The usual focus setting selected with a

The usual focus setting selected with a π Hilbert phase plate is Gaussian focus as shown in Fig. 1D. However it becomes clear from Fig. 1(C) and (B) that the pass band can be made broader by slight underfocussing. At a defocus value of about −50nm the dip within the pass band reaches about 0.5 as seen in Fig. 1B. It seems that for thick specimens (around 50nm) and in the case of uncertain defocus values the highest resolution can be achieved by aiming at an underfocus of around 25nm or slightly higher (Fig. 1C). The point resolution for this imaging mode would be about 3Å for thin specimen.
The situation for the π/2 Hilbert phase plate is somewhat different since the best WP1130 Supplier transfer is achieved at an underfocus of 62.7nm (Scherzer defocus) as seen in Fig. 1G. Slightly higher defocus leads to a dip in the CTF which finally goes all the way to zero 14.2nm above Scherzer defocus (at extended Scherzer) as seen in Fig. 1F. Defocus values lower than Scherzer simply lead to a slightly narrower pass band. For specimens up to about 20nm thickness one would therefore aim at an average underfocus of about 60nm in order to reach the highest possible resolution. For even thicker specimens and in the case of high uncertainty in focusing one would aim correspondingly lower. In an optimally focused image of a thin specimen (about 10nm thickness) the point resolution achievable in this imaging mode should be about 2.5Å as can be estimated from Fig. 1G.
The obvious drawback of the π/2 Hilbert phase plate lies in the oscillations of the CTF between 1 and seen for example in Fig. 1G. The corresponding weighting of amplitudes can in principle easily be corrected for if necessary, but this was not done in the following simulations. By applying CTF-correction and combining images recorded at varying defocus the resolution can be improved additionally beyond the point resolution limit.

Simulations and discussion
The electrostatic potential distribution inside trypsin inhibitor (Protein Data Bank entry 4pti with added hydrogen atoms), a small protein with a diameter of about 2.5–3nm, was calculated using Matlab-code written by Shang and Sigworth, which treats the molecule as a collection of neutral atoms as described in [3].
All further steps of the electron microscopic imaging simulation including Fourier ring correlations were implemented in Khoros [4].
The plots of Figs. 1 and 3 were produced using Origin™ 7.5 (www.originlab.com).
All imaging simulations were done for a 300kV microscope with a C of 2mm. When the electron wave travels down the column and hits the specimen it first enters a layer of vitrified water, which is modelled as a constant electrostatic potential distribution of about 4.9V. The upper and lower surfaces of the water layer are assumed to be flat and orthogonal to the direction of the electron beam (z-axis). Therefore the water layer does not affect the phase of the electron wave locally and does therefore not contribute to the image. To account for this, 4.9V (the potential of water) were subtracted from the phantom before calculating a projection and simulating images.
All simulations were carried out with images measuring 400 by 400 pixels with a pixel size of 0.5Å, but the results shown in Fig. 2 have been cropped to 200 by 200 pixels without changing the pixel size.
Fig. 2 shows the simulated images after correction for the anisotropy as discussed above and the corresponding true projection of the specimen. The two images are displayed with quantitatively correct grey values to be able to visually compare contrast levels. Fig. 2a shows an image generated with a π Hilbert phase plate at Gaussian focus. Fig. 2b shows an image generated with a π/2 Hilbert phase plate at Scherzer defocus. Fig. 2c shows the true projection of the phantom for comparison. Clearly the contrast for the two imaging modes is comparable whereas Fig. 2b shows a higher level of detail. For both Hilbert phase contrast images a gap of 1 pixel (g=0.5 pixels), corresponding to a cut-on periodicity of 40nm, was chosen. The only effect of a, maybe more realistic, larger gap is loss of contrast at low spatial frequencies, but the findings that the two phase contrast images have similar contrast and the π/2 Hilbert phase contrast image at Scherzer defocus shows slightly more detail are still both valid (results not shown). This can be better appreciated from Fourier ring correlation curves (FRCs) between the images and the true projection as shown in Fig. 3. Two different gaps were chosen, g=0.5 pixels, corresponding to a cut-on periodicity of 40nm, in the left column and 4.5 pixels, corresponding to a cut-on periodicity of about 4.4nm, in the right column of Fig. 3. From above the plots show the FRC between the true projection of the specimen and the π Hilbert phase contrast image at Gaussian focus, the π/2 Hilbert phase contrast image at Scherzer defocus and the π/2 Hilbert phase contrast image at extended Scherzer defocus respectively.