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Diffractive imaging

Diffractive imaging

"Diffractive imaging" is a method to reconstruct a structure image of a specimen from a diffraction pattern of the specimen. Since the diffraction pattern is less influenced by lens aberrations, the resolution of the structure image obtained is determined by the maximum diffraction angle of the diffraction pattern, thus a higher-resolution structure image (amplitude image and phase image) than a HREM image (taken by using lenses) can be obtained. This method has actively been studied in the field of X-ray diffractometry, the method being called "Coherent Diffractive Imaging." In electron microscopy, the method is called "Diffractive Imaging" or "Diffractive Microscopy." The method has been applied to carbon nanotubes, etc., and a spatial resolution around 0.1 nm has been obtained. In addition, the method can be applied to not only crystals but also non-periodic structure specimens such as a single-molecule specimen. To reconstruct the image, the Fourier repetitive phase recovery method is used. That is, the magnitudes of the diffraction amplitudes are calculated by the square root of the diffraction intensities taken from a specimen, and random initial phases are given to the diffraction amplitudes. Then, the diffraction amplitudes with the phases are Fourier-transformed to obtain an approximate structure image. The obtained image exhibits structures even in areas exceeding the specimen area from which the diffraction pattern was taken. When the external shape of the specimen is clearly determined, the image intensities from areas except for the external dimension are set to be 0 (zero) (real-space constraint conditions). (On the other hand, when it is difficult to accurately determine the external shape, a specimen area (called "support") that is slightly larger than the external dimension is defined, and the image intensities from areas exceeding the support are set to be 0. Then, the image is inverse-Fourier-transformed to a diffraction pattern. The diffraction amplitudes that do not match the experimental amplitudes are replaced with the experimental values (reciprocal-space constraint conditions), and then Fourier transform is applied again to obtain a structure image. As this procedure is repeated, the true phases of the specimen are gradually recovered and finally, a true structure image of the specimen is obtained. The number of iterations until the recovery of the true phases exceeds several thousands of times. The accuracy of the obtained image is influenced by the parameters such as inelastic scattering contained around the origin of the diffraction pattern, noise from the detection system including its electronic circuits, and the external dimension of the support. It should be noted that when taking the original diffraction pattern, an area of twice the specimen area (support) must be illuminated with an electron beam. This means that the diffraction pattern is sampled with steps twice as small as those corresponding to the original specimen size, enabling us to extract all information contained in the specimen, which is called the over-sampling condition. In an actual experiment, it is needed to create an area where no specimen exists around the specimen and to record a diffraction pattern so as to satisfy the over-sampling condition.

Diffractive_imaging

Conceptual diagram of the Fourier repetitive phase recovery method for diffractive imaging. A crystal structure image of a specimen is retrieved from an experimental diffraction data with a good accuracy by applying Fourier transform to the reciprocal space data and then inverse Fourier transform to the obtained real space data repeatedly with giving constraint conditions at each step. To be precise, 1) First, a wave field in the reciprocal space is created by using the diffraction amplitudes obtained from a diffraction pattern and giving random initial phases to the diffraction amplitudes. 2) The wave field formed in the reciprocal space is Fourier-transformed to an image in the real space. 3) The intensities of the image outside the support area are set to be 0 (zero). 4) The corrected image is inverse Fourier-transformed to a wave field in the reciprocal space. 5) The amplitudes of the wave field are replaced to those of the experimental diffraction pattern. Repetition of this cycle enables us to accurately retrieve the phases of diffraction spots in the reciprocal space and the wave field in the real space. 6) Finally, an accurate structure image of the crystalline specimen is reconstructed.