Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field

Accelerated phase retrieval using adaptive support and statistical fringe processing of phase estimates

A technique for accelerated multiple-plane phase retrieval is demonstrated by creating adaptive support through the statistical analysis of phase estimates. Its technical advantage arises from, what we believe to be, the first time use of both phase estimates and a statistical metric, enabling the fast generation of noise-robust support masks. This results in a fourfold improvement in convergence speed when compared to the conventional multiple-plane method. Evaluating data fitting performance with fewer intensity recordings showed that using four or more recordings resulted in accurate fitting, three recordings caused overfitting, and two recordings led to underfitting for the test object waves used. In principle, the adaptive support strategy based on the statistical analysis of phase estimates may be applied to other iterative phase retrieval methods.

Twin-stagnation-free phase retrieval with vortex phase illumination

The recovery of a complex-valued exit wavefront from its Fourier transform magnitude is challenging due to the stagnation problems associated with iterative phase retrieval algorithms. Among the various stagnation artifacts, the twin-image stagnation is the most difficult to address. The upright object and its inverted and complex-conjugated twin correspond to the identical Fourier magnitude data and hence appear simultaneously in the iterative solution. We show that the twin stagnation problem can be eliminated completely if a coherent beam with charge-1 vortex phase is used for illumination. Unlike the usual plane wave illumination case, a charge-1 vortex illumination intentionally introduces an isolated zero near the zero spatial frequency region, where maximal energy in the Fourier space is usually concentrated for most natural objects. The early iterations of iterative phase retrieval algorithms are observed to develop a clockwise or anti-clockwise vortex in the vicinity of this isolated zero. Once the Fourier transform of the solution latches onto a specific vortex profile in the neighborhood of this intentionally introduced intensity zero in early iterations, the solution quickly adjusts to the corresponding twin (upright or inverted) and further iterations are not observed to bring the other twin into the reconstruction. Our simulation studies with the well-known hybrid input-output (HIO) algorithm show that the solution always converges to one of the twins within a few hundred iterations when vortex phase illumination is used. Using a clockwise or anti-clockwise vortex phase as an initial guess is also seen to deterministically lead to a solution consisting of the corresponding twin. The resultant solution still has some faint residual artifacts that can be addressed via the recently introduced complexity guidance methodology. There is an additional vortex phase in the final solution that can simply be subtracted out to obtain the original test object. The near guaranteed convergence to a twin-stagnation-free solution with vortex illumination as described here is potentially valuable for deploying practical imaging systems that work based on the iterative phase retrieval algorithms.

Enhanced Terahertz Phase Retrieval Imaging by Unequal Spaced Measurement

Terahertz lensless phase retrieval imaging is a promising technique for non-destructive inspection applications. In the conventional multiple-plane phase retrieval method, the convergence speed due to wave propagations and measures with equal interval distance is slow and leads to stagnation. To address this drawback, we propose a nonlinear unequal spaced measurement scheme in which the interval space between adjacent measurement planes is gradually increasing, it can significantly increase the diversity of the intensity with a smaller number of required images. Both the simulation and experimental results demonstrate that our method enables quantitative phase and amplitude imaging with a faster speed and better image quality, while also being computationally efficient and robust to noise.

Single-exposure multi-wavelength diffraction imaging with blazed grating

Multi-wavelength diffraction imaging is a lensless, high-resolution imaging technology. To avoid multiple exposures and enable high-speed data collection, here an innovative setup for the single-exposure multi-wavelength diffraction imaging based on a blazed grating is proposed. Since the blazed angle varies with the wavelength, the diffraction patterns for the individual wavelengths can be separated from each other and recorded in a single measurement at one time. A method of high-precision position alignment between different wavelength patterns is proposed in our system to achieve good image quality and high resolution. Experiments on a phase-only USAF resolution target and biological samples were carried out to verify the effectiveness of our proposed method. This proposed setup has such advantages as a simpler structure, fast recording, and algorithm robustness.

Simple phase retrieval method based on two intensity measurements on a single plane

In this work, a simple phase retrieval method is proposed by observing two intensity patterns on a single plane, which are generated with and without a lens. Rigorous theoretical derivations show that the two fields constitute the Fourier transform pairs, and a modified Gerchberg-Saxton algorithm is proposed to recover the phase patterns from the Fourier pairs. The proposed method does not require the intensity patterns to be measured on two different planes along the propagation distance, and this is quite beneficial in a system with a phase tuning element like a spatial light modulator, which can form a virtual lens by creating a parabola-like phase distribution. Experiments are conducted to demonstrate the effectiveness of the proposed phase retrieval method.

Speckle patterns formed by broadband terahertz radiation and their applications for ghost imaging

Speckle patterns can be very promising for many applications due to their unique properties. This paper presents the possibility of numerically and experimentally formation of speckle patterns using broadband THz radiation. Strong dependence of the statistical parameters of speckles, such as size and sharpness on the parameters of the diffuser are demonstrated: the correlation length and the mean square deviation of the phase surface inhomogeneity. As the surface correlation length is increasing, the speckle size also increases and its sharpness goes down. Alternatively, the magnification of the standard deviation of the surface height leads to the speckle size diminishing and growth of the speckle sharpness. The dimensions of the experimentally formed speckles correspond to the results of numerical simulation. The possibility of utilizing formed speckle patterns for the implementation of the ghost imaging technique has been demonstrated by methods of numerical modeling.

Noniterative sub-pixel shifting super-resolution lensless digital holography

Lensless digital holography (LDH) is gaining considerable attention lately due to a simple experimental setup, wide field-of-view, and three-dimensional (3D) imaging capability. Since the resolution of LDH is limited by the Nyquist frequency of a detector array, the major drawback of LDH is resolution, and a lot of efforts were made to enhance the resolution of LDH. Here we propose and demonstrate a fast noniterative sub-pixel shifting super-resolution technique that can effectively enhance the resolution of LDH by a factor of two. We provide detailed frequency-domain formulae for our noniterative frequency-domain super-resolution method. The validity of our proposed method is experimentally demonstrated both for scattering and phase objects.

Polarization-Dependent All-Dielectric Metasurface for Single-Shot Quantitative Phase Imaging

Phase retrieval is a noninterferometric quantitative phase imaging technique that has become an essential tool in optical metrology and label-free microscopy. Phase retrieval techniques require multiple intensity measurements traditionally recorded by camera or sample translation, which limits their applicability mostly to static objects. In this work, we propose the use of a single polarization-dependent all-dielectric metasurface to facilitate the simultaneous recording of two images, which are utilized in phase calculation based on the transport-of-intensity equation. The metasurface acts as a multifunctional device that splits two orthogonal polarization components and adds a propagation phase shift onto one of them. As a proof-of-principle, we demonstrate the technique in the wavefront sensing of technical samples using a standard imaging setup. Our metasurface-based approach fosters a fast and compact configuration that can be integrated into commercial imaging systems.

Complexity-guided Fourier phase retrieval from noisy data

Reconstruction of a stable and good quality solution from noisy single-shot Fourier intensity data is a challenging problem for phase retrieval algorithms. We examine behavior of the solution provided by the hybrid input-output (HIO) algorithm for noisy data, from the perspective of the complexity guidance methodology that was introduced by us in an earlier paper [J. Opt. Soc. Am. A36, 202 (2019)JOAOD60740-323210.1364/JOSAA.36.000202]. We find that for noisy data, the complexity of the solution outside the support keeps increasing as the HIO iterations progress. Based on this observation, a strategy for controlling the solution complexity within and outside the support during the HIO iterations is proposed and tested. In particular, we actively track and control the growth of complexity of the solution outside the support region with iterations. This in turn provides us with guidance regarding the level to which the complexity of the solution within the support region needs to be adjusted, such that the total solution complexity is equal to that estimated from raw Fourier intensity data. In our studies, Poisson noise with mean photon counts per pixel in the Fourier intensity data ranges over four orders of magnitude. We observe that the performance of the proposed strategy is noise robust in the sense that with increasing noise, the quality of the phase solution degrades gradually. For higher noise levels, the solution loses textural details while retaining the main object features. Our numerical experiments show that the proposed strategy can uniformly handle pure phase objects, mixed amplitude-phase objects, and the case of dc blocked Fourier intensity data. The results may find a number of applications where single-shot Fourier phase retrieval is critical to the success of corresponding applications.

Enhancing Multi-Distance Phase Retrieval via Unequal Interval Measurements

In the conventional methods of multi-distance phase retrieval, the diffraction intensity patterns are recorded at equal intervals, which can induce slow convergence or stagnation in the subsequent reconstruction process. To solve this problem, a measurement method with unequal intervals is proposed in this paper. The interval spacings between adjacent measurement planes are decreased gradually. A large gap accelerates retrieval progress, and a short span helps to recover detailed information. The proposed approach makes full use of the available measured dataset and simultaneously generates variations in diversity amplitude, which is a crucial issue for the techniques of multi-image phase retrieval. Both computational simulations and experiments are performed. The results demonstrate that this method can improve the convergence speed by 2 to 3 times and enhance the quality of reconstruction results in comparison to that of the conventional methods.

High-fidelity pixel-super-resolved complex field reconstruction via adaptive smoothing

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.

High-speed coherent diffraction imaging by varying curvature of illumination with a focus tunable lens

A high-speed coherent diffraction imaging method is proposed by varying the curvature of illumination with a focus tunable lens. The imaging setup is free of conventional mechanical translation and takes only milliseconds to refocus by changing the electric signal applied on the lens. It is more compact and also an inexpensive alternative to coherent diffraction imaging with computerized translational stages. A detector that is kept at a fixed distance from the sample records diffraction patterns each time the spherical wavefront illuminations on the sample is changed with a control current. The complex wavefront of the object is then quantitatively recovered from the diffraction intensity measurements using an iterative phase retrieval algorithm. The feasibility of the proposed method is experimentally verified using various samples. Extremely short response time of the focus tunable lens makes the proposed method highly suitable for applications that requires high speed imaging.

Terahertz phase retrieval imaging in reflection

Terahertz phase retrieval is a promising technique able to assess the complex diffracted wave properties through an iterative processing algorithm. In this Letter, we demonstrate the implementation of this technique in reflection geometry with a continuous wave acquisition system working at 0.287 THz. To ensure a high signal-to-noise ratio in the measured dataset, we proposed a double parallel recording scheme with one detector and two lock-in amplifiers operating with the complimentary sensitivity setting. This provided a higher numerical aperture than conventional raster-scanning focal plane imaging. A specialized digital interferometric postprocessing procedure was applied to obtain a surface height map from the reconstructed phase distribution in the object's irradiated area.

High-resolution quantitative phase imaging based on a spatial light modulator and incremental binary random sampling

We propose a single-beam high-resolution quantitative phase imaging method based on a spatial light modulator (SLM) and an incremental binary random sampling (IBRS) algorithm. In this method, the image of the test object presents on the image sensor through an optical microscopy system composed of an objective lens and a collimating lens. A transmittance SLM displaying a group of well-designed IBRS patterns is inserted in the optical microscopy system to modulate the object wavefront. The phase information of the object image can be quantitatively retrieved from the recorded intensities using the IBRS algorithm and the amplitude obtained directly from the diffraction intensity. The IBRS algorithm employed in our method has higher accuracy for phase retrieval compared with our previously proposed complementary random sampling algorithm, which is confirmed by simulations. Further, we demonstrate experimentally the feasibility of our method through several examples: phase imaging of immersion oil droplets with a diffraction-limited lateral resolution of 1.54 µm and a few microbiological specimens with 0.70 µm. Experimental results reveal that our proposed method provides a feasible single-beam technique for quantitative phase imaging with a high spatial resolution.

Holography and Coherent Diffraction Imaging with Low-(30-250 eV) and High-(80-300 keV) Energy Electrons: History, Principles, and Recent Trends

In this paper, we present the theoretical background to electron scattering in an atomic potential and the differences between low- and high-energy electrons interacting with matter. We discuss several interferometric techniques that can be realized with low- and high-energy electrons and which can be applied to the imaging of non-crystalline samples and individual macromolecules, including in-line holography, point projection microscopy, off-axis holography, and coherent diffraction imaging. The advantages of using low- and high-energy electrons for particular experiments are examined, and experimental schemes for holography and coherent diffraction imaging are compared.

Single exposure lensless subpixel phase imaging: optical system design, modelling, and experimental study

Design and optimization of lensless phase-retrieval optical system with phase modulation of free-space propagation wavefront is proposed for subpixel imaging to achieve super-resolution reconstruction. Contrary to the traditional super-resolution phase-retrieval, the method in this paper requires a single observation only and uses the advanced Super-Resolution Sparse Phase Amplitude Retrieval (SR-SPAR) iterative technique which contains optimized sparsity based filters and multi-scale filters. The successful object imaging relies on modulation of the object wavefront with a random phase-mask, which generates coded diffracted intensity pattern, allowing us to extract subpixel information. The system's noise-robustness was investigated and verified. The super-resolution phase-imaging is demonstrated by simulations and physical experiments. The simulations included high quality reconstructions with super-resolution factor of 5, and acceptable at factor up to 9. By physical experiments 3 μm details were resolved, which are 2.3 times smaller than the resolution following from the Nyquist-Shannon sampling theorem.

Accelerated convergence extended ptychographical iterative engine using multiple axial intensity constraints

The extended ptychographical iterative engine (ePIE) is widely applied in the field of ptychographic imaging due to its great flexibility and computational efficiency. A technique of ePIE with multiple axial intensity constraints, which is called MAIC-PIE, is proposed to drastically improve the convergence speed and reduce the calculation time. This technique requires that the diffracted light from the sample is propagated to the multiple individual axial planes, which can be achieved by using the beam splitter and multiple CCDs. In this technique, an additional intensity constraint is involved in the iterative process that makes for building the reasonable guesses of the probe and object in the first few iterations and accelerating the convergence. Simulations and experiments have verified that MAIC-PIE behaves good performance with fast convergence. The great performance and limited computational complexity make it a very attractive and promising technique for ptychographic imaging.

Phase retrieval using axial diffraction patterns and a ptychographic iterative engine

We propose a phase retrieval method using axial diffraction patterns under planar and spherical wave illuminations. The proposed method uses a ptychographic iterative engine (PIE) for the phase retrieval algorithm. The proposed approach uses multiple diffraction patterns. Thus, adjusting the alignment of each diffraction pattern is mandatory, and we propose a method to adjust the alignment. In addition, a random selection of the measured diffraction patterns is used to further accelerate the convergence of the PIE-based optimization. To confirm the effectiveness of the proposed method, we compare the conventional and proposed methods using a simulation and optical experiments.

Enhanced multiple-plane phase retrieval using adaptive support

In the single-plane phase retrieval method, the use of a fixed object support is not efficient and could lead to inaccurate reconstructions. While there have been adaptive support strategies for the single-plane method, numerical processing is slow because such strategies are based in the space domain. Here, an adaptive support strategy based in the Fourier domain in conjunction with the multiple-plane phase retrieval method is presented and demonstrated through simulations and experiments. Optimizations of Fourier filter size and mask threshold parameters result in 3× faster convergence compared to the conventional multiple-plane method for the test object waves used. The proposed strategy offers fast, automated determination of the object support and affords the use of fewer intensity patterns.

THz coherent lensless imaging

Imaging with THz radiation has proved an important tool for both fundamental science and industrial use. Here we review a class of THz imaging implementations, named coherent lensless imaging, that reconstruct the coherent response of arbitrary samples with a minimized experimental setup based only on a coherent source and a camera. After discussing the appropriate sources and detectors to perform them, we detail the fundamental principles and implementations of THz digital holography and phase retrieval. These techniques owe a lot to imaging with different wavelengths, yet innovative concepts are also being developed in the THz range and are ready to be applied in other spectral ranges. This makes our review useful for both the THz and imaging communities, and we hope it will foster their interaction.

Iterative phase retrieval for digital holography: tutorial

This paper provides a tutorial of iterative phase retrieval algorithms based on the Gerchberg-Saxton (GS) algorithm applied in digital holography. In addition, a novel GS-based algorithm that allows reconstruction of 3D samples is demonstrated. The GS-based algorithms recover a complex-valued wavefront using wavefront back-and-forth propagation between two planes with constraints superimposed in these two planes. Iterative phase retrieval allows quantitatively correct and twin-image-free reconstructions of object amplitude and phase distributions from its in-line hologram. The present work derives the quantitative criteria on how many holograms are required to reconstruct a complex-valued object distribution, be it a 2D or 3D sample. It is shown that for a sample that can be approximated as a 2D sample, a single-shot in-line hologram is sufficient to reconstruct the absorption and phase distributions of the sample. Previously, the GS-based algorithms have been successfully employed to reconstruct samples that are limited to a 2D plane. However, realistic physical objects always have some finite thickness and therefore are 3D rather than 2D objects. This study demonstrates that 3D samples, including 3D phase objects, can be reconstructed from two or more holograms. It is shown that in principle, two holograms are sufficient to recover the entire wavefront diffracted by a 3D sample distribution. In this method, the reconstruction is performed by applying iterative phase retrieval between the planes where intensity was measured. The recovered complex-valued wavefront is then propagated back to the sample planes, thus reconstructing the 3D distribution of the sample. This method can be applied for 3D samples such as 3D distribution of particles, thick biological samples, and other 3D phase objects. Examples of reconstructions of 3D objects, including phase objects, are provided. Resolution enhancement obtained by iterative extrapolation of holograms is also discussed.

Fourier single-pixel reconstruction of a complex amplitude optical field

Based on a Fourier single-pixel imaging (SPI) technique and interference between an unknown field and a reference beam, we implement amplitude and phase reconstruction of the unknown complex field. In this Letter, we use a chessboard pattern to divide the unknown field into the signal and reference parts. A high-speed digital micro-mirror device is used to modulate the relative phase between the reference and signal fields, and the SPI method is used to acquire the Fourier spectrum of the signal field. We experimentally reconstruct a 103×103-pixel complex amplitude field with a resolution of 68.4 μm. The single-pixel real-time wavefront detection is also implemented in the rate of four frames per second.

Accelerated single-beam multiple-intensity reconstruction using unordered propagations

In conventional multiple-plane phase retrieval method, the wave propagations that proceed in the same ordered sequence could slow down convergence or lead to stagnation. In this Letter, a novel, to the best of our knowledge, algorithmic technique to accelerate phase retrieval using an unordered sequence of propagations is demonstrated experimentally. The main advantage of the technique is the significant increase in the change in amplitude, a key and essential element for a successful iterative phase retrieval. For N planes, the number of possible ways to implement the sequence of propagations is increased by a factor of N!(N-1)!, providing diversity for the change in the amplitude. Compared to the conventional algorithm, the technique performed 2 times faster and reduced the number of needed intensity patterns for the test object wave used. The technique may be adopted in the other multiple intensity-based phase retrieval methods.

Diffusion-based single-shot diffraction tomography

Holographic microscopy is a powerful technique for noninvasive label-free biomedical imaging. Most holographic methods utilize reference light and/or multiple measurements to observe both the amplitude and phase of a light wave passing through a specimen. However, such fundamental requirements degrade the spatial resolution due to the use of a reference carrier, cause difficulties for real-time imaging of dynamic biological events, and make the optical setups bulky. Here, we realized reference-free, single-shot holographic tomography by just inserting a diffuser into the optical path in a conventional microscope setup to generate randomly structured illumination. A three-dimensional complex amplitude field was reconstructed from a single scattered intensity image by means of sparsity-constrained multislice phase retrieval.

Near-field Fourier ptychography: super-resolution phase retrieval via speckle illumination

High spatial resolution is the goal of many imaging systems. While designing a high-resolution lens with diffraction-limited performance over a large field of view remains a difficult task, creating a complex speckle pattern with wavelength-limited spatial features is easily accomplished with a simple random diffuser. With this observation and the concept of near-field ptychography, we report a new imaging modality, termed near-field Fourier ptychography, to address high-resolution imaging challenges in both microscopic and macroscopic imaging settings. 'Near-field' refers to placing the object at a short defocus distance with a large Fresnel number. We project a speckle pattern with fine spatial features on the object instead of directly resolving the spatial features via a high-resolution lens. We then translate the object (or speckle) to different positions and acquire the corresponding images by using a low-resolution lens. A ptychographic phase retrieval process is used to recover the complex object, the unknown speckle pattern, and the coherent transfer function at the same time. In a microscopic imaging setup, we use a 0.12 numerical aperture (NA) lens to achieve an NA of 0.85 in the reconstruction process. In a macroscale photographic imaging setup, we achieve ~7-fold resolution gain by using a photographic lens. The collection optics do not determine the final achievable resolution; rather, the speckle pattern's feature size does. This is similar to our recent demonstration in fluorescence imaging settings (Guo et al., Biomed. Opt. Express, 9(1), 2018). The reported imaging modality can be employed in light, coherent X-ray, and transmission electron imaging systems to increase resolution and provide quantitative absorption and object phase contrast.

Variable Wavefront Curvature Phase Retrieval Compared to Off-Axis Holography and Its Useful Application to Support Intraoperative Tissue Discrimination

Quantitative phase imaging can reveal morphological features without having to stain the biological sample. This property has important implications for intraoperative applications since the time spent during histopathology can be reduced from a few minutes to a few seconds. However, most common quantitative phase imaging techniques are based on the interferometric principle, which makes them more prone to disturbing environmental influences, such as temperature drift and air turbulence. In the last decade, with the advance of computing power, many different iterative quantitative phase imaging techniques, which only require the recording of the diffracted wavefield, and therefore offer increased robustness towards environmental disturbances, have been proposed. These are particularly well-suited for the application outside the well-controlled lab environment such as an operating theatre. The optical performance of our developed iterative phase retrieval method based on variable wavefront curvature will be evaluated by reference to off-axis digital holography and applied for intraoperative discrimination of tissue.

3D Imaging Based on Depth Measurement Technologies

Three-dimensional (3D) imaging has attracted more and more interest because of its widespread applications, especially in information and life science. These techniques can be broadly divided into two types: ray-based and wavefront-based 3D imaging. Issues such as imaging quality and system complexity of these techniques limit the applications significantly, and therefore many investigations have focused on 3D imaging from depth measurements. This paper presents an overview of 3D imaging from depth measurements, and provides a summary of the connection between the ray-based and wavefront-based 3D imaging techniques.

A fast-converging iterative method based on weighted feedback for multi-distance phase retrieval

Multiple distance phase retrieval methods hold great promise for imaging and measurement due to their less expensive and compact setup. As one of their implementations, the amplitude-phase retrieval algorithm (APR) can achieve stable and high-accuracy reconstruction. However, it suffers from the slow convergence and the stagnant issue. Here we propose an iterative modality named as weighted feedback to solve this problem. With the plug-ins of single and double feedback, two augmented approaches, i.e. the APRSF and APRDF algorithms, are demonstrated to increase the convergence speed with a factor of two and three in experiments. Furthermore, the APRDF algorithm can extend the multiple distance phase retrieval to the partially coherent illumination and enhance the imaging contrast of both amplitude and phase, which actually relaxes the light source requirement. Thus the weighted feedback enables a fast-converging and high-contrast imaging scheme for the iterative phase retrieval.

Wavefront sensing with a thin diffuser

We propose and implement a broadband, compact, and low-cost wavefront sensing scheme by simply placing a thin diffuser in the close vicinity of a camera. The local wavefront gradient is determined from the local translation of the speckle pattern. The translation vector map is computed thanks to a fast diffeomorphic image registration algorithm and integrated to reconstruct the wavefront profile. The simple translation of speckle grains under local wavefront tip/tilt is ensured by the so-called "memory effect" of the diffuser. Quantitative wavefront measurements are experimentally demonstrated, both for the few first Zernike polynomials and for phase-imaging applications requiring high resolution. We finally provided a theoretical description of the resolution limit that is supported experimentally.

Phase retrieval based on coded splitting modulation

A new coded splitting imaging technique is proposed to reconstruct the complex amplitude of a light field iteratively using a single-shot measurement. In this technique, a specially designed coded splitting plate is adopted to diffract the illuminating beam into multiple beams of different orders and code their wavefronts independently and differently. From the diffraction pattern array recorded on the detector plane, both the modulus and phase distributions of the illuminating beam can be reconstructed iteratively using known transmission functions of different orders of the coded splitting plate. The feasibility of the proposed technique is verified both numerically and experimentally.

Coherent laser phase retrieval in the presence of measurement imperfections and incoherent light

Phase retrieval is a powerful numerical method that can be used to determine the wavefront of laser beams based only on intensity measurements, without the use of expensive, low-resolution specialized wavefront sensors such as Shack-Hartmann sensors. However, phase retrieval techniques generally suffer from poor convergence and fidelity when the input measurements contain electronic or optical noise and/or an incoherent intensity contribution overlapped with the otherwise spatially coherent laser beam. Here, we present an implementation of a modified version of the standard multiple-plane Gerchberg-Saxton algorithm and demonstrate that it is highly successful at extracting the intensity profile and wavefront of the spatially coherent part of the light from various lasers, including tapered laser diodes, at a very high fidelity despite the presence of incoherent light and noise.

Axial multi-image phase retrieval under tilt illumination

As a coherent diffractive imaging technique, axial multi-image phase retrieval utilizes a series of diffraction patterns on the basis of axial movement diversity to reconstruct full object wave field. Theoretically, fast convergence and high-accuracy of axial multi-image phase retrieval are demonstrated. In experiment, its retrieval suffers from the tilt illumination, in which diffraction patterns will shift in the lateral direction as the receiver traverses along the axis. In this case, the reconstructed result will be blurry or even mistaken. To solve this problem, we introduce cross-correlation calibration to derive the oblique angle and employ tilt diffraction into axial phase retrieval to recover a target, which is successfully demonstrated in simulation and experiment. Also, our method could provide a useful guidance for measuring how obliquely the incident light illuminates in an optical system.

Iterative phase retrieval based on variable wavefront curvature

An alternative phase retrieval technique is discussed in this paper, which offers some advantages for the obtained resolution and reconstruction procedure. In contrast to commonly applied iterative phase retrieval routines, diffraction patterns with varying distance between the illumination source and the object are recorded. This has the same effect as changing the object sensor distance, albeit offering the advantage of preserving the resolution. Moreover, it is possible to employ the direct Fresnel propagation method without having to worry about different pixel sizes in the reconstruction plane. In addition, the influence of speckle decorrelation has carefully been studied and considered for the experimental implementation.

Wavefront sensing based on a spatial light modulator and incremental binary random sampling

A wavefront sensing method based on a spatial light modulator (SLM) and an incremental binary random sampling (IBRS) algorithm is proposed. In this method, the recording setup is built just by a transmittance SLM and an image sensor. The tested wavefront incident to the SLM plane can be quantitatively retrieved from the diffraction intensities of the wavefront passed through the SLM displaying a IBRS pattern. Because only two modulation states (opaque and transparent) of the SLM are used, the method does not need to know the concrete modulation function of the SLM in advance. In addition by introducing the concept of the incremental random sampling into wavefront sensing, the adaptability of phase retrieving based on the diffraction intensities is significantly improved. To the best of our knowledge, no previous study has used this concept for the same purpose. Some experimental results are given for demonstrating the feasibility of our method.

Spatial-light-modulator-based adaptive optical system for the use of multiple phase retrieval methods

We propose an adaptive optical setup using a spatial light modulator (SLM), which is suitable to perform different phase retrieval methods with varying optical features and without mechanical movement. By this approach, it is possible to test many different phase retrieval methods and their parameters (optical and algorithmic) using one stable setup and without hardware adaption. We show exemplary results for the well-known transport of intensity equation (TIE) method and a new iterative adaptive phase retrieval method, where the object phase is canceled by an inverse phase written into part of the SLM. The measurement results are compared to white light interferometric measurements.

Sparsity-based multi-height phase recovery in holographic microscopy

High-resolution imaging of densely connected samples such as pathology slides using digital in-line holographic microscopy requires the acquisition of several holograms, e.g., at >6–8 different sample-to-sensor distances, to achieve robust phase recovery and coherent imaging of specimen. Reducing the number of these holographic measurements would normally result in reconstruction artifacts and loss of image quality, which would be detrimental especially for biomedical and diagnostics-related applications. Inspired by the fact that most natural images are sparse in some domain, here we introduce a sparsity-based phase reconstruction technique implemented in wavelet domain to achieve at least 2-fold reduction in the number of holographic measurements for coherent imaging of densely connected samples with minimal impact on the reconstructed image quality, quantified using a structural similarity index. We demonstrated the success of this approach by imaging Papanicolaou smears and breast cancer tissue slides over a large field-of-view of ~20 mm² using 2 in-line holograms that are acquired at different sample-to-sensor distances and processed using sparsity-based multi-height phase recovery. This new phase recovery approach that makes use of sparsity can also be extended to other coherent imaging schemes, involving e.g., multiple illumination angles or wavelengths to increase the throughput and speed of coherent imaging.

Application of the speckle-based phase retrieval method in reconstructing two unknown interfering wavefronts

We numerically demonstrate a novel method to simultaneously reconstruct two unknown interfering wavefronts. The speckle-based phase retrieval technique is applied to derive the interference field. The derived interference field along with the phase-shifting concept is used for calculating the interfering wavefronts. Our results show the success of this method even under noisy conditions.

Coherent diffraction imaging by moving a lens

A moveable lens is used for determining amplitude and phase on the object plane. The extended fractional Fourier transform is introduced to address the single lens imaging. We put forward a fast algorithm for the transform by convolution. Combined with parallel iterative phase retrieval algorithm, it is applied to reconstruct the complex amplitude of the object. Compared with inline holography, the implementation of our method is simple and easy. Without the oversampling operation, the computational load is less. Also the proposed method has a superiority of accuracy over the direct focusing measurement for the imaging of small size objects.

Enhanced intensity variation for multiple-plane phase retrieval using a spatial light modulator as a convenient tunable diffuser

In the multiple-plane phase retrieval method, a tedious-to-fabricate phase diffuser plate is used to increase the axial intensity variation for a nonstagnating iterative reconstruction of a smooth object wavefront. Here we show that a spatial light modulator (SLM) can be used as an easily controllable diffuser for phase retrieval. The polarization modulation at the SLM facilitates independent formation of orthogonally polarized scattered and specularly reflected beams. Through an analyzer, the polarization states are filtered enabling beam interference, thereby efficiently encoding the phase information in the axially diverse speckle intensity measurements. The technique is described using wave propagation and Jones calculus, and demonstrated experimentally on technical and biological samples.

Two-dimensional phase unwrapping using the transport of intensity equation

We report a method for two-dimensional phase unwrapping based on the transport of intensity equation (TIE). Given a wrapped phase profile, we generate an auxiliary complex field and propagate it to small distances to simulate two intensity images on closely spaced planes. Using the longitudinal intensity derivative of the auxiliary field as an input, the TIE is solved by employing the regularized Fourier-transform-based approach. The resultant phase profile is automatically in the unwrapped form, as it has been obtained as a solution of a partial differential equation rather than as an argument of a complex-valued function. Our simulations and experimental results suggest that this approach is fast and accurate and provides a simple and practical solution for routine phase unwrapping tasks in interferometry and digital holography.

Propagation phasor approach for holographic image reconstruction

To achieve high-resolution and wide field-of-view, digital holographic imaging techniques need to tackle two major challenges: phase recovery and spatial undersampling. Previously, these challenges were separately addressed using phase retrieval and pixel super-resolution algorithms, which utilize the diversity of different imaging parameters. Although existing holographic imaging methods can achieve large space-bandwidth-products by performing pixel super-resolution and phase retrieval sequentially, they require large amounts of data, which might be a limitation in high-speed or cost-effective imaging applications. Here we report a propagation phasor approach, which for the first time combines phase retrieval and pixel super-resolution into a unified mathematical framework and enables the synthesis of new holographic image reconstruction methods with significantly improved data efficiency. In this approach, twin image and spatial aliasing signals, along with other digital artifacts, are interpreted as noise terms that are modulated by phasors that analytically depend on the lateral displacement between hologram and sensor planes, sample-to-sensor distance, wavelength, and the illumination angle. Compared to previous holographic reconstruction techniques, this new framework results in five- to seven-fold reduced number of raw measurements, while still achieving a competitive resolution and space-bandwidth-product. We also demonstrated the success of this approach by imaging biological specimens including Papanicolaou and blood smears.

Cross-talk compensation of a spatial light modulator for iterative phase retrieval applications

Beam-propagation-based phase recovery approaches, also known as phase retrieval methods, retrieve the amplitude and the phase of arbitrary complex-valued fields. We present and experimentally demonstrate a simple and robust iterative method using a liquid crystal spatial light modulator located at an object diffraction plane. M random phase masks are applied between the object and the image sensor using the modulator, and then M diffraction patterns are collected in the Fourier plane. An iterative algorithm using these patterns and simulating the propagation of the light between the two planes allow us to recover the object wavefront. The use of this type of dynamic modulator makes the experimental setup simpler and more flexible. We need no a priori knowledge about the object field, and the convergence rate is high. Simulation results show that the method exhibits high immunity to noise and does not suffer any stagnation problem. However, experimental results have shown that the technique is sensitive to the cross talk of the modulator. We propose a method for compensating these modulator defects that are validated by experimental results.

Wide-field computational imaging of pathology slides using lens-free on-chip microscopy

Optical examination of microscale features in pathology slides is one of the gold standards to diagnose disease. However, the use of conventional light microscopes is partially limited owing to their relatively high cost, bulkiness of lens-based optics, small field of view (FOV), and requirements for lateral scanning and three-dimensional (3D) focus adjustment. We illustrate the performance of a computational lens-free, holographic on-chip microscope that uses the transport-of-intensity equation, multi-height iterative phase retrieval, and rotational field transformations to perform wide-FOV imaging of pathology samples with comparable image quality to a traditional transmission lens-based microscope. The holographically reconstructed image can be digitally focused at any depth within the object FOV (after image capture) without the need for mechanical focus adjustment and is also digitally corrected for artifacts arising from uncontrolled tilting and height variations between the sample and sensor planes. Using this lens-free on-chip microscope, we successfully imaged invasive carcinoma cells within human breast sections, Papanicolaou smears revealing a high-grade squamous intraepithelial lesion, and sickle cell anemia blood smears over a FOV of 20.5 mm(2). The resulting wide-field lens-free images had sufficient image resolution and contrast for clinical evaluation, as demonstrated by a pathologist's blinded diagnosis of breast cancer tissue samples, achieving an overall accuracy of ~99%. By providing high-resolution images of large-area pathology samples with 3D digital focus adjustment, lens-free on-chip microscopy can be useful in resource-limited and point-of-care settings.

Complex wavefront reconstruction from multiple-image planes produced by a focus tunable lens

We propose, through simulations and experiments, a wavefront reconstruction technique using a focus-tunable lens and a phase-retrieval technique. A collimated beam illuminates a complex object (amplitude and phase), and a diffuser then modulates the outgoing wavefront. Finally the diffracted complex field reaches the focus-tunable lens, and a CMOS camera positioned at a fixed plane registers the subjective speckle distribution produced by the lens (one pattern for each focal length). We have demonstrated that a tunable lens can replace the translation stage used in the conventional single-beam, multiple-intensity reconstruction algorithm. In other words, through iterations with a modified version of this algorithm, the speckle images produced by different focal lengths can be successfully employed to recover the initial complex object. With no movable elements, (speckle) image sampling can be performed at high frame rates, which is suitable for dynamical reconstruction applications.

Synthetic aperture microscopy based on referenceless phase retrieval with an electrically tunable lens

Phase imaging microscopy, based either on holography or nonholographic methods such as phase retrieval, has seen considerable attention recently. Phase retrieval offers the advantage of being free of a reference arm and enables a more stable and compact setup. We present an optical setup that provides enhanced resolution by implementing synthetic aperture imaging based on phase retrieval using an electrically tunable lens (ETL). The ETL is a more compact and less expensive alternative to computerized translation stages and spatial light modulators. Before applying phase retrieval, we discuss a general calibration algorithm, which performs image registration, corrects for magnifications, and determines the axial locations of image planes. Finally, we obtain resolution-enhanced images of a phase grating and of cells to demonstrate the practical application of our technique.

Vibrational phase imaging in wide-field CARS for nonresonant background suppression

Coherent Anti-Stokes Raman Scattering (CARS) microscopy is a valuable tool for label-free imaging of biological samples. As a major drawback quantification based on CARS images is compromised by the appearance of a nonresonant background. In this paper we propose and demonstrate a wide-field CARS vibrational phase imaging scheme that allows for nonresonant background suppression. Several CARS images at a few consecutive planes perpendicular to the propagation direction were recorded to reconstruct a phase map utilizing the iteration phase retrieval method. Experimental results verify that the CARS background is efficiently suppressed by the phase imaging approach, as compared to traditional CARS imaging without background correction. The proposed background correction method is robust against environmental disturbance, since the experimental implementation of the suggested detection scheme requires no reference beam.

Recent advances in digital holography [invited]

This article presents an overview of recent advances in the field of digital holography, ranging from holographic techniques designed to increase the resolution of microscopic images, holographic imaging using incoherent illumination, phase retrieval with incoherent illumination, imaging of occluded objects, and the holographic recording of depth-extended objects using a frequency-comb laser, to the design of an infrastructure for remote laboratories for digital-holographic microscopy and metrology. The paper refers to current trends in digital holography and explains them using new results that were recently achieved at the Institute for Applied Optics of the University Stuttgart.

Geometric super-resolved imaging based upon axial scanning and phase retrieval

In this paper, we propose a new geometric super-resolving approach that overcomes the geometric resolution reduction caused by the spatially large pixels of the detector array. The improvement process is obtained by applying an axial scanning procedure. In the scanning process, several images are captured corresponding to focus applied at several axial planes. By applying an iterative Gerchberg-Saxton-based algorithm, we managed to retrieve the phase and to reconstruct the original high-resolution image from the captured set of low-resolution images. In addition, the paper also presents a numerically efficient algorithm to compute the free space Fresnel integral.