Modal-based adaptive autofocusing phase retrieval for wavefront measurement

The wavefront measurement accuracy of a grating array based zonal wavefront sensor (GAWS) can be affected by the non-uniform focal spot array and unwanted orders in the detector plane. The non-uniform focal spot array is the outcome of the non-uniform nature of the incident illumination beam’s intensity profile. This paper describes a method that dynamically modulates the laser beam’s intensity using computer generated holography, making the focal spot array uniform and eliminating unwanted spots in a detector plane, thereby enhancing the accuracy of the wavefront measurement. Here, we present proof-of-principle simulation results that demonstrate the working of the proposed improvements in GAWS.

As an expansion of coherent diffraction imaging, ptychographic iterative engine (PIE) not only inherits advantages such as ultra-high resolution and compact optical system, but also expands the field of view in quantitative imaging, thus PIE is widely used in short wavelength imaging such as X-ray and electron beam imaging, and then extended to visible light field. However, PIE requires coherent illumination for both phase and amplitude retrieval, while traditional X-ray or electron beam sources often cannot satisfy this strict coherent condition, which leads to poor-quality information retrieval with low signal-to-noise ratio. Though several proposed methods such as multiple wavelength and multi-mode algorithms can eliminate incoherency influence to some extent, various details such as quantitative spectrum of illuminating source should be obtained before information retrieval, which complicates computing procedures. In addition, it is hard to acquire the spectrum of the illuminating source in most cases. In order to acquire high-quality information based on PIE with partially coherent illumination, a newly designed enhanced phase retrieval method for weakly scattering samples in PIE with partially coherent illumination is presented in this paper, in which only the bright field of the diffraction patterns is used in the iterative procedures mimicking the coherent cases especially for weakly scattering samples without any prior illuminating details. The bright field area can be regarded as purely coherent diffraction patterns composed of a strengthened zeroth order beam and a weakened diffracted beam. While the dark-field area generated by interference of diffracted beams cannot satisfy the requirement for coherence, therefore, dark-field diffraction patterns should be excluded in sample information extraction and only the bright field is used for phase retrieval via iterative process. Compared with the proposed multiple wavelength and multi-mode algorithms, the proposed method can simplify sample reconstructing procedures due to needing no prior knowledge. Moreover, in order to enhance the information of weakly scattering samples in retrieval, high order iteration method is also introduced in the paper. Since the bright field can be regarded as purely coherent diffraction patterns composed of a strengthened zeroth order beam and a weakened diffracted beam. For weakly scattering sample, the weakened diffracted beam is much lower than zeroth order beam, thus it is difficult to acquire high-contrast information with classical PIE algorithms. Introducing high order iterative tactic, the contrast of weakly scattering sample is obviously improved and the details of weakly scattering sample can be retrieved clearly. Both theoretical analysis and numerical simulations are illustrated in detail, proving the robustness and availability of the designed method: high-contrast phase information can be obtained with the proposed method, while traditional phase retrieval algorithm almost loses all details of the sample. In order to mimic the real experimental situation, a 30 dB white noise is added into the simulation, the details of weakly scattering sample phase information can also be retrieved clearly by using the bright field of the diffraction patterns with high order iteration method. With the newly designed enhanced phase retrieval method for weakly scattering samples with partially coherent illumination, sample retrieval via PIE can not only use ordinary X-ray source or electron beam as illumination source, thereby avoiding the dependence on complete coherent source, but also obviously improve the retrieval quality of the sample characteristics, which widely expands the application fields of the PIE.

The effective acquisition of a distorted wavefront is the foundation of studying how to suppress and correct aero-optical effects. The background-oriented schlieren (BOS) technique, as a commonly quantitative measurement method of flow field, has great application potential in aero-optical effect measurement. With consideration of the thickness change of the refractive index field, introduction of near-field correction and construction of the double telecentric configuration, wavefront measurement accuracy was improved. A standard plano-convex lens was utilized to quantitatively evaluate the improved results, and the effectiveness of the improvement was verified. Determination methods of spatial resolution, sensitivity, and dynamic measurement range of wavefront measurement based on BOS were studied. The influences of cross-correlation interrogation window size and step length on the wavefront reconstruction accuracy were investigated. The step length should be less than the size of the interrogation window to make the reconstructed results have a satisfactory accuracy. Wavefront results of supersonic film at different positions and cases were obtained. The fitting results show that there is a good linear relationship between the root mean square of the optical path difference (OPDrms) and ρ1/ρ∞ at different locations. Under different experimental cases, the aero-optical distortion tends to become stronger first and then weaker along the flow direction, and the aero-optical distortion around Position 3 is relatively strong. The large aperture approximation can be used to predict Strehl ratio values effectively in a considerable range by OPDrms.

Fourier phase retrieval problem is equivalent to the recovery of a two-dimensional image from its autocorrelation measurements. This problem is generally nonlinear and nonconvex. Good initialization and prior information about the support or sparsity of the target image are often critical for a robust recovery. In this paper, we show that the presence of a known reference image can help us solve the nonlinear phase retrieval problem as a sequence of small linear inverse problems. Instead of recovering the entire image at once, our sequential method recovers a small number of rows or columns by solving a linear deconvolution problem at every step. Existing methods for the reference-based (holographic) phase retrieval either assume that the reference and target images are sufficiently separated so that the recovery problem is linear or recover the image via nonlinear optimization. In contrast, our proposed method does not require the separation condition. We performed an extensive set of simulations to demonstrate that our proposed method can successfully recover images from autocorrelation data under different settings of reference placement and noise.

We introduce a novel phase retrieval method for astronomical applications based on the Nijboer-Zernike (NZ) theory of diffraction. We present a generalized NZ phase retrieval process that is not limited to small and symmetric aberrations and can therefore be directly applied to astronomical imaging instruments. We describe a practical demonstration of this novel method that was recently performed using data taken on-sky with NAOS-CONICA, the adaptive optics system of the Very Large Telescope. This demonstration presents the first online on-sky phase retrieval results ever obtained, and allows us to plan subsequent refinements on a well-tested basis. Among the potential refinements, and within the framework of high-contrast imaging of extra-solar planetary systems (which requires exquisite wavefront quality), we introduce an extension of the generalized NZ to the high-dynamic range case, and particularly to its use with the vector vortex coronagraph. This induces conjugated phase ramps applied to the orthogonal circular polarizations, which can be used to instantaneously retrieve the complex amplitude of the field, yielding a real-time calibration of the wavefront that does not need any other modulation such as focus or other deformable mirror probe patterns. Paper II (Riaud et al. 2012, A&A, 545, A151) presents the mathematical and practical details of the new method.

We present a phase coefficient optimization technique for determining the vortex charge of a high-power orbital angular momentum beam. This high-power vortex beam is generated by transmitting a Gaussian-like beam emitted by a gyrotron with an output power of 25 kW through a spiral phase plate (SPP), which introduces a vorticity at the center of the beam characterized by the geometrical parameters of the SPP. One rigorous, intensity-measurement-based, phase retrieval technique, known as the Gerchberg–Saxton algorithm, does not converge to a correct solution because of the presence of a phase singularity in the high-power vortex beam. Here, we introduce a new phase retrieval algorithm that chooses an appropriate initial phase estimate. This technique yields successful vortex charge determination for both low- and high-power vortex beams. The retrieved intensity profiles show 99.51% and 99.27% intensity regeneration at measurement planes with optimized initial phase estimates for the low- and high-power beams, respectively.

‘Ghosted’ fringe patterns simultaneously reflected from both the upper and lower sides of a transparent target in the fringe reflection technique are captured for transparent surface 3D shape measurement, but the phase retrieval from the captured ‘ghosted’ fringe patterns is still not solved. A novel method is proposed to solve this issue by using two sets of phase-shifted fringe patterns with slightly different frequencies. The nonlinear least-squares method is used to estimate the fringe phase and modulation from both front and rear interfaces. Several simulations are done to show the feasibility of the proposed method. The influence of fringe noise on the algorithm is studied as well, which indicates that the proposed method is able to retrieve the phase from double-sided reflective fringe patterns with fringe noise equivalent to that in practical measurements. The merits and limitations of the method are discussed and recommendations for future studies are made.

Pixel super-resolution (PSR) techniques have been developed to overcome the sampling limit in lensless digital holographic imaging. However, the inherent non-convexity of the PSR phase retrieval problem can potentially degrade reconstruction quality by causing the iterations to tend toward a false local minimum. Furthermore, the ill posedness of the up-sampling procedure renders PSR algorithms highly susceptible to noise. In this Letter, we propose a heuristic PSR algorithm with adaptive smoothing (AS-PSR) to achieve high-fidelity reconstruction. By automatically adjusting the intensity constraints on the estimated field, the algorithm can effectively locate the optimal solution and converge with high reconstruction quality, pushing the resolution toward the diffraction limit. The proposed method is verified experimentally within a coherent modulation phase retrieval framework, achieving a twofold improvement in resolution. The AS-PSR algorithm can be further applied to other phase retrieval methods based on alternating projection.

This paper presents a non-iterative phase retrieval method from randomly phase-shifted fringe images. By combining the hyperaccurate least squares ellipse fitting method with the subspace method (usually called the principal component analysis), a fast and accurate phase retrieval algorithm is realized. The proposed method is simple, flexible, and accurate. It can be easily coded without iteration, initial guess, or tuning parameter. Its flexibility comes from the fact that totally random phase-shifting steps and any number of fringe images greater than two are acceptable without any specific treatment. Finally, it is accurate because the hyperaccurate least squares method and the modified subspace method enable phase retrieval with a small error as shown by the simulations. A MATLAB code, which is used in the experimental section, is provided within the paper to demonstrate its simplicity and easiness.

In dynamic 3D measurements, the recovering carrier signal as well as phase retrieval are important issues for single fringe image analysis. Local mean decomposition is a powerful signal demodulation tool, but it usually encounters an obstacle, namely, the mode-mixing problem. Then different components, especially noise and carrier signals, are all probably mixed in one of the decomposition results, thus confusing its physical meaning. Utilizing the characteristics of original noise, we design a pair of differential signals based on the conditions that two mixed components should meet to be separated completely and then add them to the original signal. Re-decomposing the newly formed signal, the differential signal, along with the original noise, will be separated from the carrier signal, leaving very little negative impact due to the characteristics of the same amplitude and opposite polarity of the differential signal. With the mode-mixing problem of high-frequency components being resolved, the decomposition of the following low-frequency components becomes more reasonable, facilitating the fringe pattern analysis and further phase retrieval. The proposed method is suitably used for the signal, even though it is not stable. Experiments illustrate the efficiency of this novel adaptive method.

We consider the problem of reconstructing two signals from the autocorrelation and cross-correlation measurements. This inverse problem is a fundamental one in signal processing, and arises in many applications, including phase retrieval and blind channel estimation. In a typical phase retrieval setup, only the autocorrelation measurements are obtainable. We show that, when the measurements are obtained using three simple “masks”, phase retrieval reduces to the aforementioned reconstruction problem. The classic solution to this problem is based on finding common factors between the $z$ -transforms of the autocorrelation and cross-correlation vectors. This solution has enjoyed limited practical success, mainly due to the fact that it is not sufficiently stable in the noisy setting. In this paper, inspired by the success of convex programming in provably and stably solving various quadratic constrained problems, we develop a semidefinite programming-based algorithm and provide theoretical guarantees. In particular, we show that almost all signals can be uniquely recovered by this algorithm (up to a global phase). Comparative numerical studies demonstrate that the proposed method significantly outperforms the classic method in the noisy setting.

While supervised deep learning has been a prominent tool for solving many image restoration problems, there is an increasing interest on studying self-supervised or un-supervised methods to address the challenges and costs of collecting truth images. Based on the neuralization of a Bayesian estimator of the problem, this paper presents a self-supervised deep learning approach to general image restoration problems. The key ingredient of the neuralized estimator is an adaptive stochastic gradient Langevin dy-namics algorithm for efficiently sampling the posterior distri-bution of network weights. The proposed method is applied on two image restoration problems: compressed sensing and phase retrieval. The experiments on these applications showed that the proposed method not only outperformed existing non-learning and unsupervised solutions in terms of image restoration quality, but also is more computationally efficient.

For ground-based segmented telescopes, the active control system primarily adjusts three out-of-plane DOFs: tip, tilt, and piston. In contrast, in-plane DOFs rely on high-precision mechanical alignment. A systematic analysis of the effects of in-plane motion on the optical performance of ring segmented mirror systems is presented, and an analytical relationship between in-plane motion and the higher-order residual aberrations after tilt compensation is derived. Conventional alignment techniques based on three-dimensional metrology do not provide the necessary in-plane motion detection accuracy required for the CGST, which operates in the visible spectrum and demands higher surface precision than infrared telescopes. Previous measurements using laser trackers indicate that the achievable accuracy is insufficient for CGST alignment. A method based on wavefront aberration has been proposed to enhance the accuracy of in-plane motion measurements and alignments. This approach establishes a correlation between wavefront aberrations and in-plane motion. Wavefronts are obtained through phase retrieval and fitted using orthogonal Zernike polynomials to develop the mapping model. During the alignment process, the effect induced by Clocking is compensated by adjusting DX. Simulations using parameters from the CGST show that, with a point spread function image SNR above 20 dB, the residual wavefront aberration after compensation meets the diffraction-limited requirements in the visible range. The results confirm the potential of this method for achieving high-precision in-plane DOFs alignment in segmented solar telescopes.

In this paper, we address the problem of phase retrieval to recover a signal from the magnitude of its Fourier transform. In many applications of phase retrieval, the signals encountered are naturally sparse. In this work, we consider the case where the signal is sparse under the assumption that few components are nonzero. We exploit further the sparse nature of the signals and propose a two stage sparse phase retrieval algorithm. A simple iterative minimization algorithm recovers a sparse signal from measurements of its Fourier transform (or other linear transform) magnitude based on the minimization of a block l1 norm. We show in the experiments that the proposed algorithm achieves a competitive performance. It is robust to noise and scalable in practical implementation. The proposed method converges to a more accurate and stable solution than other existing techniques for synthetic signals. For speech signals, experiments show that the voice quality of reconstructed speech signals is almost as good as the original signals.

The goal of phase retrieval is to extract the phase of an optical wave field from intensity measurements. The transport-of-intensity equation (TIE)-based method is a popular deterministic solution and has been applied in various fields such as optical microscopy, electron microscopy, and x-ray imaging. For macro-imaging, a camera is often used to capture the images, and thus the phase modulation of the lens should be considered. A new formulation is proposed to extend TIE for phase retrieval in a lens-based wave propagation model. To obtain the defocus step, a data-collection-reading device is designed by equipping a camera with a micrometer caliper. Simulation and real experiments are conducted to test the proposed method.