to-valley (PV) value between the OP-PDI and the ZYGO interferometer measurement is 0.0028λ, with an RMS value difference of 0.0011λ; this verifies the feasibility of the scheme proposed in this paper. The experimental results show that the proposed OP-PDPP is an effective tool for high-precision dynamic measurement.

High precision wavefront correction method in interferometer testing.

A wavefront correction method is proposed for high-precision optic surfacing, addressing the discrepancy between wavefront and real surface errors in Fizeau interferometer testing. We believe this to be a proposed novel method that encompasses optical surface function parameters fitting, lateral distortion correction, misalignment error removal, and sag surface error calculation. The method's error has been thoroughly analyzed, including aspects of function parameters fitting, ray tracing, and interpolation. The effectiveness of the method was demonstrated by correcting the wavefront of an off-axis parabolic mirror in null testing configurations, significantly reducing artificially created annular errors and improving off-axis direction errors from 0.23λ to 0.05λ (λ=632.8 nm), with the PV of aspheric departures exceeding 8.5 mm.

Active-compensation of systematic error for the 1000 mm aperture flat interferometer.

Large-aperture elements would induce unnegligible systematic errors due to material inhomogeneity, manufacturing or gravity, that are difficult to correct in an extreme large aperture flat interferometer and result in reference wavefront distortion. We propose an active-compensation method for systematic errors by employing a deformable mirror into the interferometer to modulate reference wavefront. A mapping relationship between sag of the deformable mirror and reference wavefront error is derived by theory of matrix optics, and two interferometer optical paths are designed for whether the deformable mirror is located at the pupil or not. The algorithm for calculating and controlling the sag of a deformable mirror can eliminate the need for the deformable mirror to be positioned at the pupil in order to achieve controllable modulation of the wavefront. This algorithm has been validated through the intentional introduction of systematic errors into the 1000 mm aperture flat interferometer for effective compensation. Moreover, the optimization algorithm in Ansys Zemax is utilized to calculate the optimal solution for surface shape of the deformable mirror, treating it as a nominal value. The algorithm error is on the order of 10-6 mm, falling within the acceptable tolerance range for the deformable mirror's surface shape.

Wavefront Changes during a Sustained Reading Task in Presbyopic Eyes.

The objective of this study was to assess the effect of sustained reading on the temporal changes in the wavefront error in the presbyopic eye. The wavefront aberration of the eyes was measured using an IRX3 Shack-Hartmann aberrometer before and after (immediately, 5 min, and 10 min after) a reading task. Temporal changes in C20, C40, and C3-1 coefficient values of the eyes were plotted, showing a predominant number of V-shaped patterns (for C40 and C3-1) and inverse V-shaped patterns (for C20) among the study group, and the percentages (between 27 and 73%) were reported. The median of the total RMS of aberrations and the RMS of HOA (higher-order aberrations), which included comatic (3rd order) and spherical-like aberrations (4th and 6th order), increased immediately after finishing the near-vision reading task and then decreased. The median of RMS of comatic aberrations had a similar pattern of variations, while the median of RMS of spherical-like aberrations displayed an opposite pattern. Simulating the aberration changes due to lens decentration caused by relaxed zonules during 4 D accommodation in an eye model demonstrated that the expected range of changes for the vertical coma and spherical aberrations are in the order of 0.001 and 0.01 μm, respectively, which could justify why the observed changes were not statistically significant. The observed dynamic changes in HOA might be linked to the biomechanical characteristics and alterations in the displacement of the crystalline lens following prolonged near-vision tasks in presbyopic people. Although some predominant patterns under some conditions were shown, they exhibit considerable inter-subject and inter-ocular variability. This might be due to slight misalignments while fixating on the internal extended object in the aberrometer.

Comparison of polystyrene and hydrogel microcarriers for optical imaging of adherent cells.

The ability to observe and monitor cell density and morphology has been imperative for assessing the health of a cell culture and for producing high quality, high yield cell cultures for decades. Microcarrier-based cultures, used for large-scale cellular expansion processes, are not compatible with traditional visualization-based methods, such as widefield microscopy, due to their thickness and material composition. Here, we assess the optical imaging compatibilities of commercial polystyrene microcarriers versus custom-fabricated gelatin methacryloyl (gelMA) microcarriers for non-destructive and non-invasive visualization of the entire microcarrier surface, direct cell enumeration, and sub-cellular visualization of mesenchymal stem/stromal cells. Mie scattering and wavefront error simulations of the polystyrene and gelMA microcarriers were performed to assess the potential for elastic scattering-based imaging of adherent cells. A Zeiss Z.1 light-sheet microscope was adapted to perform light-sheet tomography using label-free elastic scattering contrast from planar side illumination to achieve optical sectioning and permit non-invasive and non-destructive, in toto, three-dimensional, high-resolution visualization of cells cultured on microcarriers. The polystyrene microcarrier prevents visualization of cells on the distal half of the microcarrier using either fluorescence or elastic scattering contrast, whereas the gelMA microcarrier allows for high fidelity visualization of cell morphology and quantification of cell density using light-sheet fluorescence microscopy and tomography. The combination of optical-quality gelMA microcarriers and label-free light-sheet tomography will facilitate enhanced control of bioreactor-microcarrier cell culture processes.