Abstract: Generation of arbitrarily complex intensity profiles by using phase-only Fourier-domain pulse shaping has thus far been performed by using optimization algorithms to find the optimal phase profile. We present an alternative method based on the Gerchberg–Saxton (GS) algorithm that converges at least several hundred times faster and is independent of the number of phase points and gray levels. The numerical and experimental performance of the GS algorithm is characterized and compared against a genetic algorithm. An application of amplified GS-synthesized waveforms to large-amplitude coherent phonon generation and destruction is demonstrated.

Abstract: Phase-contrast X-ray imaging including tomographic configuration using a crystal X-ray interferometer is reviewed. The imaging principle based on phase retrieval is described, and some demonstrations showing its high sensitivity are presented on the observations of animal and human cancerous tissues.

Abstract: Phase can be retrieved from intensity measurements with the intensity transport equation. Three-dimensional image formation of weak phase objects based on this method is investigated. It is shown that, although the refractive index of a thin object can be measured, the three-dimensional variation of refractive index of an arbitrary object cannot, in general, be reconstructed, as spatial frequencies with a zero-axial component are not detected. However, this may not be a problem if regions with known refractive index are present in the sample.

Abstract: We introduced recently phase measurements usually performed in interferometry to the domain of image processing and intelligent vision [IEEE Trans. Instrum. Meas.49, 867 (2000)]. Our purpose is to sense with a high accuracy the position, orientation, and displacement of two-dimensional (2D) surfaces observed by a static vision system. We report on significant improvements of the method. Experimental measurements reveal a peak-valley noise of approximately 10-2 CCD pixel, corresponding approximately to a 10-3 period of the phase reference pattern. Then the observation of 10 µm scaled features enables an accuracy of a few nm in the position sensing of the phase reference pattern for the extended 2D measurement range.

Abstract: Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behavior of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyze the average-case convergence at fixed points. Optimal values of the gamma parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces.

Abstract: We present the formulation of the moment method applied to the determination of phase profiles of microwave beams from known amplitudes. While traditional approaches to this problem employ an iterative error-reduction algorithm, the irradiance moment technique calculates a two-dimensional polynomial phasefront based on the moments of weighted intensity measurements. This novel formulation has the very important advantage of quantifying measurement error, thus allowing for its possible reduction. The validity of the irradiance moment approach is tested and confirmed by examining a simple case of an ideal Gaussian beam with and without measurement errors. The effectiveness of this approach is further demonstrated by applying intensity measurements from cold-test gyrotron data to produce a phasefront solution calculated via the irradiance moment technique. The accuracy of these results is shown to be comparable with that obtained from the previously developed iteration method.

Abstract: We analyze a quantum search protocol to retrieve phase information from a Rydberg-atom data register using a subpicosecond half-cycle electric field pulse. Calculations show that the half-cycle pulse can perform the phaseretrieval only within a range of peak field values. By varying the phases of the constituent orbitals of the Rydberg wave packet register, we demonstrate coherent control of the phase retrieval process. By specially programming the phases of the orbitals comprising the initial wave packet, we show that it is possible to use the search method as a way to synthesize single energy eigenstates.

Abstract: A self-calibrating algorithm for phase-shift interferometry is described that is able to cancel the effect of accidental relative tilts that may occur during phase stepping. The algorithm is able to retrieve both the phase steps and the tilts that accompany them. Only three phase-shifted interferograms are needed, and no other information about the intentional phase shifts or possible tilts has to be supplied. This purpose is achieved by division of the interferogram space into blocks on which a previously reported self-calibrating algorithm is applied and the actual values of the local phase shifts are calculated. The information thus obtained is used for extracting the global shift and tilt values. Further improvement in the results is achieved by means of a fitting routine.

Abstract: A new technique for the least-mean-squares (LMS) phase-unwrapping method is developed that incorporates the concept of branch cuts between phase singularities (residues), which are usually associated with the path-following gradient integration technique. These branch cuts are introduced by decomposition of the least-mean-squares unwrapped phase into two separate components. The first results from the transverse part of the wrapped phase gradient, which is induced by residues of the original phase, and the second component is due to a potential component, independent of the residues. This decomposition allows the reconstruction of phase patterns with a high level of accuracy and consistency with the initial (wrapped) phase, even when only partial knowledge of the placement of branch cuts between residues is available. We show how the residue-induced phase, ignored by conventional LMS phase estimators, is reconstructed for a given boundary-value problem. The method is illustrated with interferometric quality-control measurements of optical fiber-connector terminations and also with synthetic aperture radar interferometry. These experiments demonstrate the high accuracy of the method in practical situations in which only a limited number of branch cuts are available.

Abstract: We demonstrate the feasibility of implementing the generalized phase-contrast (GPC) method in a planar-integrated micro-optics platform. An entire 4-f diffractive lens system with a Fourier plane phase-only filter has been designed and manufactured in a 50-mm-diameter fused-silica substrate by use of a four-level multimask lithography. Experimental results show successful conversion of an input phase distribution into a high-contrast intensity distribution. The diffraction efficiency of the optical components is found to limit the operating range of the GPC system.

Abstract: The main thrust of this project was to develop a process including data-gathering and processing techniques that would consistently and without fail discover the phase distribution of a wave front. In the course of the work, new thoughts on data gathering evolved to the point where a novel type of intensity picture bearing little or no resemblance to the wave front or its phase distribution was required for mathematical processing to achieve the phase distribution. For convenience these information images have been dubbed phasorgrams. Phasorgrams are recorded in the diffractive plane or in the image plane of an application device. Additional data, while helpful, are not required, thus eliminating the need for data from two Fourier conjugate planes as in the well-known method of Gerchberg and Saxton. The mathematical processing of these data is carried out by an iterative algorithm which also is new. This algorithm, given good data, has not failed to achieve the phase distribution. Contemporary devices hav...

Abstract: We demonstrate quantitative phase imaging of a neutral atomic beam by using a noninterferometric technique. A collimated thermal atomic beam is phase shifted by an off-resonant traveling laser beam with both a Gaussian and a TEM01 profile and with both red and blue detuning of as much as 50 GHz. Phase variations of more than 1000 rad were recovered from velocity-selective measurements of the propagation of the atomic beam and were found to be in quantitative agreement with theoretical predictions based on independently measured phase object intensity profiles and detunings.

Abstract: A novel technique called a two-dimensional digital phase-locked loop (DPLL) for fringe pattern demodulation is presented. This algorithm is more suitable for demodulation of fringe patterns with varying phase in two directions than the existing DPLL techniques that assume that the phase of the fringe patterns varies only in one direction. The two-dimensional DPLL technique assumes that the phase of a fringe pattern is continuous in both directions and takes advantage of the phase continuity; consequently, the algorithm has better noise performance than the existing DPLL schemes. The two-dimensional DPLL algorithm is also suitable for demodulation of fringe patterns with low sampling rates, and it outperforms the Fourier fringe analysis technique in this aspect.

Abstract: The amplitude and the phase of a multimode vertical-cavity surface-emitting laser (VCSEL) are recovered by a simple tomographic procedure based on the ambiguity function. The results are of considerable importance for any field of optics in which VCSELs are employed.

Abstract: Quasi-monochromatic light will form laser speckle upon reflection from a rough object. This laser speckle provides information about the shape of the illuminated object. In a prior paper [J. Opt. Soc. Am. A19, 444 (2002)], it was shown that two intensities of two speckle patterns and their interference are sufficient to produce an unambiguous (except for object translation) band-limited image of the object, based on a root-matching technique described therein, in the absence of measurement error and in the case of distinct roots of the field polynomials and their complex conjugates. On the other hand, algorithms based on the root-matching technique are found to be slow and sensitive to noise. So motivated, several other techniques are applied to the problem, including phase retrieval, expectation maximization, and statistical maximization. The phase-retrieval and expectation-maximization techniques proved to be most effective for reconstructions of complex objects larger than 10 pixels across, and high-quality images were formed by using three independent sets of two-field data (three frames of two-wavelength data), each comprising two speckle intensity patterns and their interference. Two additional results of note are reported. First, the expectation-maximization algorithm produced relatively good images when three or more frames each of only one speckle intensity pattern (data at just one wavelength) were used and second, the phase-retrieval algorithm when only the object autocorrelation was used also produced relatively good images for the chosen test object.

Abstract: The state of the art for solving the phase retrieval problem in two dimensions relies heavily on the algorithms proposed by Gerchbercy, Saxton, and Fienup. Despite the widespread use of these algorithms, current mathematical theory cannot explain their remarkable success. It is already known that the Gerchberg-Saxton algorithm is a nonconvex version of method of alternating projections. In this paper, we show that two other prominent phase retrieval methods also have well known counterparts in the world of convex optimization algorithms: Fienup's basic input-output algorithm corresponds to Dykstra's algorithm, and Fienup's hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. This work provides a theoretical framework to better understand and, potentially, improve existing phase recovery algorithms.

Abstract: We consider the reconstruction of an optical wavefront by means of the interference of a light beam with a diffracted copy of itself. We present the basis of the method and a numerical algorithm for phase retrieval from the experimental data.

Abstract: The Misell (1973) phase retrieval algorithm for microwave holography requires the capability to axially move an antenna's subreflector or feed to obtain a defocused far-field magnitude pattern. A microwave lens may be used as an alternative method of defocusing an antenna. A metal plate lens has been designed and constructed for use in phase retrieval holography at the Georgia Tech Woodbury Research Facility. Its phase variation across the aperture was chosen to have a parabolic profile in order to imitate the effect of subreflector or feed translation. A two-dimensional (2-D) finite-difference time-domain simulation has been performed to characterize the lens. Measurements, using the measured aperture phase of the lens transfer function as a phase correction term in the Misell algorithm, show the results to be comparable with that of phase coherent holography. Other types of microwave lenses such as a Fresnel zone plate, a bed of circular waveguides, and a standard dielectric lens are also being studied.

Abstract: The technique of the multiple phases retrieval algorithm (MPRA) for designing optical security and verification systems is proposed in this paper. This technique is based on a 4-f optical correlator, which is a common architecture for optical image encryption and verification systems. In the proposed systems, however, two or more phase masks are iteratively retrieved by using the MPRA to obtain the target image. The convergent speed of the iteration process in the MPRA is significantly increased and the recovered image is much more similar to the target one than those in previous approaches. Moreover, the system security is increased since only the pair-wise retrieved phase masks can correctly recover the target images. To avoid carrying two phase keys, one of the phase mask serves as the key and the other phase mask can be stored in the database of the security system as an active lock. Finally, according to our simulation results, the misalignment effects for the phase mask in the Fourier plane are more series than that in the input plane.

Abstract: A novel technique called a two-frame digital phase-locked loop for fringe pattern demodulation is presented. In this scheme, two fringe patterns with different spatial carrier frequencies are grabbed for an object. A digital phase-locked loop algorithm tracks and demodulates the phase difference between both fringe patterns by employing the wrapped phase components of one of the fringe patterns as a reference to demodulate the second fringe pattern. The desired phase information can be extracted from the demodulated phase difference. We tested the algorithm experimentally using real fringe patterns. The technique is shown to be suitable for noncontact measurement of objects with rapid surface variations, and it outperforms the Fourier fringe analysis technique in this aspect. Phase maps produced with this algorithm are noisy in comparison with phase maps generated with the Fourier fringe analysis technique.

Abstract: Phase retrieval is a nonlinear technique used to recover the phase in the Fourier domain using intensity measurements at the image plane and additional constraints. We describe a method to solve the phase retrieval problem using linear iterations near the solution, which provides both analytical insight into phase retrieval and numerical results. The algorithm finds the maximum a posteriori estimate of the phase using prior information about the statistics of the noise and the phase, and it was found to converge well in practice. When phase retrieval is performed on data from subdivided apertures, there is a loss of information regarding the relative piston terms of the subapertures. This error is quantified. We find that there is a smaller wavefront error when estimating the phase from a full aperture rather than from a subdivided aperture. Using a combination of intensity measurements from full and subdivided apertures results in a small improvement at very high photon levels only.

Abstract: The control of two deformable mirrors for compensation of time-varying fluctuations in the complex field that results from wave propagation through a turbulent medium is considered. Iterative vector space projection methods are utilized to determine the control commands to be applied to the two deformable mirrors. Convergence of the iterative algorithm is accelerated when the algorithm is initialized, at each measurement period, with the values for the phase commands obtained from the previous measurement period. Furthermore, it is found that, if the sample frequency is sufficiently greater than the Greenwood frequency, then only a single iterative step at each measurement period is required to obtain good compensation of both amplitude and phase fluctuations.

Abstract: The main properties of the power filtering operation in the fractional Fourier domain and its relationship to the differentiation operation are considered. The application of linear power filtering for solving the phase retrieval problem from intensity distributions only is proposed. The optical configuration for the experimental realization of the method is discussed.

Abstract: The noniterative phase-retrieval method by use of Gaussian filtering is applied to the reconstruction of phase objects from experimental far-field intensities. In this method, the complex amplitude of transmitted light through an object is reconstructed from three far-field intensities, which are measured with the modulation of the object by laterally shifted and unshifted Gaussian filters. In the experiment, the amplitude of a Gaussian beam illuminating objects is utilized as a Gaussian filter, and, as the phase objects, a converging lens with a small exit pupil and a plastic fiber immersed in optical adhesive are used. The experimental results show that the Gaussian beam of a laser is capable of retrieving the phases of those objects with the accuracy of the range from ∼1/10 to 1/4 of the laser’s wavelength.