Timur E. Gureyev

Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object

We demonstrate simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. Subject to the assumptions explicitly stated in the derivation, the algorithm solves the twin‐image problem of in‐line holography and is capable of analysing data obtained using X‐ray microscopy, electron microscopy, neutron microscopy or visible‐light microscopy, especially as they relate to defocus and point projection methods. Our simple, robust, non‐iterative and computationally efficient method is applied to data obtained using an X‐ray phase contrast ultramicroscope.

X-ray phase-contrast microscopy and microtomography

In-line phase contrast enables weakly absorbing specimens to be imaged successfully with x-rays, and greatly enhances the visibility of fine scale structure in more strongly absorbing specimens. This type of phase contrast requires a spatially coherent beam, a condition that can be met by a microfocus x-ray source. We have developed an x-ray microscope, based on such a source, which is capable of high resolution phase-contrast imaging and tomography. Phase retrieval enables quantitative information to be recovered from phase-contrast microscope images of homogeneous samples of known composition and density, and improves the quality of tomographic reconstructions.

PHASE RETRIEVAL WITH THE TRANSPORT-OF-INTENSITY EQUATION. II. ORTHOGONAL SERIES SOLUTION FOR NONUNIFORM ILLUMINATION

In a previous paper [ J. Opt. Soc. Am. A12, 1932 ( 1995)] we presented a method for phase recovery with the transport-of-intensity equation by use of a series expansion. Here we develop a different method for the solution of this equation, which allows recovery of the phase in the case of nonuniform illumination. Though also based on the orthogonal series expansion, the new method does not require any separate boundary conditions and can be more easily adjusted for apertures of various shapes. The discussion is primarily for the case of a circular aperture and Zernike polynomials, but we also outline the solution for a rectangular aperture and Fourier harmonics. The latter example may have some substantial advantages, given the availability of the fast Fourier transform.

Rapid quantitative phase imaging using the transport of intensity equation

Abstract A method for digital phase imaging is proposed. It requires the measurement of intensity in two adjacent planes orthogonal to the optical axis. The phase is subsequently recovered by a Fast Fourier Transform of the Transport of Intensity Equation. The algorithm can be used when the illuminating beam has an arbitrary intensity distribution and is limited by a rectangular aperture. The phase retrieval formulae take on an especially simple and numerically efficient form in the case of uniform illumination. Several simulated examples are presented to confirm the viability of the algorithm.

Quantitative X‐ray projection microscopy: phase‐contrast and multi‐spectral imaging

We outline a new approach to X‐ray projection microscopy in a scanning electron microscope (SEM), which exploits phase contrast to boost the quality and information content of images. These developments have been made possible by the combination of a high‐brightness field‐emission gun (FEG)‐based SEM, direct detection CCD technology and new phase retrieval algorithms. Using this approach we have been able to obtain spatial resolution of < 0.2 µm and have demonstrated novel features such as: (i) phase‐contrast enhanced visibility of high spatial frequency image features (e.g. edges and boundaries) over a wide energy range; (ii) energy‐resolved imaging to simultaneously produce multiple quasi‐monochromatic images using broad‐band polychromatic illumination; (iii) easy implementation of microtomography; (iv) rapid and robust phase/amplitude‐retrieval algorithms to enable new real‐time and quantitative modes of microscopic imaging. These algorithms can also be applied successfully to recover object–plane information from intermediate‐field images, unlocking the potentially greater contrast and resolution of the intermediate‐field regime. Widespread applications are envisaged for fields such as materials science, biological and biomedical research and microelectronics device inspection. Some illustrative examples are presented. The quantitative methods described here are also very relevant to projection microscopy using other sources of radiation, such as visible light and electrons.

Hard x-ray quantitative non-interferometric phase-contrast microscopy

We report the results of quantitative phase-contrast imaging experiments using synchrotron radiation, in-line imaging geometry and a non-interferometric phase retrieval technique. This quantitative imaging method is fast, simple, robust, does not require sophisticated x-ray optical elements and can potentially provide submicron spatial resolution over a field of view of the order of centimetres. In the present experiment a spatial resolution of approximately 0.8 m has been achieved in images of a polystyrene sphere using 19.6 keV x-rays. We demonstrate that appropriate processing of phase-contrast images obtained in the in-line geometry can reveal important new information about the internal structure of weakly absorbing organic samples. We believe that this technique will also be useful in phase-contrast tomography.

Refracting Röntgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging

Absorption-contrast x-ray imaging serves to visualize the variation in x-ray attenuation within the volume of a given sample, whereas phase contrast allows one to visualize variations in x-ray refractive index. The former imaging mechanism has been well known and widely utilized since the time of Rontgen’s Nobel prize winning work, whereas the latter mechanism—sought for, but not found, by Rontgen himself—has laid the foundation for a revolution in x-ray imaging which is the central topic of this review. We consider the physical imaging mechanisms underlying both absorption contrast and phase contrast, together with the associated inverse problem of how one may obtain quantitative two- or three-dimensional information regarding a sample, given one or more phase-contrast images of the same. Practical questions are considered, regarding optimized phase-contrast imaging geometries as a function of detector resolution, source size, x-ray spectrum, and dose. Experimental examples pertaining to biomedical appli...

Composite techniques for phase retrieval in the Fresnel region

Abstract In free-space propagation of optical beams, the linear contribution of the object-plane phase to the intensity distribution in the Fresnel region is described by the Transport of Intensity equation (TIE). In the near-Fresnel region, i.e. where the Fresnel number is larger than unity, the linear term can often be dominant. Therefore in this region the TIE can provide a good initial approximation which can then be refined by other techniques, e.g. by Gerchberg–Saxton–Fienup (GSF) type algorithms. Multigrid methods with the TIE applied on coarser grids and GSF algorithms applied on finer grids are shown to be particularly effective in this context.

Phase-and-amplitude computer tomography

A tomographic technique is proposed for reconstruction under specified conditions of the three-dimensional distribution of complex refractive index in a sample from a single projection image per view angle, where the images display both absorption contrast and propagation-induced phase contrast. The algorithm achieves high numerical stability as a consequence of the complementary nature of the absorption and phase contrast transfer functions. The method is pertinent to biomedical imaging and nondestructive testing of samples exhibiting weak absorption contrast.

On the optimization of experimental parameters for x-ray in-line phase-contrast imaging

General principles and results for the optimization of the performance of x-ray in-line phase-contrast imaging systems for spatially incoherent sources are investigated. In particular, formulas expressing the dependence of image contrast, spatial resolution, and signal-to-noise ratio on instrumental parameters including source size, detector resolution, geometrical factors, x-ray energy as well as sample properties are derived for different models of sample features. The results for some special cases of interest are presented. Optimization procedures are proposed that are expected to be useful in the design of imaging systems seeking to exploit x-ray in-line phase contrast.