A new symmetrized solution for phase retrieval using the transport of intensity equation

Volkov, V. V.; Zhu, Yimei; Graef, Marc De

doi:10.1016/s0968-4328(02)00017-3

Cited by 156 publications

(114 citation statements)

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“…The right-hand side derivative in this equation can be determined numerically from a through-focus series of Lorentz images. The phase ϕ is then determined by solving the equation by means of fast Fourier transforms [1,7]. This method is exact when the defocus is kept small [8].…”

Section: Lorentz Microscopy: a Brief Introductionmentioning

confidence: 99%

Recent Progress in Lorentz Transmission Electron Microscopy

Graef

2009

ESOMAT 2009 - 8th European Symposium on Martensitic Transformations

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Abstract. After a brief review of the basic methods of Lorentz transmission electron microscopy (LTEM), including the Transport-of-Intensity formalism for phase reconstruction, we present a few examples of the application of LTEM to multiferroic materials, in this case ferromagnetic shape memory alloys. We discuss observations of magnetic domain walls pinned to anti-phase boundaries in Ni2MnGa, and domain wall behavior under an in-situ applied magnetic field in Fe-Pd-Co. Lorentz Microscopy: a Brief IntroductionIt is one of the primary goals of transmission electron microscopy (TEM) to determine quantitatively the structure and chemistry of solids. For a periodic structure, this involves determination of the content of the unit cell (atom types and positions). For magnetic materials, this includes determination of the local magnetization state. The characterization of magnetic microstructures is a complex problem, since it involves, in principle, the determination of a three-dimensional (3-D) vector field, i.e., the magnetization M(r), or the induction B(r).Lorentz TEM (or LTEM) relies on the fact that a high energy beam electron is deflected by the magnetic induction inside and around the sample. The deflection angle for a uniformly magnetized foil can be computed from a simple momentum balance [1] and is given by θ L = C L B ⊥ t, with C L = eλ/h, λ the electron wavelength, B ⊥ the induction component normal to the beam, and t the foil thickness. This beam deflection is usually about two orders of magnitude smaller than a typical Bragg angle, which means that the magnetization state in a foil must be studied by analyzing the fine-structure or splitting of the transmitted beam. The primary method to detect small deflections is to defocus the imaging lens, i.e., to look at a plane below or above the sample, so that the lateral shift of the deflected electrons can be measured; the larger the defocus, the larger the lateral shift and, hence, the easier the measurement. However, the larger the defocus, the more blurred the resulting image becomes, so that there is a practical limit to the amount of defocus that will produce interpretable magnetic contrast.The Lorentz deflection angle can be converted into a magnetic phase gradient [2] by means of the relation ∇ϕ m = 2πkθ L e, where k = 1/λ and e is a unit vector. Integrating this relation and substituting typical numerical values shows that magnetic films should normally be regarded as strong, albeit slowly varying, phase objects. It was shown by Aharonov and Bohm [3] that the total phase shift of an electron wave traveling through a magnetic material consists of two contributions (r ⊥ is a vector in the plane normal to the beam): a

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Section: Lorentz Microscopy: a Brief Introductionmentioning

confidence: 99%

Recent Progress in Lorentz Transmission Electron Microscopy

Graef

2009

ESOMAT 2009 - 8th European Symposium on Martensitic Transformations

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“…This claim is supported by the fact that the FT based methods possess an intrinsic Periodic BC, and thus, only by putting the aperture on the on-focus plane will not modify the boundary property of the FT-TIE solver. However, it is shown in reference [14] that the FT-TIE solver can be provided with the DBC and NBC through appropriate mirror padding of captured intensities. Figure 2a shows the in-focus intensity distribution I 0 with an aperture.…”

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“…Fig. 2b shows the DBC mirror padding scheme [14] for the captured intensities. The right-top and left-bottom intensities in Fig.…”

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“…In order to impose the required NBC in the FT-TIE solver, it is necessary to apply the mirror padding shown in Fig 2b to captured intensities, and thus, these images will be used to estimate the axial intensity derivative [7,9]. Later on, the padded I 0 and ∂ z I will be used as inputs of the FT-TIE method [14]. Fig.…”

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“…2. Mirror padding scheme [14] for imposing BC in the FT-TIE solver. a) Intensity distribution in the object plane with an aperture.…”

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Solution to the Boundary problem for Fourier and Multigrid transport equation of intensity based solvers

Martínez-Carranza

Falaggis

Kozacki

2015

Photon. Lett. Poland

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Abstract-In this work we present a solution for the boundary problem for phase retrieval techniques based on the Transport of Intensity Equation (TIE). The solution presented here is based on the Neumann Boundary condition and the mirror padding scheme of the captured intensities. The obtained results are derived for the widely used Fourier Transform based TIE solver, but it is shown that they can also be applied to Multigrid based techniques.

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