Using coherent X-ray ptychography to probe medium-range order

Characterization of microscopic structural order and in particular medium range order (MRO) in amorphous materials is challenging. A new technique is demonstrated that allows analysis of MRO using X-rays. Diffraction data were collected from a sample consisting of densely packed polystyrene-latex micro-spheres. Ptychography is used to reconstruct the sample transmission function and fluctuation microscopy applied to characterize structural order producing a detailed `fluctuation map' allowing analysis of the sample at two distinct length scales. Independent verification is provided via X-ray diffractometry. Simulations of dense random packing of spheres have also been used to explore the origin of the structural order measured.

Simultaneous X-ray fluorescence and ptychographic microscopy of Cyclotella meneghiniana

Scanning X-ray fluorescence microscopy (XFM) is a particularly useful method for studying the spatial distribution of trace metals in biological samples. Here we demonstrate the utility of combining coherent diffractive imaging (CDI) with XFM for imaging biological samples to simultaneously produce high-resolution and high-contrast transmission images and quantitative elemental maps. The reconstructed transmission function yields morphological details which contextualise the elemental maps. We report enhancement of the spatial resolution in both the transmission and fluorescence images beyond that of the X-ray optics. The freshwater diatom Cyclotella meneghiniana was imaged to demonstrate the benefits of combining these techniques that have complementary contrast mechanisms.

The soft x-ray instrument for materials studies at the linac coherent light source x-ray free-electron laser

The soft x-ray materials science instrument is the second operational beamline at the linac coherent light source x-ray free electron laser. The instrument operates with a photon energy range of 480-2000 eV and features a grating monochromator as well as bendable refocusing mirrors. A broad range of experimental stations may be installed to study diverse scientific topics such as: ultrafast chemistry, surface science, highly correlated electron systems, matter under extreme conditions, and laboratory astrophysics. Preliminary commissioning results are presented including the first soft x-ray single-shot energy spectrum from a free electron laser.

An in-vacuum x-ray diffraction microscope for use in the 0.7-2.9 keV range

A dedicated in-vacuum coherent x-ray diffraction microscope was installed at the 2-ID-B beamline of the Advanced Photon Source for use with 0.7-2.9 keV x-rays. The instrument can accommodate three common implementations of diffractive imaging; plane wave illumination; defocused-probe (Fresnel diffractive imaging) and scanning (ptychography) using either a pinhole, focused or defocused probe. The microscope design includes active feedback to limit motion of the optics with respect to the sample. Upper bounds on the relative optics-to-sample displacement have been measured to be 5.8 nm(v) and 4.4 nm(h) rms/h using capacitance micrometry and 27 nm/h using x-ray point projection imaging. The stability of the measurement platform and in-vacuum operation allows for long exposure times, high signal-to-noise and large dynamic range two-dimensional intensity measurements to be acquired. Finally, we illustrate the microscope's stability with a recent experimental result.

Coherence properties of individual femtosecond pulses of an x-ray free-electron laser

Measurements of the spatial and temporal coherence of single, femtosecond x-ray pulses generated by the first hard x-ray free-electron laser, the Linac Coherent Light Source, are presented. Single-shot measurements were performed at 780 eV x-ray photon energy using apertures containing double pinholes in "diffract-and-destroy" mode. We determined a coherence length of 17  μm in the vertical direction, which is approximately the size of the focused Linac Coherent Light Source beam in the same direction. The analysis of the diffraction patterns produced by the pinholes with the largest separation yields an estimate of the temporal coherence time of 0.55 fs. We find that the total degree of transverse coherence is 56% and that the x-ray pulses are adequately described by two transverse coherent modes in each direction. This leads us to the conclusion that 78% of the total power is contained in the dominant mode.

A new approach for structure analysis of two-dimensional membrane protein crystals using X-ray powder diffraction data

The application of powder diffraction methods to problems in structural biology is generally regarded as intractable because of the large number of unresolved, overlapping X-ray reflections. Here, we use information about unit cell lattice parameters, space group transformations, and chemical composition as a priori information in a bootstrap process that resolves the ambiguities associated with overlapping reflections. The measured ratios of reflections that can be resolved experimentally are used to refine the position, the shape, and the orientation of low-resolution molecular structures within the unit cell, in leading to the resolution of the overlapping reflections. The molecular model is then made progressively more sophisticated as additional diffraction information is included in the analysis. We apply our method to the recovery of the structure of the bacteriorhodopsin molecule (bR) to a resolution of 7 Å using experimental data obtained from two-dimensional purple membrane crystals. The approach can be used to determine the structure factors directly or to provide reliable low-resolution phase information that can be refined further by the conventional methods of protein crystallography.

Fresnel coherent diffraction tomography

Tomographic coherent imaging requires the reconstruction of a series of two-dimensional projections of the object. We show that using the solution for the image of one projection as the starting point for the reconstruction of the next projection offers a reliable and rapid approach to the image reconstruction. The method is demonstrated on simulated and experimental data. This technique also simplifies reconstructions using data with curved incident wavefronts.

Use of a complex constraint in coherent diffractive imaging

We demonstrate use of a complex constraint based on the interaction of x-rays with matter for reconstructing images from coherent X-ray diffraction. We show the complementary information provided by the phase and magnitude of the reconstructed wavefield greatly improves the quality of the resulting estimate of the transmission function of an object without the need for a priori information about the object composition.

Diffractive imaging using partially coherent x rays

The measured spatial coherence characteristics of the illumination used in a diffractive imaging experiment are incorporated in an algorithm that reconstructs the complex transmission function of an object from experimental x-ray diffraction data using 1.4 keV x rays. Conventional coherent diffractive imaging, which assumes full spatial coherence, is a limiting case of our approach. Even in cases in which the deviation from full spatial coherence is small, we demonstrate a significant improvement in the quality of wave field reconstructions. Our formulation is applicable to x-ray and electron diffraction imaging techniques provided that the spatial coherence properties of the illumination are known or can be measured.

Noninterferometric two-dimensional fourier-transform spectroscopy of multilevel systems

We demonstrate a technique that determines the phase of the photon-echo emission from spectrally resolved intensity data without requiring phase-stabilized input pulses. The full complex polarization of the emission is determined from spectral intensity measurements. The validity of this technique is demonstrated using simulated data, and is then applied to the analysis of two-color data obtained from the light-harvesting molecule lycopene.

Non-iterative solution of the phase retrieval problem using a single diffraction measurement

Coherent diffractive imaging is a method by which iterative methods are employed to recover image information about a finite object from its coherent diffraction pattern. We employ methods borrowed from density functional theory to show that an image can be recovered in a single non-iterative step for a finite sample subject to phase-curved illumination. The result also yields a new approach to quantitative x-ray phase-contrast imaging.

Quantitative phase measurement in coherent diffraction imaging

We demonstrate high spatial resolution phase retrieval of a non-periodic gold nano-structure using the method of Fresnel coherent diffractive imaging. The result is quantitative to better than 10% and does not rely on any a priori knowledge of the sample.

Experimental measurement of the four-dimensional coherence function for an undulator x-ray source

A full measurement of the four-dimensional coherence function from an undulator beam line is reported. The analysis is based on the observation that the data are consistent with a coherence function that is mathematically separable. The effective source size can be altered by changing the width of the exit slit, and the complete coherence function is presented for two settings. We find, to within experimental error, that the four-dimensional complex degree of coherence can be described as a real Gaussian function that depends only on the difference of the spatial coordinates.

Recovering the complete coherence function of a generalized Schell model field

We prove that in the absence of phase singularities, a generalized Schell model partially coherent field is fully defined by its intensity in three planes. We discuss the implications of this result for the problem of characterizing wave fields.

Fresnel coherent diffractive imaging

We present an x-ray coherent diffractive imaging experiment utilizing a nonplanar incident wave and demonstrate success by reconstructing a nonperiodic gold sample at 24 nm resolution. Favorable effects of the curved beam illumination are identified.

Iterative image reconstruction algorithms using wave-front intensity and phase variation

Iterative algorithms that reconstruct images from far-field x-ray diffraction data are plagued with convergence difficulties. An iterative image reconstruction algorithm is described that ameliorates these convergence difficulties through the use of diffraction data obtained with illumination modulated in both intensity and phase.

Refractive index profiling of axially symmetric optical fibers: a new technique

We present a new technique for determining the refractive index profiles of axially symmetric optical fibers based on imaging phase gradients introduced into a transmitted optical field by a fiber sample. An image of the phase gradients within the field is obtained using a new non-interferometric technique based on bright field microscopy. This provides sufficient information to reconstruct the refractive index profile using the inverse Abel transform. The technique is robust, rapid and possesses high spatial resolution and we demonstrate its application to the reconstruction of the refractive index profiles of a single-mode and a multimode optical fiber.

Astigmatic electron diffraction imaging: a novel mode for structure determination

In a conventional transmission electron microscope, stigmators are used to correct for the effects of axial astigmatism in the diffraction lens. It seems feasible that these same stigmators could also be used to form a series of 'astigmatic' diffraction patterns. It is shown how this series of diffraction patterns could then be used to perform exit-surface wavefunction reconstruction. This has the advantage that the diffraction patterns are not resolution limited by the objective aperture as are images when performing exit-surface wavefunction reconstruction from a focal series. A scheme for carrying out phase reconstruction from a series of astigmatic diffraction patterns in an electron microscope is presented.

Diffraction with wavefront curvature: a path to unique phase recovery

Modern X-ray optics is able to produce very tightly focused beams. The size of these focused spots is comparable to the scale of large molecules and therefore to the lattice spacing of crystals of these molecules. In this case, the phase of the illuminating beam may vary on the scale of the lattice and conventional diffraction theory needs to be modified. In this paper, coherent diffraction by non-planar beams is considered and it is shown that it is possible to uniquely recover the phase of the diffraction pattern.

Synchrotron beam coherence: a spatially resolved measurement

We report a precise and spatially resolved measurement of the complex degree of coherence of a one-dimensional 1.5-keV beam produced by a third-generation synchrotron source. The method of phase-space tomography is used, which requires only measurements of the x-ray intensity. We find that the field is statistically stationary to within experimental error, the correlations are very well approximated by a Gaussian distribution, and the measured coherence length is in excellent agreement with expectations.

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.

Quantitative phase amplitude microscopy IV: imaging thick specimens

The ability to image phase distributions with high spatial resolution is a key capability of microscopy systems. Consequently, the development and use of phase microscopy has been an important aspect of microscopy research and development. Most phase microscopy is based on a form of interference. Some phase imaging techniques, such as differential interference microscopy or phase microscopy, have a low coherence requirement, which enables high-resolution imaging but in effect prevents the acquisition of quantitative phase information. These techniques are therefore used mainly for phase visualization. On the other hand, interference microscopy and holography are able to yield quantitative phase measurements but cannot offer the highest resolution. A new approach to phase microscopy, quantitative phase-amplitude microscopy (QPAM) has recently been proposed that relies on observing the manner in which intensity images change with small defocuses and using these intensity changes to recover the phase. The method is easily understood when an object is thin, meaning its thickness is much less than the depth of field of the imaging system. However, in practice, objects will not often be thin, leading to the question of what precisely is being measured when QPAM is applied to a thick object. The optical transfer function formalism previously developed uses three-dimensional (3D) optical transfer functions under the Born approximation. In this paper we use the 3D optical transfer function approach of Streibl not for the analysis of 3D imaging methods, such as tomography, but rather for the problem of analysing 2D phase images of thick objects. We go on to test the theoretical predictions experimentally. The two are found to be in excellent agreement and we show that the 3D imaging properties of QPAM can be reliably predicted using the optical transfer function formalism.

Quantitative phase-amplitude microscopy. III. The effects of noise

We explore the effect of noise on images obtained using quantitative phase-amplitude microscopy - a new microscopy technique based on the determination of phase from the intensity evolution of propagating radiation. We compare the predictions with experimental results and also propose an approach that allows good-quality quantitative phase retrieval to be obtained even for very noisy data.

Unique phase recovery for nonperiodic objects

It is well known that the loss of phase information at detection means that a diffraction pattern may be consistent with a multitude of physically different structures. This Letter shows that it is possible to perform unique structural determination in the absence of a priori information using x-ray fields with phase curvature. We argue that significant phase curvature is already available using modern x-ray optics and we demonstrate an algorithm that allows the phase to be recovered uniquely and reliably.

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.

Measurement of the spatial coherence function of undulator radiation using a phase mask

A measurement of the horizontal coherence function of 7.9 keV radiation from an undulator beam line at the Advanced Photon Source is reported. X-ray diffraction from a phase-shifting mask was used, and the coherence function was measured as a function of the width of beam-conditioning slits in the beam line. The coherence distribution is found to be best described by a Lorentzian function.

Refractive-index profiling of optical fibers with axial symmetry by use of quantitative phase microscopy

The application of quantitative phase microscopy to refractive-index profiling of optical fibers is demonstrated. Phase images of axially symmetric optical fibers immersed in index-matching fluid are obtained, and the inverse Abel transform is used to obtain the radial refractive-index profile. This technique is straightforward, nondestructive, repeatable, and accurate. Excellent agreement, to within approximately 0.0005, between this method and the index profile obtained with a commercial profiler is obtained.

Precision measurement of the electromagnetic fields in the focal region of a high-numerical-aperture lens using a tapered fiber probe

We present a measurement of the intensity around the focus of a N.A.-0.95 lens using a tapered optical fiber probe. An asymmetry introduced by the vector nature of the incident polarized light is evident, although it is inconsistent with that predicted theoretically by considering the magnitude squared of the electric field. The sensitivity of the probe to different components of the electromagnetic field is considered, and it is shown that the measurement is consistent with vector diffraction theory when the probe properties are taken into account.

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.

Quantitative phase-amplitude microscopy I: optical microscopy

In this paper, the application of a new optical microscopy method (quantitative phase-amplitude microscopy) to biological imaging is explored, and the issue of resolution and image quality is examined. The paper begins by presenting a theoretical analysis of the method using the optical transfer function formalism of Streibl (1985). The effect of coherence on the formation of the phase image is explored, and it is shown that the resolution of the method is not compromised over that of a conventional bright-field image. It is shown that the signal-to-noise ratio of the phase recovery, however, does depend on the degree of coherence in the illumination. Streibl (1985) notes that partially coherent image formation is a non-linear process because of the intermingling of amplitude and phase information. The work presented here shows that the quantitative phase-amplitude microscopy method acts to linearize the image formation process, and that the phase and amplitude information is properly described using a transfer function analysis. The theoretical conclusions are tested experimentally using an optical microscope and the theoretical deductions are confirmed. Samples for microscopy influence both the phase and amplitude of the light wave and it is demonstrated that the new phase recovery method can separate the amplitude and phase information, something not possible using traditional phase microscopy. In the case of a coherent wave, knowledge of the phase and amplitude constitutes complete information that can be used to emulate other forms of microscopy. This capacity is demonstrated by recovering the phase of a sample and using the data to emulate a differential interference contrast image.

Phase measurement of waves that obey nonlinear equations

We consider the problem of phase retrieval for classical and quantum wave fields that obey a wide class of nonlinear wave equations. This problem is tackled by means of a suitable generalization of existing methods based on the linear transport-of-intensity equation. The extension of these ideas to systems of coupled nonlinear wave equations is also given.

High-resolution phase imaging of phase singularities in the focal region of a lens

Subwavelength-resolution phase images of phase dislocations at the focal region of a 20x , 0.4-N.A. lens have been obtained by use of an optical fiber interferometer with a tapered probe in one arm. A phase-stepping algorithm is used to determine a quantitative value of the phase at each point in the scan, clearly showing the presence of edge dislocations between the Airy rings of the diffraction pattern near the lens focus, as well as four isolated screw-type singularties caused by astigmatism in the lens.

Phase retrieval from images in the presence of first-order vortices

We discuss retrieval of the phase of quantum-mechanical and classical wave fields in the presence of first-order vortices. A practical method of phase retrieval is demonstrated which is robust in the presence of noise. Conditions for the uniqueness of the retrieved phase are discussed and we show that determination of the phase in a given plane requires a series of at least three two-dimensional intensity images at different propagation distances. The method is applicable to a wide range of scenarios such as the imaging of imperfect crystals, quantitative determination of the strength of vortex filaments in high-temperature superconductors, and x-ray and electron holography.

Noninterferometric quantitative phase imaging with soft x rays

We demonstrate quantitative noninterferometric x-ray phase-amplitude measurement. We present results from two experimental geometries. The first geometry uses x rays diverging from a point source to produce high-resolution holograms of submicrometer-sized objects. The measured phase of the projected image agrees with the geometrically determined phase to within +/-7%. The second geometry uses a direct imaging microscope setup that allows the formation of a magnified image with a zone-plate lens. Here a direct measure of the object phase is made and agrees with that of the magnified object to better than +/-10%. In both cases the accuracy of the phase is limited by the pixel resolution.

Quantitative phase-sensitive imaging in a transmission electron microscope

This paper presents a new technique for forming quantitative phase and amplitude electron images applicable to a conventional transmission electron microscope. With magnetised cobalt microstructures used as a test object, we use electron holography to obtain an independent measurement of the phase shift. After a suitable calibration of the microscope, we obtain quantitative agreement of the phase shift imposed on the 200 keV electrons passing through the sample.

Confocal laser scanning ophthalmoscope and spherical harmonics used as a possible aid to detect glaucoma

We present a procedure whereby the confocal laser scanning ophthalmoscope can be used to extract information about the three-dimensional structure of the central excavated area or the cup of the optic nerve head of the eye. The data are analyzed in terms of spherical harmonics. It is hypothesized that the shape of the cup of the optic nerve head for a normal eye can be parameterized by a specific set of spherical harmonic coefficients and is different from the set of coefficients describing a glaucomatous eye. The sets of coefficients are analyzed by using multivariate statistics and can in turn be used to classify new observations. Preliminary results indicate that there are significant differences in the coefficients and that the procedure might have potential as a diagnostic aid for the detection or the screening of glaucoma.

Three-dimensional phase imaging with a scanning optical-fiber interferometer

We describe a quantitative method for measuring the phase of a propagating wave field in three dimensions by use of a scanning optical-fiber interferometer. Because phase modulation in the reference arm is exploited, this technique is insensitive to large variations in the intensity of the field being studied and is therefore highly suitable for measurement of phase within spatially confined optical beams. It uses only a single detector and is not reliant on lock-in electronics. The technique is applied to the measurement of the near field of a cleaved optical fiber and is shown to produce results in good agreement with theory.

Quantitative optical phase microscopy

We present a new method for the extraction of quantitative phase data from microscopic phase samples by use of partially coherent illumination and an ordinary transmission microscope. The technique produces quantitative images of the phase profile of the sample without phase unwrapping. The technique is able to recover phase even in the presence of amplitude modulation, making it significantly more powerful than existing methods of phase microscopy. We demonstrate the technique by providing quantitatively correct phase images of well-characterized test samples and show that the results obtained for more-complex samples correlate with structures observed with Nomarski differential interference contrast techniques.

Lobster-eye x-ray optics: a rapid evaluation of the image distribution

A method for calculating the image distribution for one-dimensional lobster-eye optics is presented. This method gives the image distribution exactly for certain cases and offers improved speed over the method of ray tracing. Examples of the use of the algorithm are given. We show that the algorithm gives the same results as detailed ray-tracing codes. Extension of the method to the two-dimensional case is discussed.

Use of a confocal laser scanning ophthalmoscope to detect glaucomatous cupping of the optic disc

PURPOSE

The aim of the present study was to test the hypothesis that the three-dimensional (3-D) topography of the optic disc contains sufficient information to diagnose glaucoma with a high degree of reliability.

METHODS

The Zeiss 'Confocal Laser Scanning Ophthalmoscope' (CLSO) was used to obtain digitized samples of 3-D images of optic nerve heads from the following three groups of patients: (i) 40 normals with normal optic discs, normal Humphrey 24-2 full threshold visual fields and an intraocular pressure (IOP) < or = 21 mmHg, (ii) 20 established glaucoma patients with cupping, glaucomatous field loss and an IOP > 21 mmHg; and (iii) 20 early glaucoma patients with early cupping, field loss with a mean defect less than -10 dB and IOP > 21 mmHg. The cupping in these patients was paramaterized by spherical harmonics and was classified by multivariate statistical analysis.

RESULTS

Of 40 glaucoma patients, 39 were correctly classified. Of 40 normal patients one was classified as glaucoma.

CONCLUSIONS

This study demonstrates that the CLSO, with the use of spherical harmonics, can differentiate glaucomatous from normal optic discs in this selected group of patients without the need for a skilled observer. This constitutes a promising technique to be tested on a large, unselected body of patients as a screening tool.

Protein Crystal Diffraction Patterns Using a Capillary-Focused Synchrotron X-ray Beam

A paraboloidally tapered glass monocapillary was used to focus an 8 keV monochromated synchrotron bending-magnet X-ray beam into a 40 (+/-5) mum focal spot located 45 (+/-5) mm from the exit of the capillary. This focal spot had a measured intensity gain of 120 (+/-10) times the intensity present in an equivalent cross section of the unfocused beam from the monochromator. This focused beam was used to obtain oscillation diffraction patterns on image plates from a hen egg-white lysozyme protein crystal in two distinct geometries: one with the specimen crystal at the capillary exit and the other with the crystal at the beam focus. In the first geometry, focused Bragg reflections were observed at the focal plane. In the second geometry, diverging Bragg reflections of high intensity from a small crystal volume were observed. Image-plate diffraction patterns for these two geometries were compared with exposures with equivalent integrated diffracted intensities obtained using a 100 x 100 mum unfocused X-ray beam with the same crystal. The use of the focused beam resulted in a reduction in the exposure time required to produce equivalent patterns by a factor of between 70 and 100.

X-ray focusing with lobster-eye optics: a comparison of theory with experiment

We report an experimental investigation and comparison with simulation of the x-ray focusing of a flat, square profile microchannel plate. We use x rays with an energy of ~1.5 keV from a laser-produced plasma. The images were recorded with x-ray film. We find the focal structure to be consistent with theoretical expectations. The angular resolution of the focus is 0.96 mrad, which is a major improvement over previous results. The measured peak intensity gain is 27 ± 4, which is ~33% of that for a perfect optic.