TY - JOUR
AU - Allman,, Brendan
AB - Abstract Since the Transport Intensity Equation (TIE) has been applied to electron microscopy only recently, there are controversial discussions in the literature regarding the theoretical concepts underlying the equation and the practical techniques to solve the equation. In this report we explored some of the issues regarding the TIE, especially bearing electron microscopy in mind, and clarified that: (i) the TIE for electrons exactly corresponds to the Schrödinger equation for high-energy electrons in free space, and thus the TIE does not assume weak scattering; (ii) the TIE can give phase information at any distance from the specimen, not limited to a new field; (iii) information transfer in the TIE for each spatial frequency g will be multiplied by g2 and thus low frequency components will be dumped more with respect to high frequency components; (vi) the intensity derivative with respect to the direction of wave propagation is well approximated by using a set of three symmetric images; and (v) a substantially larger defocus distance than expected before can be used for high-resolution electron microscopy. In the second part of this report we applied the TIE down to atomic resolution images to obtain phase information and verified the following points experimentally: (i) although low frequency components are attenuated in the TIE, all frequencies will be recovered satisfactorily except the very low frequencies; and (ii) using a reconstructed phase and the measured image intensity we can correct effectively the defects of imaging, such as spherical aberrations as well as partial coherence. transfer of intensity equation, HREM, phase measurement, quantitative phase imaging, aberration correction Introduction Phase measurement is important for electron microscopy, since many samples in electron microscopy are phase objects as in the case of optical microscopy. In optical microscopy a Zernike phase plate is employed, but it has been difficult to realize a corresponding phase plate in electron microscopy. Therefore, an imaging condition proposed by Scherzer [1] has been used in high-resolution electron microscopy, where an approximate phase plate is realized by introducing a defocus in the presence of a spherical aberration. Recently, through-focus reconstruction techniques were developed for high-resolution electron microscopy and spherical aberration was corrected using a reconstructed wave front [2–4]. About 20 years ago, Teague [5] derived an equation for wave propagation in terms of a phase and intensity distribution under the small angle (paraxial) approximation and showed that the phase distribution may be determined by measuring only the intensity distribution. We call this equation the Transport Intensity Equation (TIE). Van Dyck and Coene [6] obtained a similar equation in terms of a phase and amplitude distribution. The TIE was recently applied successfully in medium resolution to observe static potential distributions of biological and non-biological samples [7] or measure magnetic fields [8]. However, there is no application of the TIE for atomic resolution images, because it is believed that an image set obtained with a very small defocus step is required [6] and therefore, signal change may be below noise level. There seems to be some confusion regarding the theoretical concepts underlying the equation and the practical techniques to solve the equation. It is important to note that the practical application of the TIE depends on an algorithm to solve the equation. The aims of this paper are to explore some of the issues regarding the TIE, especially bearing electron microscopy in mind, and to demonstrate that TIE is applicable to atomic resolution images by an illustration with a real example. Derivation of the TIE for high-energy electrons The transfer of intensity equation for electrons should be derived from the Schrödinger equation that controls the behavior of electrons. Actually, it was shown by Barty et al. [9] that the TIE is obtained from the following Schrödinger equation in vacuum under small angle approximation: \[\left(2ik\frac{{\partial}}{{\partial}z}\ +\ {\nabla}_{xy}^{2}\ +\ 2k^{2}\right){\psi}\ \left(xyz\right)\ =\ 0,\] (1) where the wave propagates along the z-direction, and k is a wave vector (k = 2π/λ, where λ is a wave length) and ∇xy2 a 2D Laplacian. Here, we express a complex wave function ψ by using the two real functions, I and ϕ, which represent the intensity and phase distributions, respectively, in the following form \[{\psi}\ \left(xyz\right)\ {\equiv}\ \sqrt{I\left(xyz\right)}\ \mathrm{exp}\ \left\{i{\phi}\ \left(xyz\right)\right\}\ \mathrm{exp}\ \left\{i\mathbf{kr}\right\}.\] (2) Then, the Schrödinger equation becomes two equations corresponding to the real and imaginary parts. This imaginary part is exactly equal to the following TIE: \[2\frac{{\pi}}{{\lambda}}\ \frac{{\partial}}{{\partial}z}\ I\ \left(xyz\right)\ =\ {-}{\nabla}_{xy}\ {\bullet}\ \left(I\ \left(xyz\right){\nabla}_{xy}\ {\phi}\ \left(xyz\right)\right).\] (3) This equation relates the phase distribution with the intensity distribution and its derivative with respect to the direction of wave propagation. It may be noted that the TIE does not assume any scattering model, and thus the TIE will be used even for a strong phase object. Since the small angle approximation for the Schrödinger equation is the basis of electron microscopy using high-energy electrons, the TIE may be universally applied in the field of electron microscopy. Van Dyck and Coene [6] derived a similar equation in terms of the phase and amplitude from the same form of the Schrödinger equation. When neglecting aberrations, except those such as spherical aberration and astigmatism, the equivalent form of the TIE could be derived by using the transfer function in the small defocus limit [10]. However, we have to note that the defocus is not necessarily limited to a small value from the specimen surface, since we can include a phase term due to the defocus to the object phase function ϕ. Therefore, an in-focus image is not necessarily an image taken at the specimen exit surface, and thus the TIE is not limited to measure the phase distribution at the near field region. It is more important to note that the TIE is a different form of the Schrödinger equation, and the small defocus is not a prerequisite for deriving the TIE. Information transfer in the TIE It is instructive to consider an information transfer in the TIE for each spatial frequency. We note that the effect of wave propagation in Fourier space is simply described by a multiplication of a well-known defocus term: \[{\Psi}\ \left(\mathbf{g};\ z_{0}\ +\ z\right)\ =\ {\Psi}\ \left(\mathbf{g};\ z_{0}\right)\ \mathrm{exp}\ \left\{{-}i{\pi}\ {\lambda}\ z\ \mathbf{g}^{2}\right\},\] (4) where g is a spatial frequency and a scattering angle is given by α = λg. Thus, the derivative of the wave function with respect to z is simply given by \[\frac{{\partial}{\Psi}\left(\mathbf{g}\right)}{{\partial}z}\ =\ {-}i{\pi}{\lambda}\mathbf{g}^{2}{\Psi}\ \left(\mathbf{g},\ z\right).\] (5) Since an image intensity is a product of the complex wave-function and its complex conjugate, we can write down an intensity derivative in Fourier space using the convolution theorem as, \begin{eqnarray*}&&F\left(\frac{{\partial}I}{{\partial}z}\right)\ =\ \frac{{\partial}}{{\partial}z}\ F\left({\psi}\ {\cdot}\ {\psi}^{{\ast}}\right)\\&&=\ \frac{{\partial}}{{\partial}z}\ {\Psi}\left(\mathbf{g}\right)\ {\otimes}\ {\Psi}^{{\ast}}\ \left({-}\mathbf{g}\right)\ +\ {\Psi}\ \left(\mathbf{g}\right)\ {\otimes}\ \frac{{\partial}}{{\partial}z}\ {\Psi}^{{\ast}}\left({-}\mathbf{g}\right)\\&&=\ \left({-}i{\pi}\ {\lambda}\ g^{2}{\Psi}\ \left(\mathbf{g}\right)\right)\ {\otimes}\ {\Psi}^{{\ast}}\left({-}\mathbf{g}\right)\ +\ {\Psi}\ \left(\mathbf{g}\right)\ {\otimes}\ \left(+i{\pi}{\lambda}\ g^{2}{\Psi}^{{\ast}}\left({-}\mathbf{g}\right)\right).\end{eqnarray*} (6) This clearly indicates that low frequency components will be dumped with respect to high frequency components, since each component of the image is multiplied by g2 in the TIE relation. This non-isotropic information transfer in terms of image frequency is a very important feature, when we try to retrieve phase information from the real images based on the TIE technique. Numerical solution of the TIE Since the TIE is a partial differential equation, we have to devise a practical way to solve the equation. Here, we introduce the algorithm proposed by Paganin and Nugent [11], which we have applied to atomic resolution images. This is a two-step solution based on two successive Poisson's equations. At first we introduce an intermediate function Φ, whose gradient corresponds to a product of the intensity and a gradient of the phase as follows: \[{\nabla}_{xy}{\Phi}\ \left(xyz\right)\ {\equiv}\ I\left(xyz\right){\nabla}_{xy}{\phi}\ \left(xyz\right).\] (7) Then, the TIE becomes a following Poisson's equation \[{\nabla}_{xy}^{2}{\Phi}\ \left(xyz\right)\ =\ {-}\frac{2{\pi}}{{\lambda}}\ \frac{{\partial}}{{\partial}z}I\left(xyz\right).\] (8) In the first step we solve this Poisson's equation to obtain the intermediate function Φ. Then, taking the 2D divergence of the gradient of Φ, after dividing it by the intensity I, we can get the second Poisson's equation for the phase function itself \[{\nabla}_{xy}^{2}{\phi}\ \left(xyz\right)\ =\ {\nabla}_{xy}\ {\bullet}\ \left(\frac{1}{I\left(xyz\right)}\ {\nabla}_{xy}{\Phi}\ \left(xyz\right)\right).\] (9) Thus, we have the following formal expression for the solution of the phase function \[{\phi}\left(xyz\right)\ =\ {-}\frac{2{\pi}}{{\lambda}}\ {\nabla}_{xy}^{{-}2}{\nabla}_{xy}\ {\bullet}\ \left(\frac{1}{I\left(xyz\right)}\ {\nabla}_{xy}{\nabla}_{xy}^{{-}2}\frac{{\partial}}{{\partial}z}\ I\left(xyz\right)\right),\] (10) where, \({\nabla}_{xy}^{{-}2}\) is a formal inverse Laplacian operator. These two Poisson's equations are solved in the usual way by using Fourier transforms in the implementation of the Quantitative Phase Imaging [12]. It may be noted that an inverse Laplacian operator in Fourier space may be written as, \[{\nabla}_{xy}^{{-}2}\ =\ {-}F^{{-}1}g^{{-}2}F\] (11) where \(g^{2}\ =\ g_{x}^{2}\ +\ g_{y}^{2}\) ⁠. Thus, the factor g2 multiplied with each frequency component of the intensity derivative [see eq. (6)] will be canceled out during the step to solve the first Poisson's equation [eq. (8)]. Therefore, the decrease of contribution of low frequency components in the TIE will be theoretically recovered if we can ignore low frequency noise. A low-frequency component is usually stronger than a high-frequency component and thus less affected by noise. Nevertheless, the recovery of very low frequencies may not be ideal due to the presence of weak noise. The problem of boundary conditions encountered in a numerical solution of the partial differential equation is always a nuisance. However, this problem is implicitly bypassed by a periodic continuation at the boundary assumed in Fourier transform. We have to note that this algorithm only works when we can provide an intensity derivative at the plane where we want to elucidate a phase distribution. Evaluation of an intensity derivative One of the difficulties of using the TIE is the estimation of an intensity derivative with respect to z as accurately as possible. The intensity derivative is a mathematical concept, and in practice we have to replace it with a finite difference calculated by using measured intensities. Figure 1 shows schematically an experimental situation for the application of the TIE, where we measure image intensities at a few consecutive planes. Here, we have to estimate the intensity derivative at the center plane on which an intensity distribution is observed. An accurate estimate of the intensity derivative should be obtained mathematically by a difference of image intensities measured at sufficiently close intervals. However, this does not work in practice, since non-negligible noise always exists in the images. Therefore, a large defocus step is preferable to increase the signal over the noise in the intensity difference. Alas, the images taken with a large defocus step will give a poor estimate of the derivative. Fig. 1 Open in new tabDownload slide An experimental situation where we measure image intensities at a few consecutive planes along the direction of wave propagation. Here, we have to estimate the intensity derivative at the center plane on which an intensity distribution is observed. Fig. 1 Open in new tabDownload slide An experimental situation where we measure image intensities at a few consecutive planes along the direction of wave propagation. Here, we have to estimate the intensity derivative at the center plane on which an intensity distribution is observed. We analyze this problem by using the following Taylor expansion of the intensity with respect to z, the direction of wave propagation: \[I\left(xy,\ z\ +\ {\varepsilon}\right)\ =\ I\left(xyz\right)\ +\ \frac{{\partial}\ I}{{\partial}z}\ {\varepsilon}\ +\ \frac{{\partial}^{2}I}{{\partial}z^{2}}\frac{{\varepsilon}^{2}}{2{!}}\ +\ \frac{{\partial}^{3}I}{{\partial}z^{3}}\frac{{\varepsilon}^{3}}{3{!}}\ +\ \frac{{\partial}^{4}I}{{\partial}z^{4}}\frac{{\varepsilon}^{4}}{4{!}}\ +\ O\left({\varepsilon}^{5}\right).\] (12) It is clear that a simple intensity difference I(z+ε)−I(z) between two consecutive planes gives a poor estimate of the derivative, where an error is the first order of the focus distance O(ε). For the symmetric three-image case as shown in Fig. 1, where the three images are recorded with the same defocus distance, the intensity derivative may be given by an intensity difference between the two side planes as follows: \[\frac{I\left(z\ +\ {\varepsilon}\right)\ {-}\ I\left(z\ {-}\ {\varepsilon}\right)}{2{\varepsilon}}\ =\ \frac{{\partial}\ I}{{\partial}z}\ +\ O\left({\varepsilon}^{2}\right),\] (13) where, all the even order terms of two in the Taylor expansions for I(z + ε) and I(z − ε) are canceled out and thus the simple difference will give the derivative, where an error is the second order of the focus distance. This may be understood when we note that a slope between the two side points exactly gives the derivative of intensity at the center point within the parabolic approximation using three equidistant data points. When the defocus distances from the center (in-focus) plane to two side planes are not exactly equal, the derivative at the center plane is still estimated as follows by using the intensities measured at all the three planes \[\frac{\left\{I\left(z\right)\ {-}\ I\left(z\ {-}\ {\varepsilon}_{1}\right)\right\}{\varepsilon}_{2}^{2}\ +\ \left\{I\left(z\ +\ {\varepsilon}_{2}\right)\ {-}\ I\left(z\right)\right\}{\varepsilon}_{1}^{2}}{{\varepsilon}_{1}{\varepsilon}_{2}\left({\varepsilon}_{1}\ +\ {\varepsilon}_{2}\right)}\ =\ \frac{{\partial}I}{{\partial}z}\ +\ O\left({\varepsilon}_{1}{\varepsilon}_{2}\right)\] (14) where ε1 and ε2 are two defocus distances from the center image plane. When the defocus asymmetry becomes large, accuracy of estimating the derivative degrades to the one attained by the consecutive two images. We may be able to use more than three images to estimate the intensity derivative with a better accuracy. Here, we assume five images taken with the same defocus distance. Then, we have the following expression for the intensity derivative at the center plane: \[\frac{{\partial}I}{{\partial}z}\ =\ \frac{4}{3}\ \frac{I\left(z\ +\ {\varepsilon}\right)\ {-}\ I\left(z\ {-}\ {\varepsilon}\right)}{2{\varepsilon}}\ {-}\ \frac{1}{3}\ \frac{I\left(z\ +\ 2{\varepsilon}\right)\ {-}\ I\left(z\ {-}\ 2{\varepsilon}\right)}{4{\varepsilon}}\ +\ O\left({\varepsilon}^{4}\right).\] (15) In this case, an error becomes the forth order of the focus distance. Although the expression of the derivative becomes complex, we can use five images taken with non-equal focus distances. Upper limit of the defocus distance An upper limit of the defocus distance that gives a good estimate of the derivative thus depends on the number of images used to estimate the derivative. We will estimate the upper limit of the defocus distance by evaluating a residual error term between a derivative and a finite difference formula. For the three-image case the error term is a third order derivative of the intensity as follows: \[O\left({\epsilon}^{2}\right)\ =\ \frac{{\epsilon}^{2}}{3{!}}\ \frac{{\partial}^{3}I}{{\partial}z^{3}}\ =\ \frac{{\epsilon}^{2}}{3{!}}\ \left(\frac{{\partial}^{3}\ {\psi}}{{\partial}\ z^{3}}\ {\cdot}\ {\psi}^{{\ast}}\ +\ {\psi}\ {\cdot}\ \frac{{\partial}^{3}\ {\psi}^{{\ast}}}{{\partial}z^{3}}\right).\] (16) Then, its expression in Fourier space becomes, \begin{eqnarray*}&&F\left(O\left({\varepsilon}^{2}\right)\right)\ =\ \frac{{\varepsilon}^{2}}{3{!}}\ \left[\left[\left({-}i\ {\pi}\ {\lambda}\ g^{2}\right)^{3}\ {\Psi}_{g}\right]\ {\otimes}\ {\Psi}_{{-}g}^{{\ast}}\right.\ \\&&\left.\ +\ {\Psi}_{g}\ {\otimes}\ \left[\left(+\ i{\pi}{\lambda}\ g^{2}\right)^{3}\ {\Psi}_{{-}g}^{{\ast}}\right]\right].\end{eqnarray*} (17) When we compare this error term with the intensity derivative [eq: (6)], the error term may be ignored if the following condition is satisfied: \[\left({\pi}\ {\lambda}\ {\epsilon}\ g_{\mathrm{max}}^{2}\right)^{2}/\ 3{!}\ {\leq}\ c\ {\ll}\ 1\] (18) where gmax is the highest spatial frequency included in the image, and c is a small number, say 0.25. If this condition is satisfied by the highest spatial frequency, the intensity derivative in real space may also be evaluated with a good accuracy by the difference of the intensities at the two side planes. Van Dyck and Coene [6] derived the following upper defocus limit assuming a linear approximation to the defocus phase term for the wave function: \[{\pi}\ {\lambda}\ {\epsilon}\ g_{\mathrm{max}}^{2}\ {\leq}\ c\ {\ll}\ 1\] (19) Then, they estimated that the upper defocus limit for a resolution of 0.14 nm assuming 400 kV electrons is 1.0 nm. (We may note here that the upper defocus limit was erroneously quoted as 0.1 nm in their paper.) Our estimate of the defocus difference between the two side images of the symmetric three-image case is written as \[{\pi}\ {\lambda}\ \left(2\ {\epsilon}\right)\ g_{\mathrm{max}}^{2}\ {\leq}\ 2\ \sqrt{c\ {\cdot}\ 3{!}}\] (20) which is almost 10 times larger then the corrected estimate of Van Dyck and Coene. (Thus, an apparent improvement of the defocus limit is almost one hundred times.) Table 1 will give you an idea on the defocus limits for some resolutions and accelerating voltages. Our upper defocus limit for a resolution of 0.14 nm assuming 400 kV electrons is 9.5 nm. This large defocus step will give a sufficient change in intensity even of high-resolution images. Therefore, the TIE will be applicable even to atomic resolution images as shown in the next section. Table 1. Typical defocus upper limits for the three-image case dmin πλ(2ε) 100 kV 200 kV 400 kV 0.14 nm 4.90×10−2 4.2 nm 6.2 nm 9.5 nm 0.2 9.80 8.4 12.4 19.0 0.3 2.20×10−1 19.0 28.0 42.7 1 2.45 211 311 474 10 2.45×102 21.1 μm 31.1 μm 47.4 μm 100 2.45×104 2.11 mm 3.11 mm 4.74 mm dmin πλ(2ε) 100 kV 200 kV 400 kV 0.14 nm 4.90×10−2 4.2 nm 6.2 nm 9.5 nm 0.2 9.80 8.4 12.4 19.0 0.3 2.20×10−1 19.0 28.0 42.7 1 2.45 211 311 474 10 2.45×102 21.1 μm 31.1 μm 47.4 μm 100 2.45×104 2.11 mm 3.11 mm 4.74 mm This table shows typical upper limits of defocus distance (2ε) between two side images for some resolutions dmin and accelerating voltages. Here, we assume an upper error limit as c = 0.25. For instance, we can use two images taken with 9.5 nm defocus distance for 0.14 nm resolution assuming 400 kV electrons. Open in new tab Table 1. Typical defocus upper limits for the three-image case dmin πλ(2ε) 100 kV 200 kV 400 kV 0.14 nm 4.90×10−2 4.2 nm 6.2 nm 9.5 nm 0.2 9.80 8.4 12.4 19.0 0.3 2.20×10−1 19.0 28.0 42.7 1 2.45 211 311 474 10 2.45×102 21.1 μm 31.1 μm 47.4 μm 100 2.45×104 2.11 mm 3.11 mm 4.74 mm dmin πλ(2ε) 100 kV 200 kV 400 kV 0.14 nm 4.90×10−2 4.2 nm 6.2 nm 9.5 nm 0.2 9.80 8.4 12.4 19.0 0.3 2.20×10−1 19.0 28.0 42.7 1 2.45 211 311 474 10 2.45×102 21.1 μm 31.1 μm 47.4 μm 100 2.45×104 2.11 mm 3.11 mm 4.74 mm This table shows typical upper limits of defocus distance (2ε) between two side images for some resolutions dmin and accelerating voltages. Here, we assume an upper error limit as c = 0.25. For instance, we can use two images taken with 9.5 nm defocus distance for 0.14 nm resolution assuming 400 kV electrons. Open in new tab Application of the TIE to atomic resolution images To demonstrate an applicability of the TIE to measure a phase distribution at atomic resolution, we used high-resolution images of Si3N4 obtained at NCEM using a Philips CM300 equipped with a field emission gun [13]. A series of through-focus images was taken at a rather large under-focus from 280 to 240 nm to prevent attenuation at high spatial frequency due to spatial coherency. A sampling interval is 0.02 nm, and the original image size (1024×1024 pixels) is ∼20 nm. Figure 2 shows a whole area of the in-focus (center) image and center parts of over- and under-focus images selected from the focal series by skipping every two images. Since an original focus step is ∼2 nm, the defocus distance between the under and over-focus images is almost 11.6 nm. A reconstructed phase as obtained is shown in Fig. 3a. There is a big slowly varying feature, whose phase excursion extends over 55 rad. Since the phase variation due to atomic structure is ∼0.2 rad (see below), its structure is hardly recognized due to this slowly varying feature. Handling of the slowly varying feature is thus an important issue of the TIE as discussed recently by Beleggia et al. [14]. Fig. 2 Open in new tabDownload slide High-resolution images of Si3N4 obtained at NCEM using a Philips CM300 equipped with a field emission gun [13]. Here, we show three images selected from the focal series of 20 images taken with 1.93 nm defocus step. A sampling interval is 0.02 nm, and the original image size (1024×1024 pixels) is ∼20 nm. (a) A whole area of the in-focus (center) image, and (b) and (c), center parts of over- and under-focus images. The defocus of the center image is 273.52 nm under-focus, and the defocus distance between the under and over-focus images is 11.58 nm. Fig. 2 Open in new tabDownload slide High-resolution images of Si3N4 obtained at NCEM using a Philips CM300 equipped with a field emission gun [13]. Here, we show three images selected from the focal series of 20 images taken with 1.93 nm defocus step. A sampling interval is 0.02 nm, and the original image size (1024×1024 pixels) is ∼20 nm. (a) A whole area of the in-focus (center) image, and (b) and (c), center parts of over- and under-focus images. The defocus of the center image is 273.52 nm under-focus, and the defocus distance between the under and over-focus images is 11.58 nm. Fig. 3 Open in new tabDownload slide Reconstructed phase images. (a) Reconstructed phase as obtained according to the two-step solution. Here, there is a big slowly varying feature, whose phase excursion extends over 55 rad, and the phase variation due to atomic structure is hardly recognized. (b) Reconstructed phase after applying a Tikhonov-type filter for the inverse Laplacian operator. The phase map becomes almost flat, and an atomic structure can now be recognized. This is the phase distribution at the plane of the center image, namely ∼270 nm under-focus from the sample. Fig. 3 Open in new tabDownload slide Reconstructed phase images. (a) Reconstructed phase as obtained according to the two-step solution. Here, there is a big slowly varying feature, whose phase excursion extends over 55 rad, and the phase variation due to atomic structure is hardly recognized. (b) Reconstructed phase after applying a Tikhonov-type filter for the inverse Laplacian operator. The phase map becomes almost flat, and an atomic structure can now be recognized. This is the phase distribution at the plane of the center image, namely ∼270 nm under-focus from the sample. The slowly varying feature results from low frequency noise. Here, we suppress very low frequencies using a Tikhonov-type filter for the inverse Laplacian operator \[T\left(g\right)\ =\ g^{2}/\left(g^{2}\ +\ g_{c}^{2}\right)^{2},\] (21) with gc = 0.5 nm−1. Then, an effective contribution of a 0.5 nm−1 spatial frequency is decreased to 25%, when we take into account the multiplication factor g2 in eq. (6). Using this filter an atomic structure is recognizable as shown in Fig. 3b. The phase variation over the whole area is almost flat, and its standard variation is 0.27 rad. A Fourier filter assuming a periodic structure can further reduce remaining inhomogeneous low-frequency features. This is the phase distribution at the plane of the center image, namely ∼270 nm under-focus from the sample. Using this phase distribution and the observed image intensity, we can now reconstruct a complex wave front at this image plane. When we back-propagate the wave front to the specimen plane, we will be able to get the wave front at the specimen exit plane. During the back-propagation we can correct spherical aberration as well as transfer attenuation due to temporal and spatial partial coherency. Figure 4a shows the phase distribution thus obtained at the specimen exit surface (zero-defocus). Here, a Fourier filtering was employed to remove small heterogeneous contrast. On the other hand, Fig. 4b shows the phase distribution at the plane where an amplitude variation is minimum. Here, a standard variation of the phase distribution is 0.21 rad. The phase map at the minimum amplitude is closely related to the structure model, and we can detect even a nitrogen atom that bridges the hexagonal rings. Both of these phase maps shown in Fig. 4 bear a striking resemblance to the results reported by Ziegler et al. [13], where the whole set of images is used to retrieve the wave field with the MAL technique [2]. This result shows that the TIE does not lose high-frequency information contrary to the conclusions given by Beleggia et al. [14]. Fig. 4 Open in new tabDownload slide Two phase distributions obtained by back-propagating the wave front from the observed center image plane. (a) Phase distribution at the specimen exit surface (zero-defocus). (b) Phase distribution at the plane where an amplitude variation becomes minimum. During the back-propagation we corrected spherical aberration as well as transfer attenuation due to temporal and spatial partial coherency. Then, Fourier filtering was employed to remove small heterogeneous contrast. The phase map at the minimum amplitude is closely related to the structure model. Fig. 4 Open in new tabDownload slide Two phase distributions obtained by back-propagating the wave front from the observed center image plane. (a) Phase distribution at the specimen exit surface (zero-defocus). (b) Phase distribution at the plane where an amplitude variation becomes minimum. During the back-propagation we corrected spherical aberration as well as transfer attenuation due to temporal and spatial partial coherency. Then, Fourier filtering was employed to remove small heterogeneous contrast. The phase map at the minimum amplitude is closely related to the structure model. The whole image processing presented here was performed on DigitalMicrograph [15] using a plug-in, QPt for DigitalMicrograph [16], developed on the basis of the Quantitative Phase Imaging [12]. Conclusions Since the TIE has been applied to electron microscopy only recently, there are controversial discussions in the literature. In this report, we have clarified the theoretical and practical aspects of the TIE, especially bearing electron microscopy in mind. The following issues regarding the TIE have been discussed: (i) the TIE for electrons exactly corresponds to the Schrödinger equation for high-energy electrons in free space, and thus the TIE can be applied not only to a weak phase object but also a strongly scattering object; (ii) the TIE can give phase information at any distance from the specimen, not limited to the near field; (iii) information transfer in the TIE for each spatial frequency g will be multiplied by g2, and thus low frequency components will be dumped more with respect to high frequency components; (vi) the intensity derivative with respect to the direction of wave propagation is well approximated by using a set of three symmetric images than the two images; and (v) similarly, the upper limit of defocus distance will be increased substantially by using the three symmetric images. With a substantially larger defocus value than expected before, high-resolution electron micrographs will give sufficient intensity change. In the second part of this report we have applied the TIE down to atomic resolution images to obtain phase information, and verified the following points experimentally: (i) although low frequency components are attenuated in the TIE, all frequencies will be recovered satisfactory except very low frequencies; and (ii) using a reconstructed phase and the measured image intensity we can effectively correct the defects of imaging such as spherical aberrations as well as partial coherency. The authors greatly acknowledge Christian Kisielowsky for kindly providing us with Si3N4 images and Les Allen for valuable information. References 1 Schrzer O ( 1949 ) The theoretical resolution limit of the electron microscope. J. Appl. Phys. 20 : 20 –29. 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TI - Phase measurement of atomic resolution image using transport of intensity equation
JF - Journal of Electron Microscopy
DO - 10.1093/jmicro/dfi024
DA - 2005-06-01
UR - https://www.deepdyve.com/lp/oxford-university-press/phase-measurement-of-atomic-resolution-image-using-transport-of-d00LhFARjE
SP - 191
EP - 197
VL - 54
IS - 3
DP - DeepDyve
ER -