Spatial incoherence in phase retrieval based on focus variation

We investigate the effect of spatial incoherence on two methods of phase retrieval based on focus variation: the transport of intensity equation and iterative wave function reconstruction. Spatial incoherence provides an upper bound on the defocus step size which should be used in each case. The requirement that phase information manifests itself in sufficient variation in the defocused images provides a lower bound on the defocus step size which should be used in each case. The scaling of these upper and lower bounds with object size and imaging resolution differs in such a way that, given the spatial incoherence properties of the source, for sufficiently low resolutions neither technique can retrieve phase information. The regions of applicability of the two techniques are discussed.     

Electron microscope object reconstruction: Retrieval of local variations in mixed type potentials. Part I: Theoretical preliminaries

The direct object retrieval via the linearized inversion of the dynamical scattering matrix is extended using a second order perturbation theory and including mixed type potentials. The higher order perturbation increases the confidence region extracting object thickness and bending directly out of amplitude and phase of an electron wave without using trial-and-error iterative matching. Applying parameterization of a mixed type total scattering potential as a priori information enables a simple extension of the structure retrieval procedure to reconstruct local variations of the object potential, too.     

A method for correcting the effect of specimen drift on coherent diffractive imaging

Coherent diffractive imaging involves the inversion of a diffraction pattern to find the wave function at the exit-surface plane of the specimen. It is a promising technique for imaging, for example, nanoparticles with electrons and biological molecules with X-rays. If the illumination is not a plane wave of infinite extent, then a relative drift between the illumination and the object introduces errors into the diffraction pattern; an issue which is often overlooked. This may be of particular importance for applications with electron microscopes which use nanoscale probes. Here we show that beams which are uniform over a sufficiently large region can be used to pose a phase retrieval problem that is immune from specimen drift, provided suitable analysis of the diffraction data is undertaken. The method only applies to objects contained within a support that is smaller than a uniform region of the beam.     

Imaging a dense nanodot assembly by phase retrieval from TEM images

Phase retrieval is a classical inverse problem in many fields dealing with waves that is becoming of increasing interest in transmission electron microscopy (TEM). A non-interferometric approach is here applied to TEM images. Phase retrieval possibilities given by the transport intensity equation are compared to the ones deriving from the weak phase object approximation. In the limit of small angles, both methods lead to a similar equation between the phase and a set of defocus images. This equation can be solved by an image processing equivalent to using a specific filter in Fourier space. This processing leads to phase images with a spatial resolution here essentially limited by the defocus amount between images. A dense assembly of silicon nanodots is used as a model case to illustrate the interest of this approximate phase retrieval method which can be carried out on standard equipment. The dot heights estimated using the phase images are found to be in good agreement with ones measured by atomic force microscopy. Since image noise and large defocus values may strongly affect the solution given by the approximate method, an iterative phase retrieval method is also used as a test for working conditions.     

On the transport of intensity technique for phase retrieval

The Transport of Intensity technique is becoming a viable alternative to electron holography for phase retrieval in Transmission Electron Microscopy. However, several issues are still to be clarified in order to ascertain the applicability of the technique; among them, the controversy regarding its geometrical or wave-optical nature, as related to the phase detection limit. We show here that the Transport of Intensity is a wave-optical technique that works in a special regime of small defocus where the image intensity is linear with the defocus parameter. By a simple analytical example we show that the Transport of Intensity correctly reconstructs the electron optical phase shift even when the phase is smaller than pi, a value defining the boundary between the geometrical and wave approaches. Another example is given, the reconstruction of a phase jump, accompanied with experimental support showing that phase retrieval by Electron Holography and Transport of Intensity techniques yields results in good agreement.     

Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector

A method for x-ray phase contrast imaging is introduced in which only one absorption grating and a microfocus x-ray source in a tabletop setup are used. The method is based on precise subpixel position determination of the x-ray pattern projected by the grating directly from the pattern image. For retrieval of the phase gradient and absorption image (both images obtained from one exposure), it is necessary to measure only one projection of the investigated object. Thus, our method is greatly simplified compared with the phase-stepping method and our method can significantly reduce the time-consuming scanning and possibly the unnecessary dose. Furthermore, the technique works with a fully polychromatic spectrum and gives ample variability in object magnification. Consequently, the approach can open the way to further widespread application of phase contrast imaging, e.g., into clinical practice. The experimental results on a simple testing object as well as on complex biological samples are presented.     

Extended ptychography in the transmission electron microscope: possibilities and limitations

The extended-ptychographical iterative engine (e-PIE) is a recently developed powerful phase retrieval algorithm which can be used to measure the phase transfer function of a specimen and overcome conventional lens resolution limits. The major improvement over PIE is the ability to reconstruct simultaneously both the object and illumination functions, robustness to noise and speed of convergence. The technique has proven to be successful at optical and X-ray wavelengths and we describe here experimental results in transmission electron microscopy supported by corresponding simulations. These simulations show the possibilities - even with strong phase objects - and limitations of ptychography; in particular issues arising from poorly-defined probe positions.     

Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy.

We discuss phase retrieval and the correction of images for aberrations, in particular defocus and spherical aberration, in high-resolution transmission electron microscopy. Non-interferometric phase retrieval requires at least two intensity measurements in different planes. Vortices in the phase may occur in the image plane or the other planes involved in the phase retrieval. We discuss the performance of various methods of phase retrieval in that case. After retrieval of the phase, the aberrations can be corrected in the Fraunhofer diffraction space (the wave function in the diffraction space is related to that in the image space by a Fourier transform). The aberration-corrected image is obtained from the aberration-corrected wave function in the diffraction plane by inverse Fourier transformation.     

Object wave reconstruction by phase-plate transmission electron microscopy

A method is described for the reconstruction of the amplitude and phase of the object exit wave function by phase-plate transmission electron microscopy. The proposed method can be considered as in-line holography and requires three images, taken with different phase shifts between undiffracted and diffracted electrons induced by a suitable phase-shifting device. The proposed method is applicable for arbitrary object exit wave functions and non-linear image formation. Verification of the method is performed for examples of a simulated crystalline object wave function and a wave function acquired with off-axis holography. The impact of noise on the reconstruction of the wave function is investigated.     

Retrieval of object information by inverse problems in electron diffraction

The imaging of crystal defects by high-resolution transmission electron microscopy or with the help of the electron diffraction contrast technique is well known and routinely used. However, a direct and phenomenological analysis of electron micrographs is mostly not possible, but requires the application of image simulation and matching techniques. The trial-and-error matching technique is the indirect solution to the direct scattering problem applied to analyse the nature of the object under investigation. Alternatively, inverse problems as direct solutions of electron scattering equations can be deduced using either an invertible linearized eigenvalue system or a discretized form of the diffraction equations. This analysis is based on the knowledge of the complex electron wave at the exit plane of an object reconstructed for the surrounding of single reflections by electron holography or other wave reconstruction techniques. In principle, it enables directly the retrieval of the local thickness and orientation of a sample as well as the refinement of potential coefficients or the determination of the atomic displacements, caused by a crystal lattice defect, relative to the atom positions of the perfect lattice. Considering especially the sample orientation as perturbation the solution is given by a generalized and regularized Moore–Penrose inverse, where the resulting numerical algorithms imply ill-posed inverse problems.     

Phase information recovery based on the methods of phase shifting interferometry with small angles between interfering beams

This paper describes the method for recovering digital holograms obtained at small angles between interfering wave fields. The technique for obtaining data on the phase of the wavefront reflected from the object is under consideration.     

基于相位恢复技术的大型光学镜面面形在位检测技术

根据目前大镜加工在位检测的需求和面临的困难,把基于衍射图像检测的相位恢复技术应用到大镜面形的在位检测,建立大镜加工在位检测系统.对影响测量系统精度的主要因素以及系统的抗振动性能进行了分析和仿真,结果表明该测量系统具有较好的抗振动能力,精度接近标准球面波光源的精度.基于GS迭代算法开发了测量软件,对( 500 mm球面镜进行在位测量.把相位恢复测量结果与高精度的ZYGO干涉仪测量结果进行比较分析,结果表明在面形误差分布及误差的峰谷值(PV)和方均根值(RMS)上,两者具有较高程度的一致性,这说明相位恢复测量方法的可行性和准确性.由该方法构成的在位测量系统具有光路简单、精度较高、抗振动能力强和操作简单等特点,完全满足大镜的在位测量需求.     

From thickness dependent exit waves to projected potential: Thickness derivative approach

In HREM, due to multiple scattering, the exit wave of the object is nonlinear thickness dependent so that there is no one-to-one relation between object structure and the exit wave. This feature hampers the direct retrieval of structural information from exit waves. In this paper we discuss the possibility to restore the object structure in a direct way using exit waves of different thicknesses. It is theoretically shown that the amplitude of the thickness derivative exit wave |∂ψ/∂z| may directly reflect the project potential in a simple way. Image simulations show that it can be applied to restore the projected potential.     

Optical 3D profilometer for in-process measurement of microsurface based on phase retrieval technique

We have proposed an optical method that can be applied to in-process or in situ measurement of the microsurface profile. The present method is based on optically performed spectral analysis and the phase retrieval technique. Spectral information of a surface profile is obtained by measuring the Fraunhofer diffraction intensity. The phase retrieval technique is used to reconstruct the surface profile from the measured spectrum. We have developed an instrument on the basis of the general principles of the present method, and measured the surface of a reference standard having rectangular pockets 44 nm deep at intervals of 10 μm. The measured surface profile was in good agreement with the nominal dimensions of the specimen as well as the surface profile obtained by atomic force microscopy (AFM).     

位相复原技术在光学成像质量测评中的应用

对于大口径或特殊波段成像的光学系统,采用常规测试手段难以实现对光学系统成像质量的评估。而光学系统波像差是表征光学系统成像质量的重要指标之一。本文基于信息光学基础理论采用傅立叶变换和迭代算法通过对光学成像系统星点图像（光强点扩散函数PSF）计算分析,从而实现对光学成像系统位相的复原获得波像差信息。本文论述了有关的理论依据和分析计算公式;并研制了位相复原分析计算软件。通过计算机仿真与误差分析论证了研制的位相复原计算方法的正确性,复原误差小于5%。提出了在实际应用中采用此方法时减少复原误差的一种途径。通过实际采集的光学系统星点图像的位相复原实验验证了位相复原分析软件的适用性。此方法为非常规光学成像系统波相差的评估工作提供了新途径。     

Flow rate measurement by kinematic wave detection in vertically upward, bubbly two-phase flows

This paper describes a technique for measuring the flow rates of both phases in one-dimensional, vertically upward, bubbly oil–water flows. Measurements of the speed of naturally occurring kinematic waves, obtained using an impedance cross-correlation flow meter, were combined with measurements of the disperse phase volume fraction and an appropriate kinematic wave model to yield predictions of the flux density of both the oil and the water. A systematic error of −3.16%, and an additional random error with a standard deviation of 2.58%, was observed in the predicted oil flux density (and the predicted oil volumetric flow rate). A systematic error of +4.41%, and an additional random error with a standard deviation of 4.83%, was observed in the predicted water flux density (and the predicted water volumetric flow rate). The impedance cross-correlation flow meter has no moving parts and is of robust construction, making the technique described here suitable for implementation in vertical oil wells.     

Envelope pulsed ultrasonic distance measurement system based upon amplitude modulation and phase modulation

A novel microcomputer-based ultrasonic distance measurement system is presented. This study proposes an efficient algorithm which combines both the amplitude modulation (AM) and the phase modulation (PM) of the pulse-echo technique. The proposed system can reduce error caused by inertia delay and amplitude attenuation effect when using the AM and PM envelope square wave form (APESW). The APESW ultrasonic driving wave form causes a phase inversion phenomenon in the relative wave form of the receiver. The phase inversion phenomenon sufficiently identifies the "measurement pulse" in the received wave forms, which can be used for accurate time-of-flight (TOF) measurement. In addition, combining a countertechnique to compute the phase shifts of the last cycle for TOF, the presented system can obtain distance resolution of 0.1% of the wavelength corresponding to the 40 kHz frequency of the ultrasonic wave. The standard uncertainty of the proposed distance measurement system is found to be 0.2 mm at a range of 50-500 mm. The APESW signal generator and phase detector of this measuring system are designed on a complex programmable logic device, which is used to govern the TOF measurement and send the data to a personal computer for distance calibration and examination. The main advantages of this APESW system are high resolution, low cost, narrow bandwidth requirement, and ease of implementation.     

Structure determination at the atomic level from dynamical electron diffraction data under systematic row conditions.

We discuss a method to obtain structural information on crystals at the atomic level in high-resolution transmission electron microscopy from dynamical diffraction data under systematic row conditions. Working at a fixed incident energy and within an N-beam approximation, data is required at a well defined set of N incident beam orientations to determine the scattering matrix, one orientation for each column in the matrix. At each orientation the corresponding column of the scattering-matrix is obtained by Fourier transformation of the exit surface wave function. Thus, in addition to each exit surface image, we must recover the phase of the wave function for that orientation in the image plane. We show that retrieval of the phase using algorithms based on conservation of flux, which assume continuity of the phase, can yield incorrect solutions for the phase. This is because singularities can occur in the phase of the wave field at points where the intensity is zero, which can lead to edge dislocations in the phase. We demonstrate, using a model example, how these edge dislocations arise. We will show that phase retrieval from a through focal series of measurements or using the Gerchberg-Saxton algorithm (starting from measurements of an image and the corresponding diffraction pattern), correctly retrieves the phase and hence the exit surface wave function for all the orientations required to obtain the scattering-matrix. The dynamical (multiple) scattering can then be inverted to uniquely obtain the projected potential.     

Performance limits of electron holography

Transmission electron microscopy is wave optics. The object exit wave contains the full object information. However, in the usual intensity images, recorded either in real space or in Fourier space, the phases are missing. In many applications at medium and at high resolution, electron holography has shown its unique ability of solving the “missing phase problem” and utilizing the recovered phase for complete interpretation of the object structure. The question is “What are the performance limits?” with respect to field of view, lateral resolution and signal resolution. In this article, the performance limits are derived and discussed.     

Hologram simulation for off-axis electron holography

Hologram simulation for electron holography using an electron biprism is described. An electron hologram is superimposed by Fresnel fringes originating from the electron biprism, which affects both the amplitude and the phase of the object wave and the reference wave. In this simulation, we consider the effects of Fresnel diffraction as well as the electron-wave phase shift due to the electromagnetic field produced by the specimen. We also take into account the phase shift due to the inner potential of the specimen, the amplitude modulation due to the absorption of the incident electrons by the specimen, reference-wave distortion caused by the electromagnetic fields, coherency of the electron wave, and quantum noise of the detected electrons. Simulated and experimentally obtained holograms and reconstructed images are compared for the cases of a charged latex spherical particle and a single magnetic-domain spherical particle placed on a carbon film.