Abstract: We describe a novel, high-speed pulsed terahertz (THz) Fourier imaging system based on compressed sensing (CS), a new signal processing theory, which allows image reconstruction with fewer samples than traditionally required. Using CS, we successfully reconstruct a 64 x 64 image of an object with pixel size 1.4 mm using a randomly chosen subset of the 4096 pixels, which defines the image in the Fourier plane, and observe improved reconstruction quality when we apply phase correction. For our chosen image, only about 12% of the pixels are required for reassembling the image. In combination with phase retrieval, our system has the capability to reconstruct images with only a small subset of Fourier amplitude measurements and thus has potential application in THz imaging with cw sources.

Abstract: An iterative phase retrieval method is proposed, which uses a sequence of diffraction intensity patterns recorded at different wavelengths. This method has a rapid convergence, and a high immunity to noise and environmental disturbance. The wrap-free phase measurement range is also extended based on the principle of two-wavelength interferometry. Simulation and experimental results are presented to demonstrate the approach.

Abstract: A windowed Fourier ridges (WFR) algorithm and a windowed Fourier filtering (WFF) algorithm have been proposed for fringe pattern analysis and have been demonstrated to be versatile and effective. Theoretical analyses of their performances are of interest. Local frequency and phase extraction errors by the WFR and WFF algorithms are analyzed in this paper. Effectiveness of the WFR and WFF algorithms will thus be theoretically proven. Consider four phase-shifted fringe patterns with local quadric phase [c(20)=c(02)=0.005 rad/(pixel)(2)], and assume that the noise in these fringe patterns have mean values of zero and standard deviations the same as the fringe amplitude. If the phase is directly obtained using the four-step phase-shifting algorithm, the phase error has a mean of zero and a standard deviation of 0.7 rad. However, when using the WFR algorithm with a window size of sigma(x)=sigma(y)=10 pixels, the local frequency extraction error has a mean of zero and a standard deviation of less than 0.01 rad/pixel and the phase extraction error in the WFR algorithm has a mean of zero and a standard deviation of about 0.02 rad. When using the WFF algorithm with the same window size, the phase extraction error has a mean of zero and a standard deviation of less than 0.04 rad and the local frequency extraction error also has a mean of zero and a standard deviation of less than 0.01 rad/pixel. Thus, an unbiased estimation with very low standard deviation is achievable for local frequencies and phase distributions through windowed Fourier transforms. Algorithms applied to different fringe patterns, different noise models, and different dimensions are discussed. The theoretical analyses are verified by numerical simulations.

Abstract: We report on progress in algorithms for iterative phase retrieval. The theory of convex optimization is used to develop and to gain insight into counterparts for the nonconvex problem of phase retrieval. We propose a relaxation of averaged alternating reflectors and determine the fixed point set of the related operator in the convex case. A numerical study supports our theoretical observations and demonstrates the effectiveness of the algorithm compared to the current state of the art.

Abstract: We propose a windowed Fourier-filtered and quality-guided phase-unwrapping algorithm that is an extension and improvement of our previous phase-unwrapping algorithm based on windowed Fourier transform [Opt. Laser Technol.37, 458 (2005)OLTCAS0030-399210.1016/j.optlastec.2004.07.007, Key Eng. Mater.326-328, 67 (2006)KEMAEY1013-9826]. First, the filtered amplitude is used as a real-valued quality map, rather than a binary mask, which makes the phase-unwrapping algorithm more tolerant to low-quality regions in a wrapped-phase map, and the process is more automatic. Second, the window size selection is considered, which enables the algorithm to be adapted to tackle different phase-unwrapping problems. A large window size is useful for removing noise, building long barriers along phase discontinuities, and identifying invalid regions, while a small window size is useful for preserving local features, such as small regions and high-quality narrow channels. Eight typical examples in Ghiglia and Pritt's excellent book Two-Dimensional Phase Unwrapping: Theory, Algorithm and Software (Wiley, 1998) are used to evaluate the proposed algorithm. The proposed algorithm is able to unwrap all these examples successfully. The windowed Fourier ridges algorithm, another algorithm based on windowed Fourier transform, is also tested and found to be useful in building barriers along phase discontinuities.

Abstract: A combination of spatial interference patterns and spectral interferometry are used to find the global phase for non-collinear two-dimensional Fourier-transform (2DFT) spectra. Results are compared with those using the spectrally resolved transient absorption (STRA) method to find the global phase when excitation is with co-linear polarization. Additionally cross-linear polarized 2DFT spectra are correctly “phased” using the all-optical technique, where the SRTA is not applicable.

Abstract: In this paper, a novel modified Fourier transform method is proposed, which employs a fringe image and a flat image to eliminate the background and in the mean time facilitate the retrieval of the absolute phase map Both the fringe and flat patterns are projected onto the object by a digital video projector With the subtraction of the flat image from the fringe image, the background is completely removed and the spectrum overlapping in the frequency domain is prevented The flat image is also employed for hole and shadow detection Two cross-shaped markers are embedded in the flat and fringe image respectively for absolute phase retrieval Experimental results showed that the proposed method produced better shape measurement results when measuring fast moving or changing objects, compared to the phase shifting method The proposed method has the potential to boost the speed of our real-time 3-D shape measurement system to 120 fps with better measurement accuracy

Abstract: The resolution of X-ray diffraction microscopy is limited by the maximum dose that can be delivered prior to sample damage. In the proposed serial crystallography method, the damage problem is addressed by distributing the total dose over many identical hydrated macromolecules running continuously in a single-file train across a continuous X-ray beam, and resolution is then limited only by the available molecular and X-ray fluxes and molecular alignment. Orientation of the diffracting molecules is achieved by laser alignment. The incident X-ray fluence (energy/area) is evaluated that is required to obtain a given resolution from (i) an analytical model, giving the count rate at the maximum scattering angle for a model protein, (ii) explicit simulation of diffraction patterns for a GroEL–GroES protein complex, and (iii) the spatial frequency cut-off of the transfer function following iterative solution of the phase problem, and reconstruction of an electron density map in the projection approximation. These calculations include counting shot noise and multiple starts of the phasing algorithm. The results indicate counting time and the number of proteins needed within the beam at any instant for a given resolution and X-ray flux. An inverse fourth-power dependence of exposure time on resolution is confirmed, with important implications for all coherent X-ray imaging. It is found that multiple single-file protein beams will be needed for sub-nanometer resolution on current third-generation synchrotrons, but not on fourth-generation designs, where reconstruction of secondary protein structure at a resolution of 7 A should be possible with relatively short exposures.

Abstract: We propose a novel phase retrieval algorithm in Hankel (or called Fourier-Bessel) transform domains by using Monte-Carlo method. Based on the proposed algorithm, we investigate the generation of Gaussian-like beams, such as hollow Gaussian beam, Bessel-Gaussian beam and Laguerre-Gaussian beam, with double phase filtering operations. The phase distributions of filters are determined by employing the proposed phase retrieval algorithm. The advantage of the method is that the total energy of the beam is conservative. Numerical simulations are shown to demonstrate the validity of the scheme.

Abstract: High-resolution synchrotron-based x-ray phase-contrast images of in vitro blood flow have been collected. We demonstrate that the application of a single-image phase retrieval algorithm to the images leads to improved detail in particle image velocimetry correlation peaks. Out-of-plane variation in flow has been extracted from the peaks leading to the three-dimensional reconstruction of velocity across an axially symmetric cylinder.

Abstract: A Fresnel coherent diffraction imaging experiment with hard x rays is here presented, using two planar crossed waveguides as optical elements, leading to a virtual pointlike source. The coherent wave field obtained with this setup is used to illuminate a micrometric single object having the shape of a butterfly. A digital two-dimensional in-line holographic reconstruction of the unknown object at low resolution (200 nm) has been obtained directly via fast Fourier transform (FFT) of the raw data. The object and its twin image are well separated because suitable geometrical conditions are satisfied. A good estimate of the incident wave field phase has been extracted directly from the FFT of the raw data. A partial object reconstruction with 50 nm spatial resolution was achieved by fast iterative phase retrieval, the major limitation for a full reconstruction being the nonideal structure of the guided beam. The method offers a route for fast and reliable phase retrieval in x-ray coherent diffraction.

Abstract: We describe a quantitative phase imaging process using an x-ray laboratory-based source with an extremely broad bandwidth spectrum. The thickness of a homogeneous object can be retrieved by using separately spectrally weighted values for the attenuation coefficient and the decrement of the real part of the refractive index. This method is valid for a wide range of object types, including objects with an absorption edge in the used energy range. The accessibility of conventional x-ray laboratory sources makes this method very useful for quantitative phase retrieval of homogeneous objects. We demonstrate the application of this method for quantitative phase retrieval imaging in tomographic measurements.

Abstract: The phase diversity technique is a useful tool to measure and pre-compensate for quasi-static aberrations, in particular non-common path aberrations, in an adaptive optics corrected imaging system. In this paper, we propose and validate by simulations an extension of the phase diversity technique that uses long exposure adaptive optics corrected images for sensing quasi-static aberrations during the scientific observation, in particular for high-contrast imaging. The principle of the method is that, for a sufficiently long exposure time, the residual turbulence is averaged into a convolutive component of the image and that phase diversity estimates the sole static aberrations of interest. The advantages of such a procedure, compared to the processing of shortexposure image pairs, are that the separation between static aberrations and turbulence-induced ones is performed by the long-exposure itself and not numerically, that only one image pair must be processed, that the estimation benefits from the high SNR of long-exposure images, and that only the static aberrations of interest are to be estimated. Long-exposure phase diversity can also be used as a phasing sensor for a segmented aperture telescope. Thus, it may be particularly useful for future planet finder projects such as EPICS on the European ELT.

Abstract: The signal extraction method based on intensity measurements in two close fractional Fourier domains is examined by using the phase space formalism. The fractional order separation has a lower bound and an upper bound that depend on the signal at hand and the noise in the optical system used for measurement. On the basis of a theoretical analysis, it is shown that for a given optical system a judicious choice of fractional order separation requires some a priori knowledge of the signal bandwidth. We also present some experimental results in support of the analysis.

Abstract: A phase diversity wavefront sensor includes an optical system including at least one optical element for receiving a light beam; a diffractive optical element having a diffractive pattern defining a filter function, the diffractive optical element being arranged to produce, in conjunction with the optical system, images from the light beam associated with at least two diffraction orders; and a detector for detecting the images and outputting image data corresponding to the detected images. In one embodiment, the optical system, diffractive optical element, and detector are arranged to provide telecentric, pupil plane images of the light beam. A processor receives the image data from the detector, and executes a Gerchberg-Saxton phase retrieval algorithm to measure the wavefront of the light beam.

Abstract: We present an optical phase measurement method based on the Hilbert transform for the analysis of a time series of speckle interferograms modulated by a temporal carrier. We discuss the influence of nonmodulating pixels, modulation loss, and noise that affect the bias and modulation intensities of the interferometric signal and propose the application of the empirical mode decomposition method for its minimization. We also show the equivalence between the phase recovery approaches that are based on the Hilbert and the Fourier transforms. Finally, we present a numerical comparison between these methods using computer-simulated speckle interferograms modulated with a temporal carrier.

Abstract: Atomic coherence phenomena are usually investigated using single beam techniques without spatial resolution. Here we demonstrate state-selective imaging of cold $^{85}\text{R}\text{b}$ atoms in a three-level ladder system, where the atomic refractive index is sensitive to the quantum coherence state of the atoms. We use a phase-sensitive diffraction contrast imaging (DCI) technique which depends on the complex refractive index of the atom cloud. A semiclassical model allows us to analytically calculate the detuning-dependent refractive index of the system. The predicted Autler-Townes splitting and our experimental measurements are in excellent agreement. DCI provided a quantitative image of the distribution of the excited-state fraction, and compared with on-resonance absorption and blue cascade fluorescence techniques, was found to be experimentally simple and robust.

Abstract: Given a network of stations with incomplete and possibly imprecise inter-station range measurements, it is required to find the relative positions of the stations. The authors show that for a planar geometry the problem can be couched using complex numbers. It then becomes evident that location estimation is equivalent to the celebrated problem of phase retrieval. Although the equations are quadratic, the proposed solution is based on solving a set of linear equations. For precise measurements, the exact solution is obtained with a small number of operations. For noisy measurements, the method provides an excellent initial point for the application of the Gerchberg-Saxton iterations that are usually associated with phase retrieval. Proof of convergence is provided for the iterations. Small error analysis of the algorithm proves that it is statistically efficient and therefore for small measurement errors achieves the Crame-r-Rao lower bound. The authors provide a compact, matrix form expression for the Crame-r-Rao bound and evaluation of the computational load. Numerical examples are provided to corroborate the results.

Abstract: A method of retrieving phase information from input intensity information, representative of a target image, in which a Fourier transform is performed on data and the result used in forming a phase estimate, the phase estimate being inverse Fourier transformed, thereby producing magnitude and phase replay, and wherein not only is the phase reply component but also data derived from the magnitude replay component, iteratively fed back.

Abstract: The paper presents an algorithm able to retrieve the phase in speckle interferometry by a single intensity pattern acquired in a deformed state, provided that the integrated speckle field is resolved in the reference condition in terms of mean intensity, modulation amplitude and phase. The proposed approach, called throughout the paper "one-step", can be applied for studying phenomena whose rapid evolution does not allow the application of a standard phase-shifting procedure, which, on the other hand, must be applied at the beginning of the experiment. The approach was proved by an experimental test reported at the end of the paper.

Abstract: A novel information security system based on multiple-phase retrieval by an iterative gyrator transform algorithm is proposed. In this method, a series of phase masks are designed and located in the input plane and the gyrator planes, and the phase distributions of all the masks are adjusted simultaneously in each iteration. It can achieve fast convergence and high quality of the recovered image and can provide a higher degree of freedom in key space with more parameters as supplementary keys. Furthermore, the security level of this method is greatly improved by the sensitivity of recovered images with the angles of gyrator transform. Numerical simulations are presented to verify its validity and efficiency.

Abstract: We present an on-shot focal-spot characterization technique based on a phase-retrieval scheme that retrieves near-field phase from multiplane focal-spot measurements in an experimental target chamber. The technique is easy to implement inside a target chamber and is demonstrated in a multiterawatt laser system. It is also found that phase retrieval can quantitatively detect residual angular dispersion coming from the pulse compressor misalignment.

Abstract: Coherent x-ray scattering in a grazing-incidence geometry was used to image SiGe nanoislands grown by liquid phase epitaxy on Si(001). Due to their narrow size distribution, identical shape, and orientation, the total scattered intensity obtained in this geometry represents a coherent diffraction pattern from an average island. Iterative phase retrieval techniques were used for the reconstruction of the projected electron density of hundred nanometer size islands. So far other x-ray imaging methods do not achieve the obtained spatial resolution of 10\char21{}15 nm for islands of this size.

Abstract: X-ray coherence evolution in the imaging process plays a key role for x-ray phase-sensitive imaging. In this work we present a phase-space formulation for the phase-sensitive imaging. The theory is reformulated in terms of the cross-spectral density and associated Wigner distribution. The phase-space formulation enables an explicit and quantitative account of partial coherence effects on phase-sensitive imaging. The presented formulas for x-ray spectral density at the detector can be used for performing accurate phase retrieval and optimizing the phase-contrast visibility. The concept of phase-space shearing length derived from this phase-space formulation clarifies the spatial coherence requirement for phase-sensitive imaging with incoherent sources. The theory has been applied to x-ray Talbot interferometric imaging as well. The peak coherence condition derived reveals new insights into three-grating-based Talbot-interferometric imaging and gratings-based x-ray dark-field imaging.