Abstract: An algorithm is described for reconstructing a symmetric three-dimensional image from its Fourier intensity that is sampled below the Nyquist rate. The study is motivated by an image reconstruction problem in macromolecular x-ray crystallography. Application of the algorithm to simulated crystallographic problems shows that it converges to the correct solution, with no initial phase information, where algorithms currently used in crystallography fail. The algorithm is potentially useful in a variety of situations in macromolecular crystallography. The results presented also lend support to the possibility of ab initio phase retrieval in macromolecular crystallography when sufficient a priori information is available. Other applications in image reconstruction are possible.

Abstract: A phase-retrieval procedure based on the maximum-entropy method is applied to infrared–visible sum-frequency generation spectroscopy. Several typical effects on the error phase are also examined with the aid of a known theoretical model. The interference between the resonant and the nonresonant parts changes the behavior of the error phase. Even in some cases when the nonresonant part is complex, the error phase becomes like a step function. This result contradicts the smoothness assumption for the error phase, and the whole phase-retrieval procedure breaks down in these cases. A comparison of the results of phase-retrieval procedures between infrared–visible sum-frequency generation and coherent anti-Stokes Raman scattering spectra is made. Some ideas that worked well in previous analyses of coherent anti-Stokes Raman scattering spectra become inapplicable in the infrared–visible sum-frequency generation spectra in spite of the resemblance of their line shape functions.

Abstract: Phase and amplitude images have been reconstructed from data collected in a scanning transmission x-ray microscope by applying the method of Wigner-distribution deconvolution. This required collecting coherent microdiffraction patterns at each point of a twodimensional scan of an object and then deconvolving the four-dimensional Wigner-distribution function of the lens from the data set. The process essentially analyses the interference which occurs in the microdiffraction plane and which modulates as the object is scanned. The imageprocessing steps required to deconvolve experimental data are described. These steps result in the reconstructions of diffraction-limited phased images, to a spatial-frequency cutoff of 1/45 nm -1 . The estimated accuracy of the images is 0.05 rad in phase and 10% in amplitude. Data were collected at an x-ray wavelength of 3.1 nm.

Abstract: Structure factors with known magnitude and phase are constructed for seven different molecular crystals in non-centrosymmetric space groups and used to calibrate the retrieval of structure-factor phase information with several different scattering factor models. Monopole models are confirmed to be inadequate, yielding poor estimates of phases and underestimating the Fourier deformation electron density by as much as 50%. Multipole models are generally successful, with the conspicuous exception of hexamethylenetetramine (HMT). The results confirm the known difficulty of phasing structure-factor magnitudes for HMT, but reveal a broad spectrum of behaviour of phase retrieval in various non-centrosymmetric space groups, depending partly on the number of reflections with restricted phases. For several systems (urea, hydrogen peroxide, l-alanine and borazine) the results confirm that multipole refinements of scattering factor models against experimental data are capable of yielding highly accurate phases and hence reliable electron distributions. For the exceptional case of HMT a simple eigenvalue filtering technique enables retrieval of phases and electron densities with an accuracy comparable to the best of the other non-centrosymmetric systems studied.

Abstract: The effect of focus anisoplanatism upon the performance of an astronomical laser guide star (LGS) adaptive optics (AO) system can in principle be reduced if the lowest order wavefront aberrations are sensed and corrected using a natural guide star (NGS). For this approach to be useful, the noise performance of the wavefront sensor (WFS) used for the NGS measurements must be optimized to enable operation with the dimmest possible source. Two candidate sensors for this application are the Shack-Hartmann sensor and “phase-diverse phase retrieval,” a comparatively novel approach in which the phase distortion is estimated from two or more well-sampled, full-aperture images of the NGS measured with known adjustments applied to the phase profile. We present analysis and simulation results on the noise-limited performance of these two methods for a sample LGS AO observing scenario. The common parameters for this comparison are the NGS signal level, the sensing wavelength, the second-order statistics of the phase distortion, and the RMS detector read noise. Free parameters for the two approaches are the Shack-Hartmann subaperture geometry, the focus biases used for the phase-diversity measurements, and the algorithms used to estimate the wavefront. We find that phase-diverse phase retrieval provides consistently superior wavefront estimation accuracy when the NGS signal level is high. For lower NGS signal levels on the order of 103 photodetection events, the Shack-Hartmann (phase diversity) approach is preferred at a RMS detector read noise level of 5 (0) electrons/pixel.

Abstract: A novel algorithm for a reconstruction of the phase shift profile produced by a transparent object in a coherent wave, from a set of two recorded intensity distributions is presented. Contrary to well known algorithms of in-line holography, the method works under the near-field condition where the size of the first Fresnel zone is much less than the characteristic size of the object's details. Such a condition is typical for an in-line holography experimental setup with the use of coherent high energy x-rays (E > 20 keV) produced by synchrotron radiation sources of the third generation like ESRF. The novel algorithm is fast and insensitive to a partial loss of coherence or weak detector resolution. The method can be applied to x-ray refraction tomography.

Abstract: Publisher Summary This chapter discusses dispersion relations and phase retrieval in optical spectroscopy. Kramers–Kronig (KK) relations have their origin in the principle of causality and are therefore connected to fundamental physics. They have been used in data inversion of linear optical constants. The KK relations also hold in most cases of nonlinear optical spectra. The shortcoming of KK analysis is the need for data extrapolation. The maximum entropy model helps in phase retrieval problems arising in both linear and nonlinear optical spectroscopies. There is no need for data extrapolation. Sum rules provide information about the microscopic properties of systems. Sum rules of general validity can be found for linear and nonlinear optical constants. The chapter discusses the Kramers-Kronig relations appearing in linear and nonlinear optical spectroscopy. In addition, the phase retrieval in linear reflection spectroscopy and in nonlinear optics with the aid of the maximum entropy procedure is considered. The chapter also reviews mathematical methods for optical spectrum analysis.

Abstract: A fiber-optic vibration sensor based on polarization and phase-step interferometry is reported. Left- and right-circularly polarized light coming back from the reference and signal arms of a modified Michelson fiber interferometer is processed with an array of five linear analyzers separated angular steps of /spl pi//4 (rad) from each other. Thus, five interference patterns are acquired simultaneously and the dynamical phase retrieval problem is reduced to five-step interferometry in the time-domain. A vibration sensor as described above was built by the authors and its performance was investigated.

Abstract: We demonstrate a powerful new tool for real-time imaging of ultrafast phase shifts without computation. Additionally, this technique can, with interferogram analysis and iterative phase retrieval, recover the intensity and phase of three pulses in a single shot. Results of ultrafast time-resolved coherent spectroscopy of cross-phase modulation in fused silica and ultrafast ionization fronts are reported.

Abstract: The Gerchberg-Saxton (GS) algorithm and its generalizations have been the main tools for phase retrieval. Unfortunately, it has been observed that the reconstruction using these algorithms does not always converge to the correct result even if the desired solution satisfies the uniqueness condition. In this paper, we propose a new deautocorrelation algorithm and a few auxiliary techniques. We recommend that a combination of the iterative Fourier transform (IFT) algorithm with our new algorithm and techniques can improve the probability of success of phase retrieval. A pragmatic procedure is illustrated. Different reconstruction examples that are difficult to reconstructed using the single IFT algorithm are used to show the robustness and effectiveness of the new combination of algorithms. If the given Fourier modulus data contain no noise, it is sometimes possible to get a perfect reconstruction. Even when the signal-to-noise ratio (SNR) of the Fourier modulus data is only 10 dB, a meaningful result remains reachable for our examples. A concept concerning the intrinsic ambiguity of phase retrieval is suggested. We emphasize the necessity of verification of the solution, since the available phase retrieval algorithms are incompetent for distinguishing between an intrinsically ambiguous solution and the true solution.

Abstract: Microwave antenna imaging techniques are a practical and popular method for antenna diagnostic analysis. Phase retrieval methods, however, are just beginning to emerge as an alternative microwave antenna measurements technique when phase cannot be directly measured. This article focuses on recent advances in microwave antenna imaging, diagnostic techniques, and phase retrieval methods for bi-polar planar near-field antenna measurements. An overview of the bi-polar planar near-field technique is included. The application of optimal sampling interpolation, holographic imaging and diagnostics, and iterative Fourier phase retrieval for the bi-polar planar near-field modality is explored in detail. Experimental results for a waveguide-fed slot array antenna are presented to illustrate these methods. © 1997 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 8, 396–406, 1997

Abstract: Antenna diagnostics techniques from phaseless measurements of the near field are receiving more and more attention, because they are the only viable methods at millimeter and submillimeter frequencies for the testing of antennas, with a diameter of some thousands of wavelengths and with low sidelobe levels. Two techniques for predicting the radiation pattern of the antenna from intensity measurements of the near field are currently under consideration. The first one is the Gabor holography which employs the interference intensity distribution between the antenna under test and a known reference source. However, such a technique presents some drawbacks concerning the measurement procedure. The second technique is phase retrieval, which requires a knowledge of the amplitude information of the near field over two scanning surfaces. Since the reference wave is not required, such a method is not affected by stability problems and the achievable accuracy is limited only by the low intensity drift. However, the phase retrieval method requires a knowledge of the intensity distributions over two scanning surfaces; this can be overcome when enhanced holographic techniques, which require three measurements of intensity distributions over a single surface, are employed. In any case, for the phase retrieval method the problem of the determination of a complex function, i.e. the near field, from amplitude data arises. The performance of the two approaches are compared with respect to the local minima problem by investigating the different mathematical properties.

Abstract: Aberrations of the coherent wavefront are analyzed using a phase Zernike filter. Developed iterative methods allow us to design a filter that decomposes the analyzed light field into a set of diffraction orders with amplitudes proportional to the circular Zernike polynomials. We also apply the algorithm to the calculation of the light field phase from measurements of the modules of decomposition coefficients. Operation of a 25-channel filter is simulated.

Abstract: The paper discusses how phase retrieval can be employed as a robust technique for estimating the wavefront distortion using a lenslet array. It is shown that the principal problem of phase retrieval, namely divergence is considerably reduced when the aperture is subdivided. The results obtained compare favorably with the alternative approach of phase diversity. Furthermore, the introduction of prior information, in the form of statistical information of the distortion, is shown to considerably enhance the success of the phase retrieval.

Abstract: The usefulness of stochastic algorithms to retrieve the exit-plane wave function from periodic high-resolution electron microscopic images is investigated. In contrast to “classical” focal series reconstruction methods which need approximately twenty input images, fully non-linear reconstructions of periodic wave functions are possible in most cases from only two images using stochastic algorithms. The efficiency and accuracy of two different algorithms, a Simulated Annealing algorithm and a genetic algorithm, are compared with each other. Simulated high-resolution images of the intermetallic alloy Ni 4 Mo and experimental images of the high-T c superconductor YBa 2 Cu 3 O 7 are used as a test input.

Abstract: Phase retrieval methods have just recently evolved into a practical and accurate means for recovering phase information from phaseless near-field antenna measurements. The appeal of phaseless near-field measurements is due to, for example, the inability to accurately measure phase at high frequencies or the prohibitive cost of vector measurement equipment. A phase retrieval algorithm for bi-polar planar near-field measurements has recently been proposed, implemented, and successfully demonstrated using measured bipolar near-field data. In this paper, focus is turned to the antenna diagnostics problem. Measurement results are presented for a waveguide-fed slot array antenna in which an anomaly has been imposed on the aperture distribution. The phase retrieval algorithm reported previously is then applied, with no a priori knowledge of the imposed anomaly, and results are compared to the case when the phase is retained in the conventional near-field data processing.

Abstract: We demonstrate a powerful new tool for real-time imaging of ultrafast phase shifts without computation. Additionally, this technique can, with interferogram analysis and iterative phase retrieval, recover the intensity and phase of three pulses in a single shot. Results of ultrafast time-resolved coherent spectroscopy of cross-phase modulation in fused silica and ultrafast ionization fronts are reported.

Abstract: In this paper a new method for 2D complex wave reconstruction is developed by measurement of 3D intensity in a first order optical system under partially coherent illumination. The system includes two cylindrical lenses oriented along two perpendicular axes in a plane perpendicular to the propagation axis. When the parameters of the 2D lens system are set to satisfy the conditions of fractional Fourier transform independently in both dimensions, the input complex wave can be determined by measurements of intensities in a set of planes, and the mutual intensity is calculated through integrals of distributional coordinates and the fractional orders of fractional Fourier transform. As a result, the case of partially coherent illumination of Gaussian Schell-model is studied. In the numerical simulation it is found that phase structure in the input planes might be retrieved by this method in this system. The approach may provide an alternative way of recovering the phase object.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Abstract: A high-resolution phase-retrieval method for an imaging system with scanning illumination that is capable of reconstructing the modulus and phase of an object without a holographic reference wave is proposed. Reconstruction involves the synthesis of the reconstructed objects from the data of zero- and higher-frequency components of two Fourier intensity measurements: the Fourier intensity of the product of the object and a probe beam that is scanned across the object and the Fourier intensity of the product of the object and a probe beam that is modulated with an exponential filter. This method improves on the resolution of a reconstructed object by previous methods that make use of the data of only the zero-frequency components of the two Fourier intensities. In addition, phase retrieval in the scanning-illumination system with a probe of unknown modulus can be treated by use of the synthetic procedure, provided the phase of the probe function is a constant or a known distribution. Computer-simulated examples of the reconstruction of one- and two-dimensional complex objects demonstrate that reconstruction is robust.

Abstract: We discuss the problem associated with obtaining an image of an object from the magnitude of its Fourier transform. This problem arises in many imaging applications. We discuss some new ideas developed to address this problem and describe constraints on the object function that can lead to its Fourier transform having only real zeros, thereby eliminating the phase retrieval problem.

Abstract: There are several important remote sensing applications where the development of Ground Penetrating Synthetic Aperture Radar (GPENSAR) is the logical approach, e.g., searching for buried military facilities, minefield mapping, survey of underground pipelines. Penetration of sufficient soil depth for useful results require a SAR to operate at VHF/UHF frequencies, e.g., 200 - 300 MHz. At these frequencies a satellite SAR will encounter substantial distortion in the double passage of the SAR signal through the ionosphere. One of the ionospheric distortions is equivalent the phase aberrations caused in imaging through the turbulent atmosphere, and the problem of phase retrieval for the GPENSAR becomes a necessity. For GPENSR there are imaging concepts that exploit dual polarization radiation of the SAR pulse. The phase retrieval problem then becomes one of compensation for the phase aberrations induced in each of the polarization components returned to the satellite receiver. We discuss the use of the two polarizations to cancel the ionospheric phase aberrations. Unfortunately, the resulting signal has only relative phase of the two polarizations. We discuss an algorithm for the retrieval of the absolute phase. The algorithm is based on an optimization approach. Although phase retrieval by optimization is difficult because of local minima, the retrieval of absolute phase in the dual polarization case is substantially less difficult, because the two polarizations constrain the solution sufficiently to eliminate many local minima.

Abstract: Our objective in this report is to develop methods to determine the output pupil wavefront using intensity measurements directly from the science detector. This wavefront can then be used to determine a reference wavefront which will precorrect for the non-common-path aberrations and produce the desired wavefront at the science detector. We describe two phase retrieval algorithms that can be used and a set of simulation studies of AO system calibration. We present the initial experimental results of applying this technique in calibration of the Lick Observatory laser guidestar AO system in a later paper.

Abstract: Complex (magnitude and phase) measurements of the near field of a radiating antenna over a known surface (usually a plane, cylinder, or sphere) can be used to determine its far-field radiation pattern using near-field to far-field Fourier transformations. Standard gain horn antennas or open-ended waveguide are often used to probe the near field. This requires the time-consuming positioning of the probe antenna to several thousand positions in the radiating near field of the antenna under test. Experimental errors are introduced into the near-field measurements by mechanical probe position inaccuracies and electrical probe interactions with the antenna under test and probe correction errors. A minimally perturbing infrared (IR) imaging technique can also be used to map the near fields of the antenna. This measurement technique is much simpler and easier to use than the probe method and eliminates probe position errors and probe correction errors. Current IR imaging techniques, which have been successfully used to rapidly map the relative magnitude of a radiating field at many locations (mXn camera pixels per image captured) over a surface, however, suffer from an inability to determine phase information. This paper describes a method for determining the absolute magnitude and relative phase data from these phaseless IR measurements by using techniques derived from optical holography. Form a pair of microwave holograms and magnitude only measurements, the complete near field (magnitude and phase) of the antenna can be determined. Once obtained, this data can then be used to determine the antenna far field pattern by conventional Fourier near-field to far-field transformation techniques.

Abstract: A two deformable mirror concept for correcting amplitude effects in laser beam projection through the turbulent atmosphere is presented. This system uses a deformable mirror and a Fourier transforming mirror to adjust the amplitude of the wave front in the telescope pupil. A second deformable mirror is used to correct the phase of the wave front before it leaves the aperture. The phase applied to the deformable mirror used for controlling the beam amplitude is obtained using a technique based on the Fienup phase retrieval algorithm. One dimensional simulation results are presented which indicate that the average on-axis energy is improved by 15% or more on average, and the 90% encircled energy radius can be reduced by up to 30% on average.

Abstract: The advantages of spatial phase shifting (SPS) compared with temporal phase shifting in the field of electronic speckle pattern interferometry are described. Some periodic phase reconstruction errors occurring in SPS are discussed. It is shown that these errors become minimal for a spatial phase-shift angle of 2π/3. Furthermore, a modified phase reconstruction formula is presented by which the noise in the reconstructed phase map is reduced.

Abstract: A simplified method for aberration measurement in confocal microscopy is proposed. The method is based on analysis of the Fourier spectrum of the confocal interference axial response. With...

Abstract: A design for diffractive phase elements (DPE’s) that implement wavelength demultiplexing and annular focusing simultaneously is presented based on the general theory of amplitude phase retrieval. The optical system considered is illuminated by a beam of polychromatic light consisting of several wavelengths. With the use of an iterative algorithm the pattern of a surface-relief DPE can be determined by solving the relevant equations. Computer simulations are carried out for several model examples. The calculation results show that the designed DPE’s can satisfactorily achieve the desired functions.