Journal of Microscopy, Vol. 259, Issue 1 2015, pp. 74–78
doi: 10.1111/jmi.12250
Received 26 November 2014; accepted 24 February 2015
Improved Zernike-type phase contrast for transmission electron microscopy P.J.B. KOECK Royal Institute of Technology, School of Technology and Health and Karolinska Institutet, Department of Biosciences and Nutrition at Novum, 14183 Huddinge, Sweden
Key words. Boersch phase plate, cryotransmission electron microscopy, phase contrast, weak phase object approximation, Zernike phase plate.
Summary Zernike phase contrast has been recognized as a means of recording high-resolution images with high contrast using a transmission electron microscope. This imaging mode can be used to image typical phase objects such as unstained biological molecules or cryosections of biological tissue. According to the original proposal discussed in Danev and Nagayama (2001) and references therein, the Zernike phase plate applies a phase shift of π/2 to all scattered electron beams outside a given scattering angle and an image is recorded at Gaussian focus or slight underfocus (below Scherzer defocus). Alternatively, a phase shift of -π/2 is applied to the central beam using the Boersch phase plate. The resulting image will have an almost perfect contrast transfer function (close to 1) from a given lowest spatial frequency up to a maximum resolution determined by the wave length, the amount of defocus and the spherical aberration of the microscope. In this paper, I present theory and simulations showing that this maximum spatial frequency can be increased considerably without loss of contrast by using a Zernike or Boersch phase plate that leads to a phase shift between scattered and unscattered electrons of only π /4, and recording images at Scherzer defocus. The maximum resolution can be improved even more by imaging at extended Scherzer defocus, though at the cost of contrast loss at lower spatial frequencies.
Abbreviations CTF contrast transfer function TEM transmission electron microscopy FRC Fourier ring correlation Introduction and theory The Fourier transform of the exit wave modified by the lens aberration function of a transmission electron microscope and Correspondence to: P.J.B. Koeck. Tel: +46-8-524-81073; fax: +46-8-524-81135; e-mail: [email protected]
a Zernike-type phase plate mounted in the back focal plane can be written as follows: −i χ (u) )e H (u) pc u = e x (u −i χ (u) u e H (u) , (1) ≈ δ u + iσ F with π C s λ3 u 4 2 for u > c . for u < c
χ (u) = π D λu 2 + H (u) =
eis 1
Here, D is the defocus value, which is negative for underfocus, λ is the relativistic wave length of the electron wave, u is the scalar spatial frequency, u is the spatial frequency vector and Cs is the spherical aberration constant. This form of the lens aberration function χ(u) is valid in the absence of astigmatism and without a resolution-limiting aperture. e x (u ) is the Fourier transform of the exit wave ψe x (x ), which is simply modelled as a plane wave multiplied by the phase factor e i σ o(x) . o (x ) is the projection along the z-axis of the electrostatic potential distribution produced by the specimen being imaged. F (u ) is the Fourier transform of o (x ). σ is the positive valued interaction constant given by σ = 2π meλ . h2 h is the Planck constant, m is the relativistic mass and e is the charge of the electron. H (u) describes a phase shift of s applied to the part of the focal plane corresponding to spatial frequency vectors with magnitudes larger than a small value c given, for example, by the radius of the hole in the thin film Zernike phase plate. The image recorded on a detector is given by the absolute square of the modified exit wave in direct space ∗ x = ψ pc x ψ pc x i m pc
F u H (u) e −i χ(u) e 2πi xu d u x d u y · ≈ 1 + iσ
C 2015 The Authors C 2015 Royal Microscopical Society Journal of Microscopy
IMPROVED ZERNIKE PHASE CONTRAST
1 − iσ
with H ∗ (u) =
F
∗ u H (u) e i χ (u) e 2πi xu d u x d u y
e −i s 1
for u > c for u < c
75
.
To arrive at the above expression for the image we have changed the sign of the integration variables in the second integral and used the fact that both H(u) and χ(u) have circular symmetry and F (u ) is Hermitian. For contributions to the phase contrast image with spatial frequencies above c this can be written For u > c F i s −i χ (u) i m pc x = 1 + iσ u e e − e −i s e i χ (u)
e 2πi xud u x d u y .
(2)
Clearly for u < c the imaging properties are the same as for a microscope without Zernike phase plate. Only spatial frequencies larger than c are affected by the phase shift. For specific values of s, these results can be summarized in a more concise way as follows: For s = π/2 transfer of spatial frequencies larger than c can be summarized as F x = 1 − 2σ u cos (χ (u)) e 2πi xud u x d u y (3) i m pc for u > c. This describes the previously proposed (Danev & Nagayama, 2001) Zernike phase contrast at Gaussian focus (choosing D = 0), which we will refer to as the π/2 Zernike phase plate. For s = π/4, transfer of spatial frequencies larger than c can be summarized as √ F i m pc x = 1 − 2σ u (cos (χ (u))
−si n (χ (u)))e 2πi xud u x d u y
(4)
for u > c. This describes the improved Zernike phase contrast at Scherzer or extended Scherzer defocus suggested in this paper. We will refer to this as the π/4 Zernike phase plate in the following. The same results also apply to the Boersch phase plate since only the relative phase shift between unscattered and scattered electrons matters. The resulting contrast transfer functions (CTFs) for Fourier components larger than c for a 300 kV transmission electron microscopy (TEM) with Cs = 2 mm are shown in Figures 1 A–D as follows: Figure 1(A): for s = π/2 and Gaussian focus; Figure 1(B): for s = π/2 and Scherzer defocus (62.7 nm); Figure 1(C): for s = π/4 and Scherzer defocus (62.7 nm); Figure 1(D): for s = π/4 and extended Scherzer defocus (76.9 nm). All plots are normalized so that their maxima are equal to 1. C 2015 The Authors C 2015 Royal Microscopical Society, 259, 74–78 Journal of Microscopy
Fig. 1. Contrast transfer functions for spatial frequencies above the cuton for Zernike phase contrast. (A) A π /2 Zernike phase plate at Gaussian focus, (B) A π /2 Zernike phase plate at Scherzer defocus, (C) A π /4 Zernike phase plate at Scherzer defocus, (D) A π /4 Zernike phase plate at extended Scherzer defocus.
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For spatial frequencies below c, contrast transfer is described by the usual phase CTF sin(χ(u)). Clearly, the first pass-band is much wider for a π/4 Zernike phase plate at Scherzer defocus (Fig. 1 C) than it is for a π/2 Zernike phase plate at Gaussian focus (Fig. 1 A). The CTF re˚ in Figure 1(C), mains positive up to a resolution of about 2.5 A ˚ resolution in whereas it goes through zero at around 3.5 A Figure 1(A). As seen in Figure 1(D), the resolution range can be extended a bit more by recording at extended Scherzer defocus, however, at the cost of losing contrast at resolutions ˚ These considerations become especially imaround 3 to 5 A. portant when imaging slightly thicker specimens (about 10 to 20 nm thickness) since then the defocus variation around Scherzer becomes quite noticeable. A practical guideline could be that one specimen surface should be just below extended Scherzer and no part of the specimen at a defocus value higher than extended Scherzer in order to achieve the highest possible resolution without contrast inversions. A broadening of the first pass-band to approximately that of a π/4 Zernike phase plate at Scherzer defocus can also be achieved for the π/2 Zernike phase plate by imaging at Scherzer defocus (compare Fig. 1B to 1C). However, this also leads to a pronounced dip at lower frequencies. Obviously, the oscillation of the CTF seen in Figure 1(C) can 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 To estimate the image quality achievable by the proposed Zernike phase contrast imaging mode, I simulated phase contrast images of small protein hexamers imaged in top view. Amplitude contrast was not included in the image simulation since we are only interested in a comparison between different imaging modes. The electrostatic potential distribution inside trypsin inhibitor (Protein data Bank entry 4 pti with added hydrogen atoms), a small protein with a diameter of about 2.5–3 nm, was calculated using Matlab code written by Shang and Sigworth, which treats the molecule as a collection of neutral atoms as described in Shang and Sigworth (2012). All further steps of the electron microscopic imaging simulation including Fourier ring correlations (FRCs) were implemented in Khoros (Konstantinides & Rasure, 1994). The plots of Figures 1 and 3 were produced using OriginTM 7.5 (www.originlab.com). The phantoms used as specimens for image simulation were generated by arranging six copies of the electrostatic potential distribution in an oligomer with C6 symmetry. All imaging simulations were done for a 300 kV microscope with a Cs of 2 mm.
The voxel size of the phantom and the pixel size of the pro˚ giving a Nyquist limit of 1 A. ˚ jection and images are 0.5 A, The Zernike phase plate is assumed to have a hole of radius ˚ (corre0.5 μm leading to a cut-on frequency of 0.0125 1/A ˚ periodicity) for an objective lens with 2 mm sponding to 80 A focal length. As pointed out by (Danev & Nagayama, 2011) and (Malac et al., 2008), the choice of cut-on frequency is important for the signal-to-noise ratio (SNR) and the fidelity of the images. In the presented simulations, I made no attempt to choose an optimized value for the cut-on frequency. However, since I used the same cut-on frequency for all simulations, the results are suitable for comparison of the discussed imaging modes. 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 4.9 V for all voxels occupied by water. In the vicinity of the protein, the potential drops below this value to take into account the finite distance between water molecules and the atoms forming the surface of the protein. For details, see Shang & Sigworth (2012). 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, 4.9 V (the potential of water) were subtracted from the phantom before calculating a projection and simulating images. Images were calculated using Eq. (1) without approximating for weak phase objects to arrive at the Fourier transform of the exit wave modified by the lens aberration function and the phase plate and then taking the absolute square in real space. Figure 2 shows the simulated images and the corresponding projection of the specimen. The three images are displayed with quantitatively correct grey values to be able to visually compare contrast levels and they have been contrast inverted to facilitate comparison with the projection of the phantom. Figure 2(A) shows an image generated with a π/2 Zernike phase plate at Gaussian focus. Figure 2(B) shows an image generated with a π/4 Zernike phase plate at Scherzer defocus. Figure 2(C) shows an image generated with a π/4 Zernike phase plate at extended Scherzer defocus. Figure 2(D) shows the projection of the phantom for comparison. Clearly, the contrast for the three imaging modes is comparable whereas Figure 2(B) shows the highest level of detail. To quantify this resolution improvement, I also calculated FRC curves (see e.g. Frank, 2006, p. 249 ff and references given there) between the images and the projection for several cases as shown in Figure 3. From above, the plots show the FRC between the projection of the specimen and the π/2 Zernike phase contrast image at Gaussian focus (A), the π/4 Zernike phase contrast image at Scherzer defocus (B) and the π/4 Zernike phase contrast image at extended Scherzer defocus (C), respectively.
C 2015 The Authors C 2015 Royal Microscopical Society, 259, 74–78 Journal of Microscopy
IMPROVED ZERNIKE PHASE CONTRAST
77
Fig. 2. (A) Simulated image for π /2 Zernike phase contrast at Gaussian focus, (B) Simulated image for π /4 Zernike phase contrast at Scherzer defocus, (C) Simulated image for π /4 Zernike phase contrast at extended Scherzer defocus, (D) Projection of the electrostatic potential. The size of the image is 10 nm × 10 nm.
In agreement with Figure 1, the FRCs show correlation close ˚ resolution for the π/4 Zernike to 1 up to and beyond 2.5 A phase contrast images at Scherzer and extended Scherzer defocus. For the π/2 Zernike phase contrast image at Gaussian ˚ focus the FRC drops to zero at a resolution of about 3.5 A. Here, it should also be pointed out that I made no attempt to optimize the phase-shift of the phase plate with respect to the phase shift from the specimen as discussed in Danev & Nagayama (2011). This would, of course, be possible but for the comparison between two imaging modes made here it does not seem necessary, since both simulations are made with the same phantom and, apart from the phase plate’s phase shift and the defocus, all parameters chosen are identical. Conclusions and outlook I present a modified Zernike-type phase contrast imaging mode using a phase plate that produces a relative phase shift between scattered and unscattered electrons of π/4 rather than the previously suggested phase shift of π/2. This phase shift could be realized either by a regular thin film Zernike phase plate (Danev & Nagayama, 2001), some type of hole-free thin film phase plate that relies on beam-induced charges (Malac et al., 2012; Danev et al., 2014) or by an electrostatic phase plate (Majorovits et al., 2007; Schultheiss et al., 2010). Images are then recorded at Scherzer or extended Scherzer defocus rather than at or close to Gaussian focus. This results in a C 2015 The Authors C 2015 Royal Microscopical Society, 259, 74–78 Journal of Microscopy
Fig. 3. FRC between projection and simulated image for (A) π /2 Zernike phase contrast at Gaussian focus, (B) π /4 Zernike phase contrast at Scherzer defocus, (C) π /4 Zernike phase contrast at extended Scherzer defocus
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pass-band without contrast inversions that extends to considerably higher resolution while maintaining the high contrast for all spatial frequencies up to this limit. Another advantage lies in the reduced amplitude loss and electron scattering in a thin film phase plate that is only half the thickness (Nagayama & Danev, 2008). Since the phase-plate only applies a phase shift of π/4, it can be kept very thin (about 11 nm for a carbon film phase plate and 300 keV electrons (Barton et al., 2008). Clearly, the presented ideas can also be applied in the case of the Volta potential phase plate (Danev et al., 2014) provided a material can be found that produces the correct phase shift for a microscope with a given acceleration potential. I suggest this imaging mode particularly for structure determination of proteins with a mass below about 200 kDa, which have been difficult to image with sufficient contrast. However, this imaging mode is completely general in nature and not limited to any particular type of specimen. Acknowledgement I acknowledge Hans Hebert for valuable comments on the project and Fred Sigworth for help with his Matlab code. References Barton, B., Joos, F. & Schr¨oder, R.R. (2008) Improved specimen reconstruction by Hilbert phase contrast tomography. J. Struct. Biol. 164, 210–220. Danev, R. & Nagayama, K. (2001) Transmission electron microscopy with Zernike phase plate. Ultramicroscopy 88, 243–252.
Danev, R. & Nagayama, K. (2011) Optimizing the phase shift and the cut-on periodicity of phase plates for TEM. Ultramicroscopy 111, 1305– 1315. Danev, R., Buijsse, B., Khoshouei, M., Plitzko, J.M. & Baumeister, W. (2014) Volta potential phase plate for in-focus phase contrast transmission electron microscopy. PNAS 111, 15635–15640. Frank, J. (2006) Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Oxford University Press, New York. Konstantinides, K. & Rasure, J.R. (1994) The Khoros software development environment for image and signal processing. IEEE T. Image Process. 3, 243–252. Majorovits, E., Barton, B., Schultheiß, K., P´erez-Willard, F., Gerthsen, D. & Schr¨oder, R.R. (2007) Optimizing phase contrast in transmission electron microscopy with an electrostatic (Boersch) phase plate. Ultramicroscopy 107, 213–226. Malac, M., Beleggia, M., Egerton, R. & Zhu, Y. (2008) Imaging of radiationsensitive samples in transmission electron microscopes equipped with Zernike phase plates. Ultramicroscopy 108, 126–140. Malac, M., Beleggia, M., Kawasaki, M., Li, P. & Egerton, R. (2012) Convenient contrast enhancement by a hole-free phaseplate. Ultramicroscopy 118, 77–89. Nagayama, K. & Danev, R. (2008) Phase contrast electron microscopy: development of thin-film phase plates and biological applications. Phil. Trans. R. Soc. B 363, 2153–2162. Schultheiss, K., Zach, J., Gamm, B., Dries, M., Frindt, N., Schr¨oder, R.R. & Gerthsen, D. (2010) New electrostatic phase plate for phase-contrast transmission electron microscopy and its application for wave-function reconstruction. Microsc. Microanal. 16, 785–794. Shang, Z. & Sigworth, F.J. (2012) Hydration-layer models for cryo-EM image simulation. J. Struct. Biol. 180, 10–16.
C 2015 The Authors C 2015 Royal Microscopical Society, 259, 74–78 Journal of Microscopy
doi: 10.1111/jmi.12250
Received 26 November 2014; accepted 24 February 2015
Improved Zernike-type phase contrast for transmission electron microscopy P.J.B. KOECK Royal Institute of Technology, School of Technology and Health and Karolinska Institutet, Department of Biosciences and Nutrition at Novum, 14183 Huddinge, Sweden
Key words. Boersch phase plate, cryotransmission electron microscopy, phase contrast, weak phase object approximation, Zernike phase plate.
Summary Zernike phase contrast has been recognized as a means of recording high-resolution images with high contrast using a transmission electron microscope. This imaging mode can be used to image typical phase objects such as unstained biological molecules or cryosections of biological tissue. According to the original proposal discussed in Danev and Nagayama (2001) and references therein, the Zernike phase plate applies a phase shift of π/2 to all scattered electron beams outside a given scattering angle and an image is recorded at Gaussian focus or slight underfocus (below Scherzer defocus). Alternatively, a phase shift of -π/2 is applied to the central beam using the Boersch phase plate. The resulting image will have an almost perfect contrast transfer function (close to 1) from a given lowest spatial frequency up to a maximum resolution determined by the wave length, the amount of defocus and the spherical aberration of the microscope. In this paper, I present theory and simulations showing that this maximum spatial frequency can be increased considerably without loss of contrast by using a Zernike or Boersch phase plate that leads to a phase shift between scattered and unscattered electrons of only π /4, and recording images at Scherzer defocus. The maximum resolution can be improved even more by imaging at extended Scherzer defocus, though at the cost of contrast loss at lower spatial frequencies.
Abbreviations CTF contrast transfer function TEM transmission electron microscopy FRC Fourier ring correlation Introduction and theory The Fourier transform of the exit wave modified by the lens aberration function of a transmission electron microscope and Correspondence to: P.J.B. Koeck. Tel: +46-8-524-81073; fax: +46-8-524-81135; e-mail: [email protected]
a Zernike-type phase plate mounted in the back focal plane can be written as follows: −i χ (u) )e H (u) pc u = e x (u −i χ (u) u e H (u) , (1) ≈ δ u + iσ F with π C s λ3 u 4 2 for u > c . for u < c
χ (u) = π D λu 2 + H (u) =
eis 1
Here, D is the defocus value, which is negative for underfocus, λ is the relativistic wave length of the electron wave, u is the scalar spatial frequency, u is the spatial frequency vector and Cs is the spherical aberration constant. This form of the lens aberration function χ(u) is valid in the absence of astigmatism and without a resolution-limiting aperture. e x (u ) is the Fourier transform of the exit wave ψe x (x ), which is simply modelled as a plane wave multiplied by the phase factor e i σ o(x) . o (x ) is the projection along the z-axis of the electrostatic potential distribution produced by the specimen being imaged. F (u ) is the Fourier transform of o (x ). σ is the positive valued interaction constant given by σ = 2π meλ . h2 h is the Planck constant, m is the relativistic mass and e is the charge of the electron. H (u) describes a phase shift of s applied to the part of the focal plane corresponding to spatial frequency vectors with magnitudes larger than a small value c given, for example, by the radius of the hole in the thin film Zernike phase plate. The image recorded on a detector is given by the absolute square of the modified exit wave in direct space ∗ x = ψ pc x ψ pc x i m pc
F u H (u) e −i χ(u) e 2πi xu d u x d u y · ≈ 1 + iσ
C 2015 The Authors C 2015 Royal Microscopical Society Journal of Microscopy
IMPROVED ZERNIKE PHASE CONTRAST
1 − iσ
with H ∗ (u) =
F
∗ u H (u) e i χ (u) e 2πi xu d u x d u y
e −i s 1
for u > c for u < c
75
.
To arrive at the above expression for the image we have changed the sign of the integration variables in the second integral and used the fact that both H(u) and χ(u) have circular symmetry and F (u ) is Hermitian. For contributions to the phase contrast image with spatial frequencies above c this can be written For u > c F i s −i χ (u) i m pc x = 1 + iσ u e e − e −i s e i χ (u)
e 2πi xud u x d u y .
(2)
Clearly for u < c the imaging properties are the same as for a microscope without Zernike phase plate. Only spatial frequencies larger than c are affected by the phase shift. For specific values of s, these results can be summarized in a more concise way as follows: For s = π/2 transfer of spatial frequencies larger than c can be summarized as F x = 1 − 2σ u cos (χ (u)) e 2πi xud u x d u y (3) i m pc for u > c. This describes the previously proposed (Danev & Nagayama, 2001) Zernike phase contrast at Gaussian focus (choosing D = 0), which we will refer to as the π/2 Zernike phase plate. For s = π/4, transfer of spatial frequencies larger than c can be summarized as √ F i m pc x = 1 − 2σ u (cos (χ (u))
−si n (χ (u)))e 2πi xud u x d u y
(4)
for u > c. This describes the improved Zernike phase contrast at Scherzer or extended Scherzer defocus suggested in this paper. We will refer to this as the π/4 Zernike phase plate in the following. The same results also apply to the Boersch phase plate since only the relative phase shift between unscattered and scattered electrons matters. The resulting contrast transfer functions (CTFs) for Fourier components larger than c for a 300 kV transmission electron microscopy (TEM) with Cs = 2 mm are shown in Figures 1 A–D as follows: Figure 1(A): for s = π/2 and Gaussian focus; Figure 1(B): for s = π/2 and Scherzer defocus (62.7 nm); Figure 1(C): for s = π/4 and Scherzer defocus (62.7 nm); Figure 1(D): for s = π/4 and extended Scherzer defocus (76.9 nm). All plots are normalized so that their maxima are equal to 1. C 2015 The Authors C 2015 Royal Microscopical Society, 259, 74–78 Journal of Microscopy
Fig. 1. Contrast transfer functions for spatial frequencies above the cuton for Zernike phase contrast. (A) A π /2 Zernike phase plate at Gaussian focus, (B) A π /2 Zernike phase plate at Scherzer defocus, (C) A π /4 Zernike phase plate at Scherzer defocus, (D) A π /4 Zernike phase plate at extended Scherzer defocus.
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For spatial frequencies below c, contrast transfer is described by the usual phase CTF sin(χ(u)). Clearly, the first pass-band is much wider for a π/4 Zernike phase plate at Scherzer defocus (Fig. 1 C) than it is for a π/2 Zernike phase plate at Gaussian focus (Fig. 1 A). The CTF re˚ in Figure 1(C), mains positive up to a resolution of about 2.5 A ˚ resolution in whereas it goes through zero at around 3.5 A Figure 1(A). As seen in Figure 1(D), the resolution range can be extended a bit more by recording at extended Scherzer defocus, however, at the cost of losing contrast at resolutions ˚ These considerations become especially imaround 3 to 5 A. portant when imaging slightly thicker specimens (about 10 to 20 nm thickness) since then the defocus variation around Scherzer becomes quite noticeable. A practical guideline could be that one specimen surface should be just below extended Scherzer and no part of the specimen at a defocus value higher than extended Scherzer in order to achieve the highest possible resolution without contrast inversions. A broadening of the first pass-band to approximately that of a π/4 Zernike phase plate at Scherzer defocus can also be achieved for the π/2 Zernike phase plate by imaging at Scherzer defocus (compare Fig. 1B to 1C). However, this also leads to a pronounced dip at lower frequencies. Obviously, the oscillation of the CTF seen in Figure 1(C) can 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 To estimate the image quality achievable by the proposed Zernike phase contrast imaging mode, I simulated phase contrast images of small protein hexamers imaged in top view. Amplitude contrast was not included in the image simulation since we are only interested in a comparison between different imaging modes. The electrostatic potential distribution inside trypsin inhibitor (Protein data Bank entry 4 pti with added hydrogen atoms), a small protein with a diameter of about 2.5–3 nm, was calculated using Matlab code written by Shang and Sigworth, which treats the molecule as a collection of neutral atoms as described in Shang and Sigworth (2012). All further steps of the electron microscopic imaging simulation including Fourier ring correlations (FRCs) were implemented in Khoros (Konstantinides & Rasure, 1994). The plots of Figures 1 and 3 were produced using OriginTM 7.5 (www.originlab.com). The phantoms used as specimens for image simulation were generated by arranging six copies of the electrostatic potential distribution in an oligomer with C6 symmetry. All imaging simulations were done for a 300 kV microscope with a Cs of 2 mm.
The voxel size of the phantom and the pixel size of the pro˚ giving a Nyquist limit of 1 A. ˚ jection and images are 0.5 A, The Zernike phase plate is assumed to have a hole of radius ˚ (corre0.5 μm leading to a cut-on frequency of 0.0125 1/A ˚ periodicity) for an objective lens with 2 mm sponding to 80 A focal length. As pointed out by (Danev & Nagayama, 2011) and (Malac et al., 2008), the choice of cut-on frequency is important for the signal-to-noise ratio (SNR) and the fidelity of the images. In the presented simulations, I made no attempt to choose an optimized value for the cut-on frequency. However, since I used the same cut-on frequency for all simulations, the results are suitable for comparison of the discussed imaging modes. 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 4.9 V for all voxels occupied by water. In the vicinity of the protein, the potential drops below this value to take into account the finite distance between water molecules and the atoms forming the surface of the protein. For details, see Shang & Sigworth (2012). 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, 4.9 V (the potential of water) were subtracted from the phantom before calculating a projection and simulating images. Images were calculated using Eq. (1) without approximating for weak phase objects to arrive at the Fourier transform of the exit wave modified by the lens aberration function and the phase plate and then taking the absolute square in real space. Figure 2 shows the simulated images and the corresponding projection of the specimen. The three images are displayed with quantitatively correct grey values to be able to visually compare contrast levels and they have been contrast inverted to facilitate comparison with the projection of the phantom. Figure 2(A) shows an image generated with a π/2 Zernike phase plate at Gaussian focus. Figure 2(B) shows an image generated with a π/4 Zernike phase plate at Scherzer defocus. Figure 2(C) shows an image generated with a π/4 Zernike phase plate at extended Scherzer defocus. Figure 2(D) shows the projection of the phantom for comparison. Clearly, the contrast for the three imaging modes is comparable whereas Figure 2(B) shows the highest level of detail. To quantify this resolution improvement, I also calculated FRC curves (see e.g. Frank, 2006, p. 249 ff and references given there) between the images and the projection for several cases as shown in Figure 3. From above, the plots show the FRC between the projection of the specimen and the π/2 Zernike phase contrast image at Gaussian focus (A), the π/4 Zernike phase contrast image at Scherzer defocus (B) and the π/4 Zernike phase contrast image at extended Scherzer defocus (C), respectively.
C 2015 The Authors C 2015 Royal Microscopical Society, 259, 74–78 Journal of Microscopy
IMPROVED ZERNIKE PHASE CONTRAST
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Fig. 2. (A) Simulated image for π /2 Zernike phase contrast at Gaussian focus, (B) Simulated image for π /4 Zernike phase contrast at Scherzer defocus, (C) Simulated image for π /4 Zernike phase contrast at extended Scherzer defocus, (D) Projection of the electrostatic potential. The size of the image is 10 nm × 10 nm.
In agreement with Figure 1, the FRCs show correlation close ˚ resolution for the π/4 Zernike to 1 up to and beyond 2.5 A phase contrast images at Scherzer and extended Scherzer defocus. For the π/2 Zernike phase contrast image at Gaussian ˚ focus the FRC drops to zero at a resolution of about 3.5 A. Here, it should also be pointed out that I made no attempt to optimize the phase-shift of the phase plate with respect to the phase shift from the specimen as discussed in Danev & Nagayama (2011). This would, of course, be possible but for the comparison between two imaging modes made here it does not seem necessary, since both simulations are made with the same phantom and, apart from the phase plate’s phase shift and the defocus, all parameters chosen are identical. Conclusions and outlook I present a modified Zernike-type phase contrast imaging mode using a phase plate that produces a relative phase shift between scattered and unscattered electrons of π/4 rather than the previously suggested phase shift of π/2. This phase shift could be realized either by a regular thin film Zernike phase plate (Danev & Nagayama, 2001), some type of hole-free thin film phase plate that relies on beam-induced charges (Malac et al., 2012; Danev et al., 2014) or by an electrostatic phase plate (Majorovits et al., 2007; Schultheiss et al., 2010). Images are then recorded at Scherzer or extended Scherzer defocus rather than at or close to Gaussian focus. This results in a C 2015 The Authors C 2015 Royal Microscopical Society, 259, 74–78 Journal of Microscopy
Fig. 3. FRC between projection and simulated image for (A) π /2 Zernike phase contrast at Gaussian focus, (B) π /4 Zernike phase contrast at Scherzer defocus, (C) π /4 Zernike phase contrast at extended Scherzer defocus
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pass-band without contrast inversions that extends to considerably higher resolution while maintaining the high contrast for all spatial frequencies up to this limit. Another advantage lies in the reduced amplitude loss and electron scattering in a thin film phase plate that is only half the thickness (Nagayama & Danev, 2008). Since the phase-plate only applies a phase shift of π/4, it can be kept very thin (about 11 nm for a carbon film phase plate and 300 keV electrons (Barton et al., 2008). Clearly, the presented ideas can also be applied in the case of the Volta potential phase plate (Danev et al., 2014) provided a material can be found that produces the correct phase shift for a microscope with a given acceleration potential. I suggest this imaging mode particularly for structure determination of proteins with a mass below about 200 kDa, which have been difficult to image with sufficient contrast. However, this imaging mode is completely general in nature and not limited to any particular type of specimen. Acknowledgement I acknowledge Hans Hebert for valuable comments on the project and Fred Sigworth for help with his Matlab code. References Barton, B., Joos, F. & Schr¨oder, R.R. (2008) Improved specimen reconstruction by Hilbert phase contrast tomography. J. Struct. Biol. 164, 210–220. Danev, R. & Nagayama, K. (2001) Transmission electron microscopy with Zernike phase plate. Ultramicroscopy 88, 243–252.
Danev, R. & Nagayama, K. (2011) Optimizing the phase shift and the cut-on periodicity of phase plates for TEM. Ultramicroscopy 111, 1305– 1315. Danev, R., Buijsse, B., Khoshouei, M., Plitzko, J.M. & Baumeister, W. (2014) Volta potential phase plate for in-focus phase contrast transmission electron microscopy. PNAS 111, 15635–15640. Frank, J. (2006) Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Oxford University Press, New York. Konstantinides, K. & Rasure, J.R. (1994) The Khoros software development environment for image and signal processing. IEEE T. Image Process. 3, 243–252. Majorovits, E., Barton, B., Schultheiß, K., P´erez-Willard, F., Gerthsen, D. & Schr¨oder, R.R. (2007) Optimizing phase contrast in transmission electron microscopy with an electrostatic (Boersch) phase plate. Ultramicroscopy 107, 213–226. Malac, M., Beleggia, M., Egerton, R. & Zhu, Y. (2008) Imaging of radiationsensitive samples in transmission electron microscopes equipped with Zernike phase plates. Ultramicroscopy 108, 126–140. Malac, M., Beleggia, M., Kawasaki, M., Li, P. & Egerton, R. (2012) Convenient contrast enhancement by a hole-free phaseplate. Ultramicroscopy 118, 77–89. Nagayama, K. & Danev, R. (2008) Phase contrast electron microscopy: development of thin-film phase plates and biological applications. Phil. Trans. R. Soc. B 363, 2153–2162. Schultheiss, K., Zach, J., Gamm, B., Dries, M., Frindt, N., Schr¨oder, R.R. & Gerthsen, D. (2010) New electrostatic phase plate for phase-contrast transmission electron microscopy and its application for wave-function reconstruction. Microsc. Microanal. 16, 785–794. Shang, Z. & Sigworth, F.J. (2012) Hydration-layer models for cryo-EM image simulation. J. Struct. Biol. 180, 10–16.
C 2015 The Authors C 2015 Royal Microscopical Society, 259, 74–78 Journal of Microscopy
