Phase radiography with neutrons

Phase-contrast neutron imaging compared with wave propagation and McStas simulations

Propagation-based phase contrast, for example in the form of edge enhancement contrast, is well established within X-ray imaging but is not widely used in neutron imaging. This technique can help increase the contrast of low-attenuation samples but may confuse quantitative absorption measurements. Therefore, it is important to understand the experimental parameters that cause and amplify or dampen this effect in order to optimize future experiments properly. Two simulation approaches have been investigated, a wave-based simulation and a particle-based simulation conducted in McStas [Willendrup & Lefmann (2020). J. Neutron Res. 22, 1-16], and they are compared with experimental data. The experiment was done on a sample of metal foils with weakly and strongly neutron absorbing layers, which were measured while varying the rotation angle and propagation distance from the sample. The experimental data show multiple signals: attenuation, phase contrast and reflection. The wave model reproduces the sample attenuation and the phase peaks but it does not reproduce the behavior of these peaks as a function of rotation angle. The McStas simulation agrees better with the experimental data, as it reproduces attenuation, phase peaks and reflection, as well as the change in these signals as a function of rotation angle and distance. This suggests that the McStas simulation approach, where the particle description of the neutron facilitates the incorporation of multiple effects, is the most convenient way of modeling edge enhancement in neutron imaging.

Neutron phase imaging by a Talbot-Lau interferometer at Kyoto University Reactor

We have developed a new neutron phase imaging system with a Talbot-Lau interferometer for utilization at the CN-3 port of the Kyoto University Reactor. To achieve efficient differential-phase imaging and visibility (dark-field) imaging at this beamline, we adopted a relatively shorter design wavelength of 2.7 Å. By fabricating neutron absorption gratings with thick gadolinium absorbers, we were able to obtain clear moiré fringes with a high visibility of 55% for thermal neutrons. As a demonstration of its imaging capabilities for expanded actual utilization in the medium-sized sources, we observed additively manufactured rods of Inconel 718. Using visibility imaging, we successfully examined variations in the size of defects in the rods caused by hot isostatic pressing process. In addition, we conducted tomography measurements of the rods, which allowed us to reveal the spatial distribution of defects at sub-micrometer scales.

Polychromatic neutron phase-contrast imaging of weakly absorbing samples enabled by phase retrieval

The use of a phase-retrieval technique for propagation-based phase-contrast neutron imaging with a polychromatic beam is demonstrated. This enables imaging of samples with low absorption contrast and/or improving the signal-to-noise ratio to facilitate e.g. time-resolved measurements. A metal sample, designed to be close to a phase pure object, and a bone sample with canals partially filled with D2O were used for demonstrating the technique. These samples were imaged with a polychromatic neutron beam followed by phase retrieval. For both samples the signal-to-noise ratios were significantly improved and, in the case of the bone sample, the phase retrieval allowed for separation of bone and D2O, which is important for example for in situ flow experiments. The use of deuteration contrast avoids the use of chemical contrast enhancement and makes neutron imaging an interesting complementary method to X-ray imaging of bone.

Physics-informed neural network for phase imaging based on transport of intensity equation

Non-interferometric quantitative phase imaging based on Transport of Intensity Equation (TIE) has been widely used in bio-medical imaging. However, analytic TIE phase retrieval is prone to low-spatial frequency noise amplification, which is caused by the illposedness of inversion at the origin of the spectrum. There are also retrieval ambiguities resulting from the lack of sensitivity to the curl component of the Poynting vector occurring with strong absorption. Here, we establish a physics-informed neural network (PINN) to address these issues, by integrating the forward and inverse physics models into a cascaded deep neural network. We demonstrate that the proposed PINN is efficiently trained using a small set of sample data, enabling the conversion of noise-corrupted 2-shot TIE phase retrievals to high quality phase images under partially coherent LED illumination. The efficacy of the proposed approach is demonstrated by both simulation using a standard image database and experiment using human buccal epitehlial cells. In particular, high image quality (SSIM = 0.919) is achieved experimentally using a reduced size of labeled data (140 image pairs). We discuss the robustness of the proposed approach against insufficient training data, and demonstrate that the parallel architecture of PINN is efficient for transfer learning.

Performance analysis of phase retrieval using transport of intensity with digital holography [Invited]

The performance of direct and unwrapped phase retrieval, which combines digital holography with the transport of intensity, is examined in detail in this paper. In this technique, digital holography is used to numerically reconstruct the intensities at different planes around the image plane, and phase retrieval is achieved by the transport of intensity. Digital holography with transport of intensity is examined for inline and off-axis geometries. The effect of twin images in the inline case is evaluated. Phase-shifting digital holography with transport of intensity is introduced. The performance of digital holography with transport of intensity is compared with traditional off-axis single- and dual-wavelength techniques, which employ standard phase unwrapping algorithms. Simulations and experiments are performed to determine and compare the accuracy of phase retrieval through a mean-squared-error figure of merit as well as the computational speeds of the various methods.

Neutron sub-micrometre tomography from scattering data

Neutrons are valuable probes for various material samples across many areas of research. Neutron imaging typically has a spatial resolution of larger than 20 µm, whereas neutron scattering is sensitive to smaller features but does not provide a real-space image of the sample. A computed-tomography technique is demonstrated that uses neutron-scattering data to generate an image of a periodic sample with a spatial resolution of ∼300 nm. The achieved resolution is over an order of magnitude smaller than the resolution of other forms of neutron tomography. This method consists of measuring neutron diffraction using a double-crystal diffractometer as a function of sample rotation and then using a phase-retrieval algorithm followed by tomographic reconstruction to generate a map of the sample's scattering-length density. Topological features found in the reconstructions are confirmed with scanning electron micrographs. This technique should be applicable to any sample that generates clear neutron-diffraction patterns, including nanofabricated samples, biological membranes and magnetic materials, such as skyrmion lattices.

Recent Progress in X-ray and Neutron Phase Imaging with Gratings

Under the JST-ERATO project in progress to develop X-ray and neutron phase-imaging methods together, recent achievements have been selected and reviewed after describing the merit and the principle of the phase imaging method. For X-ray phase imaging, recent developments of four-dimensional phase tomography and phase microscopy at SPring-8, Japan are mainly presented. For neutron phase imaging, an approach in combination with the time-of-flight method developed at J-PARC, Japan is described with the description of new Gd grating fabrication.

Achromatic Non-Interferometric Single Grating Neutron Dark-Field Imaging

We demonstrate a simple single grating beam modulation technique, which enables the use of a highly intense neutron beam for differential phase and dark-field contrast imaging and thus spatially resolved structural correlation measurements in full analogy to interferometric methods. In contrast to these interferometric approaches our method is intrinsically achromatic and provides unprecedented flexibility in the choice of experimental parameters. In particular the method enables straight forward application of quantitative dark-field contrast imaging in time-of-flight mode at pulsed neutron sources. Utilizing merely a macroscopic absorption mask unparalleled length scales become accessible. We present results of quantitative dark-field contrast imaging combining microstructural small angle scattering analyses with real space imaging for a variety of materials.

X-ray Fokker-Planck equation for paraxial imaging

The Fokker-Planck equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation. This forms a natural generalisation of the transport-of-intensity equation. The x-ray Fokker-Planck equation can simultaneously account for both propagation-based phase contrast, and the diffusive effects of sample-induced small-angle x-ray scattering, when forming an x-ray image of a thin sample. Two derivations are given for the Fokker-Planck equation associated with x-ray imaging, together with a Kramers-Moyal generalisation thereof. Both equations are underpinned by the concept of unresolved speckle due to unresolved sample micro-structure. These equations may be applied to the forward problem of modelling image formation in the presence of both coherent and diffusive energy transport. They may also be used to formulate associated inverse problems of retrieving the phase shifts due to a sample placed in an x-ray beam, together with the diffusive properties of the sample. The domain of applicability for the Fokker-Planck and Kramers-Moyal equations for paraxial imaging is at least as broad as that of the transport-of-intensity equation which they generalise, hence the technique is also expected to be useful for paraxial imaging using visible light, electrons and neutrons.

Full-field neutron microscopy based on refractive optics

Placing a compound refractive lens (CRL) as an objective in a neutron beam generates new possibilities for 2D and 3D nondestructive mapping of the structure, strain and magnetic domains within extended objects. A condenser setup is introduced that allows correction for the lateral chromatic aberration. More generally, for full-field microscopy the loss in performance caused by the chromatic aberration can be more than offset by introducing arrays of CRLs and exploiting the fact that the field of view can be much larger than the physical aperture of the CRL. Comments are made on the manufacture of such devices. The potential use is illustrated by comparisons between state-of-the-art instrumentation and suggested approaches for bright-field microscopy, small-angle neutron scattering microscopy, grain mapping and mapping of stresses. Options are discussed for depth-resolved imaging inspired by confocal light microscopy. Finally, experimental demonstrations are given of some of the basic properties of neutron full-field imaging for a single CRL.

Neutron scattering in the biological sciences: progress and prospects

The scattering of neutrons can be used to provide information on the structure and dynamics of biological systems on multiple length and time scales. Pursuant to a National Science Foundation-funded workshop in February 2018, recent developments in this field are reviewed here, as well as future prospects that can be expected given recent advances in sources, instrumentation and computational power and methods. Crystallography, solution scattering, dynamics, membranes, labeling and imaging are examined. For the extraction of maximum information, the incorporation of judicious specific deuterium labeling, the integration of several types of experiment, and interpretation using high-performance computer simulation models are often found to be particularly powerful.

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.

Continuous cropping of endangered therapeutic plants via electron beam soil-treatment and neutron tomography

Various medicinal plants are threatened with extinction owing to their over-exploitation and the prevalence of soil borne pathogens. In this study, soils infected with root-rot pathogens, which prevent continuous-cropping, were treated with an electron beam. The level of soil-borne fungus was reduced to ≤0.01% by soil electron beam treatment without appreciable effects on the levels of antagonistic microorganism or on the physicochemical properties of the soil. The survival rate of 4-year-old plant was higher in electron beam-treated soil (81.0%) than in fumigated (62.5%), virgin (78%), or untreated-replanting soil (0%). Additionally, under various soils conditions, neutron tomography permitted the monitoring of plant health and the detection of root pathological changes over a period of 4-6 years by quantitatively measuring root water content in situ. These methods allow continual cropping on the same soil without pesticide treatment. This is a major step toward the environmentally friendly production of endangered therapeutic herbs.

Optical convolution for quantitative phase retrieval using the transport of intensity equation

Propagation-based phase imaging using the transport of intensity equation (TIE) allows rapid, deterministic phase retrieval from defocused images. However, computational solutions to the TIE suffer from significant low-frequency noise artifacts and are unique up to the application of boundary conditions on the phase. We demonstrate that quantitative phase can be imaged directly at the detector for a class of pure-phase samples by appropriately patterning the illumination to solve the TIE through an optical convolution with the source. This can reduce noise artifacts, obviates the need for user-supplied boundary conditions and is demonstrated via simulation and experiment.

Source diversity for transport of intensity phase imaging

The transport of intensity equation (TIE) is a phase retrieval method that relies on measurements of the intensity of a paraxial field under propagation between two or more closely spaced planes. A limitation of TIE is its susceptibility to low frequency noise artifacts in the reconstructed phase. Under Köhler illumination, when both illumination power and exposure time are limited, the use of larger sources can improve low-frequency performance although it introduces blurring. Appropriately combining intensity measurements taken with a diversity of source sizes can improve both low- and high-frequency performance in phase reconstruction.

CONRAD-2: the new neutron imaging instrument at the Helmholtz-Zentrum Berlin

The construction of the new neutron imaging instrument at the BER-2 research reactor of the Helmholtz-Zentrum Berlin has greatly increased the potential of the facility. The redesign of the facility included improvements of the neutron extraction and transportation systems, more effective shielding, and innovative instrumentation. The cold neutron flux at the neutron guide exit was increased by more than one order of magnitude, which allowed for an implementation of methods that require monochromatic or polarized beams, thus enabling the exploitation of nonconventional contrast mechanisms such as phase, diffraction and magnetic contrasts. The improved instrument design also facilitates the development of high-resolution neutron tomography by providing an increased beam intensity at the sample position.

Iterative optimum frequency combination method for high efficiency phase imaging of absorptive objects based on phase transfer function

In this work, an optimum frequency combination (OFC) method is proposed to reconstruct high quality phase information of the complex light field, which is really valuable for many objects such as optical elements and cells. It is shown that the difference image between two symmetrical separated, larger defocused planes contains a lot of lower frequency components of the phase distribution and the higher frequency components can be easily observed in the difference image between two nearly focused planes. Based on the phase transfer function (PTF), our method combines different frequency components with high Signal-to-Noise Ratio (SNR) together to estimate a more accurate frequency spectrum of the object's phase distribution without any complicated linear or nonlinear regression. Then, we can directly reconstruct a high-quality phase map through inverse Fourier transform. What's more, in order to compensate the phase discrepancy resulted from strong absorption in the intensity, an iterative compensation algorithm is proposed. Both the simulation and experimental results demonstrate that our iterative OFC (IOFC) method can give a computationally efficient and noise-robust phase reconstruction for absorptive phase objects with higher accuracy and fewer defocus planes.

Experimental evaluation of the 'transport-of-intensity' equation for magnetic phase reconstruction in Lorentz transmission electron microscopy

The 'transport-of-intensity' equation (TIE) is a general phase reconstruction methodology that can be applied to Lorentz transmission electron microscopy (TEM) through the use of Fresnel-contrast (defocused) images. We present an experimental study to test the application of the TIE for quantitative magnetic mapping in Lorentz TEM without aberration correction by examining sub-micrometer sized Ni80Fe20 (Permalloy) elements. For a JEOL JEM 2100F adapted for Lorentz microscopy, we find that quantitative magnetic phase reconstructions are possible for defoci distances ranging between approximately 200 μm and 800 μm. The lower limit originates from competing sources of image intensity variations in Fresnel-contrast images, namely structural defects and diffraction contrast. The upper defocus limit is due to a numerical error in the estimation of the intensity derivative based on three images. For magnetic domains, we show quantitative reconstructions of the product of the magnetic induction vector and thickness in element sizes down to approximately 100 nm in lateral size and 5 nm thick resulting in a minimal detection of 5Tnm. Three types of magnetic structures are tested in terms of phase reconstruction: vortex cores, domain walls, and element edges. We quantify vortex core structures at a diameter of 12 nm while the structures of domain walls and element edges are characterized qualitatively. Finally, we show by image simulations that the conclusions of this experimental study are relevant to other Lorentz TEM in which spherical aberration and defocus are dominant aberrations.

Physical unsharp mask with structured detection

We present a method to implement physically an unsharp mask filter in an imaging system. The idea is based on the use of a spatially periodic variation in the detection efficiency of an area detector. Such a "structured detection" introduces harmonic peaks in the Fourier spectrum associated with the image, enabling the use of higher spatial frequencies that would otherwise be inaccessible, due to the system point spread function. The result is an effective deblurring of the image implemented in hardware, i.e., before the detection noise occurs. The method is first demonstrated via a numerical simulation and then validated using experimental neutron imaging data.

Transport of intensity phase retrieval of arbitrary wave fields including vortices

The phase problem can be considered as one of the cornerstones of quantum mechanics intimately connected to the detection process and the uncertainty relation. The latter impose fundamental limits on the manifold phase reconstruction schemes invented to date, in particular, at small magnitudes of the quantum wave. Here, we show that a rigorous solution of the transport of intensity reconstruction (TIE) scheme in terms of a linear elliptic partial differential equation for the phase provides reconstructions even in the presence of wave zeros if particular boundary conditions are given. We furthermore discuss how partial coherence hampers phase reconstruction and show that a modified version of the TIE reconstructs the curl-free current density at arbitrary (in)coherence. Our results open the way for TIE-based phase retrieval of arbitrary wave fields, eventually containing zeros such as phase vortices.

The transport of intensity equation for optical path length recovery using partially coherent illumination

We investigate the measurement of a thin sample's optical thickness using the transport of intensity equation (TIE) and demonstrate a version of the TIE, valid for partially coherent illumination, that allows the measurement of a sample's optical path length by the removal of illumination effects.

Phase contrast synchrotron microtomography: improving noninvasive investigations of fossil embryos in ovo

Fossil embryos are paramount for our understanding of the development of extinct species. However, although thousands of fossil amniote eggs are known, very few embryos in ovo have been described. First reports of fossil embryos were based on broken eggs, where the embryonic remains were already exposed, because destructive methods on complete eggs were avoided. Investigations of complete eggs therefore required nondestructive approaches, such as X-ray microtomography (μCT). However, due to the general low density contrast between fossilized bones and infilling matrix, only a few specimens have been reported using these techniques. Using propagation phase contrast X-ray synchrotron microtomography (PPC-SR-μCT), we report here the discovery of three well-preserved embryos in Early Cretaceous eggs from Thailand. By scanning these eggs using different imaging techniques, we show that vastly different interpretations can be made regarding the preservation state and/or the developmental stage of these embryos. PPC-SR-μCT also revealed differential contrast between bone categories, presumably reflecting the ossification pattern of these embryos. Applying such an approach to large-scale studies of fossil eggs could lead to more discoveries and detailed studies of fossil embryos, providing important developmental and phylogenetic information on extinct and extant amniotes.

Neutron phase contrast imaging beamline at CIRUS, reactor, India

This paper presents the development of neutron phase contrast imaging facility at medium flux research reactor, CIRUS, India. The approach adopted for this study is innovative in the sense that both conventional and phase contrast imaging can be performed within same experiment hutch without any major modification in the experimental hutch or collimator.

Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes

A method based on the transport of intensity equation (TIE) for phase retrieval is presented, which can retrieve the optical phase from intensity measurements in multiple unequally-spaced planes in the near-field region. In this method, the intensity derivative in the TIE is represented by a linear combination of intensity measurements, and the coefficient of the combination can be expressed by explicitly analytical form related to the defocused distances. The proposed formula is a generalization of the TIE with high order intensity derivatives. The numerical experiments demonstrate that the proposed method can improve the accuracy of phase retrieval with higher-order intensity derivatives and is more convenient for practical application.

Optimisation of phase imaging geometry

Certain phase retrieval methods use knowledge about the free space propagation of a wave to phase a paraxial beam passing through one or more measurement planes. This approach has been widely applied and has been shown to quantitatively retrieve the refractive index profile of a sample. The quality of the phase retrieval will depend on a range of factors including sample feature size, propagation distance, measurement plane separation, wavelength and noise. Here we describe an optimisation study for two-plane phase retrieval using a laboratory-based X-ray source that considers all of these factors. We discuss our results in the context of a three-dimensional reconstruction of a sample refractive index profile.

Neutron dark-field tomography

We report how a grating interferometer yields neutron dark-field scatter images for tomographic investigations. The image contrast is based on ultrasmall-angle scattering. It provides otherwise inaccessible spatially resolved information about the distribution of micrometer and submicrometer sized structural formations. Three complementary sets of tomographic data corresponding to attenuation, differential phase, and small-angle scattering can be obtained from one measurement. The method is compatible with conventional imaging and provides significantly higher efficiency than existing techniques.

Design, fabrication, and characterization of diffraction gratings for neutron phase contrast imaging

We have developed a neutron phase contrast imaging method based on a grating interferometer setup. The principal constituents are two absorption gratings made of gadolinium and a phase modulating grating made of silicon. The design parameters of the setup, such as periodicity, structure heights of the gratings, and the distances between the gratings, are calculated. The fabrication of each grating is described in detail. The produced diffraction gratings were finally characterized within the setup, by locally evaluating the produced contrast (visibility) in each detector pixel, resulting in a visibility map over the whole grating size. An averaged value of 23% is achieved.

X-ray noninterferometric phase imaging: a unified picture

A unified theory of noninterferometric phase recovery based on the so-called ambiguity function is introduced. The theory is used to analyze previously published techniques and unify them with the methods of phase-space tomography applicable to partially coherent data. The theory is then used to propose some new approaches to noninterferometric phase recovery.

Neutron phase imaging and tomography

We report how a setup consisting of three gratings yields quantitative two- and three-dimensional images depicting the quantum-mechanical phase shifts of neutron de Broglie wave packets induced by the influence of macroscopic objects. Since our approach requires only a little spatial and chromatic coherence it provides a more than 2 orders of magnitude higher efficiency than existing techniques. This dramatically reduces the required measurement time for computed phase tomography and opens up the way for three-dimensional investigations of previously inaccessible quantum-mechanical phase interactions of neutrons with matter.

Modern Neutron Imaging: Radiography, Tomography, Dynamic and Phase Contrast Imaging with Neutrons

This paper gives a review about the current state of the art in neutron imaging like neutron radiography, neutron tomography, stroboscopic imaging and phase contrast imaging. The different techniques are described and compared to X-rays.

Error analysis and correction in wavefront reconstruction from the transport-of-intensity equation

Wavefront reconstruction from the transport-of-intensity equation (TIE) is a well-posed inverse problem given smooth signals and appropriate boundary conditions. However, in practice experimental errors lead to an ill-condition problem. A quantitative analysis of the effects of experimental errors is presented in simulations and experimental tests. The relative importance of numerical, misalignment, quantization, and photodetection errors are shown. It is proved that reduction of photodetection noise by wavelet filtering significantly improves the accuracy of wavefront reconstruction from simulated and experimental data.

Advances in neutron radiographic techniques and applications: a method for nondestructive testing

A brief history of neutron radiography is presented to set the stage for a discussion of significant neutron radiographic developments and an assessment of future directions for neutron radiography. Specific advances are seen in the use of modern, high dynamic range imaging methods (image plates and flat panels) and for high contrast techniques such as phase contrast, and phase-sensitive imaging. Competition for neutron radiographic inspection may develop as these techniques offer application prospects for X-ray methods.

Thermal and cold neutron phase-contrast radiography

In this paper, we will discuss a phase-contrast imaging method that avoids the complications of interferometry to provide phase contrast in weakly absorbing samples. A transversely coherent neutron beam is used with the traditional radiography scheme. Images taken with this scheme show dramatic intensity variations due to sharp changes in the neutron wave refractive index. With some numerical processing these images may be used to reconstruct a quantitative phase radiograph of specimens imaged with this technique.

Regular and alternant spiral waves of contractile motion on rat ventricle cell cultures

We demonstrate that meandering as well as regular spiral waves can form in a well-controlled culture layer of rat ventricle cells and that the meandering spiral wave, in particular, can generate an alternant rhythm. These observations are made possible by a newly developed, noninvasive phase contrast macro-optics that is simple but highly effective in visualizing the contractile motion of the populations of cardiac cells.

Resolution extension and exit wave reconstruction in complex HREM

Direct methods in real and reciprocal space are developed for structural reversion. The direct method in real space involves the use of a novel method to retrieve the phase in the image plane using transport of intensity equation/maximum entropy method (TIE/MEM) and exit wave reconstruction by self-consistent propagation. Since the exit wave is restored from the complex signal in the image planes, no image model between the exit wave and image is assumed. The structural information in the reconstructed exit wave is then further extended by a "complex" maximum entropy method as a direct method in reciprocal space to extrapolate the phase to higher frequencies.

Quantitative phase radiography with polychromatic neutrons

We develop and experimentally demonstrate a formalism that allows accurate phase imaging using neutron sources producing highly polychromatic beams. The results of measurements from a rectangular block of silicon compare favorably with theoretical simulations based upon the known composition and geometry of the block. The increased flux and reduced exposure times will permit a simple extension of the technique to tomographic phase imaging.

Phase imaging and nanoscale currents in phase objects imaged with fast electrons

We derive the magnetic transport-of-intensity equation (MTIE) that links defocused contrast of magnetic nanoobjects imaged by partially coherent electron waves to their micromagnetic parameters. This provides Maxwell's explanation for observable contrast in terms of the Z component of currents existing around vortices in superconductors and domain vortices and walls in ferromagnets. The solution of the MTIE via Fourier transform is used for quantitative mapping of magnetic flux and projected induction in magnetic and superconducting materials imaged by Lorentz microscopy.

Noniterative two-dimensional phase-retrieval method from two Fourier intensities by use of an exponential filter

A noniterative method of retrieving the two dimensional phase of a wave field from two intensity measurements is proposed. In the measurements, one records two far-field intensities of the wave field modulated and unmodulated with an exponential filter. The phase retrieval method is based on the solution of the simultaneous equations with unknown coefficients of the two-dimensional discrete Fourier transform for the phase. Then there is no need for the information about the wave field, which is used in iterative phase-retrieval methods. The usefulness of this method is shown in computer-simulated examples of the reconstruction of two-dimensional complex amplitude objects.

Phase retrieval from experimental far-field intensities by use of a Gaussian beam

The noniterative phase-retrieval method by use of Gaussian filtering is applied to the reconstruction of phase objects from experimental far-field intensities. In this method, the complex amplitude of transmitted light through an object is reconstructed from three far-field intensities, which are measured with the modulation of the object by laterally shifted and unshifted Gaussian filters. In the experiment, the amplitude of a Gaussian beam illuminating objects is utilized as a Gaussian filter, and, as the phase objects, a converging lens with a small exit pupil and a plastic fiber immersed in optical adhesive are used. The experimental results show that the Gaussian beam of a laser is capable of retrieving the phases of those objects with the accuracy of the range from approximately 1/10 to 1/4 of the laser's wavelength.

Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM

Although phase contrast microscopy is widespread in optical microscopy, it has not been as widely adopted in transmission electron microscopy (TEM), which has therefore to a large extent relied on staining techniques to yield sufficient contrast. Those methods of phase contrast that are used in biological electron microscopy have been limited by factors such as the need for small phase shifts in very thin samples, the requirement for difficult experimental conditions, or the use of complex data analysis methods. We here demonstrate a simple method for quantitative TEM phase microscopy that is suitable for large phase shifts and requires only two images. We present a TEM phase image of unstained Radula sp. (liverwort spore). We show how the image may be transformed into the differential interference contrast image format familiar from optical microscopy. The phase images contain features not visible with the other imaging modalities. The resulting technique should permit phase contrast TEM to be performed almost as readily as phase contrast optical microscopy.

Theory of singular-phase reconstruction for an optical speckle field in the turbulent atmosphere

Analytical expressions are derived and computational algorithms are constructed for retrieving optical-field phase distribution under strong scintillation. The input data for the phase reconstruction are the wave-front slopes registered by a Hartmann sensor or shearing interferometer. The theory is based on representing the slope-vector field as the sum of its potential and solenoid components; it introduces the concept of phase-source and phase-vortex density and uses strict integral expressions relating these quantities to the wave-front slopes. To overcome the difficulties arising from the singular character of phase distribution, use is made of regularization of the wave-front slopes. The slopes can be measured with an ideal point wave-front sensor. It is shown that the slopes measured at the output of a nonideal sensor can be treated as regularized values of these slopes. Numerical simulation of phase unwrapping from the reference values of the wave-front slopes has shown that the algorithm designed for visualization of local phase singularities and those for phase reconstruction are very helpful in eliminating the measurement noise.

Computational aberration correction for an arbitrary linear imaging system

We show that aberration corrections can be made in any arbitrary linear imaging system provided the aberrations are well characterized and at least one of these aberrations can be independently varied in a well-controlled manner. We derive a generalization of the Schrödinger equation for wave propagation in aberration space assuming forward scattering. Transport equations in aberration space are derived. A general iterative algorithm which can retrieve the phase, and is robust in the presence of noise, is also derived. This is demonstrated using simulated data pertinent to electron microscopy, from a series of images with differing spherical aberration.