Zernike Modes Research Articles

Image metric-based multi-observation single-step deep deterministic policy gradient for sensorless adaptive optics.

Sensorless adaptive optics (SAO) has been widely used across diverse fields such as astronomy, microscopy, and ophthalmology. Recent advances have proved the feasibility of using the deep deterministic policy gradient (DDPG) for image metric-based SAO, achieving fast correction speeds compared to the coordinate search Zernike mode hill climbing (ZMHC) method. In this work, we present a multi-observation single-step DDPG (MOSS-DDPG) optimization framework for SAO on a confocal scanning laser ophthalmoscope (SLO) system with particular consideration for applications in preclinical retinal imaging. MOSS-DDPG optimizes N target Zernike coefficients in a single-step manner based on 2N + 1 observations of the image sharpness metric values. Through in silico simulations, MOSS-DDPG has demonstrated the capability to quickly achieve diffraction-limited resolution performance with long short-term memory (LSTM) network implementation. In situ tests suggest that knowledge learned through simulation adapts swiftly to imperfections in the real system by transfer learning, exhibiting comparable in situ performance to the ZMHC method with a greater than tenfold reduction in the required number of iterations.

Wavefront reconstruction based on ASH-Net with less lenslets SHWFS

Generally, the number of lenslets in Shack–Hartman wavefront sensor (SHWFS) determines its spatial sampling frequency. However, increasing the number of lenslets is limited by the data processing delay time and low sensitivity in the adaptive optics system. In this paper, a wavefront reconstruction method based on ASH-Net for SHWFS is presented to recover the distorted wavefront from the light spots array with high accuracy with a spatial sampling frequency than traditionally required. The results show that the maximum number of Zernike modes recovered by the method is 36, 54, and 70 for 4 × 4, 6 × 6, and 8 × 8 lenslets arrays, respectively. Residual wavefront root mean square values are as low as about 0.02 μm, corresponding to a Strehl Ratio of 0.99. This method breaks the limitation that the number of reconstructed Zernike modes in the traditional wavefront reconstruction algorithm must be less than the number of effective lenslets. Experiments in lab are used to validate the method. Additionally, fewer lenslets is valuable to improve the detection capability of SHWFS for faint targets. This is important for applications of adaptive optics in areas such as astronomy and retinal imaging.

Prediction of effective size of a propagated laser beam with a known phase profile and its experimental validation

In this paper we demonstrate that the size of a laser beam, which increases with propagation due to diffraction induced divergence, carrying a given phase profile, can be predicted and the same can be validated experimentally. Since an arbitrary phase profile is also a linear combination of a set of Zernike polynomials hence the divergence corresponding to each Zernike mode can be utilized to estimate the effective size of the beam carrying the arbitrary phase profile. Numerical simulation using the scalar diffraction theory shows how a laser beam with a certain Zernike mode describing its phase profile diverges. Observations from numerical simulations lead to construction of a functional form that can predict the effective size of a laser beam carrying a specific Zernike mode in its phase profile, at different propagation distances. In this paper we develop an experimental arrangement to holographically incorporate a specific phase profile into an incident laser beam. We utilize the programmable control of the beam size for a given propagation distance to achieve different Fresnel numbers. We further develop a detector assembly which can be used to measure the effective size of the beam in the observation plane. We then show experimentally that the effective size of a laser beam propagating distances corresponding to different Fresnel numbers and carrying a known phase profile, described either by a single Zernike mode or a random combination of several Zernike modes, can be very well predicted by the functional form even in a realistic scenario.

Nonlinear Wave Front Reconstruction from a Pyramid Sensor using Neural Networks

The pyramid wave front sensor (PyWFS) has become increasingly popular to use in adaptive optics (AO) systems due to its high sensitivity. The main drawback of the PyWFS is that it is inherently nonlinear, which means that classic linear wave front reconstruction techniques face a significant reduction in performance at high wave front errors, particularly when the pyramid is unmodulated. In this paper, we consider the potential use of neural networks (NNs) to replace the widely used matrix vector multiplication (MVM) control. We aim to test the hypothesis that the NN's ability to model nonlinearities will give it a distinct advantage over MVM control. We compare the performance of a MVM linear reconstructor against a dense NN, using daytime data acquired on the Subaru Coronagraphic Extreme Adaptive Optics system (SCExAO) instrument. In a first set of experiments, we produce wavefronts generated from 14 Zernike modes and the PyWFS responses at different modulation radii (25, 50, 75, and 100 mas). We find that the NN allows for a far more precise wave front reconstruction at all modulations, with differences in performance increasing in the regime where the PyWFS nonlinearity becomes significant. In a second set of experiments, we generate a data set of atmosphere-like wavefronts, and confirm that the NN outperforms the linear reconstructor. The SCExAO real-time computer software is used as baseline for the latter. These results suggest that NNs are well positioned to improve upon linear reconstructors and stand to bring about a leap forward in AO performance in the near future.

Remote focusing with dynamic aberration elimination by model-based adaptive optics

Remote focusing using a deformable mirror (DM) provides a rapid and responsive technique for performing axial scanning. However, to maintain the laser focusing quality, the aberrations within the system must be adequately controlled. Accordingly, the present study proposes a low control complexity model-based adaptive optics (AO) approach for remote focusing with zero aberrations. Based on the Zernike coefficients measured by a self-built Shack-Hartmann wavefront sensor (SHWS), 15 independent closed-loop controllers (one for each Zernike mode according to Wyant expansion scheme) execute a three-step DM identification process to determine the control vectors required to actuate the DM in such a way as to restore the Zernike coefficients of the wavefront to their required values. The experimental results show that the controllers converge within four time steps (20 ms) and reduce the converged wavefront variance by nearly 475 times. Moreover, the controller used to manipulate the Z3 (defocus) Zernike coefficient converges within just three time steps (15 ms). Given a static disturbance of the wavefront, the proposed AO control method enables the defocus laser spot to be driven precisely along a target sinusoidal trajectory while maintaining all the other Zernike modes at zero. The control system shows a similar performance in compensating the dynamic aberrations produced by a low-frequency thermal disturbance within the system. Overall, the results indicate that the proposed model-based AO controller facilitates remote focusing with simultaneous aberration elimination and improves the central intensity of the focus laser spot by around 3.1 times compared to the case in which the controller is not applied.

Laser beam wavelength determination algorithm using a digital micromirror device

An approach for wavelength characterization whereby dynamic Fresnel zone plate (FZP) patterns are written to a digital micromirror device (DMD) and used to focus incoming light onto a camera is proposed. The incoming wavelength is then assessed by scanning the focal length. Rectangular basis Zernike modes are implemented to match the rectangular geometry of the DMD, we believe for the first time. A procedure is developed to correct for inherent aberrations of the optical system by adjusting the Zernike mode amplitudes, thus improving focal spot quality. The aberration corrected FZPs are used to assess the focusing conditions needed for four lasers of different visible wavelengths ranging from blue (405 nm) to near infrared (760 nm). The system is able to produce focused beams with a width of twice the diffraction limit and achieve a wavelength resolution of 3 nm. The operational wavelength of the system is ultimately limited by the spectral bandwidth of the detector.

Study on the orthogonality property of Zernike modes in light beams undergoing free space propagation.

This paper presents a study to investigate how the orthogonality property of Zernike modes gets modified as a light beam carrying the mode in its phase propagates through free space. We use a numerical simulation using scalar diffraction theory to generate propagated light beams carrying the commonly encountered Zernike modes. We present our results in terms of the inner product and orthogonality contrast matrix at propagation distances ranging from near field to far field regions. Our study will help in the understanding of how far the Zernike modes describing the phase profile of a light beam in a given plane remain approximately orthogonal to one another on propagation.

Effect of fixational eye movements in corneal topography measurements with optical coherence tomography.

There is an increasing interest in applying optical coherence tomography (OCT) to quantify the topography of ocular structures. However, in its most usual configuration, OCT data is acquired sequentially while a beam is scanned through the region of interest, and the presence of fixational eye movements can affect the accuracy of the technique. Several scan patterns and motion correction algorithms have been proposed to minimize this effect, but there is no consensus on the ideal parameters to obtain a correct topography. We have acquired corneal OCT images with raster and radial patterns, and modeled the data acquisition in the presence of eye movements. The simulations replicate the experimental variability in shape (radius of curvature and Zernike polynomials), corneal power, astigmatism, and calculated wavefront aberrations. The variability of the Zernike modes is highly dependent on the scan pattern, with higher variability in the direction of the slow scan axis. The model can be a useful tool to design motion correction algorithms and to determine the variability with different scan patterns.

Fiber coupling efficiency in ocean with adaptive optics corrections

Underwater optical wireless communication (UOWC) is a very promising technology that enables high-speed data transfer through the use of laser beams in an oceanic turbulent medium. The high-tech fiber optical devices, which are already available in the market, can be integrated with the UOWC systems. When integration is achieved, oceanic turbulence, which distorts the wavefront of the propagating laser beam, plays an important role in reducing the fiber coupling efficiency (FCE), which in turn results in reducing the light power received from the fiber optical components. In this paper, we propose the use of the adaptive optics technique in a UOWC system to mitigate the effects of oceanic turbulence and boost the FCE. For this reason, the field correlation for a Gaussian laser beam is derived by using the Huygens–Fresnel principle. This way, the light power over the coupling lens and the light power accepted by the fiber core are formulated under the effect of adaptive optics corrections, which are represented by the number of Zernike modes. The results demonstrate that under the oceanic turbulence effect, the FCE of the UOWC system employing adaptive optics is always larger than that of the UOWC system employing no adaptive optics.

Performance Characterization of Deep-Phase-Retrieval Shack-Hartmann Wavefront Sensors

Shack-Hartmann wavefront sensor (SHWFS) is the most popular wavefront sensor in adaptive optics systems. The Deep-Phase-Retrieval Wavefront Reconstruction (DPRWR) method, which was proposed by our group previously, is a kind of deep learning-based wavefront reconstruction method. It can extract more information from the SHWFS images to accurately obtain more Zernike mode coefficients. However, the application limits, performance upper bound, and noise immunity have not been investigated in detail in previous reports. In this paper, sub-aperture spot sampling, bit depth, number of reconstructed mode coefficients, and noise intensities are analyzed by simulations and experiments to investigate the influence of changes in these parameters on the performance of DPRWR. This work aims to optimize the configuration of DPRWR for better measurement accuracy, spatial resolution, and robustness.

Quantum limits on localizing point objects against a uniformly bright disk

We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the two-dimensional location of a point source against a uniformly bright disk of known center and radius in the ideal photon-counting limit. We present both a perturbative calculation of the QFI in powers of the background-to-source brightness ratio and a numerically exact calculation of the QFI in the eigenbasis of the one-photon density operator. A related problem of the quantum limit on estimating the location of a small-area brightness hole in an otherwise uniformly bright disk, a problem of potential interest to the extrasolar planet detection community, is also treated perturbatively in powers of the ratio of the areas of the hole and the background disk. We then numerically evaluate the Cram\'er-Rao lower bound (CRB) for wavefront projections in three separate bases, those comprising Zernike, Fourier-Bessel, and localized point-source modes, for unbiased estimation of the two position coordinates of the point source and of the brightness hole center, respectively, for the two problems. By comparing these CRBs with the corresponding quantum-limited minimum error variances, given by inverting the QFI matrix, and with the CRBs associated with direct imaging, we assess the maximum efficiency of these wavefront projections in performing such estimations.

Sixty-Nine-Element Voice Coil Deformable Mirror for Visible Light Communication

To overcome the atmospheric turbulence aberration, and improve the quality of light beam in visible light communication (VLC), a compact 69-element deformable mirror (DM) using micro voice coil actuators was designed based on systematic theoretical analysis. The structural parameters of the micro voice coil actuator were optimized by electromagnetic theory and the finite element method. The DM was optimized from the aspects of thermal deformation, response time, coupling coefficient, and other parameters. Finally, wavefront fitting and residual calculation were completed according to the influence function. The optimized voice coil deformable mirror (VCDM) has a large phase stroke, good thermal stability, a short response time of less than 0.7 ms, and a large first resonance of 2045 Hz. The fitting residuals of the VCDM for the first 10 Zernike modes with a PV value of 8 μm are all below 10 nm (RMS). Compared with a similar DM, the obtained results from our compact VCDM indicate that it has a higher wavefront fitting precision. VCDM corrected complex random aberrations in the VLC scenario and improved the coupling efficiency of the signal beam, proving that the compact VCDM with high performance and low cost has a good application prospect in VLC systems.

Wavefront Control Strategies for Large Active Thin Shell Primaries with Unimorph Actuators

This paper presents various aspects of the wavefront control strategies for an ultra-lightweight composite reflector made of polymers for the large primary of a space telescope, and the shape control is made by a set of patterned unimorph strain actuators attached to the reflector. It starts with an analytical investigation of the mechanical behaviors of a strain-actuated curved shell, resulting in the accurate prediction of typical features, such as the damped wave deformation at the transition between electrodes and the limited morphing amplitude of a “print-through” actuation, which indicates that the curvature-induced rigidity deteriorates the performances of the forming accuracy of the active reflector and the morphing stroke of the actuators. The morphing capabilities are evaluated with both petal-like segmented and monolithic configured reflectors by numerical tests on forming target shapes of Zernike modes with various patternings of electrodes, and the structural dynamics are examined. Finally, a compound control strategy is proposed, which uses a deformable relay mirror to compensate for the residual surface error corrected partially by the active unimorph primary mirror, showing a great relaxation of the shape error budget of the thin-shell primary, especially for on-axis observation.

High-dimensional spatial mode sorting and optical circuit design using multi-plane light conversion

Multi-plane light converters (MPLCs) are an emerging class of optical devices capable of converting a set of input spatial light modes to a new target set of output modes. This operation represents a linear optical transformation—a much sought after capability in photonics. MPLCs have potential applications in both the classical and quantum optics domains, in fields ranging from optical communications to optical computing and imaging. They consist of a series of diffractive optical elements (the “planes”), typically separated by a free space. The phase delays imparted by each plane are determined by the process of inverse-design, most often using an adjoint algorithm known as the wavefront matching method (WMM), which optimizes the correlation between the target and actual MPLC outputs. In this work, we investigate high mode capacity MPLCs to create arbitrary spatial mode sorters and linear optical circuits. We focus on designs possessing low numbers of phase planes to render these MPLCs experimentally feasible. To best control light in this scenario, we develop a new inverse-design algorithm, based on gradient ascent with a specifically tailored objective function, and show how, in the low-plane limit, it converges to MPLC designs with a substantially lower modal cross-talk and higher fidelity than those achievable using the WMM. We experimentally demonstrate several prototype few-plane high-dimensional spatial mode sorters, operating on up to 55 modes, capable of sorting photons based on their Zernike mode or orbital angular momentum state, or an arbitrarily randomized spatial mode basis. We discuss the advantages and drawbacks of these proof-of-principle prototypes and describe future improvements. Our work points to a bright future for high-dimensional MPLC-based technologies.

Zernike mode rescaling extends capabilities of adaptive optics for microscopy

Zernike polynomials are widely used mathematical models of experimentally observed optical aberrations, and they have found widespread use in adaptive optic realizations that are used to correct wavefront aberrations. However, Zernike aberrations lose their orthogonality when used in combination with Gaussian beams and, as a consequence, start to cross-couple between each other, a phenomenon that does not occur for Zernike aberrations in plane waves. Here, we describe how the aberration radius (i.e. the radius of the beam relative to the active aperture of an active optical element) influences this cross-coupling of Zernike aberrations in a way that is distinct from simple truncation or balancing. Furthermore, we show that this effect can actually be harnessed to allow efficient compensation of higher-order aberrations using only low-order Zernike modes. This finding has important practical implications, as it suggests the possibility of using adaptive optics devices with low element numbers to compensate aberrations that would normally require more complex and expensive devices.

Method of wavefront phase retrieval from wavefront curvature sensing using membrane modes.

Wavefront phase retrieval is one of the most critical problems in adaptive optics. Here, phase retrieval by solving the transport of intensity equation using membrane vibration modes is proposed. Our study shows that the wavefront curvature sensing signal on the pupil can be expanded as a set of corresponding membrane vibration modes. The analytic expressions of the reconstructed phase are given. The coefficients of the functions are obtained by the integral over the pupil and boundary. Several representative Zernike circular and annular polynomials are respectively fitted by eigenfunctions and membrane modes in the absence of noise. In addition, wavefront recovery from noisy curvature data of the simulated atmospheric turbulence phase based on Zernike modes and Kolmogorov spectrum is demonstrated to verify the accuracy and robustness of the proposed method.

Emulating atmospheric turbulence effects on a micro-mirror array: assessing the DMD for use with free-space-to-fibre optical connections

Emulated atmospheric turbulence effects constructed from a set of 22 Zernike modes have been written upon a DMD micromirror array operating as a binary amplitude spatial light modulator. Sequences of aberrated frames with controlled amounts of turbulence have been produced and can be operated at controlled rates that can exceed 1 kHz rates which can be seen in strong turbulence. In this case 400 Hz was chosen and the scintillation levels observed for the same turbulence sequence with single, few and multi-mode fibres at a receiver. Resulting scintillation levels are consistent with standard turbulence models. Increased received intensity and reduced scintillation was observed with larger core fibres and related to aberration-induced focal spot size. Correlation between the received intensity variation and the amplitude variations for individual Zernike modes demonstrates specifically the effect of atmospheric induced beam wander when focusing into a receive fibre. The DMD is thus shown to be able to generate both the necessary frequency content and range of scintillation required for atmospheric emulation.

Neural networks and PCA coefficients to identify and correct aberrations in adaptive optics

Context. Static and quasi-static aberrations represent a great limit for high-contrast imaging in large telescopes. Among them the most important ones are all the aberrations not corrected by the adaptive optics (AO) system, which are called non-common path aberrations (NCPA). Several techniques have been proposed to mitigate it. The typical approach is to set an offset on the AO system with exactly the opposite sign of the NCPA in order to correct for the aberrations introduced by all the optical components downstream the wave-front sensor (WFS) up to the science camera. An estimate of the NCPA can be obtained with a trial-and-error approach or by more sophisticated techniques of focal-plane wave-front sensing. Aims. In all cases, a fast procedure is desirable to limit the telescope downtime and to repeat, if needed, the correction procedure to cope with the temporal variation of the NCPA. Very recently, new approaches based on neural networks (NNs) have also been proposed as an alternative. Methods. In this work, through simulated images, we test the application of a supervised NN for the mitigation of NCPAs in high-contrast imaging at visible wavelengths and, in particular, we investigate the possibility of applying this method to fast imagers such as SHARK-VIS, the forthcoming visible-band high-contrast imager for the Large Binocular Telescope (LBT). Results. Preliminary results show a measurement accuracy of the NCPA of 2 nm root mean square (RMS) for each sensed Zernike mode in turbulence-free conditions, and 5 nm RMS per mode when the residual turbulence has a wave-front error (WFE) of approximately 42.5 nm RMS, a typical value during LBT AO system calibration. This measurement is sufficient to guarantee that, after correction, NCPA residuals in the system are negligible compared to the typical WFE > 100 nm RMS of the best AO systems at large telescopes. Conclusions. Our simulations show this method is robust even in the presence of turbulence-induced aberrations that are not labelled in the training phase of the NN. The method could thus be used in a real-world setting by offloading a corrective static offset to the AO system of a telescope to mitigate the NCPA.

A spherical wave expansion for a steerable parametric array loudspeaker using Zernike polynomials.

A steerable parametric array loudspeaker (PAL) can electronically steer highly directional audio beams in the desired direction. The challenge of modelling a steerable PAL is to obtain the audio sound pressure in both near and far fields with a low computational load. To address this issue, an extension of the spherical wave expansion is proposed in this paper. The steerable velocity profile on the radiation surface is expanded as Zernike polynomials which are an orthogonal and form a complete set over a unit circle. An expression for the radiated audio sound is then obtained using a superposition of Zernike modes. Compared to the existing methods, the proposed expansion is computationally efficient and provides a rigorous transformation of the quasilinear solution of the Westervelt equation without paraxial approximations. The proposed expansion is further extended to accommodate local effects by using an algebraic correction to the Westervelt equation. Numerical results for steering single and dual beams are presented and discussed. It is shown that the single beam can be steered in the desired direction in both near and far fields. However, dual beams cannot be well separated in the near field, which cannot be predicted by the existing far field models.

Resonant Adaptive MEMS Mirror

A novel MEMS continuous deformable mirror (DM) is presented. The mirror can be integrated into optical systems to compensate for monochromatic and chromatic aberrations. It is comprised of a 1.6 mm circular plate supported by eight evenly spaced flexural springs. Unlike traditional bias actuated DMs, it uses resonant electrostatic actuation (REA) to realize low- and high-order Zernike modes with a single drive signal. Instead of the hundreds or thousands of electrodes deployed by traditional DMs, the proposed DM employs only 49 electrodes and eliminates the need for spatial control algorithms and associated hardware, thereby providing a compact low-cost alternative. It also exploits dynamic amplification to reduce power requirements and increase the stroke by driving the DM at resonance. The DM was fabricated using a commercial silicon-on-insulator (SOI) MEMS process. Experimental modal analysis was carried out using laser Doppler vibrometry (LDV) to identify mode shapes of the DM and their natural frequencies. We are able to observe all of the lowest eight Zernike modes.