Precise method for measuring spatial coherence in TEM beams using Airy diffraction patterns

Precise method for measuring spatial coherence in TEM beams using Airy diffraction patterns Abstract We have developed a method to precisely measure spatial coherence in electron beams. The method does not require an electron biprism and can be implemented in existing analytical transmission electron microscopes equipped with a post-column energy filter. By fitting the Airy diffraction pattern of the selector aperture, various parameters such as geometric aberrations of the lens system and the point-spread function of the diffraction blurring are precisely determined. From the measurements of various beam diameters, components that are attributed to the partial spatial coherence are successfully separated from the point-spread functions. A linear relationship between the spatial coherence length and beam diameter is revealed, thus indicating that a wide range of coherence lengths can be determined by our proposed method as long as the coherence length remains >80% of the aperture diameter. A remarkable feature of this method is its ability to simultaneously determine diffraction blurring and lens aberrations. Possible applications of this method are also discussed. spatial coherence, electron beam, TEM, Airy diffraction pattern, selector aperture, lens aberration Introduction Interference phenomena of coherent electron beams are often used in both analytical and imaging methods in transmission electron microscopy (TEM). The most established methods that utilize interference in the imaging plane are off-axis electron holography and high-resolution TEM (HRTEM) observing interference fringes between transmitted and diffracted beams. Material structures are also analyzed using electron diffraction patterns formed in the diffraction plane as the result of interference between scattered waves from atoms. In interference-based methods, the partial coherence of the beams imposes various limitations. In off-axis holography, the partial coherence suppresses the visibility of the interference fringe and reduces the field of view and signal-to-noise ratio (SNR) in the holograms [1,2]. Partial coherence also causes envelope dumping of the phase-contrast transfer function (PCTF) in HRTEM [3] and blurring of focused probes in scanning TEM (STEM) [4]; these factors limit the spatial resolution, particularly in aberration-corrected microscopes. Accuracy and precision are known to be affected in methods that deal with Fraunhofer or Fresnel propagations of electron waves, such as electron diffractive imaging [5–7], focal-series reconstruction [8], the transport of intensity equation method [8] and in-line holography [2]. Thus, quantitative estimations of the electron-beam coherence are important for the correct interpretation and enhanced quality of experimental data gathered using these interference methods. The beam coherence is generally divided into temporal and spatial components. The former is the coherence in the longitudinal direction and is determined by the energy spread of the emitted electrons. The partial temporal coherence is not considered problematic in interference experiments in TEM, with the exception of chromatic damping in PCTF. The latter is the coherence in the transverse direction in the specimen plane. The spatial coherence is significantly suppressed by finite source size, beam convergence and incoherence in the source plane [2,9–12]. Such partial spatial coherence is expressed by the degree of coherence γ(X) between any two positions with a separation X in the specimen plane. The coherence function γ(X) attenuates from 1 to 0 with an increase in X and is generally approximated by a Gaussian function of X. Thus, its standard deviation (SD) is used as a numerical indicator of the partial spatial coherence and is called the spatial coherence length Lsc. If an electron biprism is equipped in a TEM in which the above-mentioned methods are implemented, the spatial coherence can be measured by analyzing the visibility of the interference fringes [1,2,12]. However, attention should be paid to uncertainties such as vibration of the biprism, drifting of the beam and/or biprism, and the effect of the detector’s modulation transfer function (MTF). Since the majority of TEMs are not equipped with biprisms, developing other methods of measuring the coherence is important from a practical viewpoint. The spatial coherence in a STEM probe is measured by observing the Ronchigram or lattice fringes in STEM images [13,14]. For TEM probes, there are some reports in which the spatial coherence was measured by Airy diffraction patterns under parallel illumination conditions only [14], or by using additional devices or special lens alignments [11,15]. As an example, we have previously measured the spatial coherence of various TEM probe diameters using a specially fabricated selector aperture (SA) with an effective diameter of 3 nm. In the study, Airy diffraction patterns from the small SA were analyzed to obtain not only Lsc but also the damping shapes of γ(X) [11]. One of the merits of using such a small SA is that it allows us to obtain individual measurements of the source size and wavefront curvature of the probe, both of which influence the Airy pattern [11,16]. It is also true that combining a small SA and an imaging aberration corrector also has great potential for nanomaterial analysis [17–19]. However, for versatility, expanding the method to SAs of a standard size would allow coherence measurements to be made with the standard microscopes used in most experiments. Thus, we have developed a method to measure the spatial coherence in a conventional analytical field-emission gun (FEG) TEM equipped with a post-column energy filter. In this method, the purpose of using the energy filter is not energy-filtering but extending the camera length. Theory When a circular aperture with an effective diameter Dap in the specimen plane is illuminated by a parallel beam originating from a point source, its Fraunhofer diffraction pattern is the so-called Airy diffraction pattern. The pattern intensity profile is expressed as follows:   IAiry(qx,qy)∝|ℱ[a(x,y)]|2=|2J1(πDapq)πDapq|2a(x,y)={=1ifx2+y2≤Dap/2=0othersq=qx2+qy2, (1)where, (x,y) and (qx,qy) are the coordinates with their origin at the optical axis in the specimen and diffraction plane, respectively. A Fourier transform is represented by ℱ, and a first-order Bessel function of the first kind by J1. As shown in Fig. 1, however, the measured diffraction pattern from a circular SA (Fig. 1a) shows significant deviation from the calculation in Eq. (1). Possible origins of the discrepancy are investigated as follows. Fig. 1. View largeDownload slide Comparison of measured and calculated Airy patterns. (a) TEM image of the SA. (b) Airy pattern calculated from the image in (a) using Eq. (1). (c) Airy pattern measured in the TEM using the SA in (a). (d) Comparison of the intensity profiles of (b) and (c). Fig. 1. View largeDownload slide Comparison of measured and calculated Airy patterns. (a) TEM image of the SA. (b) Airy pattern calculated from the image in (a) using Eq. (1). (c) Airy pattern measured in the TEM using the SA in (a). (d) Comparison of the intensity profiles of (b) and (c). Firstly, we consider lens aberrations in the TEM and the energy filter. A specimen image observed on the detector plane is blurred by the aberrations of all the lenses between the specimen and detector. In general, lens aberrations are classified as geometric or chromatic aberrations. As far as narrow areas near the optical axis are concerned, the axial geometric aberration χ(qx,qy) described as a function of spatial frequency is considered. Although this χ is representative of the whole lens system, in practice, the primary contribution comes from the objective lens (OL) that dominates the imaging of a specimen. Meanwhile, if there is no material in the specimen plane, the diffraction pattern consists only of a direct spot, which is an image of the source through the apertures. The source image is blurred by the aberrations in all the lenses between the source and detector. Analogous to the case of specimen imaging, as far as small-angle scatterings are concerned, the blurring is given by a function of position, χ′(x,y). It is considered that the main contribution to χ′ comes from the condenser lenses (CLs) and intermediate lenses (ILs) that dominate the imaging of diffraction patterns. Again, by analogy with the description of OL geometric aberrations [3], χ′ should be described as follows:   χ′(x,y)=χ′(r)≡ℜ[12C1rr⁎+12A1r⁎2+B2r2r⁎+13A2r⁎3+14C3r2r⁎2+S3r3r⁎+14A3r⁎4+⋯], (2)where R indicates the real part of a complex function and * denotes a complex conjugate. The complex-number expression r=x+iy is used for real-space coordinates in the specimen plane. A1, A2, A3, C1, B2, C3 and S3 are the coefficients of first-, second- and third-order astigmatisms, defocus, second-order axial coma, and third-order spherical and star aberrations, respectively. When a circular SA is inserted near the optical axis in the specimen plane, the diffraction intensity in Eq. (1) is modified by the aberrations as follows:   I0(qx,qy)∝|ℱ[a(x−xc,y−yc)Ibeam(x,y)×exp[−i2πλχ′(x,y)]]|2=|ℱ[a(x,y)]⊗ℱ[Ibeam(x+xc,y+yc)×exp[−i2πλχ′(x+xc,y+yc)]]|2, (3)where λ is the wavelength and ⨂ denotes the convolution operation. This is a detailed expression that includes the shift of the aperture center (xc,yc) from the optical axis and non-uniform beam intensity Ibeam(x,y) inside the SA. Generally, the effects of chromatic aberration are expressed by the focus fluctuation, which is represented by a Gaussian distribution in the first approximation. Hence, an Airy pattern affected by the chromatic aberration is expressed as follows:   Ich(qx,qy)∝∫−∞+∞I0(qx,qy;C1+ΔC1)exp(−ΔC122σch2)d(ΔC1), (4)where the C1 dependency of the integrand is explicitly written. The SD of the fluctuation σch is called the spread of focus. Since one of the origins of σch is the energy spread of the emitted electrons, the effects of temporal coherence are included in Eq. (4). The following are the remaining possible origins of the discrepancy in Fig. 1: (i) the partial spatial coherence; (ii) blurring attributed to the detector MTF; (iii) drift of the Airy pattern during the measurement; and (iv) any other electronic and mechanical instabilities in the instrument. The first factor is responsible for the blurring of the diffraction patterns induced by the illumination angle αs=λ/(2πLsc) onto the specimen plane (the glancing angle of the source). The point-spread function (PSF) is generally given by a Gaussian function with an SD of σsc=αs/λ=12πLsc. Because the other factors also cause incoherent blurring of diffraction patterns Sothers(qx,qy), the measured diffraction pattern in Fig. 1c is summarized as follows:   Imeasured(qx,qy)∝Ich(qx,qy)⊗S(qx,qy)  S(qx,qy)=exp(−q22σsc2)⊗Sothers(qx,qy) (5) Note that the diffraction intensity is modified differently in Eqs. (3–5). This is demonstrated in Fig. 2, where the effects of the three factors are compared. Figure 2c shows a simple enhancement of blurring with an increase in the SD of the PSF S(qx,qy) in Eq. (5), which is tentatively assumed to be a Gaussian function. However, changes of defocus aberration C1 cause complicated intensity variations, as shown in Fig. 2a. In Fig. 2b, the chromatic aberration changes the intensity profile in a similar but different manner from the simple blurring in Fig. 2c. As an example, the difference is clearly observed in the evolution of the width of the main peak. These examples strongly suggest that we can independently determine all the parameters in Eqs. (2–5) by fitting these calculations to the measured Airy pattern. Fig. 2. View largeDownload slide Comparison of intensity variations in Airy patterns induced by changes in (a) diffraction focus C1, (b) spread of focus σch and (c) SD of the PSF S(qx,qy). In all three calculations, the ideal Airy pattern given by Eq. (1) is modified only by the specified parameter. Fig. 2. View largeDownload slide Comparison of intensity variations in Airy patterns induced by changes in (a) diffraction focus C1, (b) spread of focus σch and (c) SD of the PSF S(qx,qy). In all three calculations, the ideal Airy pattern given by Eq. (1) is modified only by the specified parameter. Experimental methods We used a 200 kV thermal field-emission TEM (JEOL: JEM-ARM200F) in illumination mode called ‘TEM-L, spot size 5’, in which the Airy pattern from the SA was observed most clearly. The effective diameter of the circular SA on the specimen plane was calibrated to 127 nm using lattice fringes of crystalline silicon. After inserting the SA around the optical axis, alignment to the voltage center was conducted such that the swing of the direct spot was minimized. To record the direct spot (the Airy diffraction pattern from the SA) with a sufficiently fine sampling interval, we used the maximum camera length in the microscope (nominal value of 200 cm) and an additional magnification (nominal value ×8) in the post-column imaging energy filter (Gatan: GIF QuantumTM). The necessity of the additional magnification is discussed later. As explained later, the sampling interval of the 2 k × 2 k CCD camera was calibrated to be 0.83 μrad/pixel (3.30 × 105 m−1/pixel), which is fine enough to record the intensity profiles of the Airy patterns, as shown in Fig. 1d. After moving from the imaging mode (‘SAMag mode’ in JEOL microscopes) to the diffraction mode, we consistently found that the Airy pattern drifted in the same direction, which should be induced during the stabilization of the magnetic lenses. To minimize the elongation of the Airy pattern during exposures, we sequentially recorded many frames, each with a short exposure (typically 50 frames with an exposure of 0.5–1.0 s each). After background subtraction and mutual alignments with subpixel precision [20], they were summed to enhance the SNR. Recording an intense beam is known to change the background pattern of a CCD camera, as frequently seen in measurements of zero-loss peaks in electron energy-loss spectra. Therefore, in our experiments, the maximum count at the main peak of the Airy patterns is carefully controlled to be ∼5000, which is <10% of the dynamic range of the camera. As a result, changes in the background were negligible (<10 counts). Thus, as shown in Fig. 1c and d, the Airy patterns were successfully recorded with quantitative intensity and sufficient SNR from the tops of the main peak to the surrounding skirt regions. Although we assumed a circular shape for the SA in Eq. (1), in general, actual SAs deviate slightly from a perfect circle. For precise fitting calculations, the image of the SA (Fig. 1a) was rotated to compensate for the mutual rotation between diffraction patterns and images, the degree of which was calibrated using a streak pattern extracted from a sharp edge of a commercially available calibration specimen (an antigorite crystal). Then the magnification (nm/pixel) was adjusted so that the fringe spacing in the Fourier transform of the aperture image agreed with that of the measured Airy pattern. Moreover, for precision, this value was updated during the fitting process. Based on inverse of the final magnification value, the sampling interval for the diffraction pattern was calibrated to the value mentioned before. Unlike the shape of a(x,y), precise measurements of the intensity distribution I(x,y) inside the SA were difficult. To record the Airy patterns clearly, the beam diameter should be larger than 10 times the SA diameter, as explained later. Thus, the image of the SA was quite dark and noisy compared to that of the Airy patterns, in which most of the counts concentrate around the main peak. Reducing the beam diameter to increase the beam intensity is effective while measuring the shape a(x,y) in Fig. 1a but tends to change the intensity distribution I(x,y). Therefore, we approximated the intensity distribution as the following linear function:   Ibeam(x,y)=Iavg(1+∆Ix(x−xc)+∆Iy(y−yc)), (6)where Iavg is the average intensity inside the SA and can be calculated from the total intensity in the measured Airy pattern. ∆Ix and ∆Iy are intensity gradients in the x and y directions, respectively, and are used as fitting parameters. Unfortunately, the analytical form for the integral calculation in Eq. (7) is not known. For numerical calculations, the range from −2σch to +2σch was sampled at 11 points with regular intervals. Then, the values calculated from the 11 different ∆C1 were summed and substituted for the integral result. The sequential acquisition mentioned above effectively minimizes pattern elongation caused by diffraction drift. However, unless the exposure time becomes infinitely short, slight elongation is still possible owing to drift during each exposure. Hence, the PSF Sothers(qx,qy) and S(qx,qy) in Eq. (5) could be slightly anisotropic. For the fitting calculations, we used the following normalized anisotropic Gaussian function:   S(qx,qy)=12πσ1σ2exp[−(qxcosθ+qysinθ)22σ12]×exp[−(−qxsinθ+qycosθ)22σ22], (7)where θ is the azimuthal angle of the primary axis, and σ1 and σ2 ( σ1>σ2) are SDs along the primary and secondary axes, respectively. In Eq. (7), the secondary axis is perpendicular to the primary axis. Since σ1 reflects the elongation, σ2 should represent the intrinsic blurring without any influence from the elongation. We can now calculate an Airy pattern from a(x,y) prepared by the above-mentioned procedures and Eq. (5) using the following parameters: the geometric aberrations up to the third order ( C1, A1, B2, A2, C3, S3 and A3 in Eq. (2)), the coordinates of the aperture center ( xc and yc in Eq. (3)), the intensity gradient ( ∆Ix and ∆Iy in Eq. (6)), and the shape of the PSF ( σ1, σ2 and θ in Eq. (7)). The values of these parameters were estimated by fitting the equations to experimental patterns. The regions used for the fitting calculations are squares of 256 × 256 pixels around the main peaks, clipped from the original 2048 × 2048 data. One side of the square corresponds to 0.21 mrad. As shown in Fig. 1c and d, the fourth rings of the Airy patterns are included in the regions. Because most electrons concentrate in the main peak, typically 3/4 of the pixels in each region have electron counts <10. To evaluate such low counts properly, the fitting progress was monitored by the maximum likelihood method based on the Poisson statistics, instead of the least square method. Then the optimum parameters were found using the steepest descent method. Results Figure 3 shows an example of the fitting results. As the Airy pattern in Fig. 3a was recorded without adjusting the IL stigmator, the main peak elongates owing to considerable A1. The pattern obtained from the fitting is shown in Fig. 3b. As seen in the comparison of the intensity profiles across the main peaks (Fig. 3c and d), the measured pattern is reproduced with a high degree of precision. The fitted parameters are listed in Table 1. The aberration coefficients are given by inverse lengths, since they relate to imaging in reciprocal space. The values of the intensity gradients indicate that the maximum difference in beam intensity in the SA is only 1.9%. The coordinates of the SA center indicate that the distance between the optical axis and the SA center is ~10% of the SA radius. The elongation direction of the PSF is 91.4° counterclockwise from the x axis, which is close to that of the diffraction drift. As mentioned previously, the important parameter in this study is σ2 rather than σ1. The value for σ2, 1.13 × 10−3 nm−1, corresponds to about three pixels of the detector. Thus, the determined values for the parameters other than lens aberrations remain within reasonable ranges. Regarding the chromatic aberration, it is clear that the influence of σch, 0.0976 m−1, is vanishingly small as seen in Fig. 2b. Moreover, when the third-order aberrations and σch are fixed at zero, the optimum values for the other parameters do not change more than 2%. Similar to real-space imaging using a smaller objective aperture, it is considered that the effective SA diameter of 127 nm is small enough to suppress the effects of higher-order geometric aberrations and the chromatic aberration. To reduce calculation costs, subsequent fitting calculations were conducted without considering third-order aberrations and the spread of focus. Table 1. The parameters determined by fitting to the measured Airy pattern shown in Fig. 3a Geometric aberrations  Modulus (m−1)  Argument (deg)  C1  −409    A1  164  −154.8  B2  0.0542  85.0  A2  0.361  100.5  C3  0.000152    S3  0.00  0.0  A3  0.0000275  155.0  Chromatic aberration  σch (m−1)  0.0976  Intensity gradient  ΔIx (nm−1)  −1.18 × 10−4  ΔIy (nm−1)  9.09 × 10−5  SA center  xc (nm)  −1.38  yc (nm)  6.73  Diffraction blurring  σ1 (m−1)  1.23 × 106  σ2 (m−1)  1.13 × 106  θ (deg)  91.4  Geometric aberrations  Modulus (m−1)  Argument (deg)  C1  −409    A1  164  −154.8  B2  0.0542  85.0  A2  0.361  100.5  C3  0.000152    S3  0.00  0.0  A3  0.0000275  155.0  Chromatic aberration  σch (m−1)  0.0976  Intensity gradient  ΔIx (nm−1)  −1.18 × 10−4  ΔIy (nm−1)  9.09 × 10−5  SA center  xc (nm)  −1.38  yc (nm)  6.73  Diffraction blurring  σ1 (m−1)  1.23 × 106  σ2 (m−1)  1.13 × 106  θ (deg)  91.4  Fig. 3. View largeDownload slide An example of the fitting results. (a) Measured and (b) calculated Airy patterns. (c) and (d) Comparison of the intensity profiles in (a) and (b) in the horizontal direction (c) and the vertical direction (d). In (a) and (b), log of intensity is indicated by pseudo-color. One side of the displayed regions is twice the area of the fitting regions. (e) Lens aberrations inside the SA. (f) Profile of the PSF along the primary axis and its direction. Fig. 3. View largeDownload slide An example of the fitting results. (a) Measured and (b) calculated Airy patterns. (c) and (d) Comparison of the intensity profiles in (a) and (b) in the horizontal direction (c) and the vertical direction (d). In (a) and (b), log of intensity is indicated by pseudo-color. One side of the displayed regions is twice the area of the fitting regions. (e) Lens aberrations inside the SA. (f) Profile of the PSF along the primary axis and its direction. Among the parameters listed in Table 1, C1 and A1 can be directly controlled by users through the IL current and IL stigmator. Figure 4a shows a focal series ( C1 series) of Airy patterns taken with an interval of 10 notches on the IL focus knob. From the gradient of the least-squares-fitted line in Fig. 4b, the amount of C1 change per notch was calibrated as 12.7 ± 1.6 m−1. Judging from the residual errors (deviation of the data from the linear fit), the accuracy of the C1 measurements is about ±30 m−1. Figure 5a shows an A1 series taken with an interval of 15 notches on one of the IL stigmator knobs. Figure 5b shows changes in the modulus and argument of the complex number A1, respectively. From the gradient of the least-squares-fitted line in Fig. 5b, the A1 change per notch was calibrated as 2.56 ± 0.43 m−1. The errors in the A1 measurements are less than ±20 m−1 for the modulus and ±10° for the argument. Fig. 4. View largeDownload slide Analysis of the changes in IL focus C1. (a) C1 series taken with an interval of 10 notches of the IL focus knob. (b) Change in C1 values. Fig. 4. View largeDownload slide Analysis of the changes in IL focus C1. (a) C1 series taken with an interval of 10 notches of the IL focus knob. (b) Change in C1 values. Fig. 5. View largeDownload slide Analysis of the changes in IL astigmatism A1. (a) A1 series taken with an interval of 15 notches of the IL stigmator knob. (b) Changes in the modulus (upper panel) and argument (lower panel) of A1. Fig. 5. View largeDownload slide Analysis of the changes in IL astigmatism A1. (a) A1 series taken with an interval of 15 notches of the IL stigmator knob. (b) Changes in the modulus (upper panel) and argument (lower panel) of A1. Generally, the glancing angle of the source decreases as the beam diameter is enlarged [9,10]. Therefore, blurring of the diffraction pattern due to partial spatial coherence can be changed indirectly through the beam diameter Dbeam on the specimen plane. Figure 6 shows the relationship between σ2 and Dbeam, which was sequentially enlarged by increasing the excitation of the second condenser lens. In Fig. 6b, a plateau appears when Dbeam is larger than 5μm. This indicates that illumination inside the SA has reached perfect coherence. In other words, σsc in Eq. (5) is infinitely small in the data plateau. Therefore, the residual blurring σr corresponds to the SD along the shorter axis of the anisotropic Sothers(qx,qy) in Eq. (5). As shown by the broken line in Fig. 6b, σr is estimated to be (1.083 ± 0.008) × 106 m−1 from the five points in the plateau. The error value is the 95% confidence interval based on the standard error (0.004 × 106 m−1), whereas the error in each measurement of σ2 is estimated to be 0.02 × 106 m−1 based on the SD of the five points. Fig. 6. View largeDownload slide Analysis of the diffraction blurring as a function of the beam diameter Dbeam. (a) Example Airy patterns. (b) Change of the blurring SD, σ2. Fig. 6. View largeDownload slide Analysis of the diffraction blurring as a function of the beam diameter Dbeam. (a) Example Airy patterns. (b) Change of the blurring SD, σ2. Based on Eq. (5), σsc of the seven data points for Dbeam<5μm in Fig. 6(b) were calculated by using the following equation:   σsc2=σ22−σr2 (8)and then converted into a spatial coherence length as Lsc=1/(2πσsc). Figure 7a shows the resulting Lsc as a function of Dbeam. Assuming all the σ2 data have the same degree of accuracy (±0.02 × 106 m−1), the error bars in Fig. 7a were calculated by error propagation. The broken line, determined by the weighted least-square fitting, shows a linear relationship between Dbeam and Lsc with a proportional constant of 0.073 ± 0.005 and intercept of 26 ± 7 nm. Fig. 7. View largeDownload slide Analysis of the diffraction blurring measured in Fig. 6b. (a) Dependence of the spatial coherence length Lsc on Dbeam. The weighted least-square fitting is shown by the broken line and in the inset. (b) Spatial coherence functions γ(X) for the data I and V and for the fully coherent case. (c) Minimum degree of coherence γmin inside the SA. (d) Relative error (∆Lsc/Lsc) as a function of Lsc, normalized by the aperture diameter Dap. Fig. 7. View largeDownload slide Analysis of the diffraction blurring measured in Fig. 6b. (a) Dependence of the spatial coherence length Lsc on Dbeam. The weighted least-square fitting is shown by the broken line and in the inset. (b) Spatial coherence functions γ(X) for the data I and V and for the fully coherent case. (c) Minimum degree of coherence γmin inside the SA. (d) Relative error (∆Lsc/Lsc) as a function of Lsc, normalized by the aperture diameter Dap. In Fig. 7b, a comparison of the coherence functions γ(X) corresponding to the data I and V in Fig. 7(a) and the fully coherent case for reference is shown. The coherence length Lsc defined by the SD of each γ(X) is shown by a double-headed arrow. If we assign the notation γmin to the minimum value of γ(X) inside the SA, γmin is given by the coherence between the furthest two positions in the SA as follows:   γmin=exp[−12(DapLsc)2], (9)where Dap is the SA diameter. Figure 7c shows a plot of γmin calculated from the Lsc in Fig. 7a. The error bars were again calculated using error propagation. We observe that γmin gradually increases with an increase in Dbeam, and finally reaches 1 at Dbeam>5μm, thus indicating perfect coherence inside the SA. Discussion In Fig. 7a, judging from both error bars and deviations from the fitting line, Lsc has been measured with a precision of ±20–30 nm when Dbeam < 4μm. However, this evaluation of precision based on a range of beam diameters could be useless, because beam diameters change due to illumination modes and condenser aperture (CA) sizes in each microscope. We believe that the measurement precision of our method should be evaluated based on the ratio Lsc/Dap instead of Dbeam, as explained below. Note that the magnitudes of the error bars in Fig. 7a and c show the different dependencies on Dbeam. They are calculated by propagating the errors in Fig. 6b. Nevertheless, only those in Fig. 7a increase with an increase in Dbeam. Since the coherence information only inside the SA (solid lines in the range X≤Dap in Fig. 7b) is directly reflected in the measured Airy patterns, the precision of the estimated γmin should not be affected by Dbeam. However, Lsc are values estimated by extrapolations of the coherence functions γ(X) in X≤Dap (broken lines in Fig. 7b). For example, if the coherence inside the SA is high, like the case labeled V, Lsc becomes much larger than Dap ( LscV in Fig. 7b). Therefore, the measurement error of γmin is greatly amplified in the estimated Lsc value. This is an intuitive explanation for the error bars in Fig. 7a that tend to become large when Dbeam is large. Thus, the important parameter for the precise estimation of Lsc is the ratio of Lsc to Dap rather than Dbeam. Figure 7d shows the relative error ∆Lsc/Lsc as a function of the ratio Lsc/Dap. We can conclude that Lsc smaller than ~2.5 times Dap can be directly measured with an uncertainty of less than 10% using our method. Based on such reliable measurements, the linear fitting as shown in Fig. 7a can be executed properly. Extrapolating the fitted lines, also Lsc greater than 2.5×Dap, given by a larger Dbeam, can be estimated. The Airy pattern labeled I in Fig. 6a was measured by Dbeam~10×Dap and has the smallest Lsc among the measurements. The fringe pattern has almost disappeared owing to the blurring. In Fig. 6b, the amount of blurring is 1.74×106m−1, which is close to the second largest blurring amount in Fig. 2c. A comparison of the profiles in Fig. 2c reveals that further reduction of Dbeam would completely erase the fringe pattern. Since our method is based on precise fitting to the Airy patterns, the lack of obvious features in the patterns will abruptly reduce the fitting accuracy. In that sense, the lower limit of direct measurements of Lsc should be ~0.8×Dap, as seen in Fig. 7d. Unlike the case where Lsc is larger than 2.5×Dap, extrapolation to the region of Lsc<0.8×Dap requires careful consideration. A proportional relationship between Dbeam and Lsc has been theoretically derived under some approximations [10]. In Fig. 7a, when Dbeam becomes larger, the relation between Dbeam and Lsc approaches a proportional one. However, when Dbeam becomes smaller, deviations from the proportional relationship become noticeable due to the existence of the intercept. Since asymptotic behavior toward Dbeam=0 has not been clarified, extrapolated estimations should be avoided. Future comparisons between our results and additional measurements in STEM probes using another method [13,14] would be necessary for a deeper understanding of the phenomenon and for clarifying the origin of the intercept. However, from another perspective, the vanishing Airy fringes around Lsc/Dap=0.8 can be used to quickly confirm the coherence in each microscope and at each illumination condition. In fact, at the beginning of this study, it was possible in a short time to identify the illumination mode that gives the best coherence using the criterion of presence or absence of the fringe pattern. Another possible reason of the fringe disappearance is competition between the fringe spacing and the sampling interval of the detector pixels. In all the data presented in this study, the fringe spacing corresponds to 25–30 pixels. If the Airy patterns are recorded without additional magnification in the energy filter, the number of pixels corresponding to the spacing is only three or four. While also accounting for the detector MTF, this number of pixels would be insufficient for precise recording of the Airy patterns required for the fitting calculations. Both reducing the SA diameter and enlarging the camera length increases the fringe spacing at the detector plane. Therefore, the availability and capability of our method would be improved if the standard configuration of future microscopes included smaller SAs or larger camera lengths than those of standard contemporary TEMs. Concluding remarks In this study, we developed a method for precisely measuring spatial coherence using the Airy diffraction patterns of an SA. Based on the measurements for various Dbeam, we demonstrated that the coherence length Lsc can be measured directly in the range of 0.8≤Lsc/Dap≤2.5 with relative errors <10%. For Lsc larger than 2.5×Dap, estimation is possible based on the linear relationship between Lsc and Dbeam. We have also discussed the difficulty of estimating Lsc smaller than 80% of Dap, and the potential to solve the problem by combining our work with another method for STEM probes. Another possible application of our approach is conducting a similar analysis for STEM probes. Since STEM probes that are focused on a specimen plane are effectively Airy patterns from a CA, it may be possible to determine coherence in the probes in a future study by a similar kind of fitting calculations. An additional advantage of our method is the ability to determine lens aberrations and diffraction blurring simultaneously. We expect this advantage to be maximized when this method is combined with electron diffractive imaging [7]. In electron diffractive imaging, a small-angle scattering pattern reflecting the specimen structure is recorded in the same manner as the proposed method. Then the phase image of the specimen is reconstructed from the pattern through numerical calculations. It was recently found that the phase reconstruction precision is severely limited by the diffraction blurring and lens aberrations studied in this work [21]. Therefore, determining these factors using our method (and following processing such as deconvolution [6]) will greatly enhance the quality of the phase imaging. Combining our method not only with electron diffractive imaging but also with other interference methods independent of biprisms is expected to present new possibilities in the field of electron microscopy. Acknowledgements The authors would like to thank Enago (www.enago.jp) for the English language review. The authors are grateful to Dr. S. Ohta and Dr. Y. Kohno in JEOL Ltd. for discussion. Funding Japan Society for the Promotion of Science (JSPS), Grants-in-Aid for Scientific Research KAKENHI (grant numbers JP26286049, JP26600042, JP21760026 and JP26105009); The Public Foundation of Chubu Science and Technology Center; and Toyoaki Scholarship Foundation. References 1 Speidel R, and Kurz D ( 1977) Richtstrahlwertmessungen an einem Strahlerzeugungssystem mit Feldemissionskathode. Optik  49: 173– 185. 2 Latychevskaia T ( 2017) Spatial coherence of electron beams from field emitters and its effect on the resolution of imaged objects. Ultramicroscopy  175: 121– 129. Google Scholar CrossRef Search ADS PubMed  3 Erni R ( 2015) Aberration-Corrected Imaging in Transmission Electron Microscopy , 2nd edn, section 7.5.3 ( Imperial College Press, London). Google Scholar CrossRef Search ADS   4 Beyer A, Belz J, Knaub N, Jandieri K, and Volz K ( 2016) Influence of spatial and temporal coherences on atomic resolution high angle annular dark field imaging. Ultramicroscopy  169: 1– 10. Google Scholar CrossRef Search ADS PubMed  5 Spence J C H, Weierstall U, and Howells M ( 2004) Coherence and sampling requirements for diffractive imaging. Ultramicroscopy  101: 149– 152. Google Scholar CrossRef Search ADS PubMed  6 Kawahara K, Gohara K, Maehara Y, Dobashi T, and Kamimura O ( 2010) Beam-divergence deconvolution for diffractive imaging. Phys. Rev. B  81: 081404(R). Google Scholar CrossRef Search ADS   7 Yamasaki J, Ohta K, Morishita S, and Tanaka N ( 2012) Quantitative phase imaging of electron waves using selected-area diffraction. Appl. Phys. Lett.  101: 234105. Google Scholar CrossRef Search ADS   8 Martin A V, Chen F-R, Hsieh W-K, Kai J-J, Findlay S D, and Allen L J ( 2006) Spatial incoherence in phase retrieval based on focus variation. Ultramicroscopy  104: 914– 924. Google Scholar CrossRef Search ADS   9 Reimer L, and Kohl H ( 2008) Section 4.2 in Transmission Electron Microscopy , 5th edn, ( Springer, New York). 10 Pozzi G ( 1987) Theoretical considerations on the spatial coherence in field emission electron microscopes. Optik  77: 69– 73. 11 Morishita S, Yamasaki J, and Tanaka N ( 2013) Measurement of spatial coherence of electron beams by using a small selected-area aperture. Ultramicroscopy  129: 10– 17. Google Scholar CrossRef Search ADS PubMed  12 Kuwahara M, Kusunoki S, Nambo Y, Saitoh K, Jin X, Ujihara T, Asano H, Taketa Y, and Tanaka N ( 2014) Coherence of a spin-polarized electron beam emitted from a semiconductor photocathode in a transmission electron microscope. Appl. Phys. Lett.  105: 193101. Google Scholar CrossRef Search ADS   13 Dwyer C, Erni R, and Etheridge J ( 2008) Method to measure spatial coherence of subangstrom electron beams. Appl. Phys. Lett.  93: 021115. Google Scholar CrossRef Search ADS   14 Maunders C, Dwyer C, Tiemeijer P C, and Etheridge J ( 2011) Practical methods for the measurement of spatial coherence—a comparative study. Ultramicroscopy  111: 1437– 1446. Google Scholar CrossRef Search ADS PubMed  15 Dwyer C, Kirkland A I, Hartel P, Muller H, and Haider M ( 2007) Electron nanodiffraction using sharply focused parallel probes. Appl. Phys. Lett.  90: 151104. Google Scholar CrossRef Search ADS   16 Morishita S, Yamasaki J, and Tanaka N ( 2011) Estimation of wave fields of incident beams in a transmission electron microscope by using a small selected-area aperture. J. Electron Microsc.  60: 101– 108. Google Scholar CrossRef Search ADS   17 Yamasaki J, Sawada H, and Tanaka N ( 2005) First experiments of selected area nano-diffraction from semiconductor interfaces using a spherical aberration corrected TEM. J. Electron Microsc.  54: 123– 126. 18 Morishita S, Yamasaki J, Nakamura K, Kato T, and Tanaka N ( 2008) Diffractive imaging of the dumbbell structure in silicon by spherical-aberration-corrected electron diffraction. Appl. Phys. Lett.  93: 183103. Google Scholar CrossRef Search ADS   19 Uesugi F ( 2013) Strain mapping in selected area electron diffraction method combining a Cs-corrected TEM with a stage scanning system. Ultramicroscopy  135: 80– 83. Google Scholar CrossRef Search ADS PubMed  20 Isakozawa S, Tomonaga S, Hashimoto T, and Baba N ( 2014) High-precision image-drift-correction method for EM images with a low signal-to-noise ratio. Microscopy  63: 301– 312. Google Scholar CrossRef Search ADS PubMed  21 Yamasaki J, Shimaoka Y, and Sasaki H ( 2016) Developing quantitative phase imaging by electron diffractive imaging. AMTC Lett.  5: 236– 237. © The Author 2017. Published by Oxford University Press on behalf of The Japanese Society of Microscopy. All rights reserved. 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Precise method for measuring spatial coherence in TEM beams using Airy diffraction patterns

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Abstract

Abstract We have developed a method to precisely measure spatial coherence in electron beams. The method does not require an electron biprism and can be implemented in existing analytical transmission electron microscopes equipped with a post-column energy filter. By fitting the Airy diffraction pattern of the selector aperture, various parameters such as geometric aberrations of the lens system and the point-spread function of the diffraction blurring are precisely determined. From the measurements of various beam diameters, components that are attributed to the partial spatial coherence are successfully separated from the point-spread functions. A linear relationship between the spatial coherence length and beam diameter is revealed, thus indicating that a wide range of coherence lengths can be determined by our proposed method as long as the coherence length remains >80% of the aperture diameter. A remarkable feature of this method is its ability to simultaneously determine diffraction blurring and lens aberrations. Possible applications of this method are also discussed. spatial coherence, electron beam, TEM, Airy diffraction pattern, selector aperture, lens aberration Introduction Interference phenomena of coherent electron beams are often used in both analytical and imaging methods in transmission electron microscopy (TEM). The most established methods that utilize interference in the imaging plane are off-axis electron holography and high-resolution TEM (HRTEM) observing interference fringes between transmitted and diffracted beams. Material structures are also analyzed using electron diffraction patterns formed in the diffraction plane as the result of interference between scattered waves from atoms. In interference-based methods, the partial coherence of the beams imposes various limitations. In off-axis holography, the partial coherence suppresses the visibility of the interference fringe and reduces the field of view and signal-to-noise ratio (SNR) in the holograms [1,2]. Partial coherence also causes envelope dumping of the phase-contrast transfer function (PCTF) in HRTEM [3] and blurring of focused probes in scanning TEM (STEM) [4]; these factors limit the spatial resolution, particularly in aberration-corrected microscopes. Accuracy and precision are known to be affected in methods that deal with Fraunhofer or Fresnel propagations of electron waves, such as electron diffractive imaging [5–7], focal-series reconstruction [8], the transport of intensity equation method [8] and in-line holography [2]. Thus, quantitative estimations of the electron-beam coherence are important for the correct interpretation and enhanced quality of experimental data gathered using these interference methods. The beam coherence is generally divided into temporal and spatial components. The former is the coherence in the longitudinal direction and is determined by the energy spread of the emitted electrons. The partial temporal coherence is not considered problematic in interference experiments in TEM, with the exception of chromatic damping in PCTF. The latter is the coherence in the transverse direction in the specimen plane. The spatial coherence is significantly suppressed by finite source size, beam convergence and incoherence in the source plane [2,9–12]. Such partial spatial coherence is expressed by the degree of coherence γ(X) between any two positions with a separation X in the specimen plane. The coherence function γ(X) attenuates from 1 to 0 with an increase in X and is generally approximated by a Gaussian function of X. Thus, its standard deviation (SD) is used as a numerical indicator of the partial spatial coherence and is called the spatial coherence length Lsc. If an electron biprism is equipped in a TEM in which the above-mentioned methods are implemented, the spatial coherence can be measured by analyzing the visibility of the interference fringes [1,2,12]. However, attention should be paid to uncertainties such as vibration of the biprism, drifting of the beam and/or biprism, and the effect of the detector’s modulation transfer function (MTF). Since the majority of TEMs are not equipped with biprisms, developing other methods of measuring the coherence is important from a practical viewpoint. The spatial coherence in a STEM probe is measured by observing the Ronchigram or lattice fringes in STEM images [13,14]. For TEM probes, there are some reports in which the spatial coherence was measured by Airy diffraction patterns under parallel illumination conditions only [14], or by using additional devices or special lens alignments [11,15]. As an example, we have previously measured the spatial coherence of various TEM probe diameters using a specially fabricated selector aperture (SA) with an effective diameter of 3 nm. In the study, Airy diffraction patterns from the small SA were analyzed to obtain not only Lsc but also the damping shapes of γ(X) [11]. One of the merits of using such a small SA is that it allows us to obtain individual measurements of the source size and wavefront curvature of the probe, both of which influence the Airy pattern [11,16]. It is also true that combining a small SA and an imaging aberration corrector also has great potential for nanomaterial analysis [17–19]. However, for versatility, expanding the method to SAs of a standard size would allow coherence measurements to be made with the standard microscopes used in most experiments. Thus, we have developed a method to measure the spatial coherence in a conventional analytical field-emission gun (FEG) TEM equipped with a post-column energy filter. In this method, the purpose of using the energy filter is not energy-filtering but extending the camera length. Theory When a circular aperture with an effective diameter Dap in the specimen plane is illuminated by a parallel beam originating from a point source, its Fraunhofer diffraction pattern is the so-called Airy diffraction pattern. The pattern intensity profile is expressed as follows:   IAiry(qx,qy)∝|ℱ[a(x,y)]|2=|2J1(πDapq)πDapq|2a(x,y)={=1ifx2+y2≤Dap/2=0othersq=qx2+qy2, (1)where, (x,y) and (qx,qy) are the coordinates with their origin at the optical axis in the specimen and diffraction plane, respectively. A Fourier transform is represented by ℱ, and a first-order Bessel function of the first kind by J1. As shown in Fig. 1, however, the measured diffraction pattern from a circular SA (Fig. 1a) shows significant deviation from the calculation in Eq. (1). Possible origins of the discrepancy are investigated as follows. Fig. 1. View largeDownload slide Comparison of measured and calculated Airy patterns. (a) TEM image of the SA. (b) Airy pattern calculated from the image in (a) using Eq. (1). (c) Airy pattern measured in the TEM using the SA in (a). (d) Comparison of the intensity profiles of (b) and (c). Fig. 1. View largeDownload slide Comparison of measured and calculated Airy patterns. (a) TEM image of the SA. (b) Airy pattern calculated from the image in (a) using Eq. (1). (c) Airy pattern measured in the TEM using the SA in (a). (d) Comparison of the intensity profiles of (b) and (c). Firstly, we consider lens aberrations in the TEM and the energy filter. A specimen image observed on the detector plane is blurred by the aberrations of all the lenses between the specimen and detector. In general, lens aberrations are classified as geometric or chromatic aberrations. As far as narrow areas near the optical axis are concerned, the axial geometric aberration χ(qx,qy) described as a function of spatial frequency is considered. Although this χ is representative of the whole lens system, in practice, the primary contribution comes from the objective lens (OL) that dominates the imaging of a specimen. Meanwhile, if there is no material in the specimen plane, the diffraction pattern consists only of a direct spot, which is an image of the source through the apertures. The source image is blurred by the aberrations in all the lenses between the source and detector. Analogous to the case of specimen imaging, as far as small-angle scatterings are concerned, the blurring is given by a function of position, χ′(x,y). It is considered that the main contribution to χ′ comes from the condenser lenses (CLs) and intermediate lenses (ILs) that dominate the imaging of diffraction patterns. Again, by analogy with the description of OL geometric aberrations [3], χ′ should be described as follows:   χ′(x,y)=χ′(r)≡ℜ[12C1rr⁎+12A1r⁎2+B2r2r⁎+13A2r⁎3+14C3r2r⁎2+S3r3r⁎+14A3r⁎4+⋯], (2)where R indicates the real part of a complex function and * denotes a complex conjugate. The complex-number expression r=x+iy is used for real-space coordinates in the specimen plane. A1, A2, A3, C1, B2, C3 and S3 are the coefficients of first-, second- and third-order astigmatisms, defocus, second-order axial coma, and third-order spherical and star aberrations, respectively. When a circular SA is inserted near the optical axis in the specimen plane, the diffraction intensity in Eq. (1) is modified by the aberrations as follows:   I0(qx,qy)∝|ℱ[a(x−xc,y−yc)Ibeam(x,y)×exp[−i2πλχ′(x,y)]]|2=|ℱ[a(x,y)]⊗ℱ[Ibeam(x+xc,y+yc)×exp[−i2πλχ′(x+xc,y+yc)]]|2, (3)where λ is the wavelength and ⨂ denotes the convolution operation. This is a detailed expression that includes the shift of the aperture center (xc,yc) from the optical axis and non-uniform beam intensity Ibeam(x,y) inside the SA. Generally, the effects of chromatic aberration are expressed by the focus fluctuation, which is represented by a Gaussian distribution in the first approximation. Hence, an Airy pattern affected by the chromatic aberration is expressed as follows:   Ich(qx,qy)∝∫−∞+∞I0(qx,qy;C1+ΔC1)exp(−ΔC122σch2)d(ΔC1), (4)where the C1 dependency of the integrand is explicitly written. The SD of the fluctuation σch is called the spread of focus. Since one of the origins of σch is the energy spread of the emitted electrons, the effects of temporal coherence are included in Eq. (4). The following are the remaining possible origins of the discrepancy in Fig. 1: (i) the partial spatial coherence; (ii) blurring attributed to the detector MTF; (iii) drift of the Airy pattern during the measurement; and (iv) any other electronic and mechanical instabilities in the instrument. The first factor is responsible for the blurring of the diffraction patterns induced by the illumination angle αs=λ/(2πLsc) onto the specimen plane (the glancing angle of the source). The point-spread function (PSF) is generally given by a Gaussian function with an SD of σsc=αs/λ=12πLsc. Because the other factors also cause incoherent blurring of diffraction patterns Sothers(qx,qy), the measured diffraction pattern in Fig. 1c is summarized as follows:   Imeasured(qx,qy)∝Ich(qx,qy)⊗S(qx,qy)  S(qx,qy)=exp(−q22σsc2)⊗Sothers(qx,qy) (5) Note that the diffraction intensity is modified differently in Eqs. (3–5). This is demonstrated in Fig. 2, where the effects of the three factors are compared. Figure 2c shows a simple enhancement of blurring with an increase in the SD of the PSF S(qx,qy) in Eq. (5), which is tentatively assumed to be a Gaussian function. However, changes of defocus aberration C1 cause complicated intensity variations, as shown in Fig. 2a. In Fig. 2b, the chromatic aberration changes the intensity profile in a similar but different manner from the simple blurring in Fig. 2c. As an example, the difference is clearly observed in the evolution of the width of the main peak. These examples strongly suggest that we can independently determine all the parameters in Eqs. (2–5) by fitting these calculations to the measured Airy pattern. Fig. 2. View largeDownload slide Comparison of intensity variations in Airy patterns induced by changes in (a) diffraction focus C1, (b) spread of focus σch and (c) SD of the PSF S(qx,qy). In all three calculations, the ideal Airy pattern given by Eq. (1) is modified only by the specified parameter. Fig. 2. View largeDownload slide Comparison of intensity variations in Airy patterns induced by changes in (a) diffraction focus C1, (b) spread of focus σch and (c) SD of the PSF S(qx,qy). In all three calculations, the ideal Airy pattern given by Eq. (1) is modified only by the specified parameter. Experimental methods We used a 200 kV thermal field-emission TEM (JEOL: JEM-ARM200F) in illumination mode called ‘TEM-L, spot size 5’, in which the Airy pattern from the SA was observed most clearly. The effective diameter of the circular SA on the specimen plane was calibrated to 127 nm using lattice fringes of crystalline silicon. After inserting the SA around the optical axis, alignment to the voltage center was conducted such that the swing of the direct spot was minimized. To record the direct spot (the Airy diffraction pattern from the SA) with a sufficiently fine sampling interval, we used the maximum camera length in the microscope (nominal value of 200 cm) and an additional magnification (nominal value ×8) in the post-column imaging energy filter (Gatan: GIF QuantumTM). The necessity of the additional magnification is discussed later. As explained later, the sampling interval of the 2 k × 2 k CCD camera was calibrated to be 0.83 μrad/pixel (3.30 × 105 m−1/pixel), which is fine enough to record the intensity profiles of the Airy patterns, as shown in Fig. 1d. After moving from the imaging mode (‘SAMag mode’ in JEOL microscopes) to the diffraction mode, we consistently found that the Airy pattern drifted in the same direction, which should be induced during the stabilization of the magnetic lenses. To minimize the elongation of the Airy pattern during exposures, we sequentially recorded many frames, each with a short exposure (typically 50 frames with an exposure of 0.5–1.0 s each). After background subtraction and mutual alignments with subpixel precision [20], they were summed to enhance the SNR. Recording an intense beam is known to change the background pattern of a CCD camera, as frequently seen in measurements of zero-loss peaks in electron energy-loss spectra. Therefore, in our experiments, the maximum count at the main peak of the Airy patterns is carefully controlled to be ∼5000, which is <10% of the dynamic range of the camera. As a result, changes in the background were negligible (<10 counts). Thus, as shown in Fig. 1c and d, the Airy patterns were successfully recorded with quantitative intensity and sufficient SNR from the tops of the main peak to the surrounding skirt regions. Although we assumed a circular shape for the SA in Eq. (1), in general, actual SAs deviate slightly from a perfect circle. For precise fitting calculations, the image of the SA (Fig. 1a) was rotated to compensate for the mutual rotation between diffraction patterns and images, the degree of which was calibrated using a streak pattern extracted from a sharp edge of a commercially available calibration specimen (an antigorite crystal). Then the magnification (nm/pixel) was adjusted so that the fringe spacing in the Fourier transform of the aperture image agreed with that of the measured Airy pattern. Moreover, for precision, this value was updated during the fitting process. Based on inverse of the final magnification value, the sampling interval for the diffraction pattern was calibrated to the value mentioned before. Unlike the shape of a(x,y), precise measurements of the intensity distribution I(x,y) inside the SA were difficult. To record the Airy patterns clearly, the beam diameter should be larger than 10 times the SA diameter, as explained later. Thus, the image of the SA was quite dark and noisy compared to that of the Airy patterns, in which most of the counts concentrate around the main peak. Reducing the beam diameter to increase the beam intensity is effective while measuring the shape a(x,y) in Fig. 1a but tends to change the intensity distribution I(x,y). Therefore, we approximated the intensity distribution as the following linear function:   Ibeam(x,y)=Iavg(1+∆Ix(x−xc)+∆Iy(y−yc)), (6)where Iavg is the average intensity inside the SA and can be calculated from the total intensity in the measured Airy pattern. ∆Ix and ∆Iy are intensity gradients in the x and y directions, respectively, and are used as fitting parameters. Unfortunately, the analytical form for the integral calculation in Eq. (7) is not known. For numerical calculations, the range from −2σch to +2σch was sampled at 11 points with regular intervals. Then, the values calculated from the 11 different ∆C1 were summed and substituted for the integral result. The sequential acquisition mentioned above effectively minimizes pattern elongation caused by diffraction drift. However, unless the exposure time becomes infinitely short, slight elongation is still possible owing to drift during each exposure. Hence, the PSF Sothers(qx,qy) and S(qx,qy) in Eq. (5) could be slightly anisotropic. For the fitting calculations, we used the following normalized anisotropic Gaussian function:   S(qx,qy)=12πσ1σ2exp[−(qxcosθ+qysinθ)22σ12]×exp[−(−qxsinθ+qycosθ)22σ22], (7)where θ is the azimuthal angle of the primary axis, and σ1 and σ2 ( σ1>σ2) are SDs along the primary and secondary axes, respectively. In Eq. (7), the secondary axis is perpendicular to the primary axis. Since σ1 reflects the elongation, σ2 should represent the intrinsic blurring without any influence from the elongation. We can now calculate an Airy pattern from a(x,y) prepared by the above-mentioned procedures and Eq. (5) using the following parameters: the geometric aberrations up to the third order ( C1, A1, B2, A2, C3, S3 and A3 in Eq. (2)), the coordinates of the aperture center ( xc and yc in Eq. (3)), the intensity gradient ( ∆Ix and ∆Iy in Eq. (6)), and the shape of the PSF ( σ1, σ2 and θ in Eq. (7)). The values of these parameters were estimated by fitting the equations to experimental patterns. The regions used for the fitting calculations are squares of 256 × 256 pixels around the main peaks, clipped from the original 2048 × 2048 data. One side of the square corresponds to 0.21 mrad. As shown in Fig. 1c and d, the fourth rings of the Airy patterns are included in the regions. Because most electrons concentrate in the main peak, typically 3/4 of the pixels in each region have electron counts <10. To evaluate such low counts properly, the fitting progress was monitored by the maximum likelihood method based on the Poisson statistics, instead of the least square method. Then the optimum parameters were found using the steepest descent method. Results Figure 3 shows an example of the fitting results. As the Airy pattern in Fig. 3a was recorded without adjusting the IL stigmator, the main peak elongates owing to considerable A1. The pattern obtained from the fitting is shown in Fig. 3b. As seen in the comparison of the intensity profiles across the main peaks (Fig. 3c and d), the measured pattern is reproduced with a high degree of precision. The fitted parameters are listed in Table 1. The aberration coefficients are given by inverse lengths, since they relate to imaging in reciprocal space. The values of the intensity gradients indicate that the maximum difference in beam intensity in the SA is only 1.9%. The coordinates of the SA center indicate that the distance between the optical axis and the SA center is ~10% of the SA radius. The elongation direction of the PSF is 91.4° counterclockwise from the x axis, which is close to that of the diffraction drift. As mentioned previously, the important parameter in this study is σ2 rather than σ1. The value for σ2, 1.13 × 10−3 nm−1, corresponds to about three pixels of the detector. Thus, the determined values for the parameters other than lens aberrations remain within reasonable ranges. Regarding the chromatic aberration, it is clear that the influence of σch, 0.0976 m−1, is vanishingly small as seen in Fig. 2b. Moreover, when the third-order aberrations and σch are fixed at zero, the optimum values for the other parameters do not change more than 2%. Similar to real-space imaging using a smaller objective aperture, it is considered that the effective SA diameter of 127 nm is small enough to suppress the effects of higher-order geometric aberrations and the chromatic aberration. To reduce calculation costs, subsequent fitting calculations were conducted without considering third-order aberrations and the spread of focus. Table 1. The parameters determined by fitting to the measured Airy pattern shown in Fig. 3a Geometric aberrations  Modulus (m−1)  Argument (deg)  C1  −409    A1  164  −154.8  B2  0.0542  85.0  A2  0.361  100.5  C3  0.000152    S3  0.00  0.0  A3  0.0000275  155.0  Chromatic aberration  σch (m−1)  0.0976  Intensity gradient  ΔIx (nm−1)  −1.18 × 10−4  ΔIy (nm−1)  9.09 × 10−5  SA center  xc (nm)  −1.38  yc (nm)  6.73  Diffraction blurring  σ1 (m−1)  1.23 × 106  σ2 (m−1)  1.13 × 106  θ (deg)  91.4  Geometric aberrations  Modulus (m−1)  Argument (deg)  C1  −409    A1  164  −154.8  B2  0.0542  85.0  A2  0.361  100.5  C3  0.000152    S3  0.00  0.0  A3  0.0000275  155.0  Chromatic aberration  σch (m−1)  0.0976  Intensity gradient  ΔIx (nm−1)  −1.18 × 10−4  ΔIy (nm−1)  9.09 × 10−5  SA center  xc (nm)  −1.38  yc (nm)  6.73  Diffraction blurring  σ1 (m−1)  1.23 × 106  σ2 (m−1)  1.13 × 106  θ (deg)  91.4  Fig. 3. View largeDownload slide An example of the fitting results. (a) Measured and (b) calculated Airy patterns. (c) and (d) Comparison of the intensity profiles in (a) and (b) in the horizontal direction (c) and the vertical direction (d). In (a) and (b), log of intensity is indicated by pseudo-color. One side of the displayed regions is twice the area of the fitting regions. (e) Lens aberrations inside the SA. (f) Profile of the PSF along the primary axis and its direction. Fig. 3. View largeDownload slide An example of the fitting results. (a) Measured and (b) calculated Airy patterns. (c) and (d) Comparison of the intensity profiles in (a) and (b) in the horizontal direction (c) and the vertical direction (d). In (a) and (b), log of intensity is indicated by pseudo-color. One side of the displayed regions is twice the area of the fitting regions. (e) Lens aberrations inside the SA. (f) Profile of the PSF along the primary axis and its direction. Among the parameters listed in Table 1, C1 and A1 can be directly controlled by users through the IL current and IL stigmator. Figure 4a shows a focal series ( C1 series) of Airy patterns taken with an interval of 10 notches on the IL focus knob. From the gradient of the least-squares-fitted line in Fig. 4b, the amount of C1 change per notch was calibrated as 12.7 ± 1.6 m−1. Judging from the residual errors (deviation of the data from the linear fit), the accuracy of the C1 measurements is about ±30 m−1. Figure 5a shows an A1 series taken with an interval of 15 notches on one of the IL stigmator knobs. Figure 5b shows changes in the modulus and argument of the complex number A1, respectively. From the gradient of the least-squares-fitted line in Fig. 5b, the A1 change per notch was calibrated as 2.56 ± 0.43 m−1. The errors in the A1 measurements are less than ±20 m−1 for the modulus and ±10° for the argument. Fig. 4. View largeDownload slide Analysis of the changes in IL focus C1. (a) C1 series taken with an interval of 10 notches of the IL focus knob. (b) Change in C1 values. Fig. 4. View largeDownload slide Analysis of the changes in IL focus C1. (a) C1 series taken with an interval of 10 notches of the IL focus knob. (b) Change in C1 values. Fig. 5. View largeDownload slide Analysis of the changes in IL astigmatism A1. (a) A1 series taken with an interval of 15 notches of the IL stigmator knob. (b) Changes in the modulus (upper panel) and argument (lower panel) of A1. Fig. 5. View largeDownload slide Analysis of the changes in IL astigmatism A1. (a) A1 series taken with an interval of 15 notches of the IL stigmator knob. (b) Changes in the modulus (upper panel) and argument (lower panel) of A1. Generally, the glancing angle of the source decreases as the beam diameter is enlarged [9,10]. Therefore, blurring of the diffraction pattern due to partial spatial coherence can be changed indirectly through the beam diameter Dbeam on the specimen plane. Figure 6 shows the relationship between σ2 and Dbeam, which was sequentially enlarged by increasing the excitation of the second condenser lens. In Fig. 6b, a plateau appears when Dbeam is larger than 5μm. This indicates that illumination inside the SA has reached perfect coherence. In other words, σsc in Eq. (5) is infinitely small in the data plateau. Therefore, the residual blurring σr corresponds to the SD along the shorter axis of the anisotropic Sothers(qx,qy) in Eq. (5). As shown by the broken line in Fig. 6b, σr is estimated to be (1.083 ± 0.008) × 106 m−1 from the five points in the plateau. The error value is the 95% confidence interval based on the standard error (0.004 × 106 m−1), whereas the error in each measurement of σ2 is estimated to be 0.02 × 106 m−1 based on the SD of the five points. Fig. 6. View largeDownload slide Analysis of the diffraction blurring as a function of the beam diameter Dbeam. (a) Example Airy patterns. (b) Change of the blurring SD, σ2. Fig. 6. View largeDownload slide Analysis of the diffraction blurring as a function of the beam diameter Dbeam. (a) Example Airy patterns. (b) Change of the blurring SD, σ2. Based on Eq. (5), σsc of the seven data points for Dbeam<5μm in Fig. 6(b) were calculated by using the following equation:   σsc2=σ22−σr2 (8)and then converted into a spatial coherence length as Lsc=1/(2πσsc). Figure 7a shows the resulting Lsc as a function of Dbeam. Assuming all the σ2 data have the same degree of accuracy (±0.02 × 106 m−1), the error bars in Fig. 7a were calculated by error propagation. The broken line, determined by the weighted least-square fitting, shows a linear relationship between Dbeam and Lsc with a proportional constant of 0.073 ± 0.005 and intercept of 26 ± 7 nm. Fig. 7. View largeDownload slide Analysis of the diffraction blurring measured in Fig. 6b. (a) Dependence of the spatial coherence length Lsc on Dbeam. The weighted least-square fitting is shown by the broken line and in the inset. (b) Spatial coherence functions γ(X) for the data I and V and for the fully coherent case. (c) Minimum degree of coherence γmin inside the SA. (d) Relative error (∆Lsc/Lsc) as a function of Lsc, normalized by the aperture diameter Dap. Fig. 7. View largeDownload slide Analysis of the diffraction blurring measured in Fig. 6b. (a) Dependence of the spatial coherence length Lsc on Dbeam. The weighted least-square fitting is shown by the broken line and in the inset. (b) Spatial coherence functions γ(X) for the data I and V and for the fully coherent case. (c) Minimum degree of coherence γmin inside the SA. (d) Relative error (∆Lsc/Lsc) as a function of Lsc, normalized by the aperture diameter Dap. In Fig. 7b, a comparison of the coherence functions γ(X) corresponding to the data I and V in Fig. 7(a) and the fully coherent case for reference is shown. The coherence length Lsc defined by the SD of each γ(X) is shown by a double-headed arrow. If we assign the notation γmin to the minimum value of γ(X) inside the SA, γmin is given by the coherence between the furthest two positions in the SA as follows:   γmin=exp[−12(DapLsc)2], (9)where Dap is the SA diameter. Figure 7c shows a plot of γmin calculated from the Lsc in Fig. 7a. The error bars were again calculated using error propagation. We observe that γmin gradually increases with an increase in Dbeam, and finally reaches 1 at Dbeam>5μm, thus indicating perfect coherence inside the SA. Discussion In Fig. 7a, judging from both error bars and deviations from the fitting line, Lsc has been measured with a precision of ±20–30 nm when Dbeam < 4μm. However, this evaluation of precision based on a range of beam diameters could be useless, because beam diameters change due to illumination modes and condenser aperture (CA) sizes in each microscope. We believe that the measurement precision of our method should be evaluated based on the ratio Lsc/Dap instead of Dbeam, as explained below. Note that the magnitudes of the error bars in Fig. 7a and c show the different dependencies on Dbeam. They are calculated by propagating the errors in Fig. 6b. Nevertheless, only those in Fig. 7a increase with an increase in Dbeam. Since the coherence information only inside the SA (solid lines in the range X≤Dap in Fig. 7b) is directly reflected in the measured Airy patterns, the precision of the estimated γmin should not be affected by Dbeam. However, Lsc are values estimated by extrapolations of the coherence functions γ(X) in X≤Dap (broken lines in Fig. 7b). For example, if the coherence inside the SA is high, like the case labeled V, Lsc becomes much larger than Dap ( LscV in Fig. 7b). Therefore, the measurement error of γmin is greatly amplified in the estimated Lsc value. This is an intuitive explanation for the error bars in Fig. 7a that tend to become large when Dbeam is large. Thus, the important parameter for the precise estimation of Lsc is the ratio of Lsc to Dap rather than Dbeam. Figure 7d shows the relative error ∆Lsc/Lsc as a function of the ratio Lsc/Dap. We can conclude that Lsc smaller than ~2.5 times Dap can be directly measured with an uncertainty of less than 10% using our method. Based on such reliable measurements, the linear fitting as shown in Fig. 7a can be executed properly. Extrapolating the fitted lines, also Lsc greater than 2.5×Dap, given by a larger Dbeam, can be estimated. The Airy pattern labeled I in Fig. 6a was measured by Dbeam~10×Dap and has the smallest Lsc among the measurements. The fringe pattern has almost disappeared owing to the blurring. In Fig. 6b, the amount of blurring is 1.74×106m−1, which is close to the second largest blurring amount in Fig. 2c. A comparison of the profiles in Fig. 2c reveals that further reduction of Dbeam would completely erase the fringe pattern. Since our method is based on precise fitting to the Airy patterns, the lack of obvious features in the patterns will abruptly reduce the fitting accuracy. In that sense, the lower limit of direct measurements of Lsc should be ~0.8×Dap, as seen in Fig. 7d. Unlike the case where Lsc is larger than 2.5×Dap, extrapolation to the region of Lsc<0.8×Dap requires careful consideration. A proportional relationship between Dbeam and Lsc has been theoretically derived under some approximations [10]. In Fig. 7a, when Dbeam becomes larger, the relation between Dbeam and Lsc approaches a proportional one. However, when Dbeam becomes smaller, deviations from the proportional relationship become noticeable due to the existence of the intercept. Since asymptotic behavior toward Dbeam=0 has not been clarified, extrapolated estimations should be avoided. Future comparisons between our results and additional measurements in STEM probes using another method [13,14] would be necessary for a deeper understanding of the phenomenon and for clarifying the origin of the intercept. However, from another perspective, the vanishing Airy fringes around Lsc/Dap=0.8 can be used to quickly confirm the coherence in each microscope and at each illumination condition. In fact, at the beginning of this study, it was possible in a short time to identify the illumination mode that gives the best coherence using the criterion of presence or absence of the fringe pattern. Another possible reason of the fringe disappearance is competition between the fringe spacing and the sampling interval of the detector pixels. In all the data presented in this study, the fringe spacing corresponds to 25–30 pixels. If the Airy patterns are recorded without additional magnification in the energy filter, the number of pixels corresponding to the spacing is only three or four. While also accounting for the detector MTF, this number of pixels would be insufficient for precise recording of the Airy patterns required for the fitting calculations. Both reducing the SA diameter and enlarging the camera length increases the fringe spacing at the detector plane. Therefore, the availability and capability of our method would be improved if the standard configuration of future microscopes included smaller SAs or larger camera lengths than those of standard contemporary TEMs. Concluding remarks In this study, we developed a method for precisely measuring spatial coherence using the Airy diffraction patterns of an SA. Based on the measurements for various Dbeam, we demonstrated that the coherence length Lsc can be measured directly in the range of 0.8≤Lsc/Dap≤2.5 with relative errors <10%. For Lsc larger than 2.5×Dap, estimation is possible based on the linear relationship between Lsc and Dbeam. We have also discussed the difficulty of estimating Lsc smaller than 80% of Dap, and the potential to solve the problem by combining our work with another method for STEM probes. Another possible application of our approach is conducting a similar analysis for STEM probes. Since STEM probes that are focused on a specimen plane are effectively Airy patterns from a CA, it may be possible to determine coherence in the probes in a future study by a similar kind of fitting calculations. An additional advantage of our method is the ability to determine lens aberrations and diffraction blurring simultaneously. We expect this advantage to be maximized when this method is combined with electron diffractive imaging [7]. In electron diffractive imaging, a small-angle scattering pattern reflecting the specimen structure is recorded in the same manner as the proposed method. Then the phase image of the specimen is reconstructed from the pattern through numerical calculations. It was recently found that the phase reconstruction precision is severely limited by the diffraction blurring and lens aberrations studied in this work [21]. Therefore, determining these factors using our method (and following processing such as deconvolution [6]) will greatly enhance the quality of the phase imaging. Combining our method not only with electron diffractive imaging but also with other interference methods independent of biprisms is expected to present new possibilities in the field of electron microscopy. Acknowledgements The authors would like to thank Enago (www.enago.jp) for the English language review. The authors are grateful to Dr. S. Ohta and Dr. Y. Kohno in JEOL Ltd. for discussion. Funding Japan Society for the Promotion of Science (JSPS), Grants-in-Aid for Scientific Research KAKENHI (grant numbers JP26286049, JP26600042, JP21760026 and JP26105009); The Public Foundation of Chubu Science and Technology Center; and Toyoaki Scholarship Foundation. References 1 Speidel R, and Kurz D ( 1977) Richtstrahlwertmessungen an einem Strahlerzeugungssystem mit Feldemissionskathode. Optik  49: 173– 185. 2 Latychevskaia T ( 2017) Spatial coherence of electron beams from field emitters and its effect on the resolution of imaged objects. Ultramicroscopy  175: 121– 129. Google Scholar CrossRef Search ADS PubMed  3 Erni R ( 2015) Aberration-Corrected Imaging in Transmission Electron Microscopy , 2nd edn, section 7.5.3 ( Imperial College Press, London). Google Scholar CrossRef Search ADS   4 Beyer A, Belz J, Knaub N, Jandieri K, and Volz K ( 2016) Influence of spatial and temporal coherences on atomic resolution high angle annular dark field imaging. Ultramicroscopy  169: 1– 10. Google Scholar CrossRef Search ADS PubMed  5 Spence J C H, Weierstall U, and Howells M ( 2004) Coherence and sampling requirements for diffractive imaging. Ultramicroscopy  101: 149– 152. Google Scholar CrossRef Search ADS PubMed  6 Kawahara K, Gohara K, Maehara Y, Dobashi T, and Kamimura O ( 2010) Beam-divergence deconvolution for diffractive imaging. Phys. Rev. B  81: 081404(R). Google Scholar CrossRef Search ADS   7 Yamasaki J, Ohta K, Morishita S, and Tanaka N ( 2012) Quantitative phase imaging of electron waves using selected-area diffraction. Appl. Phys. Lett.  101: 234105. Google Scholar CrossRef Search ADS   8 Martin A V, Chen F-R, Hsieh W-K, Kai J-J, Findlay S D, and Allen L J ( 2006) Spatial incoherence in phase retrieval based on focus variation. Ultramicroscopy  104: 914– 924. Google Scholar CrossRef Search ADS   9 Reimer L, and Kohl H ( 2008) Section 4.2 in Transmission Electron Microscopy , 5th edn, ( Springer, New York). 10 Pozzi G ( 1987) Theoretical considerations on the spatial coherence in field emission electron microscopes. Optik  77: 69– 73. 11 Morishita S, Yamasaki J, and Tanaka N ( 2013) Measurement of spatial coherence of electron beams by using a small selected-area aperture. Ultramicroscopy  129: 10– 17. Google Scholar CrossRef Search ADS PubMed  12 Kuwahara M, Kusunoki S, Nambo Y, Saitoh K, Jin X, Ujihara T, Asano H, Taketa Y, and Tanaka N ( 2014) Coherence of a spin-polarized electron beam emitted from a semiconductor photocathode in a transmission electron microscope. Appl. Phys. Lett.  105: 193101. Google Scholar CrossRef Search ADS   13 Dwyer C, Erni R, and Etheridge J ( 2008) Method to measure spatial coherence of subangstrom electron beams. Appl. Phys. Lett.  93: 021115. Google Scholar CrossRef Search ADS   14 Maunders C, Dwyer C, Tiemeijer P C, and Etheridge J ( 2011) Practical methods for the measurement of spatial coherence—a comparative study. Ultramicroscopy  111: 1437– 1446. Google Scholar CrossRef Search ADS PubMed  15 Dwyer C, Kirkland A I, Hartel P, Muller H, and Haider M ( 2007) Electron nanodiffraction using sharply focused parallel probes. Appl. Phys. Lett.  90: 151104. Google Scholar CrossRef Search ADS   16 Morishita S, Yamasaki J, and Tanaka N ( 2011) Estimation of wave fields of incident beams in a transmission electron microscope by using a small selected-area aperture. J. Electron Microsc.  60: 101– 108. Google Scholar CrossRef Search ADS   17 Yamasaki J, Sawada H, and Tanaka N ( 2005) First experiments of selected area nano-diffraction from semiconductor interfaces using a spherical aberration corrected TEM. J. Electron Microsc.  54: 123– 126. 18 Morishita S, Yamasaki J, Nakamura K, Kato T, and Tanaka N ( 2008) Diffractive imaging of the dumbbell structure in silicon by spherical-aberration-corrected electron diffraction. Appl. Phys. Lett.  93: 183103. Google Scholar CrossRef Search ADS   19 Uesugi F ( 2013) Strain mapping in selected area electron diffraction method combining a Cs-corrected TEM with a stage scanning system. Ultramicroscopy  135: 80– 83. Google Scholar CrossRef Search ADS PubMed  20 Isakozawa S, Tomonaga S, Hashimoto T, and Baba N ( 2014) High-precision image-drift-correction method for EM images with a low signal-to-noise ratio. Microscopy  63: 301– 312. Google Scholar CrossRef Search ADS PubMed  21 Yamasaki J, Shimaoka Y, and Sasaki H ( 2016) Developing quantitative phase imaging by electron diffractive imaging. AMTC Lett.  5: 236– 237. © The Author 2017. Published by Oxford University Press on behalf of The Japanese Society of Microscopy. All rights reserved. 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MicroscopyOxford University Press

Published: Feb 1, 2018

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