Ultramicroscopy 154 (2015) 37–41
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Improved Hilbert phase contrast for transmission electron microscopy Philip J.B. Koeck Royal Institute of Technology, School of Technology and Health and Karolinska Institutet, Department of Bioscience and Nutrition at Novum, Huddinge 14183, Sweden
art ic l e i nf o
a b s t r a c t
Article history: Received 30 October 2014 Received in revised form 27 February 2015 Accepted 5 March 2015 Available online 7 March 2015
Hilbert phase contrast has been recognized as a means of recording high resolution images with high contrast using a transmission electron microscope. This imaging mode could be used to image typical phase objects such as unstained biological molecules or cryo sections of biological tissue. According to the original proposal by (Danev et al., 2002) the Hilbert phase plate applies a phase shift of π to approximately half the focal plane (for example the right half excluding the central beam) and an image is recorded at Gaussian focus. After correction for the inbuilt asymmetry of differential phase contrast this image will have an almost perfect contrast transfer function (close to 1) from the lowest spatial frequency up to a maximum resolution determined by the wave length and spherical aberration of the microscope. In this paper I present theory and simulations showing that this maximum spatial frequency can be increased considerably almost without loss of contrast by using a Hilbert phase plate of half the thickness, leading to a phase shift of π/2, 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. & 2015 Elsevier B.V. All rights reserved.
Keywords: Hilbert phase contrast Weak phase approximation Cryo transmission electron microscopy
1. Introduction and theory The Fourier transform of the exit wave modified by the lens aberration function of a transmission electron microscope and a Hilbert phase plate mounted in the back focal plane can be written as follows:
Ψpc (⇀ u ) = Ψex (⇀ u ) e−i χ (u) H (⇀ u ) ≈ ⎡⎣δ (⇀ u ) + i σ F (⇀ u ) ⎤⎦ e−i χ (u) H (⇀ u ) (1) with
χ (u) = π D λ u2 +
⎧ ⎪ ei s H (⇀ u) = ⎨ ⎪ ⎩1
π Cs λ 3u4 2
⎪ for u x > g ⎫ ⎬ ⎪ for u x < g ⎭
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. An envelope describing partial spatial and temporal coherence or a resolution limiting aperture are not included in the image formation model. Abbreviations: CTF, contrast transfer function; TEM, transmission electron microscopy; FRC, Fourier ring correlation http://dx.doi.org/10.1016/j.ultramic.2015.03.002 0304-3991/& 2015 Elsevier B.V. All rights reserved.
Ψex (⇀ u ) is the Fourier transform of the exit wave and Ψpc (⇀ u ) is the same modified by the lens aberration function and the phase plate. F (⇀ u ) is the Fourier transform of the projection of the electrostatic potential distribution produced by the specimen being imaged. H (⇀ u ) describes a phase shift of s applied to the part of the focal plane corresponding to spatial frequency vectors with x-components larger than a small value g known as the “gap”. s is the positive valued interaction constant given by: σ = 2 π m e λ /h2. h is the Planck constant, m is the relativistic mass and e is the charge of the electron. The image recorded on a detector is given by the absolute square of the modified exit wave in direct space ⁎ ⇀ im pc (⇀ x ) = ψpc (⇀ x ) ψpc ( x )≈
⎡ ⎣⎢1 + i σ
∫ F (⇀u ) H (⇀u ) e−i χ (u) e2 π i ⇀x ⇀u du x du y ⎤⎦⎥
⎡ ⎣⎢1 − i σ
∫ F (⇀u ) H⁎ (−⇀u ) ei χ (u) e2 π i ⇀x ⇀u du x du y ⎤⎦⎥
(2)
with
⎧ ⎪ e−i s H⁎ ( − ⇀ u) = ⎨ ⎪ ⎩ 1
⎪ for u x < − g ⎫ ⎬ ⎪ for u x > − g ⎭
Due to the asymmetry of the Hilbert phase plate three different
38
P.J.B. Koeck / Ultramicroscopy 154 (2015) 37–41
contributions to the phase contrast image have to be distinguished depending on the x-component of the spatial frequency. For u x < g : ⇀
()
im pc x = 1 + 2 σ
⇀
⇀⇀
∫ F (u ) sin [ χ (u)] e2 π i x u du x du y
(3)
∫ F (⇀u ) [ei π s e−i χ (u) − ei χ (u) ] e2 π i ⇀x ⇀u du x du y
(4)
For u x > g :
im pc (⇀ x)=1+iσ For u x < − g :
im pc (⇀ x) =1+iσ
∫ F (⇀u ) [e−i χ (u) − e−i π s ei χ (u) ] e2 π i ⇀x ⇀u du x du y
(5)
Clearly for u x < g the imaging properties are the same as for a microscope without Hilbert phase plate. Only spatial frequencies outside the gap are affected by the phase shift. For specific values of s these results can be summarized in a more concise way as follows: For s¼ π transfer of spatial frequencies outside the gap can be summarized as ⇀
()
im pc x
=1−2iσ
⇀
⇀⇀
∫ F (u ) cos (χ (u)) sgn (u x ) e2 π i x u
du x du y for
ux > g
(6)
This describes the previously proposed differential phase contrast at Gaussian focus (choosing D ¼0), which we will refer to as the π Hilbert phase plate. See also [2]. For s ¼ π/2 transfer of spatial frequencies outside the gap can be summarized as: ⇀
()
im pc x
=1+ iσ
⇀⇀ e2 π i x u
for
ux > g
⇀
∫ F (u ) (i − sgn (u x ))(cos (χ (u)) −
sin (χ (u)))
du x du y
(7)
This describes the improved differential phase contrast near Scherzer defocus suggested in this paper. We will refer to this as the π/2 Hilbert phase plate in the following. In both the above cases the anisotropy of the contrast transfer function can be removed by modifying the FT of the recorded image. In the case of s¼ π the FT is divided or multiplied by i sgn (ux) for all Fourier components outside the gap. See also [2]. Contrast transfer for Fourier components outside the gap is then described by cos(χ(u)). In the case of s ¼ π/2 the FT is divided by (1 þi sgn (ux)) for all Fourier components outside the gap. Contrast transfer for Fourier components outside the gap is then proportional to cos(χ(u)) sin (χ(u)). The resulting CTFs for Fourier components outside the gap for a 300 kV TEM with Cs ¼2 mm are shown in Fig. 1A–J for various defocus values as follows: Fig. 1A–E (left column) show the CTF for s¼ π and various defocus values. Fig. 1F–J (right column) show the CTF for s¼ π/2 and various defocus values. Inside the gap contrast transfer is described by the usual phase contrast transfer function sin(χ(u)). Fig. 1(A) and (F) shows the respective defocus value where the CTF displays a dip all the way down to zero within the first
Fig. 1. Contrast transfer functions outside the gap for differential phase contrast after correction for the imaging anisotropy. A–E (left column): The π Hilbert phase plate, F–J (right column): the π/2 Hilbert phase plate. The two columns are arranged according to corresponding defocus values such that the defocus difference is always 14.2 nm (see text for detailed explanantions): (A) 62.7 nm (Scherzer defocus), (B) 48.5 nm, (C) 25.0 nm, (D) 0 nm (Gaussian focus), (E) þ14.2 nm, (F) 76.9 nm (extended Scherzer defocus). (G) 62.7 nm (Scherzer defocus). (H) 39.2 nm. (I) 14.2 nm. (J) 0 nm (Gaussian focus).
passband. This occurs at Scherzer defocus ( 62.7 nm) for s¼ π (1 A) and at extended Scherzer defocus ( 76.9 nm) for s¼ π/2 (1F). These defocus values can be considered respective upper bounds for imaging without CTF-correction. Of course in practice one would try to stay closer to focus to avoid too much of a dip in the pass band. In the remaining plots of Fig. 1 the defocus difference between the right and the left column is kept at 14.2 nm (76.9 62.7) for easy comparison. Real specimens have finite thickness and therefore different
P.J.B. Koeck / Ultramicroscopy 154 (2015) 37–41
39
Fig. 2. (a) corrected π Hilbert phase contrast at Gaussian focus, (b) corrected π/2 Hilbert phase contrast at Scherzer defocus. For both simulated images a gap of 1 pixel (g¼ 0.5 pixels), corresponding to a cut-on periodicity of 40 nm, was chosen. See text for a detailed discussion. (c) true projection of the electrostatic potential.
layers of a specimen are imaged at different defocus values. In addition to this it is experimentally difficult to select a precise defocus setting when imaging cryo specimens. Therefore it is important to consider the behavior of an imaging method at a range of defocus values. 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 50 nm the dip within the pass band reaches about 0.5 as seen in Fig. 1B. It seems that for thick specimens (around 50 nm) and in the case of uncertain defocus values the highest resolution can be achieved by aiming at an underfocus of around 25 nm or slightly higher (Fig. 1C). The point resolution for this imaging mode would be about 3 Å for thin specimen. The situation for the π/2 Hilbert phase plate is somewhat different since the best contrast transfer is achieved at an underfocus of 62.7 nm (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 14.2 nm 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 20 nm thickness one would therefore aim at an average underfocus of about 60 nm 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 10 nm thickness) the point resolution achievable in this imaging mode should be about 2.5 Å 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 1/ 2 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.
2. Simulations and discussion To estimate the image quality achievable by the proposed differential 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 4pti 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 [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). The phantoms used as specimens for image simulation were generated by arranging 6 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. 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. 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. All simulations were carried out with images measuring 400 by 400 pixels with a pixel size of 0.5 Å, 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 ¼0.5 pixels), corresponding to a cut-on periodicity of 40 nm, 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 ¼0.5 pixels, corresponding to a cut-on periodicity of 40 nm, in the left column and 4.5 pixels, corresponding to a cut-on periodicity of about 4.4 nm,
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P.J.B. Koeck / Ultramicroscopy 154 (2015) 37–41 1.0
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Fig. 3. FRC between projection and image. From top: corrected π Hilbert phase contrast at Gaussian focus, corrected π/2 Hilbert phase contrast at Scherzer defocus, corrected π/2 Hilbert phase contrast at extended Scherzer defocus. Left column 1 pixel gap (g ¼ 0.5) corresponding to a cut-on periodicity of 40 nm, right column 9 pixel gap (g ¼4.5) corresponding to a cut-on periodicity of about 4.4 nm.
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. In agreement with Fig. 1 the FRCs show correlation close to 1 up to and beyond 2.5 Å resolution for the π/2 Hilbert phase contrast images at Scherzer and extended Scherzer defocus. For the π Hilbert phase contrast image at Gaussian focus the FRC drops to zero at a resolution of about 3.5 Å.
3. Conclusions and outlook I present a modified Hilbert phase contrast imaging mode using a phase plate that produces a phase shift of π/2 rather than the previously suggested phase shift of π [1]. Images are then recorded close to Scherzer defocus rather than Gaussian focus. This results in a 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 in a phase plate that is only half the thickness [5]. I suggest this imaging mode particularly for
P.J.B. Koeck / Ultramicroscopy 154 (2015) 37–41
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.
Acknowledgements I acknowledge Hans Hebert for valuable comments on the project and Fred Sigworth for help with his Matlab code.
References [1] R. Danev, et al., A novel phase-contrast transmission electron microscopy producing high-contrast topographic images of weak objects, J. Biol. Phys. 28
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(2002) 627–635. [2] B. Barton, F. Joos, R.R. Schröder, Improved specimen reconstruction by Hilbert phase contrast tomography, J. Struct. Biol. 164 (2008) 210–220. [3] Z. Shang, F.J. Sigworth, Hydration-layer models for cryo-EM image simulation, J. Struct. Biol. 180 (2012) 10–16. [4] K. Konstantinides, J.R. Rasure, The Khoros software development environment for image and signal processing, IEEE Trans. Image Process. 3 (3) (1994) 243–252. [5] K. Nagayama, R. Danev, Phase contrast electron microscopy: development of thin-film phase plates and biological applications, Philos. Trans. R. Soc. B 363 (2008) 2153–2162.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Improved Hilbert phase contrast for transmission electron microscopy Philip J.B. Koeck Royal Institute of Technology, School of Technology and Health and Karolinska Institutet, Department of Bioscience and Nutrition at Novum, Huddinge 14183, Sweden
art ic l e i nf o
a b s t r a c t
Article history: Received 30 October 2014 Received in revised form 27 February 2015 Accepted 5 March 2015 Available online 7 March 2015
Hilbert phase contrast has been recognized as a means of recording high resolution images with high contrast using a transmission electron microscope. This imaging mode could be used to image typical phase objects such as unstained biological molecules or cryo sections of biological tissue. According to the original proposal by (Danev et al., 2002) the Hilbert phase plate applies a phase shift of π to approximately half the focal plane (for example the right half excluding the central beam) and an image is recorded at Gaussian focus. After correction for the inbuilt asymmetry of differential phase contrast this image will have an almost perfect contrast transfer function (close to 1) from the lowest spatial frequency up to a maximum resolution determined by the wave length and spherical aberration of the microscope. In this paper I present theory and simulations showing that this maximum spatial frequency can be increased considerably almost without loss of contrast by using a Hilbert phase plate of half the thickness, leading to a phase shift of π/2, 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. & 2015 Elsevier B.V. All rights reserved.
Keywords: Hilbert phase contrast Weak phase approximation Cryo transmission electron microscopy
1. Introduction and theory The Fourier transform of the exit wave modified by the lens aberration function of a transmission electron microscope and a Hilbert phase plate mounted in the back focal plane can be written as follows:
Ψpc (⇀ u ) = Ψex (⇀ u ) e−i χ (u) H (⇀ u ) ≈ ⎡⎣δ (⇀ u ) + i σ F (⇀ u ) ⎤⎦ e−i χ (u) H (⇀ u ) (1) with
χ (u) = π D λ u2 +
⎧ ⎪ ei s H (⇀ u) = ⎨ ⎪ ⎩1
π Cs λ 3u4 2
⎪ for u x > g ⎫ ⎬ ⎪ for u x < g ⎭
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. An envelope describing partial spatial and temporal coherence or a resolution limiting aperture are not included in the image formation model. Abbreviations: CTF, contrast transfer function; TEM, transmission electron microscopy; FRC, Fourier ring correlation http://dx.doi.org/10.1016/j.ultramic.2015.03.002 0304-3991/& 2015 Elsevier B.V. All rights reserved.
Ψex (⇀ u ) is the Fourier transform of the exit wave and Ψpc (⇀ u ) is the same modified by the lens aberration function and the phase plate. F (⇀ u ) is the Fourier transform of the projection of the electrostatic potential distribution produced by the specimen being imaged. H (⇀ u ) describes a phase shift of s applied to the part of the focal plane corresponding to spatial frequency vectors with x-components larger than a small value g known as the “gap”. s is the positive valued interaction constant given by: σ = 2 π m e λ /h2. h is the Planck constant, m is the relativistic mass and e is the charge of the electron. The image recorded on a detector is given by the absolute square of the modified exit wave in direct space ⁎ ⇀ im pc (⇀ x ) = ψpc (⇀ x ) ψpc ( x )≈
⎡ ⎣⎢1 + i σ
∫ F (⇀u ) H (⇀u ) e−i χ (u) e2 π i ⇀x ⇀u du x du y ⎤⎦⎥
⎡ ⎣⎢1 − i σ
∫ F (⇀u ) H⁎ (−⇀u ) ei χ (u) e2 π i ⇀x ⇀u du x du y ⎤⎦⎥
(2)
with
⎧ ⎪ e−i s H⁎ ( − ⇀ u) = ⎨ ⎪ ⎩ 1
⎪ for u x < − g ⎫ ⎬ ⎪ for u x > − g ⎭
Due to the asymmetry of the Hilbert phase plate three different
38
P.J.B. Koeck / Ultramicroscopy 154 (2015) 37–41
contributions to the phase contrast image have to be distinguished depending on the x-component of the spatial frequency. For u x < g : ⇀
()
im pc x = 1 + 2 σ
⇀
⇀⇀
∫ F (u ) sin [ χ (u)] e2 π i x u du x du y
(3)
∫ F (⇀u ) [ei π s e−i χ (u) − ei χ (u) ] e2 π i ⇀x ⇀u du x du y
(4)
For u x > g :
im pc (⇀ x)=1+iσ For u x < − g :
im pc (⇀ x) =1+iσ
∫ F (⇀u ) [e−i χ (u) − e−i π s ei χ (u) ] e2 π i ⇀x ⇀u du x du y
(5)
Clearly for u x < g the imaging properties are the same as for a microscope without Hilbert phase plate. Only spatial frequencies outside the gap are affected by the phase shift. For specific values of s these results can be summarized in a more concise way as follows: For s¼ π transfer of spatial frequencies outside the gap can be summarized as ⇀
()
im pc x
=1−2iσ
⇀
⇀⇀
∫ F (u ) cos (χ (u)) sgn (u x ) e2 π i x u
du x du y for
ux > g
(6)
This describes the previously proposed differential phase contrast at Gaussian focus (choosing D ¼0), which we will refer to as the π Hilbert phase plate. See also [2]. For s ¼ π/2 transfer of spatial frequencies outside the gap can be summarized as: ⇀
()
im pc x
=1+ iσ
⇀⇀ e2 π i x u
for
ux > g
⇀
∫ F (u ) (i − sgn (u x ))(cos (χ (u)) −
sin (χ (u)))
du x du y
(7)
This describes the improved differential phase contrast near Scherzer defocus suggested in this paper. We will refer to this as the π/2 Hilbert phase plate in the following. In both the above cases the anisotropy of the contrast transfer function can be removed by modifying the FT of the recorded image. In the case of s¼ π the FT is divided or multiplied by i sgn (ux) for all Fourier components outside the gap. See also [2]. Contrast transfer for Fourier components outside the gap is then described by cos(χ(u)). In the case of s ¼ π/2 the FT is divided by (1 þi sgn (ux)) for all Fourier components outside the gap. Contrast transfer for Fourier components outside the gap is then proportional to cos(χ(u)) sin (χ(u)). The resulting CTFs for Fourier components outside the gap for a 300 kV TEM with Cs ¼2 mm are shown in Fig. 1A–J for various defocus values as follows: Fig. 1A–E (left column) show the CTF for s¼ π and various defocus values. Fig. 1F–J (right column) show the CTF for s¼ π/2 and various defocus values. Inside the gap contrast transfer is described by the usual phase contrast transfer function sin(χ(u)). Fig. 1(A) and (F) shows the respective defocus value where the CTF displays a dip all the way down to zero within the first
Fig. 1. Contrast transfer functions outside the gap for differential phase contrast after correction for the imaging anisotropy. A–E (left column): The π Hilbert phase plate, F–J (right column): the π/2 Hilbert phase plate. The two columns are arranged according to corresponding defocus values such that the defocus difference is always 14.2 nm (see text for detailed explanantions): (A) 62.7 nm (Scherzer defocus), (B) 48.5 nm, (C) 25.0 nm, (D) 0 nm (Gaussian focus), (E) þ14.2 nm, (F) 76.9 nm (extended Scherzer defocus). (G) 62.7 nm (Scherzer defocus). (H) 39.2 nm. (I) 14.2 nm. (J) 0 nm (Gaussian focus).
passband. This occurs at Scherzer defocus ( 62.7 nm) for s¼ π (1 A) and at extended Scherzer defocus ( 76.9 nm) for s¼ π/2 (1F). These defocus values can be considered respective upper bounds for imaging without CTF-correction. Of course in practice one would try to stay closer to focus to avoid too much of a dip in the pass band. In the remaining plots of Fig. 1 the defocus difference between the right and the left column is kept at 14.2 nm (76.9 62.7) for easy comparison. Real specimens have finite thickness and therefore different
P.J.B. Koeck / Ultramicroscopy 154 (2015) 37–41
39
Fig. 2. (a) corrected π Hilbert phase contrast at Gaussian focus, (b) corrected π/2 Hilbert phase contrast at Scherzer defocus. For both simulated images a gap of 1 pixel (g¼ 0.5 pixels), corresponding to a cut-on periodicity of 40 nm, was chosen. See text for a detailed discussion. (c) true projection of the electrostatic potential.
layers of a specimen are imaged at different defocus values. In addition to this it is experimentally difficult to select a precise defocus setting when imaging cryo specimens. Therefore it is important to consider the behavior of an imaging method at a range of defocus values. 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 50 nm the dip within the pass band reaches about 0.5 as seen in Fig. 1B. It seems that for thick specimens (around 50 nm) and in the case of uncertain defocus values the highest resolution can be achieved by aiming at an underfocus of around 25 nm or slightly higher (Fig. 1C). The point resolution for this imaging mode would be about 3 Å for thin specimen. The situation for the π/2 Hilbert phase plate is somewhat different since the best contrast transfer is achieved at an underfocus of 62.7 nm (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 14.2 nm 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 20 nm thickness one would therefore aim at an average underfocus of about 60 nm 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 10 nm thickness) the point resolution achievable in this imaging mode should be about 2.5 Å 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 1/ 2 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.
2. Simulations and discussion To estimate the image quality achievable by the proposed differential 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 4pti 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 [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). The phantoms used as specimens for image simulation were generated by arranging 6 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. 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. 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. All simulations were carried out with images measuring 400 by 400 pixels with a pixel size of 0.5 Å, 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 ¼0.5 pixels), corresponding to a cut-on periodicity of 40 nm, 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 ¼0.5 pixels, corresponding to a cut-on periodicity of 40 nm, in the left column and 4.5 pixels, corresponding to a cut-on periodicity of about 4.4 nm,
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Fig. 3. FRC between projection and image. From top: corrected π Hilbert phase contrast at Gaussian focus, corrected π/2 Hilbert phase contrast at Scherzer defocus, corrected π/2 Hilbert phase contrast at extended Scherzer defocus. Left column 1 pixel gap (g ¼ 0.5) corresponding to a cut-on periodicity of 40 nm, right column 9 pixel gap (g ¼4.5) corresponding to a cut-on periodicity of about 4.4 nm.
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. In agreement with Fig. 1 the FRCs show correlation close to 1 up to and beyond 2.5 Å resolution for the π/2 Hilbert phase contrast images at Scherzer and extended Scherzer defocus. For the π Hilbert phase contrast image at Gaussian focus the FRC drops to zero at a resolution of about 3.5 Å.
3. Conclusions and outlook I present a modified Hilbert phase contrast imaging mode using a phase plate that produces a phase shift of π/2 rather than the previously suggested phase shift of π [1]. Images are then recorded close to Scherzer defocus rather than Gaussian focus. This results in a 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 in a phase plate that is only half the thickness [5]. I suggest this imaging mode particularly for
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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.
Acknowledgements I acknowledge Hans Hebert for valuable comments on the project and Fred Sigworth for help with his Matlab code.
References [1] R. Danev, et al., A novel phase-contrast transmission electron microscopy producing high-contrast topographic images of weak objects, J. Biol. Phys. 28
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(2002) 627–635. [2] B. Barton, F. Joos, R.R. Schröder, Improved specimen reconstruction by Hilbert phase contrast tomography, J. Struct. Biol. 164 (2008) 210–220. [3] Z. Shang, F.J. Sigworth, Hydration-layer models for cryo-EM image simulation, J. Struct. Biol. 180 (2012) 10–16. [4] K. Konstantinides, J.R. Rasure, The Khoros software development environment for image and signal processing, IEEE Trans. Image Process. 3 (3) (1994) 243–252. [5] K. Nagayama, R. Danev, Phase contrast electron microscopy: development of thin-film phase plates and biological applications, Philos. Trans. R. Soc. B 363 (2008) 2153–2162.
