Phase retrieval algorithms: a comparison.
TL;DR: Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods and it is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm forThe problem of two intensity measurements converge.
Abstract: Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods. Both the problem of phase retrieval from two intensity measurements (in electron microscopy or wave front sensing) and the problem of phase retrieval from a single intensity measurement plus a non-negativity constraint (in astronomy) are considered, with emphasis on the latter. It is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm for the problem of two intensity measurements converge. The error-reduction algorithm is also shown to be closely related to the steepest-descent method. Other algorithms, including the input-output algorithm and the conjugate-gradient method, are shown to converge in practice much faster than the error-reduction algorithm. Examples are shown.
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TL;DR: Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens was proposed in this paper, where the authors extended the methodology to allow the imaging of micro-scale specimens.
Abstract: Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens
1,791 citations
TL;DR: It is suggested that lower crustal foundering occurred within the North China craton during the Late Jurassic, and thus provides constraints on the timing of lithosphere removal beneath the NorthChina craton.
Abstract: Foundering of mafic lower continental crust into underlying convecting mantle has been proposed as one means to explain the unusually evolved chemical composition of Earth's continental crust, yet direct evidence of this process has been scarce. Here we report that Late Jurassic high-magnesium andesites, dacites and adakites (siliceous lavas with high strontium and low heavy-rare-earth element and yttrium contents) from the North China craton have chemical and petrographic features consistent with their origin as partial melts of eclogite that subsequently interacted with mantle peridotite. Similar features observed in adakites and some Archaean sodium-rich granitoids of the tonalite-trondhjemite-granodiorite series have been interpreted to result from interaction of slab melts with the mantle wedge. Unlike their arc-related counterparts, however, the Chinese magmas carry inherited Archaean zircons and have neodymium and strontium isotopic compositions overlapping those of eclogite xenoliths derived from the lower crust of the North China craton. Such features cannot be produced by crustal assimilation of slab melts, given the high Mg#, nickel and chromium contents of the lavas. We infer that the Chinese lavas derive from ancient mafic lower crust that foundered into the convecting mantle and subsequently melted and interacted with peridotite. We suggest that lower crustal foundering occurred within the North China craton during the Late Jurassic, and thus provides constraints on the timing of lithosphere removal beneath the North China craton.
1,495 citations
TL;DR: In this article, the authors summarize the problem of measuring an ultrashort laser pulse and describe in detail a technique that completely characterizes a pulse in time: frequency-resolved optical gating.
Abstract: We summarize the problem of measuring an ultrashort laser pulse and describe in detail a technique that completely characterizes a pulse in time: frequency-resolved optical gating. Emphasis is placed on the choice of experimental beam geometry and the implementation of the iterative phase-retrieval algorithm that together yield an accurate measurement of the pulse time-dependent intensity and phase over a wide range of circumstances. We compare several commonly used beam geometries, displaying sample traces for each and showing where each is appropriate, and we give a detailed description of the pulse-retrieval algorithm for each of these cases.
1,447 citations
TL;DR: An imaging method, termed Fourier ptychographic microscopy (FPM), which iteratively stitches together a number of variably illuminated, low-resolution intensity images in Fourier space to produce a wide-field, high-resolution complex sample image, which can also correct for aberrations and digitally extend a microscope's depth-of-focus beyond the physical limitations of its optics.
Abstract: We report an imaging method, termed Fourier ptychographic microscopy (FPM), which iteratively stitches together a
number of variably illuminated, low-resolution intensity images in Fourier space to produce a wide-field, high-resolution
complex sample image. By adopting a wavefront correction strategy, the FPM method can also correct for aberrations and
digitally extend a microscope’s depth of focus beyond the physical limitations of its optics. As a demonstration, we built a
microscope prototype with a resolution of 0.78 µm, a field of view of ∼120 mm^2 and a resolution-invariant depth of focus
of 0.3 mm (characterized at 632 nm). Gigapixel colour images of histology slides verify successful FPM operation. The
reported imaging procedure transforms the general challenge of high-throughput, high-resolution microscopy from one
that is coupled to the physical limitations of the system’s optics to one that is solvable through computation.
1,363 citations
Cites background from "Phase retrieval algorithms: a compa..."
...& Fienup, J. R. Phase retrieval with transverse translation diversity: a nonlinear optimization approach. Optics express 16, 7264-7278 (2008). 40 Zheng, G....
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TL;DR: In this article, the propagation of phase and irradiance are derived, and a Green's function solution for the phase in terms of irradiance and perimeter phase values is given A measurement scheme is discussed, and the results of a numerical simulation are given Both circular and slit pupils are considered.
Abstract: Equations for the propagation of phase and irradiance are derived, and a Green’s function solution for the phase in terms of irradiance and perimeter phase values is given A measurement scheme is discussed, and the results of a numerical simulation are given Both circular and slit pupils are considered An appendix discusses the local validity of the parabolic-wave equation based on the factorized Helmholtz equation approach to the Rayleigh–Sommerfeld and Fresnel diffraction theories Expressions for the diffracted-wave field in the near-field region are given
1,310 citations
References
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Book•
01 Jan 1965
TL;DR: In this paper, the authors provide a broad overview of Fourier Transform and its relation with the FFT and the Hartley Transform, as well as the Laplace Transform and the Laplacian Transform.
Abstract: 1 Introduction 2 Groundwork 3 Convolution 4 Notation for Some Useful Functions 5 The Impulse Symbol 6 The Basic Theorems 7 Obtaining Transforms 8 The Two Domains 9 Waveforms, Spectra, Filters and Linearity 10 Sampling and Series 11 The Discrete Fourier Transform and the FFT 12 The Discrete Hartley Transform 13 Relatives of the Fourier Transform 14 The Laplace Transform 15 Antennas and Optics 16 Applications in Statistics 17 Random Waveforms and Noise 18 Heat Conduction and Diffusion 19 Dynamic Power Spectra 20 Tables of sinc x, sinc2x, and exp(-71x2) 21 Solutions to Selected Problems 22 Pictorial Dictionary of Fourier Transforms 23 The Life of Joseph Fourier
5,714 citations
Journal Article•
TL;DR: In this article, an algorithm is presented for the rapid solution of the phase of the complete wave function whose intensity in the diffraction and imaging planes of an imaging system are known.
5,197 citations
3,156 citations
TL;DR: A digital method for solving the phase-retrieval problem of optical-coherence theory: the reconstruction of a general object from the modulus of its Fourier transform, which should be useful for obtaining high-resolution imagery from interferometer data.
Abstract: We present a digital method for solving the phase-retrieval problem of optical-coherence theory: the reconstruction of a general object from the modulus of its Fourier transform. This technique should be useful for obtaining high-resolution imagery from interferometer data.
1,762 citations
Journal Article•
644 citations