Zernike Modes Research Articles

Adaptive optics for dynamic aberration compensation using parallel model-based controllers based on a field programmable gate array.

Adaptive optics (AO) is an effective technique for compensating the aberrations in optical systems and restoring their performance for various applications such as image formation, laser processing, and beam shaping. To reduce the controller complexity and extend the compensation capacity from static aberrations to dynamic disturbances, the present study proposes an AO system consisting of a self-built Shack-Hartmann wavefront sensor (SHWS), a deformable mirror (DM), and field programmable gate array (FPGA)-based controllers. This AO system is developed for tracking static and dynamic disturbances and tuning the controller parameters as required to achieve rapid compensation of the incoming wavefront. In the proposed system, the FPGA estimates the coefficients of the eight Zernike modes based on the SHWS with CameraLink operated at 200 Hz. The estimated coefficients are then processed by eight parallel independent discrete controllers to generate the voltage vectors to drive the DM to compensate the aberrations. To have the DM model for controller design, the voltage vectors are identified offline and are optimized by closed-loop controllers. Furthermore, the controller parameters are tuned dynamically in accordance with the main frequency of the aberration as determined by a fast Fourier transform (FFT) process. The experimental results show that the AO system provides a low complexity and effective means of compensating both static aberrations and dynamic disturbance up to 20 Hz.

Focal plane wavefront sensing using machine learning: performance of convolutional neural networks compared to fundamental limits

ABSTRACT Focal plane wavefront sensing (FPWFS) is appealing for several reasons. Notably, it offers high sensitivity and does not suffer from non-common path aberrations (NCPAs). The price to pay is a high computational burden and the need for diversity to lift any phase ambiguity. If those limitations can be overcome, FPWFS is a great solution for NCPA measurement, a key limitation for high-contrast imaging, and could be used as adaptive optics wavefront sensor. Here, we propose to use deep convolutional neural networks (CNNs) to measure NCPAs based on focal plane images. Two CNN architectures are considered: ResNet-50 and U-Net that are used, respectively, to estimate Zernike coefficients or directly the phase. The models are trained on labelled data sets and evaluated at various flux levels and for two spatial frequency contents (20 and 100 Zernike modes). In these idealized simulations, we demonstrate that the CNN-based models reach the photon noise limit in a large range of conditions. We show, for example, that the root mean squared wavefront error can be reduced to <λ/1500 for 2 × 106 photons in one iteration when estimating 20 Zernike modes. We also show that CNN-based models are sufficiently robust to varying signal-to-noise ratio, under the presence of higher order aberrations, and under different amplitudes of aberrations. Additionally, they display similar to superior performance compared to iterative phase retrieval algorithms. CNNs therefore represent a compelling way to implement FPWFS, which can leverage the high sensitivity of FPWFS over a broad range of conditions.

Experimental study on adaptive optical wavefront correction with dual mirrors in free space optical communication

The wavefront of light wave contains a tilt component after atmospheric turbulence, which increases the wavefront distortion and reduce the quality of laser communication. A fast steering mirror (FSM) must be introduced to correct the wavefront tilt. The adaptive optical closed-loop control algorithm based on Zernike mode decomposition is used to correct the wavefront distortion of light wave after indoor (0.5 m), 600 m, 1 km, 5 km, 10 km, and 100 km transmission. The experimental results show that, under the above-mentioned six different distances, the wavefront phase decreased from 9.5 µm, 15 µm, 12.5 µm, 22 µm, 35 µm, and 450–5 µm, 10 µm, 7.5 µm, 8 µm, 20 µm, and 300 µm, respectively, only after the closed-loop of the FSM; only after the closed-loop of the deformable mirror (DM), the wavefront phase decreased to 8 µm, 9 µm, 9 µm, 20 µm, 15 µm, and 300 µm, respectively. Further, when the FSM and DM are closed simultaneously, the wavefront phase decreases to 0.2 µm, 2.1 µm, 0.4 µm, 3.1 µm, 5 µm, and 200 µm. The correction effect of closed-loop of FSM and DM at the same time is better than that of either FSM or DM only.

Adaptive optics for structured illumination microscopy based on deep learning.

Structured illumination microscopy (SIM) is widely used in biological imaging for its high resolution, fast imaging speed, and simple optical setup. However, when imaging thick samples, the structured illumination patterns in SIM will suffer from optical aberrations, leading to a serious deterioration in resolution. Therefore, it is necessary to reconstruct structured illumination patterns with high quality and efficiency in deep tissue imaging. Here we demonstrate an adaptive optics (AO) correction method based on deep learning in wide-field SIM imaging system. The mapping between the coefficients of the first 15 Zernike modes and their corresponding distorted patterns is established to train the convolution neural network (CNN). The results show that the optimized CNN can predict the aberration phase within ~10.1 ms with a personal computer. The correlation index between the aberration phases and their corresponding predicted aberration phase is up to 0.9986. This method is highly robust and effective for patterns with various spatial densities and illumination conditions and able to effectively correct the imaging distortion caused by optical aberration in SIM system.

Laser processing over a large area by wavefront-controlled scanning

In the processing of laser materials, the scanning area is limited by the extent to which the laser focus can be maintained within a plane. In order to increase the processing area, one has to enlarge the size of the laser focal spot, and, therefore, reduce the peak intensity and spatial resolution. In this paper, the authors report on a method that extends the processing area by controlling the wavefront of a laser beam using adaptive optics. By adding Zernike modes to the laser beam wavefront, the axial (along the beam propagation direction) location of the focal spot can be tuned within a range larger than the confocal length (two times the Rayleigh length), while the lateral spot size (and, thus, peak intensity and resolution) is maintained throughout the tuning range. A wave propagation model that simulates the propagation of a wavefront-controlled beam is used to compare with the experimental results, which are also confirmed by Zemax calculation. Laser trepanning of circular holes with consistent size on a wide polypropylene sheet is demonstrated. This work shows the potential of tailoring the phase of a laser beam for improving the accuracy and throughput in large-area laser processing of materials.

Defocused Image Deep Learning Designed for Wavefront Reconstruction in Tomographic Pupil Image Sensors

Mathematical modelling methods have several limitations when addressing complex physics whose calculations require considerable amount of time. This is the case of adaptive optics, a series of techniques used to process and improve the resolution of astronomical images acquired from ground-based telescopes due to the aberrations introduced by the atmosphere. Usually, with adaptive optics the wavefront is measured with sensors and then reconstructed and corrected by means of a deformable mirror. An improvement in the reconstruction of the wavefront is presented in this work, using convolutional neural networks (CNN) for data obtained from the Tomographic Pupil Image Wavefront Sensor (TPI-WFS). The TPI-WFS is a modified curvature sensor, designed for measuring atmospheric turbulences with defocused wavefront images. CNNs are well-known techniques for its capacity to model and predict complex systems. The results obtained from the presented reconstructor, named Convolutional Neural Networks in Defocused Pupil Images (CRONOS), are compared with the results of Wave-Front Reconstruction (WFR) software, initially developed for the TPI-WFS measurements, based on the least-squares fit. The performance of both reconstruction techniques is tested for 153 Zernike modes and with simulated noise. In general, CRONOS showed better performance than the reconstruction from WFR in most of the turbulent profiles, with significant improvements found for the most turbulent profiles; overall, obtaining around 7% of improvements in wavefront restoration, and 18% of improvements in Strehl.

An Automatic Optimized Method for a Digital Optical Phase Conjugation System in Focusing through Scattering Media

In this paper, a reliable automatic optimized method for a digital optical phase conjugation (DOPC) system based on a multipopulation genetic algorithm (MPGA) is proposed for improving the compensation quality of DOPC. The practical implementation and compensation quality of DOPC in focusing through scattering media are greatly limited by imperfect pixel alignment, optical aberration, and mechanical error in the DOPC system. For comprehensively solving the above problems, the concept of global optimization is introduced by Zernike polynomials (Zernike modes) to characterize overall imperfections, and MPGA is used to search for the most optimal Zernike coefficient and compensate for the overall imperfections of the DOPC system. The significant optimization ability of the proposed method is verified in DOPC-related experiments for focusing through scattering media. The peak-to-background ratio (PBR) of the OPC focus increases 174 times that of the initial OPC focus. Furthermore, we evaluated the optimization results of the proposed method with a fitness function of intensity fitness and correlation coefficient fitness in MPGA. The results show that the optimized capability is excellent and more efficiently used than the correlation coefficient fitness function in the Zernike modes.

Cascading optofluidic phase modulators for performance enhancement in refractive adaptive optics

We discuss the implementation and performance of an adaptive optics (AO) system that uses two cascaded deformable phase plates (DPPs), which are transparent optofluidic phase modulators, mimicking the common woofer/tweeter-type astronomical AO systems. One of the DPPs has 25 electrodes forming a keystone pattern best suited for the correction of low-order and radially symmetric modes; the second device has 37 hexagonally packed electrodes better suited for high-order correction. We also present simulation results and experimental validation for a new open-loop control strategy enabling simultaneous control of both DPPs, which ensures optimum correction for both large-amplitude low-order, and complex combinations of low- and high-order aberrations. The resulting system can reproduce Zernike modes up to the sixth radial order with stroke and fidelity up to twice better than what is attainable with either of the DPPs individually. The performance of the new AO configuration is also verified in a custom-developed fluorescence microscope with sensorless aberration correction.

Leveraging the orthogonality of Zernike modes for robust free-space optical communication

Free-space optical communication systems exploit the properties of light beams to transfer information through a free-space link. Indeed such systems provide an exciting alternative for communication. Here we introduce information transfer through free-space using a laser beam having its phase encoded with multiple orthogonal aberration modes. We use Zernike polynomials, which form a complete basis set, to represent the aberration modes. The user information is converted to co-efficients of the Zernike modes which are summed digitally to obtain the resultant phase profile. A single phase modulating device then reads the resultant phase to shape the wavefront of the beam to be transmitted. The receiving station estimates the co-efficients of all modes in the beam from a single measurement of a wavefront sensor, to retrieve the user information. We demonstrate data transfer using multiple modes, each with multiple strengths, and external perturbation compensation using the completeness property of the modes.

Compensating atmospheric turbulence with CNNs for defocused pupil image wavefront sensors

Abstract Adaptive optics are techniques used for processing the spatial resolution of astronomical images taken from large ground-based telescopes. In this work, computational results are presented for a modified curvature sensor, the tomographic pupil image wavefront sensor (TPI-WFS), which measures the turbulence of the atmosphere, expressed in terms of an expansion over Zernike polynomials. Convolutional neural networks (CNN) are presented as an alternative to the TPI-WFS reconstruction. This technique is a machine learning model of the family of artificial neural networks, which are widely known for its performance as modeling and prediction technique in complex systems. Results obtained from the reconstruction of the networks are compared with the TPI-WFS reconstruction by estimating errors and optical measurements (root mean square error, mean structural similarity and Strehl ratio). The reconstructed wavefronts from both techniques are compared for wavefronts of 153 Zernike modes. For this case, a detailed comparison and grid search to find the most suitable neural network is performed, searching between multi-layer perceptron, CNN and recurrent networks topologies. In general, the best network was a CNN trained for TPI-WFS reconstruction, achieving better performance than the reconstruction software from TPI-WFS in most of the turbulent profiles, but the most significant improvements were found for higher turbulent profiles that have the lowest r0 values.

Differentiable model-based adaptive optics with transmitted and reflected light.

Aberrations limit optical systems in many situations, for example when imaging in biological tissue. Machine learning offers novel ways to improve imaging under such conditions by learning inverse models of aberrations. Learning requires datasets that cover a wide range of possible aberrations, which however becomes limiting for more strongly scattering samples, and does not take advantage of prior information about the imaging process. Here, we show that combining model-based adaptive optics with the optimization techniques of machine learning frameworks can find aberration corrections with a small number of measurements. Corrections are determined in a transmission configuration through a single aberrating layer and in a reflection configuration through two different layers at the same time. Additionally, corrections are not limited by a predetermined model of aberrations (such as combinations of Zernike modes). Focusing in transmission can be achieved based only on reflected light, compatible with an epidetection imaging configuration.

Integrating engineered point spread functions into the phasor-based single-molecule localization microscopy framework.

In single-molecule localization microscopy (SMLM), the use of engineered point spread functions (PSFs) provides access to three-dimensional localization information. The conventional approach of fitting PSFs with a single 2-dimensional Gaussian profile, however, often falls short in analyzing complex PSFs created by placing phase masks, deformable mirrors or spatial light modulators in the optical detection pathway. Here, we describe the integration of PSF modalities known as double-helix, saddle-point or tetra-pod into the phasor-based SMLM (pSMLM) framework enabling fast CPU based localization of single-molecule emitters with sub-pixel accuracy in three dimensions. For the double-helix PSF, pSMLM identifies the two individual lobes and uses their relative rotation for obtaining z-resolved localizations. For the analysis of saddle-point or tetra-pod PSFs, we present a novel phasor-based deconvolution approach entitled circular-tangent pSMLM. Saddle-point PSFs were experimentally realized by placing a deformable mirror in the Fourier plane and modulating the incoming wavefront with specific Zernike modes. Our pSMLM software package delivers similar precision and recall rates to the best-in-class software package (SMAP) at signal-to-noise ratios typical for organic fluorophores and achieves localization rates of up to 15kHz (double-helix) and 250kHz (saddle-point/tetra-pod) on a standard CPU. We further integrated pSMLM into an existing software package (SMALL-LABS) suitable for single-particle imaging and tracking in environments with obscuring backgrounds. Taken together, we provide a powerful hardware and software environment for advanced single-molecule studies.

Image based aberration retrieval using helical point spread functions.

A practical method for determining wavefront aberrations in optical systems based on the acquisition of an extended, unknown object is presented. The approach utilizes a conventional phase diversity approach in combination with a pupil-engineered, helical point spread function (PSF) to discriminate the aberrated PSF from the object features. The analysis of the image's power cepstrum enables an efficient retrieval of the aberration coefficients by solving a simple linear system of equations. An extensive Monte Carlo simulation is performed to demonstrate that the approach makes it possible to measure low-order Zernike modes including defocus, primary astigmatism, coma, and trefoil. The presented approach is tested experimentally by retrieving the two-dimensional aberration distribution of a test setup by imaging an extended, unknown scene.

Forward imaging neural network with correction of positional misalignment for Fourier ptychographic microscopy.

Fourier ptychographic microscopy (FPM) is a computational imaging technology used to achieve high-resolution imaging with a wide field-of-view. The existing methods of FPM suffer from the positional misalignment in the system, by which the quality of the recovered high-resolution image is determined. In this paper, a forward neural network method with correction of the positional misalignment (FNN-CP) is proposed based on TensorFlow, which consists of two models. Both the spectrum of the sample and four global position factors, which are introduced to describe the positions of the LED elements, are treated as the learnable weights in layers in the first model. By minimizing the loss function in the training process, the positional error can be corrected based on the trained position factors. In order to fit the wavefront aberrations caused by optical components in the FPM system for better recovery results, the second model is designed, in which the spectrum of the sample and coefficients of different Zernike modes are treated as the learnable weights in layers. After the training process of the second model, the wavefront aberration can be fit according to the coefficients of different Zernike modes and the high-resolution complex image can be obtained based on the trained spectrum of the sample. Both the simulation and experiment have been performed to verify the effectiveness of our proposed method. Compared with the state-of-art FPM methods based on forward neural network, FNN-CP can achieve the best reconstruction results.

Holographic wavefront sensors

A brief historical review of the first designs of holographic wavefront sensors (WFS’s) and the concepts lying in their basis is presented. The main directions in the development of these sensors are highlighted and considered. One of these directions implies a two-stage transition from the use of several separate measuring channels with holograms filtering only one Zernike mode: first to the use of one channel with a multiplexed hologram filtering several first Zernike modes, and then to filtering the entire set of Zernike modes with the aid of one combined multiplexed hologram. Another line of research in this field (related to the first one) is the optimisation of the filter hologram structure in order to reduce cross-modulation interferences, increase multiplexing level, etc. One more line of research implements principles of dynamic holography by introducing spatial light modulators into the WFS composition. Hence, the advantages of time multiplexing of holograms can be used. The approach developed by G. Andersen’s team, aimed at adapting the holographic WFS design for operation as an element of adaptive optical system with a zonal corrector, as well as an approach implying development of hybrid holographic sensors, are considered separately. The results of the authors’ studies of holographic WFS’s with application of the methods of Fourier holography (i.e., holography of focused beams, in particular, using diffuse scatterers in a hologram recording channel) are also reported.

Numerical Simulations of a Monomorph Deformable Mirror in ANSYS Software

Based on the numerical simulations in ANSYS software the design of a monomorph (unimorph) deformable mirror has been developed. The control electrodes pattern, which allows reproducing aberrations up to 5th order (the first 21 Zernike modes) on the light aperture with high fidelity, was chosen. All important characteristics of the monomorph mirror were simulated: electrode-influence functions, errors in the reproduction of given aberrations, gravitational sag of the mirror, mirror surface deformations due to ambient temperature changes, thermal deformations and thermal field due to the incident laser beam, and natural frequencies of the mirror. The results prove a high efficiency of this mirror as an element of adaptive optics systems.

Improving system performance by using adaptive optics and aperture averaging for laser communications in oceanic turbulence.

We theoretically investigate the effectiveness of adaptive optics correction for Gaussian beams affected by oceanic turbulence. Action of an idealized adaptive optics system is modeled as a perfect removal of a certain number of Zernike modes from the aberrated wavefront. We focused on direct detection systems and we used the aperture-averaged scintillation as the main metric to evaluate optical system performances. We found that, similar to laser beam propagation in atmospheric turbulence, adaptive optics is very effective in improving the performance of laser communication links if an optimum aperture size is used. For the specific cases we analyzed in this study, scintillation was reduced by a factor of ∼7 when 15 modes were removed and when the aperture size of the transceiver was large enough to capture 4-5 speckles of the oceanic turbulence-affected beam.

Characterization of laser thermal loading on microelectromechanical systems-based fast steering mirror in vacuum

Microelectromechanical systems (MEMS) have produced high-quality, high-bandwidth, small form factor, and inexpensive fast steering mirror (FSM) devices potentially suitable for a large variety of applications, such as image stabilization and beam pointing in satellite-based and ground-based, free-space optical communication systems. However, one outstanding question for this application is power handling. The absorption of the mirror substrate is low, but non-negligible, so the question remains of whether thermal loading from laser radiation on a MEMS mirror will deform its surface and, if so, to what extent. We show experimental results of optical performance changes due to thermal loading for MEMS two-axis FSM devices from Mirrorcle Technologies, Inc. Results and reproducible behavior are reported and compared in ambient versus vacuum conditions, where the benefits of convective cooling are absent. Finite element analyses corroborate the experimental results and show that the mirror substrate can deform due to thermal expansion imbalances. The deformation changes the focusing characteristics of the mirror, with a peak to valley defocus (second-order Zernike mode) of up to 50 nm when the mirrors are tested in ambient and up to approximately 450 nm when under vacuum. Such defocusing negatively impacts the link budget for laser-based satellite communications.

Depth-resolved optimization of a real-time sensorless adaptive optics optical coherence tomography.

Sensorless adaptive optics optical coherence tomography (AO-OCT) is a technology to image retinal tissue with high resolution by compensating ocular aberrations without wavefront sensors. In this Letter, a fast and robust hill-climbing algorithm is developed to optimize five Zernike modes in AO-OCT with a numerical aperture between that of conventional AO and commercial OCT systems. The merit function is generated in real time using graphics processing unit while axially tracking the retinal layer of interest. A new method is proposed to estimate the largest achievable field of view for which aberrations are corrected uniformly in sensorless AO-OCT.

Improved model-based wavefront sensorless adaptive optics for extended objects using N + 2 images

The original model-based wavefront sensorless adaptive optics (WFSless AO) for extended objects uses the low spatial frequency content of images as the metric function and employs 2N + 1 images to correct N Lukosz aberration modes. We propose an improved method that uses the same metric but requires only N + 2 images to correct N aberration modes. The N + 2 method can achieve comparable corrective accuracy but requiring much smaller number of images compared with the 2N + 1 method. The N + 2 method is not only more efficient, but also provides the flexibility of choosing arbitrary basis modes by involving the non-orthogonality between modes in a linear least-squares optimization process. It is demonstrated that the deformable mirror's influence functions modes (IFM), orthogonal mirror modes (OMM) and fitted Zernike modes (FZM) all can be used as modal basis for the N + 2 method to generate biased images. It is also proved that the N + 2 method is robust to image sampling rate and image noise, showing good prospects of applications in various imaging systems.