Complex Observation in Electron Microscopy VIII: Novel Hilbert Phase-plates to Maximize Phase-contrast Sensitivity

Phase-plate transmission electron microscopy has recently regressed since a report that Volta phase-plate phase-contrast is less sensitive than non-phase-plate phase contrast, which leads to conventional defocusing phase-contrast. What about Hilbert phase-plate phase-contrast? We report that the Hilbert phase-plate method can survive if two experiments using a pair of symmetric Hilbert phase-plates, of which phase is set to a value smaller than {\pi}, are combined. Three phase-contrast methods using the symmetric Hilbert phase-plates and Zernike phase-plate representing Volta phase-plate and Scherzer defocus respectively were compared in sensitivity theoretically relying on a contrast transfer theory and computationally on a simulator specifically designed for phase plate transmission electron microscopy. For the two phase-plate phase-contrasts, the phase that gives the highest sensitivity was searched for by changing the phase-plate phase. As a result, the symmetric Hilbert phase-plates phase-contrast was found to outperform Scherzer defocus phase-contrast in the phase around {\pi}/2. On the other hand, Zernike phase-plate phase-contrast was found considerably inferior to Scherzer defocus phase-contrast in the entire phase range from 0 to {\pi}. Furthermore, the novel Hilbert phase-plate method was compared with complex observation transmission electron microscopy, which also requires two experiments, and the origin of the higher sensitivity of symmetric Hilbert phase-plates phase-contrast was examined.