Electron interferometry with nanogratings

Abstract

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...

Asymmetric Talbot-Lau interferometry for inertial sensing

“…This can be of interest experimentally for a variety of cases [6]. Magnifying configurations have been actually realized for low energy electrons [25], using however different resonance conditions and an extreme (η = 100) magnification factor, so that the observation plane was effectively in the far field of the second gratings. As a matter of fact that configuration requires different coherence conditions than the Talbot-Lau interferometer and is referred to as a Lau interferometer [15].…”

“…On the one hand, equation (25) tells that to improve the sensitivity, the relative displacement ∆x/d 3 should be maximised. On the other hand, equation (26) shows that this quantity increases monotonically with the total flight time, as expected, but also that it tends to zero as η 1.…”

“…In the Talbot-Lau interferometer, the parameters C and ∆x appearing in equation (25) for the inertial sensitivity, strongly depend on the longitudinal speed distribution P (v) of the particle beam. This section is devoted to a numerical analysis of this dependence.…”

A Kapitza–Dirac–Talbot–Lau interferometer for highly polarizable molecules

“…In addition, coherent beam splitting at non-resonant standing light waves, often designated the KapitzaDirac effect, has been observed for all of these species [18][19][20] . Recent implementations of near-field interferometry 13,[21][22][23] underlined the particular advantages of the Talbot-Lau concept for experiments with massive objects: the required grating period scales only weakly (d ∼ √ l) with the de Broglie wavelength, and the design accepts beams of low spatial coherence, which makes high signals possible even for weak sources.A symmetric Talbot-Lau interferometer (TLI) consists of three identical gratings. The first one prepares the transverse coherence of the weakly collimated beam.…”

Electron Diffraction and Interferometry with Nano-gratings

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We used nanofabricated gratings as beam splitters for an electron interferometer [1,2,3]. This shows that nanotechnology can be used for coherent electron optics.

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