Iterative Phase Retrieval Method Research Articles

Enhanced deterministic phase retrieval using a partially developed speckle field

A technique for enhanced deterministic phase retrieval using a partially developed speckle field (PDSF) and a spatial light modulator (SLM) is demonstrated experimentally. A smooth test wavefront impinges on a phase diffuser, forming a PDSF that is directed to a 4f setup. Two defocused speckle intensity measurements are recorded at the output plane corresponding to axially-propagated representations of the PDSF in the input plane. The speckle intensity measurements are then used in a conventional transport of intensity equation (TIE) to reconstruct directly the test wavefront. The PDSF in our technique increases the dynamic range of the axial intensity derivative for smooth phase objects, resulting in a more robust solution to the TIE. The SLM setup enables a fast and accurate recording of speckle intensity. Experimental results are in good agreement with those obtained using the iterative phase retrieval and digital holographic methods of wavefront reconstruction.

Iterative retrieval of one-dimensional x ray wave field using a single intensity measurement

The problem of retrieving a complex function from the modulus of its Fourier transform has non-unique solutions in one dimension. Therefore iterative phase retrieval methods cannot in general be confidently applied to one-dimensional problems, due to the presence of ambiguities. We present a method for a posteriori reduction of the ambiguities based on the correlation analysis of the solution of a large number of runs of an iterative phase retrieval algorithm with different random starting phases. The method is applied to experimentally measured diffraction patterns from an x ray waveguide illuminated by hard x rays. We demonstrate the possibility of retrieving the complex wave field at the exit face of the waveguide and compare the result with theoretical prediction.

41.1: Three‐dimensional Electro‐holographic Retinal Display

AbstractWe use a single phase‐only spatial light modulator and an iterative phase retrieval method to directly modulate the light at multiple planes inside the eye. By changing the focal length of the eye, images at different depth planes are sharply projected onto the retina, with which we mimic the accommodation of natural vision.

Noniterative approach to the missing data problem in coherent diffraction imaging by phase retrieval

When a very intense beam is used for illuminating an object in coherent x-ray diffraction imaging, the intensities at the center of the diffraction pattern for the object are cut off by a beam stop that is utilized to block the intense beam. Until now, only iterative phase-retrieval methods have been applied to object reconstruction from a single diffraction pattern with a deficiency of central data due to a beam stop. As an alternative method, I present a noniterative solution in which an interpolation method based on the sampling theorem for the missing data is used for object reconstruction with our previously proposed phase-retrieval method using an aperture-array filter. Computer simulations demonstrate the reconstruction of a complex-amplitude object from a single diffraction pattern with a missing data area, which is generally difficult to treat with the iterative methods because a nonnegativity constraint cannot be used for such an object.

Beam-divergence deconvolution for diffractive imaging

This Rapid Communication presents a method of beam-divergence deconvolution for diffractive imaging. First, the detected diffraction intensity is formulated as a convolution between the diffraction intensity of parallel incident beams and the divergence of an incident beam. It is shown numerically that the convolution causes the reconstructed image to shrink and become blurred. Next, the algorithm of deconvolution used in the iterative Fourier phase retrieval method is applied to the convoluted diffraction intensity deteriorated by quantum noise. Numerical simulations show that the proposed algorithm recovers the deconvoluted diffraction intensity and improves the reconstructed image. Finally, the algorithm is applied to an electron-beam experiment to reconstruct a multiwall carbon nanotube. The results verified that the algorithm reduces the influence of beam divergence.

Ab initiostructure solution by iterative phase-retrieval methods: performance tests on charge flipping and low-density elimination

Comprehensive tests on the density-modification methods charge flipping [Oszlányi & Sütő (2004).Acta Cryst.A60, 134–141] and low-density elimination [Shiono & Woolfson (1992).Acta Cryst.A48, 451–456] for solving crystal structures are performed on simulated diffraction data of periodic structures and quasicrystals. A novel model-independent figure of merit, which characterizes the reliability of the retrieved phase of each reflection, is introduced and tested. The results of the performance tests show that the quality of the phase retrieval highly depends on the presence or absence of an inversion center and on the algorithm used for solving the structure. Charge flipping has a higher success rate for solving structures, while low-density elimination leads to a higher accuracy in phase retrieval. The best results can be obtained by combining the two methods,i.e.by solving a structure with charge flipping followed by a few cycles of low-density elimination. It is shown that these additional cycles dramatically improve the phases not only of the weak reflections but also of the strong ones. The results can be improved further by averaging the results of several runs and by applying a correction term that compensates for a reduction of the structure-factor amplitudes by averaging of inconsistently observed reflections. It is further shown that in most cases the retrieved phases converge to the best solution obtainable with a given method.

Experimental evaluation of a full-color compact lensless holographic display

An iterative phase retrieval method for a lensless color holographic display using a single light modulator is experimentally validated. The technique involves iterative calculation of a three-plane synthetic hologram which is displayed on a SLM simultaneously lit with three laser beams providing an RGB illumination. Static and animated two-dimensional flicker-free full color images are reconstructed at a fixed position and captured using a high resolution CMOS sensor. The image finesse, color fidelity, contrast ratio and influence of speckles are evaluated and compared with other techniques of holographic color image encoding. The results indicate the technique superior in a case of full-color real-life pictures which are correctly displayed by this ultra-compact and simple projection setup.

Phase-coded aperture for optical imaging

For the optical spectrum region, we describe a novel phase-coded aperture imaging system that can be used in a computational imaging camera. The optical design includes a phase-only screen followed by a detector array. A specific diffraction pattern forms at the detector array when the wavefront from a point source object passes through the phase screen. Since diffraction effects cannot be ignored in the optical regime, an iterative phase retrieval method is used to calculate the phase coded screen. Correlation type processing can be applied for the image recovery. Computer simulation results are presented to illustrate the excellent imaging performance of this camera.

Fast-convergent algorithm for speckle-based phase retrieval and a design for dynamic wavefront sensing

Wavefront reconstruction is carried out using sequentially recorded speckle patterns and an iterative phase retrieval method based on wave propagation. A novel fast-convergent algorithm that maintains the propagation distance in the iteration step equal to the distance between measurement planes is demonstrated. Employing the new algorithm, influences of the number of measurement planes, number of iterations, and uncertainties in the detector's transverse and axial positions on the rate of phase convergence are analyzed experimentally. A conceptual design for a dynamic wavefront sensor using arrays of beam splitters and detectors for parallel speckle recording is described.

コヒーレントＸ線回折によるナノ構造解析

X-ray crystallography can routinely determine the atomic structure of crystalline materials. The method can be extended to non-crystalline materials by using coherent X-ray diffraction. In X-ray diffraction microscopy, coherent X-ray diffraction patterns are sampled finely enough to satisfy the oversampling condition for solving the phase problem, and the iterative phase retrieval method is used for the sample image reconstruction. Recently, we succeeded in three-dimensional visualization of an unstained human chromosome by X-ray diffraction microscopy. It is the first hard X-ray tomography for cellular organelles. The reconstructed images revealed the internal axial structure, demonstrating an excellent image-contrast of the method.

Performance tests on iterative phase-retrieval methods in higher dimensions

Performance tests on iterative phase-retrieval methods in higher dimensions

An optical demonstration of ptychographical imaging for focussed-probe illumination

We demonstrate experimentally an application of ptychographical iterative phase retrieval using a focussed-probe geometry, analogous to the scanning transmission electron microscope (STEM). Unlike conventional iterative phase retrieval methods, the technique is inherently able to handle complex, softly varying illumination, due to its use of an update function rather than a hard support mask to localise the modulation of the object plane. This feature is crucial to application in electron microscopy, where hard-edged localisation of the illumination on the specimen is difficult to achieve experimentally.

Hard X Ray Holographic Diffraction Imaging

We determine the absolute electron density of a lithographically grown nanostructure with 25 nm resolution by combining hard x-ray Fourier transform holography with iterative phase retrieval methods. While holography immediately reveals an unambiguous image of the object, we deploy in addition iterative phase retrieval algorithms for pushing the resolution close to the diffraction limit. The use of hard (8 keV) x rays eliminates practically all constraints on sample environment and enables a destruction-free investigation of relatively thick or buried samples, making holographic diffraction imaging a very attractive tool for materials science. We note that the technique is ideally suited for subpicosecond imaging that will become possible with the emerging hard x-ray free-electron lasers.

Phase retrieval using multiple illumination wavelengths

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.

Ciphertext-only attack on double random phase encoding optical encryption system

Security analysis of optical encryption system based on double random phase encoding indicates that the system can be classified as a linear symmetric block-cipher cryptosystem, which may lead to a great vulnerability. Under the ciphertext-only attack (COA), an opponent can attack such a cryptosystem only on the basis of estimated support of wave function in the object plane with iterative phase retrieval methods, and subsequently deduce the phase keys in the Fourier plane easily. The ciphertext-only attack (COA) requires much less resources than other types of attacks. Estimated support of wave function in the object plane could have some translations relative to the true support, so retrieved wave function could also have translations in both the amplitude and the phase, leading to a translation of retrieved plaintext relative to original plaintext. However, attackers can take this translation as a priori knowledge to traverse estimated support in the object plane until finding the best estimated keys, which bring about the best decryption quality.

Iterative phase retrieval without support

An iterative phase retrieval method for nonperiodic objects has been developed from the charge-flipping algorithm proposed in crystallography. A combination of the hybrid input-output (HIO) algorithm and the flipping algorithm has greatly improved performance. In this combined algorithm the flipping algorithm serves to find the support (object boundary) dynamically, and the HIO part improves convergence and moves the algorithm out of local minima. It starts with a single intensity measurement in the Fourier domain and does not require a priori knowledge of the support in the image domain. This method is suitable for general image recovery from oversampled diffuse elastic x-ray and electron-diffraction intensities. The relationship between this algorithm and the output-output algorithm is elucidated.

Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm.

We propose an iterative phase retrieval method that uses a series of diffraction patterns, measured only in intensity, to solve for both amplitude and phase of the image wave function over a wide field of view and at wavelength-limited resolution. The new technique requires an aperture that is scanned to two or more positions over the object wave function. A simple implementation of the method is modeled and demonstrated, showing how the algorithm uses overlapping data in real space to resolve ambiguities in the solution. The technique opens up the possibility of practical transmission lensless microscopy at subatomic resolution using electrons, x rays, or nuclear particles.

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.

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 Defocused Images and Its Applications in Lithography

This paper discusses the application of iterative phase retrieval method to phase measurement in optical lithography. A simple algorithm was developed for reconstructing phase information from image intensity distributions measured under different defocus conditions. The phase distribution in the image plane and in the pupil plane are reconstructed from the images obtained using the micro-lithography simulation microscope, MSM100 (Zeiss). Two applications of this method in optical lithography meteorology were investigated. First, the phase transmission of phase-shifting masks was successfully measured both for alternating and for attenuating type masks. The reconstructed phase showed that the uniform background of light leaking from the opaque regions on the mask can affect the phase in the image plane. Second, the possibility of using this method in measuring aberrations of projection optics was examined.