The Question of Phase Retrieval in Optics

Walther, A.

doi:10.1080/713817747

Cited by 435 publications

(288 citation statements)

References 2 publications

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“…The possibility of inverting a set of oversampled Fourier amplitudes (without phases) depends on the dimensionality: There is no unique solution in 1D [18], but it has been argued that one may be found if oversampling is possible in 2D and 3D [19,20]. In the seminal works [12,13], the object recovered was a 2D projection of 2 non-periodic dimensions.…”

Section: A Basic Algorithmmentioning

confidence: 99%

A Bayesian Approach to Surface X-ray Diffraction

Lyman¹,

Saldin²

2006

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Section: A Basic Algorithmmentioning

confidence: 99%

A Bayesian Approach to Surface X-ray Diffraction

Lyman¹,

Saldin²

2006

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“…This is a variation on the phase problem in crystal diffraction theory (Hauptman, 1986;Walther, 1963;Karle & Hauptman, 1950). Many of the distributions derived are unphysical because they have regions of large negative probability, but some ambiguity remains.…”

Section: Fourier Expansion Of the Probability Distributionmentioning

confidence: 99%

Multilayer roughness evaluated by X-ray reflectivity

Rosen¹,

Brown²,

Gilfrich³

et al. 1988

J Appl Cryst

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A crystal diffraction theory has been developed and compared with experimental measurements in order to model the integrated reflectivity of multilayer structures. This study shows that the most important defect reducing the integrated reflectivities of the multilayer structures studied was correlated roughness (root mean square value about 7 A). The theory describes correlated roughness as a probability distribution of the substrate surface displacement. A computer simulation and an analytical solution have been used to calculate the reflectivity of multilayer structures. The calculations using a rectangular distribution of correlated displacement result in a good first-order approximation of the experimental data. A Gaussian probability distribution for the substrate surface displacement results in calculations inconsistent with the measured reflectivities, although such a distribution has been assumed in other studies. Although other defects were studied, only roughness could explain the experimental data.

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“…King shows that the phase problem may be defined as a homogeneous Riemann boundary value problem for the semi-plane (Gakhov, 1966) and that under certain circumstances one can write a dispersion relation (logarithmic Hilbert transform, Burge et al, 1976) between the intensity and the phase. Unfortunately, as was explained by Walther (1963), if F(z) has any complex zeros within the contour of integration (the upper half of the complex plane) then the calculated phase will be incorrect.…”

Section: Analyticity Of the Scattered Fieldmentioning

confidence: 99%

Phase assignment to diffraction patterns based on the analytic properties of scattered fields applied to the structure of nerve myelin

Burge¹,

Fiddy²

1981

J Appl Cryst

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An analysis of published X-ray diffraction data from nerve myelin is given based on the properties of analytic functions. Functions defined by a finite Fourier transform may be described by their distribution of zeros. This description allows a phase function to be determined from real data, which is unique in principle. A solution to the phase assignment is given and compared with corresponding published solutions derived by other methods. The strong measure of agreement for the phases of thc first nine diffraction orders, and the stability of this agreement against the efforts of experimental error, leads to the conclusion that these phases are probably correct.

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The Question of Phase Retrieval in Optics

Cited by 435 publications

References 2 publications

A Bayesian Approach to Surface X-ray Diffraction

A Bayesian Approach to Surface X-ray Diffraction

Multilayer roughness evaluated by X-ray reflectivity

Phase assignment to diffraction patterns based on the analytic properties of scattered fields applied to the structure of nerve myelin

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