Deterministic Phase Retrieval Research Articles

Accurate quantitative phase imaging by the transport of intensity equation: a mixed-transfer-function approach.

As a well-established deterministic phase retrieval approach, the transport of intensity equation (TIE) is able to recover the quantitative phase of a sample under coherent or partially coherent illumination with its through-focus intensity measurements. Nevertheless, the inherent paraxial approximation limits its validity to low-numerical-aperture imaging and slowly varying objects, precluding its application to high-resolution quantitative phase imaging (QPI). Alternatively, QPI can be achieved by phase deconvolution approaches based on the coherent contrast transfer function or partially coherent weak object transfer function (WOTF) without invoking paraxial approximation. But these methods are generally appropriate for "weakly scattering" samples in which the total phase delay induced by the object should be small. Consequently, high-resolution high-accuracy QPI of "nonweak" phase objects with fine details and large phase excursions remains a great challenge. In this Letter, we propose a mixed-transfer-function (MTF) approach to address the dilemma between measurement accuracy and imaging resolution. By effectively merging the phases reconstructed by TIE and WOTF in the frequency domain, the high-accuracy low-frequency phase "global" profile can be secured, and high-resolution high-frequency features can be well preserved simultaneously. Simulations and experimental results on a microlens array and unstained biological cells demonstrate the effectiveness of MTF.

Stability Estimates for Phase Retrieval from Discrete Gabor Measurements

Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016) and possibly severely ill-conditioned in finite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016). Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions (Alaifari in Found Comput Math 19(4):869–900, 2019). In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years (Bojarovska and Flinth in J Fourier Anal Appl 22(3):542–567, 2016; Eldar et al. in IEEE Signal Process Lett 22(5):638–642, 2014; Li et al. in IEEE Signal Process Lett 24(4):372–376, 2017; Nawab et al. in IEEE Trans Acoust Speech Signal Process 31(4):986–998, 1983).

Modeling classical wavefront sensors.

We present an image formation model for deterministic phase retrieval in propagation-based wavefront sensing, unifying analysis for classical wavefront sensors such as Shack-Hartmann (slopes tracking) and curvature sensors (based on Transport-of-Intensity Equation). We show how this model generalizes commonly seen formulas, including Transport-of-Intensity Equation, from small distances and beyond. Using this model, we analyze theoretically achievable lateral wavefront resolution in propagation-based deterministic wavefront sensing. Finally, via a prototype masked wavefront sensor, we show simultaneous bright field and phase imaging numerically recovered in real-time from a single-shot measurement.

Phase-contrast tomography from a series of single-shot measurements using a deterministic phase-retrieval method in X-ray imaging

An X-ray tomographic imaging method is proposed that is capable of reconstructing the complex refractive index of an object from a data set that involves only a single-shot image obtained at each tomographic view angle. This method is based on a deterministic phase retrieval method from a single diffraction intensity by use of an aperture array, and so the real and imaginary components, which are independent of each other, of the complex refractive index distribution can be reconstructed without object’s assumptions about negligible absorption or phase-attenuation duality, which have been frequently required in the other single-shot phase-retrieval methods for tomography. The validity of the method is shown in computer-simulated examples of three-dimensional object reconstruction.

Ultrashort pulse reconstruction using a deterministic phase retrieval method with Gaussian-envelope gates

A novel scheme of ultrashort pulse characterization is proposed that is capable of reconstructing the complex-valued envelope of a pulse in the time domain by use of a second harmonic generation system with two Gaussian gate pulses. In this scheme, we apply an analytic (noniterative) phase-retrieval method for coherent diffractive imaging in the space domain to the pulse characterization, and so the spectral phase of an unknown pulse can be retrieved from only two spectra of the unknown pulse modulated with a Gaussian gate and its time-shifted Gaussian one. Computer-simulated examples demonstrate that the pulse retrieval is robust. In particular, it is shown that the reconstruction of an unknown pulse using approximate Gaussian gates derived from replicas of the pulse itself is also effective, provided the spectrum of the unknown pulse has a gentle slope within the spectral bandwidth of the gate pulse.

Optical convolution for quantitative phase retrieval using the transport of intensity equation.

Propagation-based phase imaging using the transport of intensity equation (TIE) allows rapid, deterministic phase retrieval from defocused images. However, computational solutions to the TIE suffer from significant low-frequency noise artifacts and are unique up to the application of boundary conditions on the phase. We demonstrate that quantitative phase can be imaged directly at the detector for a class of pure-phase samples by appropriately patterning the illumination to solve the TIE through an optical convolution with the source. This can reduce noise artifacts, obviates the need for user-supplied boundary conditions and is demonstrated via simulation and experiment.

Hybrid single-beam reconstruction technique for slow and fast varying wave fields

An iterative single-beam wave field reconstruction technique that employs both non-paraxial, wave propagation based and paraxial deterministic phase retrieval techniques is presented. This approach overcomes two major obstacles that exist in the current state of the art techniques: iterative methods do not reconstruct slowly varying wave fields due to slow convergence and stagnation, and deterministic methods have paraxial limits, making the reconstructions of quickly varying object features impossible. In this work, a hybrid approach is reported that uses paraxial wave field corrections within iterative phase retrieval solvers. This technique is suitable for cases ranging from slow to fast varying wave fields, and unlike the currently available methods, can also reconstruct measurement objects with different regions of both slowly and quickly varying object features. It is further shown that this technique gives a higher accuracy than current single-beam phase retrieval techniques, and in comparison to the iterative methods, has a higher convergence speed.

Deterministic phase retrieval interferometry

An interferometry method based on reconstructed wavefronts obtained via deterministic phase retrieval with speckle illumination is demonstrated numerically and experimentally. A refractive sample is illuminated with a partially-developed speckle field and the transmitted light is detected and processed using the transport-of-intensity equation to retrieve the phases at different loading states. The difference between retrieved phase maps is then used to evaluate the refractive index change due to mechanical loading. The interferometry method affords a simple setup (because no reference beam is needed) and offers direct calculations of the phases and phase difference maps.

Strain mapping of LED devices by dark-field inline electron holography: Comparison between deterministic and iterative phase retrieval approaches

Dark-field inline electron holography has recently been established as a convenient method to map strain in semiconductor devices, combining high precision, low noise, sub-nm spatial resolution and fields-of-view larger than 1 μm. Here we compare two approaches to reconstruct the geometric phase from a transmission electron microscopy dark-field focal series and their effects on the strain measurement: the transport-of-intensity-equation (TIE) and a flux-preserving iterative approach. For this task, we used a GaN-based light emitting diode with a highly complex heterostructure as a model system. While the TIE relies on 3 images only but requires the optimization of two free parameters (defocus step and low-limit cut-off frequency), the iterative reconstruction algorithm involves no adjustable parameters and uses images recorded at 9 different planes of focus with quadratically increasing defocus values. Optimum parameters for the TIE-reconstruction could be identified. However, the iterative phase retrieval approach yields the strain values that agree best with the expected strain levels and provides also higher spatial resolution.

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.

Phase retrieval using an aperture-array filter under partially coherent illumination

Recently we have proposed a deterministic phase retrieval method using an aperture-array filter to reconstruct a complex-valued object from a single diffraction intensity pattern. We describe here the effect of quasi-monochromatic partially coherent illumination on the object reconstruction by the phase retrieval method, and then present the method of eliminating the effect of the partially coherent illumination from the diffraction intensity pattern via a simple Fourier deconvolution operation provided that the complex degree of spatial coherence of the illuminating beam is known. The usefulness of this method is shown in computer-simulated examples of the object reconstructions under Gaussian Schell-model partially coherent illumination.

Lensless coherent imaging by a deterministic phase retrieval method with an aperture-array filter

Recently we have proposed a noniterative and analytic phase retrieval method using the filter of an aperture array to reconstruct a complex-valued object from a diffraction intensity pattern [Phys. Rev. Lett.98, 223901 (2007)], but this method suffers from the restriction of the far-field condition of two distances between the object and the filter and between the filter and the detector for the intensity measurement. An improved method, which extends the adaptable condition of those distances to the region of Fresnel diffraction, is proposed here. In addition, the procedure for reducing the influence of noises on the phase retrieval is presented, in which the phase information contained in multiple groups of sampling data of a single intensity distribution is utilized. The usefulness of this method is shown in computer-simulated examples of the object reconstructions, including an object with phase vortices.

Phase contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems

We treat the problems of phase-contrast image formation, deterministic phase retrieval and aberration balancing, in the imaging of weak objects using two-dimensional shift-invariant linear imaging systems. Three classes of model sample are considered: weak phase objects, weak phase-amplitude objects and single-material weak phase-amplitude objects. For each class of sample we show how the various aberration coefficients, which characterise a given imaging system, contribute to the structure of the associated phase-contrast image. The corresponding inverse problem, of obtaining a closed-form expression for the input wave-field given one or more aberrated phase-contrast images of the same, is then examined. Two sample applications are considered: analyser-crystal phase-contrast imaging of weak objects using hard X-rays, and Zernike-type phase-contrast imaging. We close with a discussion of how coherent and incoherent aberrations may be “balanced” against one another, briefly mentioning the applications of this idea to both “deblur by defocus” and proximity-corrected X-ray lithography.

Deterministic phase retrieval from diffracted intensities speckle fields

We present a model where the phase of a wave front is obtained by processing the intensity of speckle field pattern. By using the light propagation equation and knowing the intensity of the wavefront it is possible to build a system of linear equations which can be solved to give the wanted phase information. Simulations are presented and the convergence of the solution is discussed.

Correlation between intensity and phase in monochromatic light

Analytical expressions to describe the phase gradient of monochromatic light by means of the three-dimensional intensity distribution are derived. With these formulas it is shown that the two-dimensional phase gradient in a plane can be completely determined from noninterferometric intensity measurements if the light propagates strictly in one direction. The analytical expressions are verified by means of numerical investigations on simulated speckle fields, and the results are discussed with respect to common deterministic phase retrieval approaches.

Phase-space interpretation of deterministic phase retrieval

Deterministic phase retrieval is reinterpreted in terms of phase-space optics. A novel derivation of the transport-of-intensity equation is presented based on the Wigner distribution function and the ambiguity function. The phase retrieval problem is formulated as estimating the local first-order moment of the Wigner function from intensity information. A comparison with phase-space tomography suggests a generalization of deterministic phase retrieval that provides larger flexibility for signal recovery. In addition, one particular numerical implementation of generalized deterministic phase retrieval is presented. Simulated intensity data are used to validate the method.

The holographic twin image problem: a deterministic phase solution

We describe a novel method for solving the twin image problem of in-line holography using a new technique for deterministic phase retrieval combined with numerical back-propagation of the reconstructed complex field. The technique presented here is applicable to any field described by the paraxial scalar wave equation, and is therefore equally applicable to X-ray, electron and neutron optics as well as visible light optics.

Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments

Experimental demonstrations of deterministic phase retrieval based on the Teague-Streibl irradiance transport equation are presented. A new technique is proposed, in which the transport equation is solved by the Fourier transform method for a periodic boundary condition with high spatial carrier frequency, which is created by making a light beam with unknown phase distribution pass through a grating. Quantitative phase measurements were performed by experiments without recourse to interferometry, and the results were found to be in good agreement with theory.