Gabor’s signal expansion in optics

Bastiaans, MJ Martin

doi:10.1007/978-1-4612-2016-9_15

Cited by 9 publications

(4 citation statements)

References 37 publications

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“…In order to obtain the object function O(r) and the illumination function P(r) we identify the diffracted wavefield, Eqs. (2) and 4, as an expansion of the object signal onto a sort of "elementary" 2D signals [23,31,32,34]. Therefore, from windowed-Fourier transform theory, we can directly write the inversion formula for the object as [23,[30][31][32][35][36][37]:…”

Section: Reconstruction Formulasmentioning

confidence: 99%

Elementary signals in ptychography

Silva¹,

Menzel²

2015

Opt. Express

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Ptychographic imaging has gained popularity for its high resolving power and sensitivity as well as for its ability to map simultaneously the sample's complex-valued refractive index and the illumination. Yet, despite significant progress that allows for reliable practical implementation, some of the technique's fundamentals remain poorly understood, and oftentimes successful data acquisition is either overly conservative or relies more on experimenters experience than on rational data acquisition strategies. Here, we propose a theoretical framework of ptychography, which is based on Gabor's notion of decomposition into elementary signals and the concept of frames. We demonstrate how this framework can straightforwardly be used to derive sampling requirements or to provide arguments on how to optimize the ptychographic scan. More generally, our theoretical framework can serve as a bridge between the experimental technique and the rich and well established mathematical disciplines of wavelets decomposition and spectrogram analysis.

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Section: Reconstruction Formulasmentioning

confidence: 99%

Elementary signals in ptychography

Silva¹,

Menzel²

2015

Opt. Express

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“…Several applications of Gabor expansions in optics have been described by BASTIAANS (1998). Although, this approach provides a self-consistent selection criterion for the summation of Gaussian beams, it was observed by DAUBECHIES (1990) that the Gabor expansion coefficients of BASTIAANS (1980) at critical sampling are only marginally stable.…”

Section: Extensions Of the Gaussian Beam Methodsmentioning

confidence: 99%

Calculation of Synthetic Seismograms with Gaussian Beams

Nowack¹

2003

Pure appl. geophys.

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In this paper, an overview of the calculation of synthetic seismograms using the Gaussian beam method is presented accompanied by some representative applications and new extensions of the method. Since caustics are a frequent occurrence in seismic wave propagation, modifications to ray theory are often necessary. In the Gaussian beam method, a summation of paraxial Gaussian beams is used to describe the propagation of high-frequency wave fields in smoothly varying inhomogeneous media. Since the beam components are always nonsingular, the method provides stable results over a range of beam parameters. The method has been shown, however, to perform better for some problems when different combinations of beam parameters are used. Nonetheless, with a better understanding of the method as well as new extensions, the summation of Gaussian beams will continue to be a useful tool for the modeling of high-frequency seismic waves in heterogeneous media.

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“…Clearly, such a fine-structure, highly distributed function drastically complicates the numerical integration in (20), which has to be performed many times to obtain the full set of amplitudes. This circumstance may even be a reason to reject the Gabor expansion in favour of a computationally more expensive 'frame-based' representation [16], which is also known as the oversampled Gabor expansion [10]. But for the purpose herein, since the decomposition into narrower beamlets implies the condition ρ 2 1;2 Im Q αβ (ŝ)u (1;2) α u (1;2) β < 1, one can reduce the infinite integration in (20) to the series of integrals over the h-function quasi-periods within which the relatively slow function f (ξ, η) is replaced with its truncated local Taylor expansion.…”

Section: Gb Propagation and Beamlet Decompositionmentioning

confidence: 99%

“…A more plain approach is to perform a proper discrete representation of the wavefield at once. That is what can be done with the Gabor expansion, which has been initially introduced in the context of signal processing [7] and subsequently obtained a practical footing [8] and various applications in optics [9][10][11][12][13]. The free parameters of the Gabor expansion allow to vary the beamlet width and thus may be chosen suitably small.…”

Section: Introductionmentioning

confidence: 99%

Application of Gabor expansion to beam-tracing problems in inhomogeneous plasmas

Tereshchenko¹,

Castejón²,

Cappa³

2013

Plasma Phys. Control. Fusion

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In this paper we develop a theoretical framework for the phase-space Gaussian beamlet decomposition of the beam-like wavefields in plasmas. It is based on the discrete Gabor expansion of the wavefield distribution along a pre-selected equiphase surface. In particular, we consider the situations where the Gaussian beamlets can be chosen much narrower than the parent beam, though much wider than the wavelength of the radiation. Eventual summation of the independently computed beamlets will thus give the net wavefield that properly accounts for the effects of smaller-scale inhomogeneities. While using the standard beam-tracing (BT) method for the computation of each propagating beamlet, this approach is able to improve the accuracy of the stand-alone BT technique applied to the wide parent beam in cases of strong refraction or oblique absorption. We demonstrate the potency and relatively low computational cost of the introduced method in numerical examples, which reproduce different patterns of microwave propagation in the TJ-II stellarator plasma.

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Gabor’s signal expansion in optics

Cited by 9 publications

References 37 publications

Elementary signals in ptychography

Elementary signals in ptychography

Calculation of Synthetic Seismograms with Gaussian Beams

Application of Gabor expansion to beam-tracing problems in inhomogeneous plasmas

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