Fourier Images: I - The Point Source

Cowley, J. M.; Moodie, A. F.

doi:10.1088/0370-1301/70/5/305

Cited by 261 publications

(88 citation statements)

References 7 publications

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“…This verifies that electron beams diffracted by nanostructures remain coherent after propagating farther than the Talbot length zT = 2d 2 /λ = 1.2 mm, and hence is a proof of principle for the function of a Talbot-Lau interferometer for electrons. Distorted fringes due to a phase object demonstrates an application for this new type of electron interferometer.Near-field interference effects that result in self-similar images of a periodic structure were noticed by Talbot in 1836, and later described as Fourier images [1,2,3]. One remarkable feature is that revivals in image visibility occur periodically as the plane of observation is separated from the periodic structure.…”

mentioning

confidence: 96%

Electron interferometry with nanogratings

Cronin

McMorran

2006

Phys. Rev. A

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We present an electron interferometer based on near-field diffraction from two nanostructure gratings. Lau fringes are observed with an imaging detector, and revivals in the fringe visibility occur as the separation between gratings is increased from 0 to 3 mm. This verifies that electron beams diffracted by nanostructures remain coherent after propagating farther than the Talbot length zT = 2d 2 /λ = 1.2 mm, and hence is a proof of principle for the function of a Talbot-Lau interferometer for electrons. Distorted fringes due to a phase object demonstrates an application for this new type of electron interferometer.Near-field interference effects that result in self-similar images of a periodic structure were noticed by Talbot in 1836, and later described as Fourier images [1,2,3]. One remarkable feature is that revivals in image visibility occur periodically as the plane of observation is separated from the periodic structure. The visibility of self-images is maximized at half-integer multiples of the Talbot distance z T = 2d 2 /λ, with d being the period of the structure and λ the wavelength of the light or de Broglie waves illuminating the structure. Partially coherent waves are required to observe self-images of a single grating (the Talbot effect). However, a related phenomenon (the Lau effect) occurs with incoherent light if two gratings are used [3,4,5]. Fringes are then formed behind the second grating, and the fringe visibility oscillates as a function of grating separation. These Lau fringes have maximum visibility on a distant screen when the gratings are separated by nd 2 /λ, with n being an integer. In a Talbot-Lau interferometer, Lau fringes are detected with the aid of a third grating, but the fringes can also be observed directly on a screen, thus making a Lau interferometer as shown in Figure 1.Here we present a Lau interferometer for electrons based on two nanostructure gratings that each have a period of d = 100 nm. With medium energy (5 keV) electrons that have a de Broglie wavelength of λ = 17 pm, the Talbot length is 1.16 mm. An imaging detector 80 cm beyond the gratings was used to observe the Lau fringes shown in Figure 2, and the fringe visibility as a function of grating separation is plotted in Figure 3. If the fringes are analyzed with a third grating (even a digital grating in the image processing can achieve this purpose), then this apparatus serves as a Talbot-Lau interferometer. However, even more information is gained by studying images of the Lau fringes directly.Interferometers based on the Talbot and Lau effects have found applications in light optics [3,5,6], in atom optics [7,8,9,10,11], and more recently with x-rays [12]. Yet even though electron interferometry is a mature field [13,14,15], neither Lau nor Talbot-Lau interferometer designs have been operated with electrons until now.Perhaps the chief reason that Talbot-Lau interferometers have not previously been created for electrons is that suitable periodic structures have not been available. Crystals with a lattice peri...

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confidence: 96%

Electron interferometry with nanogratings

Cronin

McMorran

2006

Phys. Rev. A

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“…In that case, we need to consider both defocusing corresponding to the period of Cowley and Moodie's Fourier images [9] and canceling the phase shift due to scattering.…”

Section: Sharp Interface Images Without Fresnel Fringesmentioning

confidence: 99%

“…For this purpose, we have investigated a new method of z-slicing in Cs-corrected TEM. The principle is a simple one based on the Fourier images studied by Cowley and Moodie [9]. The image contrast of a phase grating in TEM is expressed in the Fourier images, whose period of image intensity variation is written as…”

Section: Application To Single-wall Carbon Nanotubes Andmentioning

confidence: 99%

Present status and future prospects of spherical aberration corrected TEM/STEM for study of nanomaterials^∗

Tanaka

2008

Science and Technology of Advanced Materials

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The present status of Cs-corrected TEM/STEM is described from the viewpoint of the observation of nanomaterials. Characteristic features in TEM and STEM are explained using the experimental data obtained by our group and other research groups. Cs correction up to the 3rd-order aberration of an objective lens has already been established and research interest is focused on correcting the 5th-order spherical aberration and the chromatic aberration in combination with the development of a monochromator below an electron gun for smaller point-to-point resolution in optics. Another fundamental area of interest is the limitation of TEM and STEM resolution from the viewpoint of the scattering of electrons in crystals. The minimum size of the exit-wave function below samples undergoing TEM imaging is determined from the calculation of scattering around related atomic columns in the crystals. STEM does not have this limitation because the resolution is, in principle, determined by the probe size. One of the future prospects of Cs-corrected TEM/STEM is the possibility of extending the space around the sample holder by correcting the chromatic and spherical aberrations. This wider space will contribute to the ease of performing in situ experiments and various combinations of TEM and other analysis methods. High-resolution, in situ dynamic and 3D observations/analysis are the most important keywords in the next decade of high-resolution electron microscopy.

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“…In [1 1 2n] orientation these structures present their atom columns in pairs that have become known as "dumbbells". These arrangements of atoms are advantageous in test specimens because they reduce the "Fourier image" effects that occur with simpler projections, such as in the [100] zone [34][35][36]. Resolution can be determined for both TEM and STEM using the same criterion -separation of atom peaks in the image.…”

Section: Sub-ångström Atomic-resolution Imaging A) Microscope Resolutmentioning

confidence: 99%

HRTEM imaging of atoms at sub-Ångström resolution

O’Keefe¹,

Allard²,

Blom³

2005

Microscopy

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John Cowley and his group at Arizona State University pioneered the use of transmission electron microscopy (TEM) for high-resolution imaging. Images were achieved three decades ago showing the crystal unit cell content at better than 4Å resolution. This achievement enabled researchers to pinpoint the positions of heavy atom columns within the unit cell. Lighter atoms appear as resolution is improved to sub-Ångström levels. Currently, advanced microscopes can image the columns of the light atoms (carbon, oxygen, nitrogen) that are present in many complex structures, and even the lithium atoms present in some battery materials. SubÅngström imaging, initially achieved by focal-series reconstruction of the specimen exit surface wave, will become commonplace for next-generation electron microscopes with C S -corrected lenses and monochromated electron beams. Resolution can be quantified in terms of peak separation and inter-peak minimum, but the limits imposed on the attainable resolution by the properties of the microscope specimen need to be considered. At extreme resolution the "size" of atoms can mean that they will not be resolved even when spaced farther apart than the resolution of the microscope.

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Fourier Images: I - The Point Source

Cited by 261 publications

References 7 publications

Electron interferometry with nanogratings

Electron interferometry with nanogratings

Present status and future prospects of spherical aberration corrected TEM/STEM for study of nanomaterials^∗

HRTEM imaging of atoms at sub-Ångström resolution

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Fourier Images: I - The Point Source

Cited by 261 publications

References 7 publications

Electron interferometry with nanogratings

Electron interferometry with nanogratings

Present status and future prospects of spherical aberration corrected TEM/STEM for study of nanomaterials∗

HRTEM imaging of atoms at sub-Ångström resolution

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Product

Resources

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Present status and future prospects of spherical aberration corrected TEM/STEM for study of nanomaterials^∗