Electron holography on Fraunhofer diffraction

Microscopy, 2019, 254–260 doi: 10.1093/jmicro/dfz007 Advance Access Publication Date: 12 March 2019 Article Electron holography on Fraunhofer diffraction 1 CEMS, RIKEN (The Institute of Physical and Chemical Research), Hatoyama, Saitama 350-0395, Japan, 2Department of Materials Science, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan, 3 Department of Materials Science & Engineering, Kyoto University, Kyoto, Kyoto 606-8501, Japan, 4 Graduate School of Science and Technology, Meijo University, Nagoya, Aichi 468-8502, Japan, 5 Research & Development Group, Hitachi, Ltd., Hatoyama, Saitama 350-0395, Japan, 6Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University, Fukuoka, Fukuoka 810-0395, Japan, and 7Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University, Sendai, Miyagi 980-8577, Japan * To whom correspondence should be addressed. E-mail: Received 14 November 2018; Editorial Decision 28 January 2019; Accepted 30 January 2019 Abstract Electron holography in Fraunhofer region was realized by using an asymmetric double slit. A Fraunhofer diffraction wave from a wider slit worked as an objective wave interfered with a plane wave from a narrower slit as a reference wave under the preFraunhofer condition and recorded as a hologram. Here, the pre-Fraunhofer condition means that the following conditions are simultaneously satisfied: single-slit observations are performed under the Fraunhofer condition and the double-slit observations are performed under the Fresnel condition. Amplitude and phase distributions of the Fraunhofer diffraction wave were reconstructed from the hologram by the Fourier transform reconstruction method. The reconstructed amplitude and phase images corresponded to Fraunhofer diffraction patterns; in particular, the phase steps of π at each band pattern in the phase image were confirmed. We hope that the developed Fraunhofer electron holography can be extended to a direct phase detection method in the reciprocal space. Key words: electron holography, Fraunhofer diffraction, double slit, phase distribution, interferometry Introduction Recently several electron imaging methods have been developed by utilizing not only the real space but also the reciprocal space, such as diffractive imaging [1–3] and ptychography [4–6]. These developments are due to advancement of the following technologies: highly sensitive imaging technologies based on direct electron detection cameras [7], image-processing technologies for multiple and large-scale image data [8], and iteration algorisms for image data analysis [9,10]. Understanding of wave propagation between the real space and the reciprocal space is very important from the optical point of view, especially in © The Author(s) 2019. Published by Oxford University Press on behalf of The Japanese Society of Microscopy. 254 This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact Ken Harada1,2,*, Kodai Niitsu3, Keiko Shimada1, Tetsuji Kodama4, Tetsuya Akashi5,6, Yoshimasa A. Ono1, Daisuke Shindo1,7, Hiroyuki Shinada5, and Shigeo Mori2 Microscopy, 2019, Vol. 68, No. 3 Theoretical Figure 1 shows an illustration for explaining the concept of this study and ‘pre-Fraunhofer condition’ [22]. An asymmetric double-slit, which has one narrow opening and other slightly wider opening, is positioned at the upstream of a coherent electron wave. In the downstream side, two waves pass through the slit-openings and their propagations are illustrated. The term ‘pre-Fraunhofer condition’ indicates that the following conditions are simultaneously satisfied: each single-slit observations are performed under the Fraunhofer condition and the double-slit observations are performed under the Fresnel condition. In general, Fraunhofer/Fresnel conditions depend on wavelengths, sizes of the scattering objects, and propagation distances. When two scattering objects with different sizes are observed under appropriate conditions regarding the wavelength and the propagation distance, the following observation condition is possible: the Fraunhofer condition is realized for a smaller object and the Fresnel condition is realized for a larger object at the same time under a single experimental condition. In the present paper, an asymmetric double slit was used as shown in Figs 1 and 2, and the propagation distance was chosen for the Fraunhofer condition for each slit with the opening width as a parameter and the propagation distance was chosen for the Fresnel condition for the double slit with widths of the slit spacing as a parameter. To describe this experimental condition we coined the term ‘pre-Fraunhofer condition’. Fresnel fringes from the edges of the either opening or interference fringes due to the two passed waves were numerically obtained using Fresnel diffraction theory [11] in the wave optics. Since the Fraunhofer diffraction can be analyzed by the Fresnel diffraction theory as a case of an extremely large-distance propagation, two waves ϕright(x, Δf) and ϕleft(x, Δf) can be written as follows: ϕright (x, Δf ) = = αright2 λ Δf iπα (x, Δf ) 2 dα ∫αright1 exp 2 2 αright2 ⎛ λ Δf πα (x, Δf ) 2 ⎜ cos ∫ 2 2 αright1 ⎝ + i sin = πα (x, Δf ) 2 ⎞ ⎟ dα 2 ⎠ , λ Δf {[C (α right2 (x, Δf )) − C (α right1 (x, Δf ))] 2 + i [S (α right2 (x, Δf )) − S (α right1 (x, Δf ))]} (1) ϕleft (x, Δf ) = = = αleft2 λΔf iπα (x, Δf ) 2 exp dα ∫ αleft1 2 2 αleft2 λΔf πα (x, Δf ) 2 πα (x, Δf )2 (cos + isin ) dα ∫ α 2 2 2 left1 λΔf {[C (αleft2 (x, Δf )) − C (αleft1 (x, Δf ))] 2 + i [S (αleft2 (x, Δf )) − S (αleft1 (x, Δf ))]}, (2) where ϕright(x, Δf) and ϕleft(x, Δf) are complex amplitude distributions of the waves projected on the observation plane from the right and left slits, respectively. In addition, x indicates the coordinate of the observation plane whose origin is the midpoint of the projected image of the double slit on the observation plane; Δf is the propagation distance between the slit plane and the observation plane, i.e. image reconstructions in the real space from data in the reciprocal space, and in phase retrieval for data in the real and reciprocal spaces. Precise investigation of amplitude and phase distributions of wave fields in the reciprocal space is necessary for these purposes. In wave optics, this has already been performed theoretically as well as experimentally [11,12]. In electron microscopy, the amplitude and phase distributions of wave fields in the reciprocal space, such as diffraction planes, have been extensively discussed in phase microscopy [13,14] and vortex beam microscopy [15–19]. In particular, the phase distribution in the reciprocal space is difficult to detect directly ev (...truncated)