Overcoming resolution loss due to thermal magnetic field fluctuations from phase plates in transmission electron microscopy

Abstract

We identify thermal magnetic field fluctuations, caused by thermal electron motion ("Johnson noise") in electrically conductive materials, as a potential resolution limit in transmission electron microscopy with a phase plate. Specifically, resolution loss can occur if the electron diffraction pattern is magnified to extend phase contrast to lower spatial frequencies, and if conductive materials are placed too close to the electron beam. While our initial implementation of a laser phase plate (LPP) was significantly affected by these factors, a redesign eliminated the problem and brought the performance close to the expected level. The resolution now appears to be limited by residual Johnson noise arising from the electron beam liner tube in the region of the LPP, together with the chromatic aberration of the relay optics. These two factors can be addressed during future development of the LPP.

Figure 1.

Ray diagram of the modified Titan 80–300 keV TEM used in this work. The marginal ray (red) and paraxial ray (blue) are shown between the sample plane ($z=-0.1\text{mm}$, at the top of the Figure) and the selected area aperture plane ($z=445\text{mm}$, at the bottom of the Figure). The marginal ray is drawn so that it leaves the optical axis (vertical dotted line) in the sample plane at an angle of 1 radian. The Lorentz lens (LL) forms a magnified image of the back focal plane of the objective lens (OL) in the phase plate plane (PPP) at $z=180\text{mm}$. In that plane, the distance of the marginal ray from the optical axis, $X\text{(}z\text{)}$, is approximately 20 mm. Cross-sections of the electron beam liner tube and LPP dummy (2 mm hole diameter) are represented by the light gray and dark gray shaded regions, respectively. The dimensions of the liner tube shown in this figure have been simplified at the request of the microscope manufacturer.

Figure 2.

Benchmarking the relative performance of the modified Titan that is currently used for development of a laser phase plate. (A) Example of a portion of the refined density map of apoferritin, which was obtained without relay optics and is displayed with graphics provided in Chimera [30]. The atomic model that was docked as a rigid body into the EM density was obtained from PDB accession number 6Z6U. (B) The same portion of the map as is shown in (A), but this time obtained with relay optics. (C) Comparison of the gold-standard Fourier shell correlation (FSC) curves for data collected without relay optics (green), and with relay optics (blue). The dashed line indicates the FSC = 0.143 resolution cutoff level. (D) Plots of the inverse-squared resolution achieved as a function of the natural logarithm of the number of particles (multiply by 24 to get the number of asymmetric units) in the data set, generated by bfactor_plot.py in Relion 3.1.3 [25]; the upper (green) line is fitted to the results obtained without relay optics, and the lower (blue) line is fitted to the results obtained with relay optics. The corresponding B-factor (two divided by the slope of the best fit line) for each dataset is listed in the legend.

Figure 3.

(A) Contrast transfer function (CTF) envelope functions with relay optics measured either without inserting a laser phase plate dummy or when dummies were inserted that had electron beam hole diameters equal to 8 mm, 4 mm, or 2 mm. The colored, solid lines represent the measured CTF envelope, with the associated shaded regions representing a 50% confidence interval. The dashed lines represent a best fit to the data, modelled as the product of the envelopes due to imperfect temporal coherence and Johnson noise. The best fit electron beam energy spread $\delta E$ (full width half maximum) was 0.91 eV (50% confidence interval [0.88, 0.94] eV), somewhat higher than the manufacturer’s specification of 0.8 eV for the S-FEG field emission gun used here. The black solid line shows the theoretical temporal coherence (TC) CTF envelope function in the absence of Johnson noise. (B) The fitted values of the Johnson noise image blur variance (red dots) plotted as a function of the inverse of the dummy’s electron beam hole diameter. The error bars show the 50% confidence interval. The solid black line shows the theoretical value of the image blur variance as a function of inverse hole diameter (based on Equation 4, using manufacturer-supplied values for the marginal ray distance and liner tube diameter). The dotted black line represents a best fit to the data when a constant value is added to the theoretical model as a fit parameter (see Section 5.2).