Measurement of spatial coherence of electron beams by using a small selected-area aperture

Ultramicroscopy 129 (2013) 10–17

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Measurement of spatial coherence of electron beams by using a small selected-area aperture Shigeyuki Morishita a,n, Jun Yamasaki b, Nobuo Tanaka b a b

Department of Crystalline Materials Science, Nagoya University, Furo-cho, Nagoya 464-8603, Japan EcoTopia Science Institute, Nagoya University, Furo-cho, Nagoya 464-8603, Japan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 September 2012 Received in revised form 18 February 2013 Accepted 22 February 2013 Available online 5 March 2013

A new method for measuring the spatial coherence of an electron beam in a transmission electron microscope is proposed. In this method, an Airy pattern produced by a circular selected-area (SA) aperture with an effective diameter of several nanometers is analyzed to obtain the degree of coherence as a function of separation in the specimen plane. Using typical TEM illumination conditions, demonstrative measurements were carried out to determine the spatial coherence length, angular size of the electron source and shape of the coherence function. Based on the results, it was shown that the ratio of the spatial coherence length to the beam radius is about 5% for a condenser aperture with a diameter of 100 mm. This means that perfectly coherent illumination exists within the small SA aperture for beam diameters larger than 560 nm. As an example application of these results, the advantage of SA diffraction over nano-beam diffraction in electron diffractive imaging is discussed. The proposed method is unaffected by temporal coherence or geometric aberrations of the lenses. The possibility of carrying out future measurements using SA apertures with conventional sizes is also discussed. & 2013 Elsevier B.V. All rights reserved.

Keywords: Spatial coherence Transmission electron microscope Selected-area diffraction Electron beam Coherence length Airy pattern

1. Introduction In a transmission electron microscope (TEM), the spatial coherence of the electron beam is an important parameter that can affect many kinds of image formation and measurement methods. One well-known example is damping of the phase contrast transfer function (PCTF) in high-resolution TEM (HRTEM) imaging, which is often described in terms of the angular size of the effective electron source [1]. A consideration of spatial coherence is particularly important for methods that deal with wave fields, such as electron holography, focal-series reconstruction of exit waves [2], the transport of intensity equation (TIE) method [3,4] and electron diffractive imaging [5–7]. Theoretical estimates have been made of the degree of spatial coherence required for different methods in order to carry out reliable quantitative analysis [8–10]. Thus, practical methods of accurately measuring the spatial coherence during experiments are important in order to utilize such theoretical results. The simplest method of measuring the spatial coherence of an electron beam in a TEM should be Young’s double-slit experiment [11,12]. Although the degree of coherence can be directly measured

n Corresponding author. Present Address: JEOL Ltd., 3-1-2 Musashino, Akishima, Tokyo 196-8558, Japan. Tel.: þ 81 42 542 2227; fax: þ 81 42 546 8063. E-mail address: [email protected] (S. Morishita).

0304-3991/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2013.02.019

from the visibility of the interference fringes, a specially-fabricated double-slit apparatus must be prepared [11]. It is known that the coherence can be measured more easily when an electron biprism is attached in the TEM [13]. There is also another kind of measurements using the electron biprism, by which a direct and diffraction spots/discs are overlapped to measure the coherence [14]. As regards measurements without electron biprisms, the spatial coherence lengths were estimated from interference fringes in a nanometersized probe [15] or its Airy pattern in a diffraction mode [16] by assuming that the electron source has a Gaussian intensity distribution. Also in a scanning TEM (STEM) probe, the degrees of coherence were measured from the Ronchigrams or lattice fringes in the STEM images [17,18]. So far the spatial coherence in a typical TEM beam can be measured by using electron biprisms as mentioned above or direct imaging (DI) of the electron source [18]. The degree of coherence is a function of separation r in the specimen plane. For many TEM applications, a knowledge of the spatial coherence function, g(r), would be more appropriate than the use of a single value such as the coherence length. Changing a biprism voltage, g(r) is measured at discrete intervals based on changes in the fringe visibility [8], although special care is often required to take into account the effects of the modulation transfer function (MTF) of the recording device and drift of the biprism. On the other hand, the continuous profile of g(r) can be measured only by the DI method [18].

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In the present study, an alternative method is proposed for measuring the spatial coherence in typical TEM beams. In this method, a profile of g(r) can be obtained by analyzing the Airy patterns as shown in Fig. 1a, unlike the previous analysis of an Airy pattern in which only the coherence length was estimated by a model fitting [16]. The Airy patterns in our study are produced by a special small selected-area (SA) aperture with an effective diameter of 3 nm at the specimen plane [6,19]. Measurements using a standard SA aperture with an effective diameter of more than 100 nm and comparison with the DI method are discussed in the final part of this paper.

where FT denotes the Fourier transform, A(q) the Fourier transform of a(r), J1 a first-order Bessel function of the first kind, and q is the modulus of q. The blurred Airy pattern due to an extended source is represented by entering Eq. (2) into Eq. (1) and schematically shown in Fig. 2e. On the other hand, even due to a point source, the Airy pattern is modified if the CL is under/over-focused from the source. In such cases, the illumination wavefront is not plane but curved due to beam convergence as shown in Fig. 2c. The resultant illumination wave field at the specimen plane, b(r), is a complex function, the phase of which changes by r. Therefore the diffraction intensity from a circular SA aperture is given by a Fourier power spectrum of aðrÞ  bðrÞ. Thus, Eq. (2) becomes

2. Theory and method

Ipoint ðqÞ ¼ 9FT½aðrÞbðrÞ9 ¼ 9AðqÞ  BðqÞ9 ¼ ½AðqÞ  BðqÞ

2

Fig. 2 illustrates influences of an extended electron source and beam convergence. The illumination system is represented by a virtual single lens for simple explanations. As shown in Fig. 2a, electron beams emitted from a point source can be a parallel and perfectly coherent illumination on the specimen plane when the focal length of the CL is adjusted to the distance to the source. However, in practice, the electron beam in a TEM is emitted from an extended source. As shown in Fig. 2b, electrons emitted from different points on the extended source intersect the specimen plane at different angles, and thus their incoherent summation results in blurring of diffraction patterns. Here we assign the notations Ipoint(q) and S(q) to a diffraction intensity due to a point source and the intensity distribution of the extended source, respectively. The diffraction intensity due to an extended source, Iextend(q), is a convolution of them [20]; Iextend ðqÞ ¼ Ipoint ðqÞ  SðqÞ,

ð1Þ

where q is a two-dimensional vector in reciprocal space and  denotes the convolution operation. When a double-slit apparatus is illuminated by such a beam, the diffraction pattern, that is, Young’s fringes appear blurred compared to those due to a point source. This means that the spatial coherence of the slit separation is reduced. The blurring in the case of Airy pattern is illustrated in Fig. 2d and e. Let a(r) be a circular aperture function with a radius of a0, which has a value of 1 inside and 0 outside the aperture. When a SA aperture set in a vacuum is illuminated by a parallel beam from a point source, the diffraction intensity is described by the normalized Airy pattern 2J ð2pa0 qÞ 2 2 2 , Ipoint ðqÞ ¼ 9FT ½aðrÞ9 ¼ 9AðqÞ9 ¼ 1 2pa0 q

ð2Þ

2

½AðqÞ  BðqÞn ,

ð3Þ

where n denotes the complex conjugate. B(q) is the Fourier transform of b(r), which is delta function of q for a plane wave. The modified diffraction wave A(q)B(q) is schematically drawn in Fig. 2f, which finally induces another kind of blurring in the Airy pattern. Here we assign the notations iextend(r) and ipoint(r) to the inverse Fourier transforms of Iextend(q) and Ipoint(q), respectively. ipoint(r) is not the intensity in real space but the autocorrelation function of the wave field in real space. The inverse Fourier transforms of Eqs. (1) and (3) are given by iextend ðrÞ ¼ ipoint ðrÞ  gðrÞ

ð4Þ

and n

ipoint ðrÞ ¼ aðrÞbðrÞ  an ðrÞb ðrÞ,

ð5Þ

respectively. The inverse Fourier transform of S(q) is g(r), which is the spatial coherence function from the van Cittert–Zernike theorem [21]. Finally, g(r) in convergent illumination due to an extended source is expressed as

gðrÞ ¼

iextend ðrÞ IFT½Iextend ðqÞ ¼ , n ipoint ðrÞ aðrÞbðrÞ  an ðrÞb ðrÞ

ð6Þ

where IFT indicates the inverse Fourier transform. Although Eq. (6) has been derived here by assuming a circular SA aperture for convenience, it is clear that the same result will be obtained regardless of the shape of the aperture. In the present study, the shape and diameter of the aperture a(r) are determined by TEM observations. As described later, b(r) can be easily and precisely determined using a small SA aperture [22]. Thus, g(r) can be derived on the basis of the measured diffraction pattern intensity Iextend(q).

Fig. 1. (a) Airy pattern from a SA aperture with an effective diameter of 3 nm when illuminated by a beam 560 nm in diameter. (b) Intensity profiles across the center of the Airy patterns recorded with beam diameters of 140 and 560 nm. A calculation for perfectly coherent illumination is also shown.

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Fig. 2. Influences of an extended electron source and beam convergence. Schematics of (a) parallel illumination from a point source, (b) from an extended source, and (c) convergent illumination from a point source. The illumination system is represented here by a virtual single lens for simple explanations. (d) Normalized intensity and real part of the wave field for the Airy pattern produced by a SA aperture due to the ideal illumination in (a). (e) Influence of the partial spatial coherence. The Airy pattern intensity formed by a point source, Ipoint(q), is blurred as a result of summations of electrons emitted from different positions on an extended source in (b). (f) Influence of the wavefront curvature. The wave field for the ideal Airy pattern, A(q), is modified in the form of convolution with B(q), the Fourier transform of the converged wave in (c) at the specimen plane.

3. Experimental In the present study, a 200 kV thermal field-emission TEM (JEOL: JEM-2100F) was used, for which the energy spread of the electron beam was about 1 eV. The spherical-aberration coefficient of the objective lens (OL) was less than 1 mm owing to the presence of an imaging aberration corrector (CEOS GmbH: CETCOR) installed in the TEM. However, as discussed later, the proposed method should be valid even without such correction. In the microscope, the illumination system consists of two condenser lenses (CLs) and a condenser mini-lens (CM) located between the second CL and the OL. The first CL is used to change beam current density and degree of coherence in the CL aperture plane, which are transferred onto the specimen plane through the second CL and the CM. They control beam diameter and convergence angle in the specimen plane. In most JEOL microscopes, some typical settings for the first CL and the CM are denoted by optical parameters called spot size and a, respectively. In the present study, spot size 1, a 1 and a CL aperture with a diameter of 100 mm were used to form typical bright beams, as in conventional TEM observations. Under such illumination mode setting, the beam diameter in the specimen plane, which is a blurred projection of the CL aperture, was changed by the second CL from 140 to 1120 nm in 140 nm steps to record eight different Airy patterns from a small SA aperture. The small SA aperture was prepared using a focused ion beam (FIB). It consists of a pinhole with a nearly circular shape and a diameter of about 0.24 mm, which corresponds to about 3 nm in the specimen plane. The Airy patterns were recorded using imaging plates at a nominal camera length of 200 cm, which was calibrated later based on a diffraction pattern from a silicon crystal. With increasing the beam diameter, we increased the exposure time from 4 to 60 s to keep high S/N ratios. Fig. 1a shows an example of an Airy pattern

recorded with a beam diameter of 560 nm. Since the diameter of the Airy pattern is inversely proportional to the aperture diameter, the ring fringe pattern from the small SA aperture was recorded at a sufficiently fine sampling interval.

4. Results 4.1. Spatial coherence function As described in Section 2, both of the reduced spatial coherence and wavefront curvature of the illumination beam could induce deviations in profiles of the Airy pattern. In fact, we observed the Airy patterns in which intensity minima have finite values as shown in Fig. 1b, despite the value of zero at minima in the mathematical definition in Eq. (2). Increasing the beam diameter by use of the second CL increases the planarity of the wavefront and also the spatial coherence; in other words, it should reduce both the influences of wavefront curvature and spatial coherence on the Airy pattern. To estimate the degree of the influences, we monitored the intensity of the first minimum, hB, and of the first subsidiary peak, hA, which should make changes reflecting the blurring sensitively (see Fig. 1b). Fig. 3 shows a plot of the ratio hB/hA obtained using different beam diameters. The ratio converges to about 0.15 for diameters of more than 500 nm. The reason why it never goes to zero is the presence of background intensity due to effects such as electron backscattering at the edge of the aperture and other parts inside the microscope. In Fig. 1b, the background profile estimated by spline interpolation of the minima in the Airy pattern for a beam diameter of 560 nm is also shown. Based on the assumption that the background profile is independent of the beam diameter, the estimated profile was subtracted from all other Airy patterns

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Fig. 3. Ratio of hB to hA defined in Fig. 1b, for different beam diameters.

recorded at different beam diameters. The resulting Airy patterns are then described by Iextend(q) in Eq. (1), and performing an inverse Fourier transform gives iextend(r) in Eq. (6). In order to determine ipoint(r) in Eq. (6), the complex wave field b(r) showing the wavefront curvature must be known as shown in Eq. (5). If the beam diameter is much larger than the SA aperture, the illumination intensity and thus the amplitude of b(r) are uniform inside the aperture. The phase of b(r) is, however, sensitive to convergence of the illumination waves. In the center of a conventional TEM illumination area, the wave field is approximated as a spherical wave propagating from a point source located a distance 9Df9 away, where Df is the defocus value of the TEM illumination system. In a previous paper [22], we reported a technique to estimate Df using the small SA aperture based on the lateral shift of the Airy pattern caused by wavefront curvature. From the data set taken together with that in the present study, the Df has been measured to be  11,  22, 45 and  69 mm for beam diameters of 140, 280, 560 and 840 nm, respectively (the negative values represent underfocus) [22]. For each Df, b(r) due to the converging spherical wave is calculated as a two-dimensional complex function. With careful corrections for CL astigmatism, the b(r) should have uniform amplitude and phase decreasing with distance from the beam center (See Fig. 5a in the Ref. [22]). When a small SA aperture with a centrosymmetric shape is set at the center of the illumination beam, a(r)b(r) is also a centrosymmetric complex function. In this case, it can be mathematically proven that the autocorrelation function ipoint(r) in Eq. (5) is a centrosymmetric real function. Actually, the imaginary parts of ipoint(r) and iextend(r) measured by the above procedure are vanishingly small, since a nearly circular SA aperture was used. For the illumination beams with diameters of more than 100 nm, the wavefront curvature, that is, the phase modulation of b(r) is small inside the small SA aperture. Therefore, the real parts of ipoint(r) and iextend(r) have profiles similar to a circular cone, as is the case for the autocorrelation function of a circle. Fig. 4a shows the radial profiles of the autocorrelation cones for a beam diameter of 140 nm. In principle, the radius of the autocorrelation cone is simply twice the aperture radius, that is, around 3 nm in Fig. 4a. Fig. 4b shows the experimental spatial coherence functions g(r) only up to 90% of the distance to the edge, that is, about 2.8 nm. The reason for no display around the edge is that noise in the experimentally obtained iextend(r) is largely enhanced in the region by the division in Eq. (6). The results in Fig. 4b quantitatively show the degree of coherence increasing with beam diameter. When the beam diameter is more than 560 nm, there

Fig. 4. (a) Autocorrelation functions of wave fields inside the aperture for a beam diameter of 140 nm. They are calculated based on the measured wavefront curvature for the point source and the experimental Airy pattern for the extended source. (b) Spatial coherence functions for different beam diameters, which are experimentally obtained as continuous functions. Also shown are example fits to Gaussian functions with standard deviations ls.

is no degradation of the coherence for a separation of 3 nm. This means that the aperture is filled with perfectly coherent illumination. This result is consistent with Fig. 3, which suggests that hB/hA is constant for beam diameters of more than 500 nm. If the aperture was not circular, but instead had a different centrosymmetric shape such as elliptical, the radial profiles for ipoint(r) and iextend(r) shown in Fig. 4a would vary with azimuthal direction. Moreover, if a non-centrosymmetric aperture was used, ipoint(r) and iextend(r) would become complex functions. In both cases, however, such changes should be canceled by the division in Eq. (6). Thus, the same g(r) as shown in Fig. 4b should be finally obtained regardless of the aperture shape. 4.2. Coherence length and angular source size As seen in Fig. 4b, even for the beam diameter of 140 nm, g(r) does not decrease to less than 0.7 inside the aperture. Therefore, based on the present results, precise discussion on the whole shape of g(r) is difficult. Since S(q), and thus also g(r), are generally thought to have distributions similar to Gaussian functions [23], the peak shapes of g(r) for beam diameters of 140 and 280 nm were fitted somehow using a Gaussian function with the standard deviations of ls ¼3.7 and 7.2 nm, respectively. Defining the values as the coherence lengths, they are 5.2–5.3% of the beam radius in the present illumination setting. For beam diameters of more than 560 nm, fitting to a Gaussian function is difficult because of the almost flat nature of g(r) inside the aperture. Consequently, based on the linear relationship between coherence length and beam diameter [23], ls for the larger beam diameters are estimated on the basis of the results for 140 and 280 nm, as shown in Fig. 5. The good agreements between the experimental plots and the straight line from the origin should be collateral evidence for small errors in the results in Fig. 4(b).

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Fig. 5. Coherence length ls and angular size as of effective electron source. The experimental plots for beam diameters of 140 nm and 280 nm are fitted well to the straight line from the origin and the inversely proportional curve.

The results indicate that, under the present illumination setting in our microscope, a beam diameter of 400 nm is sufficient to achieve coherent illumination of a 10-nm-wide area, which is a typical field of view for HRTEM observations. It has been reported that most of the spatial coherence is lost when the separation is about 10% of the beam radius [24]. From the Gaussian fits in Fig. 4b, the degree of coherence is 0.17 for such a separation. All that can be stated here is that these results are consistently of the same order. The ratio should also depend on the CL aperture and the illumination lens setting. For example, it has been reported that the ratio is about 20% if a smaller CL aperture (e.g., 20–40 mm in diameter) is used instead of the 100-mm aperture used in the present study [15]. In addition to the coherence length ls, the angular size of the effective electron source is generally used as an alternative parameter denoting the spatial coherence [25]. Assuming S(q) and g(r) as Gaussian functions, the angular source size can be defined as as ¼ l/2pls based on the Fourier transform between them. Thus, the ls values measured for beam diameters of 140 and 280 nm can be converted to as ¼0.11 and 0.055 mrad, respectively. Based on these results, the inverse relationship between beam diameter and angular source size is plotted in Fig. 5. The results indicate that the angular source size in the present instrument is about 0.05 mrad for beam diameters of a few hundred nanometers, which are generally used in HRTEM observations. As mentioned in the introduction, the angular source size is an important parameter influencing the PCTF for HRTEM imaging [1]. Estimation of angular source sizes using the present method should therefore be useful for precise comparisons between simulated and experimental HRTEM images. 4.3. Spatial coherence and current density One of the important points for data acquisition in actual experiments is to achieve a high S/N ratio at a practicable exposure time. From this point of view, knowledge of beam current density at a given degree of coherence should be beneficial. Fig. 6 shows the relationship between coherence length and current density in the specimen plane. The coherence lengths for beam diameters of more than 560 nm were estimated based on the fitted line in Fig. 5. The values of beam current through the SA aperture were measured from the total electron counts of the Airy patterns after corrections for intensity fading in imaging plates [26]. The results were divided by the dimensions of the selected area in the specimen plane to be converted to the current density in Fig. 6. Considering the linear relationship between the

Fig. 6. Coherence length and current density in the specimen plane. The plots are fitted well to the curve by Eq. (7).

beam diameter d and coherence length ls, the current density J should be 2

Jp1=d p1=ls

2

ð7Þ

All the plots in Fig. 6 are fitted well to a curve based on the relationship. The coefficient of the curve generally varies with illumination modes and a beam extraction voltage for a fieldemission gun. Moreover, the actual size and the shape of the emitter are different from microscope to microscope and change over time. The present method should be useful for measuring the curve in individual microscopes, for example, to estimate the achievable spatial coherence to record images/diffractions within a practicable exposure time.

5. Discussion 5.1. Influence of spherical aberration of the objective lens It is known well that one of the drawbacks of SA diffraction is area selection errors induced by aberrations of the OL [27]. Therefore, to obtain a SA diffraction pattern from a nanometersized area using a small SA aperture, an aberration corrector for the imaging system is indispensable [19]. In the present study, however, only the direct Airy spot from a vacuum area is required. In our previous study [22], for a beam diameter of 140 nm, the angular spread due to the wavefront curvature were estimated to be 0.14 and 4.5 mrad inside the SA apertures with effective diameters of 3 and 100 nm, respectively. Even the latter value corresponds to an area selection error of less than 1 nm in a conventional TEM (e.g., a 200-kV TEM with a spherical aberration coefficient of a few millimeters). Thus, it is clear that the present method is available regardless of the presence of an aberration corrector. 5.2. Influence of temporal coherence In the present study, the spatial coherence was measured from changes in the Airy pattern, which should in principle also be influenced by temporal coherence. When an electron beam is not monochromatic but has a finite energy spread, that is, it is partially temporally coherent, the visibility of the concentric interference fringes in the Airy patterns should be reduced at larger scattering angles (see Fig. 7). The influence of temporal coherence on the Airy pattern can be estimated by comparing the

S. Morishita et al. / Ultramicroscopy 129 (2013) 10–17

Fig. 7. Schematic showing relationship between temporal coherence length and path difference. Electrons emitted from a point source are diffracted through an aperture located on the optical axis. For simplicity, the Fraunhofer diffraction in the absence of lenses is shown.

temporal coherence length and the path difference between electrons passing through different positions in the SA aperture. In the present study, the Airy patterns were recorded from areas of d ¼3 nm up to a scattering angle j ¼11 mrad. Therefore, the maximum path difference d sin j is less than 0.1 nm. On the other hand, the energy spread DE of 1 eV in the present system results in a temporal coherence length vh/DE ¼800–900 nm, where v is the velocity of 200 kV electrons and h is Planck’s constant [25]. Comparing these values, it is concluded that the influence of the temporal coherence can be neglected in the present experiments. This proves the validity of the theory described in Section 2 based on monochromatic illumination, which is also a prerequisite condition for the use of the van Cittert–Zernike theorem. Another factor relating to an energy spread is chromatic aberration, mainly associated with the intermediate lenses (ILs). The Airy pattern can be blurred by such aberrations in the same manner in which HRTEM images are blurred by chromatic aberration in the OL. Since the frequency of the ring fringes in an Airy pattern is proportional to the aperture diameter, influences of the blurring should be more prominent when a larger SA aperture is used. In fact, a significant decrease in fringe visibility was observed for a SA aperture with an effective diameter of 1220 nm. Comparing with simulated patterns, the defocus spread to form the blurred Airy pattern was estimated roughly, although the details are not described here. The result indicates that the chromatic aberrations of the ILs have almost no influence on the Airy patterns for SA apertures with effective diameters of a few hundred nanometers or smaller. 5.3. Measurements by larger SA apertures It would be more practical if the spatial coherence is measured using a conventional SA aperture with an effective diameter such as a few hundred nanometers, rather than the specially-manufactured small aperture used in the present study. This would lead to the following two difficulties. The first is the reduced size of the Airy pattern. As mentioned above, the frequency of the ring fringes is proportional to the aperture diameter. In order to record the ring

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fringe pattern at a sufficiently fine sampling interval, the camera length must be several meters. Although such a long camera length can be achieved in a TEM by switching off the OL, this is not compatible with ordinary SA diffraction. One possible solution is to carry out additional magnification using a post-column imaging filter [7], by which the TEM screen is generally magnified 10–20 times. The second problem is the influence of geometrical aberrations of the ILs. When the specimen plane is magnified at the conjugate plane by the OL, the larger the radius of the OL aperture, the more the HRTEM image is influenced by geometrical aberrations of the OL. In the same manner, when the back-focal plane of the OL is magnified by the ILs to form a diffraction pattern, aberrations of the ILs induce blurring in the Airy pattern, and this effect is more prominent for larger SA apertures. Although two-fold IL astigmatism is usually corrected in TEMs, the specific values of higher-order residual aberrations are generally unknown for any given microscope. Since such higher-order aberrations may change the Airy pattern, they should be estimated, for example, by numerical fitting taking also the background intensity into consideration. If such additional procedures are carried out, the versatility of the proposed method could be significantly enhanced even for a conventional SA aperture with a diameter of a few hundred nanometers. 5.4. Comparison with the DI method One of the characteristics in our method is detecting continuous profiles of g(r) instead of discrete sampling of the shapes. Before the present paper, the only way to measure the continuous profile was the DI method, in which direct imaging of the S(q) is conducted under parallel illumination. In terms of simple and easy measurements, the DI method is better than our method which needs a small SA aperture. However, from the viewpoint of precision, the S(q) measured in the DI method is influenced by diffraction effects arising from the CL aperture, precise elimination of which could be difficult. On the other hand, by the SA aperture we select the central area in a TEM beam, in which the diffraction effects from the CL aperture appear only near the beam edge. In a diffraction plane, the diffraction effect from the SA aperture but not from the CL aperture appears as the Airy pattern, which is convoluted with the S(q) as illustrated in Fig. 2e. Eq. (6) shows the deconvolution processing for the diffraction effects considering also the beam convergence effects. Unless we use a SA aperture, we must find another processing instead of Eq. (6) to deconvolute the effect from the CL aperture. To make matters worse, the problems described in Section 5.3 may become obvious. We may be required to judge carefully the influence of the higher-order aberrations of ILs in the S(q) measured by the DI method. Finally we can summarize that simple and easy measurements are the great advantage in the DI method and more precise measurements, in our method. For the solid verification, further precise measurements of the whole profile of g(r) by our method should be required, for example, using a beam diameter of 50 nm. 5.5. Applications of the measurements As mentioned in the introduction, the proposed technique is expected to be particularly beneficial when applied to TEM methods that are sensitive to partial spatial coherence, for example, HRTEM and methods dealing with wave fields. As an example, we consider the influence of spatial coherence on diffractive imaging based on the results obtained in the present study. During the past decade, there has been extensive interest in diffractive imaging using X-rays and, to a lesser degree, electron beams [5–7]. One reason for the fewer experimental studies involving electron beams is that they have much lower

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spatial coherence than the coherent X-rays produced in accelerator facilities [28]. In general, for electron diffractive imaging of an isolated nanostructure such as a carbon nanotube or a nanoparticle, nano-beam diffraction (NBD) rather than SA diffraction is carried out [5,15,29,30]. This involves the use of a CL aperture with a smaller diameter (e.g., 10–20 mm) than that used for conventional TEM observations (e.g., 100 mm in the present study) [5]. Although the coherence length is about 5% of the beam radius using a 100-mm CL aperture, it is about 50% using a 10-mm aperture if we use the same illumination lens setting. It has been suggested that for diffractive imaging, the spatial coherence length must be at least two times larger than the object being reconstructed [9]. This means that for an object about 10 nm in size, the beam diameter should be larger than 100 nm. This reduces the ability to find nanostructures that are sufficiently isolated enough from their surroundings. On the other hand, using SA diffraction, perfectly coherent illumination can be realized inside the aperture by enlarging the beam diameter, as shown in Fig. 4b. Diffraction patterns recorded under such conditions must have a great advantage for precise reconstructions of specimen wave fields without the need for any additional procedures to recover the spatial coherence [28,29]. This was recently proved by precise medium-resolution reconstructions of wave fields from a small-angle scattering pattern, that is, if the influence from the OL aberrations is negligible [7]. Also for lattice imaging, if we use an imaging corrector, the high degrees of the coherence should be beneficial for precise reconstructions with a spatial resolution comparable to or better than that of aberrationcorrected TEM images [6]. From this viewpoint, SA diffraction is considered to be more suitable for diffractive imaging than NBD.

6. Conclusions A new method has been proposed for measuring the spatial coherence of an electron beam in a TEM by utilizing the Airy pattern produced by a SA aperture. The method can be used to measure not only the spatial coherence length and angular size of the effective electron source, but also spatial coherence as a function of separation in the specimen plane, that is, the spatial coherence function g(r). It was shown as demonstrations under typical TEM illumination settings that the ratio of the coherence length to the beam radius was about 5% using a CL aperture with a diameter of 100 mm. This means that fully coherent illumination exists within the SA aperture 3 nm in diameter for a beam diameter larger than 560 nm. Such a quantitative evaluation of actual conditions in individual microscopes is made possible by the use of a SA aperture with an effective diameter of a few tens of nanometers or smaller. Such a small SA aperture can also be used for purposes such as high-resolution electron diffractive imaging [6,7], measurement of wavefront curvature [22] and SA diffraction analysis of nanometer-sized regions [19]. From the viewpoint of versatility, measurements using a conventional SA aperture with a diameter of a few hundred nanometers are important. Although the present study focused on the use of a very small aperture, it is hoped that this method can be extended further based on the results obtained in this study. Also precise measurements of the whole profile of g(r) by using a more converged beam are expected in the near future and will provide important insights with regard to a range of imaging methods, such HRTEM and diffractive imaging.

Acknowledgments The authors are most grateful to Dr. T. Kato of Japan Fine Ceramics Center for producing the small SA aperture. One of the

authors (J.Y.) thanks Prof. H. Lichte for valuable discussions on coherence in electron beams. This work was supported in part by a Grant-in-Aid for Scientific Research on Priority Areas [Grant number 18029011] from the Ministry of Education, Culture, Sports, Science and Technology; a Grant-in-Aid for Young Scientists (B) [Grant number 21760026] and a Grant-in-Aid for JSPS Fellows [Grant number 226455] and the JSPS Institutional Program for Young Researcher Overseas Visits from the Japan Society for the Promotion of Science.

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