Linear versus non-linear structural information limit in high-resolution transmission electron microscopy

Ultramicroscopy 110 (2010) 1404–1410

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Linear versus non-linear structural information limit in high-resolution transmission electron microscopy S. Van Aert a,, J.H. Chen b, D. Van Dyck a a b

Electron Microscopy for Materials Science (EMAT), University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium Center for High-Resolution Electron Microscopy, College of Materials Science and Engineering, Hunan University, Changsha City 410082, China

a r t i c l e in f o

a b s t r a c t

Article history: Received 17 July 2009 Received in revised form 23 April 2010 Accepted 8 July 2010

A widely used performance criterion in high-resolution transmission electron microscopy (HRTEM) is the information limit. It corresponds to the inverse of the maximum spatial object frequency that is linearly transmitted with sufficient intensity from the exit plane of the object to the image plane and is limited due to partial temporal coherence. In practice, the information limit is often measured from a diffractogram or from Young’s fringes assuming a weak phase object scattering beyond the inverse of the information limit. However, for an aberration corrected electron microscope, with an information limit in the sub-angstrom range, weak phase objects are no longer applicable since they do not scatter sufficiently in this range. Therefore, one relies on more strongly scattering objects such as crystals of heavy atoms observed along a low index zone axis. In that case, dynamical scattering becomes important such that the non-linear and linear interaction may be equally important. The non-linear interaction may then set the experimental cut-off frequency observed in a diffractogram. The goal of this paper is to quantify both the linear and the non-linear information transfer in terms of closed form analytical expressions. Whereas the cut-off frequency set by the linear transfer can be directly related with the attainable resolution, information from the non-linear transfer can only be extracted using quantitative, model-based methods. In contrast to the historic definition of the information limit depending on microscope parameters only, the expressions derived in this paper explicitly incorporate their dependence on the structure parameters as well. In order to emphasize this dependence and to distinguish from the usual information limit, the expressions derived for the inverse cut-off frequencies will be referred to as the linear and non-linear structural information limit. The present findings confirm the well-known result that partial temporal coherence has different effects on the transfer of the linear and non-linear terms, such that the non-linear imaging contributions are damped less than the linear imaging contributions at high spatial frequencies. This will be important when coherent aberrations such as spherical aberration and defocus are reduced. & 2010 Elsevier B.V. All rights reserved.

Keywords: Resolution Information limit Damping envelope function Non-linear imaging High-resolution transmission electron microscopy Aberration correction

1. Introduction A widely used performance criteria in high-resolution transmission electron microscopy (HRTEM) are the point resolution and the information limit of the electron microscope. The point resolution rs represents the smallest detail that may be interpreted directly from the image provided that the object is thin and that the defocus is adjusted to the so-called Scherzer defocus [1]. The point resolution is set by coherent aberrations. For uncorrected electron microscopes, it only depends on the spherical aberration constant Cs and the electron wavelength l, 3 according to the formula rs ¼ 0:66ðCs l Þ1=4 [2]. The information limit is defined as the inverse of the highest spatial object

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E-mail address: [email protected] (S. Van Aert). 0304-3991/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2010.07.001

frequency that can be linearly transferred by the microscope imaging system with sufficient intensity from the exit plane of the object to the image plane [3,4]. The information limit is determined by the damping envelope incorporating partial temporal coherence due to chromatic aberration, but not partial spatial coherence due to beam convergence. In principle, the beam convergence can be reduced using a smaller illuminating aperture combined with the brightness of a field emission gun. Moreover, with the introduction of a Cs-corrector [5], the effect of beam convergence on the spatial coherence has been reduced drastically and may even vanish completely when Cs is equal to zero. In addition, the Cs-corrector effectively reduces coherent aberrations such that incoherent aberrations, chromatic aberration in particular, become the limiting factors determining the point resolution of the microscope. In a sense, the point resolution equals the information limit. Conditions where the third-order spherical aberration, fifth-order spherical aberration or chromatic

S. Van Aert et al. / Ultramicroscopy 110 (2010) 1404–1410

aberration are the limiting coefficients have been explored in [6]. Chromatic aberration results from a spread in defocus values which results from fluctuations in accelerating voltage, lens current and thermal energy of the electron, where the thermal energy fluctuation is often the dominating term. The information limit due to chromatic aberration is defined as rL ¼ ðplD=2Þ1=2 , with D the defocus spread, expressed in terms of the standard deviation [2]. The information limit is often used in order to demonstrate the performance of the electron microscope for the following reason. In terms of the classical Rayleigh-like resolution criterion, it corresponds to a detail that is present in the image and that may be resolved by image processing techniques such as off-axis holography [7] and the focal-series reconstruction method [8–12]. Both techniques retrieve the exit wave, which ideally is free from any lens aberration. It is important to rule out that this one-to-one relationship between the information limit and attainable resolution only applies under linear imaging conditions for which the non-linear interferences to the image spectrum can be neglected, that is, if their amplitudes are much smaller than those of the linear interferences. Furthermore, dynamical diffraction should be negligible in comparison to kinematic scattering [3]. These conditions only hold for objects that scatter electrons weakly, such as, thin amorphous specimens and very thin crystalline specimens. Such specimens may be treated as weakphase objects. In that case, only linear imaging contributions will add to an experimental diffractogram or to Young’s fringes [13]. The diffractogram is defined as the modulus square of the Fourier transform of a HRTEM image. Young’s fringes are produced in a diffractogram of two superimposed images of the same sample area under the same conditions that have been shifted over a small distance. All information present in both images will contribute to the diffractogram and will be modulated with the corresponding shift frequency. Since the noise in the two images is different, it is reduced and will thus not be interpreted as information. The cut-off frequency of the area containing Young’s fringes can then be associated with the information limit of the system provided that the atomic scattering function of the object under study extents further than the microscope dependent damping envelope function. Under these specific conditions, the experimental cut-off frequency of a diffractogram or Young’s fringes reflects the performance of the electron microscope independently of the object under study. However, if the scattering power of the specimen would be insufficient, the experimentally measured cut-off frequency will be set by the atomic scattering function of the object under study [14]. Despite the usefulness of the information limit as a possible performance measure for resolution, a new situation arises with the presence of aberration corrected electron microscopes. For sub-angstrom resolution instruments, the instrumental properties can no longer be measured independently of the object. In this case, objects scattering beyond the information limit can no longer be considered as weak phase objects. If dynamical scattering becomes important and non-linear effects have to be included into the imaging process, it becomes much more difficult to define the information limit independently of the object [15]. Furthermore, images are nowadays more and more interpreted quantitatively rather than visually by extracting structural information from HRTEM images by fitting a model to the experimental images using a criterion of goodness-of-fit, such as, least squares, least absolute values or maximum likelihood [16–20]. In such a model-based approach [21], a model is required which describes the experimental images adequately, including the non-linear imaging contributions. From this point of view, non-linear image frequencies, which are observed beyond the

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cut-off value for linear imaging contain useful, additional information to further improve the criterion of goodness-of-fit. This is convincingly demonstrated in [22], where it is shown that the least squares sum significantly decreases if also the non-linear imaging contributions are used in the reconstruction of an exit wave by means of the so-called PAM method. In this paper, closed form analytical expressions quantifying the cut-off frequency set by the linear and non-linear information transfer will be derived. In order to emphasize their dependence on structure parameters and to distinguish from the usual information limit, the inverse of these cut-off frequencies will be referred to as the linear, rL , and non-linear structural information limit, rNL . The subscripts L and NL refer to the linear and non-linear imaging contributions, respectively. This extended definition of the information limit, which is not restricted to linear imaging, will be used in what follows and is based on [15,23,24]. However, it should be emphasized once more that, in contrast to the information limit, presence of a 1=rNL image frequency in the HRTEM image spectrum is not sufficient to demonstrate a corresponding Rayleigh resolution of rNL in a reconstructed exit wave [4]. In the attempt to demonstrate that the information limit of an aberration-corrected instrument enters the sub-angstrom range one inevitably has to use heavy scattering specimens. However, in such objects the dynamical interaction of the electron and the object is very strong, so that non-linear imaging contributions are important which may set the experimental cut-off frequency observed in an image spectrum. Such effects will be important when coherent aberrations such as spherical aberration and defocus are reduced, even at acceleration voltages of the order of 80 kV. Experimental results indeed confirm that Young’s fringes extend beyond the inverse of the information limit. In [25], for example, information transfer down to even 0.1 nm is observed at 80 kV though an information limit of only 0.16 nm is expected. In that sense, neglecting the presence of non-linear imaging contributions may lead to wrong conclusions. In order to avoid this, alternative methods to define the information limit may be used [26] or a better understanding of the effect of non-linear imaging on the image spectrum is required. Partial temporal and partial spatial coherence have different effects on the transfer of linear and non-linear terms into the image intensity spectrum, and thus produce different limits on linear and non-linear images [3]. The effects of partial coherence on linear (weak-phase object) images can be described in reciprocal space as damping envelope functions that multiply the usual contrast transfer function [13]. For specimens, for which dynamical scattering becomes important and non-linear effects have to be included into the imaging process, the frequency damping action of the coherence effects is complex and must be described in terms of the transmission cross-coefficient [4,27,28]. The transmission cross-coefficient assigns a damping factor to every pair-wise interference term that is summed to form an image intensity component. In principle, non-linear information transfer could be studied using the transmission cross-coefficient. However, this approach does not give insight in the underlying physics in terms of the atomic structure of the object. In this paper, the damping effect due to non-linear imaging contributions and the corresponding non-linear structural information limit will be derived using the simplified channelling theory to model dynamical scattering [22,29]. The advantage of this scattering theory is the existence of a real-space expression describing the exit wave in closed analytical form that is easy to interpret and can form the basis of a discussion of the relevant imaging and structure parameters. The outline of this paper is as follows. Section 2 gives a concise overview of the channelling theory, which will be used to describe

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dynamical scattering. Next, in Section 3, an expression describing the image intensity distribution will be derived. This expression can then be used to derive an object-dependent linear and nonlinear damping envelope in closed analytical form and to define the linear and non-linear structural information limit. This will be the subject of Section 4. In Section 5, the results will be discussed and in Section 6, conclusions are drawn.

3. Image intensity distribution 3.1. Aberration-free intensity distribution In the absence of any kind of aberrations, the image wave would be identical to the exit wave. From Eq. (1), it follows that the intensity distribution would then be equal to IðrÞ ¼ jcðr,zÞj2

2. Channelling theory

¼ 1 þ4c1s f1s ðrÞðc1s f1s ðrÞ1Þ sin2

with E¼

pE1s E0 l

:

ð2Þ

In Eq. (1), r ¼ ðx yÞT is a two-dimensional vector in the plane at the exit face of the object, perpendicular to the incident beam direction, z is the object thickness, E0 is the incident electron energy, and l is the electron wavelength. The function f1s ðrÞ is the lowest energy bound state of the atom column and E1s is its energy. The lowest energy bound state is a real-valued, centrally peaked, radially symmetric function, which is a two-dimensional analogue of the 1s-state of an atom. Following [29], the S-state may be approximated by a real-valued, single, quadratically normalized, parameterized Gaussian function 1



f1s ðrÞ ¼ pffiffiffiffiffiffi exp  a 2p

 r2 , 4a2

ð3Þ

where r is the Euclidean norm of the two-dimensional vector r, that is, r ¼ jrj, and a represents the column dependent width. This width is directly related to the energy of the S-state and will be smaller for heavier columns and larger for weaker columns. In practice, a is of the order of 0.1–0.4 A. The excitation coefficient c1s ˚ for plane wave incidence is given by [22] Z c1s ¼ f1s ðrÞ dr: ð4Þ Then, it follows from Eqs. (3) and (4) that pffiffiffiffiffiffi c1s ¼ 2 2pa,

ð5Þ

such that   r2 c1s f1s ðrÞ ¼ 2 exp  2 : 4a

ð6Þ

Note that the two-dimensional Fourier transform c1s F1s ðgÞ of Eq. (6), which will be needed in Section 3, is given by

Furthermore, the S-state is a real function such that it follows from Eq. (4) that c1s ¼

Z

f1s ðrÞ dr:

ð7Þ

with g being the Euclidean norm of the two-dimensional spatial frequency vector g in reciprocal space, that is, g ¼ jgj.

ð10Þ

From Eqs. (9) and (10) it follows that Z c1s f1s ðrÞðc1s f1s ðrÞ1Þ dr ¼ 0,

ð11Þ

meaning that the total intensity is indeed conserved. The intensity in the peak is thus compensated by a reduction of the intensity around the peak. This is illustrated in Fig. 1 where an intersection of the two-dimensional, radially symmetric aberration-free intensity distribution, given by Eq. (8), is shown with param˚ E¼  0.01961/A, ˚ and z¼80 A. ˚ The contribution of eters a¼0.4 A, the combined linear and non-linear intensity distribution (neglecting the unscattered incident electron beam) in Fourier space should thus be equal to 0 at g ¼0. In Section 4, it will be shown that this property is more generally valid.

6

5

4

3

2

1

0 −5

Ir-g c1s f1s ðrÞ ¼ c1s F1s ðgÞ ¼ 8pa2 expð4p2 a2 g 2 Þ

ð8Þ

The intensity thus varies periodically with depth. The function 4c1s sin2 ðEz=2Þf1s ðrÞ represents the linear contribution, whereas 2 4c1s f21s ðrÞsin2 ðEz=2Þ represents the non-linear contribution. Moreover, it can be shown that the total intensity is conserved. The Sstate is quadratically normalized such that Z f21s ðrÞ dr ¼ 1: ð9Þ

aberration−free intensity distribution

Electron channelling occurs when the incident electron beam is parallel to the atom columns of an object, such as a crystal or a particular crystal defect. Then, the electrons are trapped in the electrostatic potential of an atom column in which they scatter dynamically. In the simplified channelling theory, an expression for the exit wave of a single atom column is given by [22,29,30]     Ez Ez sin ð1Þ cðr,zÞ ¼ 1 þ 2ic1s f1s ðrÞexp i 2 2

  Ez : 2

0

5

spatial coordinate (angstrom) Fig. 1. Intersection of the two-dimensional, radially symmetric aberration-free ˚ E¼  0.01961/A, ˚ intensity distribution given by Eq. (12) with parameters a ¼0.4 A, ˚ and z¼ 80 A.

S. Van Aert et al. / Ultramicroscopy 110 (2010) 1404–1410

As stated before, the linear imaging contribution scales with

f1s ðrÞ, whereas the non-linear imaging contribution scales with f21s ðrÞ. Hence, the non-linear imaging contribution will be more sharply peaked than the linear imaging contribution and a corresponding improvement of the classical Rayleigh resolution is expected [3,31,32]. Following [14], it can be shown that the classical Rayleigh resolution is expected to be equal to 4a for pffiffiffi the linear term and 2 2a for the non-linear pffiffiffi term. Therefore, the Rayleigh resolution is expected to be 2 times better for the non-linear term than for the linear term in the absence of aberrations. The Rayleigh resolution is defined as the distance for which the ratio of the value at the central dip in the composite intensity distribution to that at the maxima on either side is equal to 0.81 [33].

3.2. Intensity distribution including coherent aberrations In this section, the aberrations of the electron microscope will be taken into account. In this paper, we will restrict the analysis to chromatic aberration only assuming Cs-correction such that partial spatial coherence and spherical aberration are of minor importance. The effect of defocus on the exit wave can be described as a convolution product of the exit wave with the point spread function t(r) of the electron microscope [34]. This gives the so-called image wave cðr,zÞ  tðrÞ. Next, an expression for the image intensity distribution I(r) is given by the modulus square image wave. Hence, it follows from Eq. (1) that IðrÞ ¼ jcðr,zÞ  tðrÞj2     Ez iEz ðc1s f1s ðrÞ  tðrÞÞexp ¼ 1 þ 2i sin 2 2   iEz  ðc1s f1s ðrÞ  t ðrÞÞexp  2   Ez 2 2 : þ 4c1s jf1s ðrÞ  tðrÞj2 sin 2

ð12Þ

In comparison with Eq. (8), the S-state f1s ðrÞ is now replaced with f1s ðrÞ  tðrÞ which makes the result more complicated. Moreover, t(r) is a complex function. Thus far, the focal spread due to chromatic aberration is not taken into account. This will be incorporated in Section 4. For a Cs-corrected microscope the spherical aberration constant Cs can be set equal to 0. Moreover, if all other higher order aberrations can be neglected, only the defocus e is important. In that case, the microscope’s transfer function T(g), defined as the two-dimensional Fourier transform of t(r), reduces to TðgÞ ¼ expðipelg 2 Þ:

ð13Þ

Next, the expression c1s f1s ðrÞ  tðrÞ, appearing in Eq. (12), will be calculated analytically. Its two-dimensional Fourier transform is given by Ir-g c1s f1s ðrÞ  tðrÞ ¼ c1s F1s ðgÞTðgÞ ¼ 8pa2 expð4p2 a2 g 2 Þexpðipelg 2 Þ     iel 2 ¼ 8pa2 exp 4p2 a2 þ g : 4p

ð14Þ

This follows from Eqs. (7) and (13). An expression for c1s f1s ðrÞ  tðrÞ is then given by inverse Fourier transforming equation (14) 0 1 B C 2 r2  expB  C: c1s f1s ðrÞ  tðrÞ ¼   @ iel iel A 2 1þ 4a 1 þ 4pa2 4pa2

ð15Þ

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2 From this result, it follows that the function c1s jf1s ðrÞ  tðrÞj2 in Eq. (12) is equal to 2 c1s jf1s ðrÞ  tðrÞj2

4

¼ 1þ

e2 l2 ð4pa2 Þ2

0 B B ! expB B @

1 r2 2a2 1 þ

e2 l2 ð4pa2 Þ2

C C !C C: A

ð16Þ

4. Linear and non-linear damping envelopes Due to chromatic aberration, the defocus e will fluctuate. The resulting image is then given by the superposition, i.e. integration, of the images corresponding to different defocus values. In this section, this will be calculated analytically resulting in the damping envelope for partial temporal coherence for linear imaging and an analytical expression for the damping envelope for non-linear imaging. In both expressions, the unavoidable resolution limiting effect of the object will be incorporated. 4.1. Linear damping envelope From Eq. (12), it follows that the linear term in the expression for the intensity distribution is given by     Ez iEz ðc1s f1s ðrÞ  tðrÞÞexp 2 2   iEz  ðc1s f1s ðrÞ  t ðrÞÞexp  : 2

IL ðrÞ ¼ 2isin

ð17Þ

Apart from a constant factor depending on the thickness z, the coherent linear imaging contribution in reciprocal space is determined by Eq. (14). However, due to partial temporal coherence the defocus e will fluctuate. Usually, a Gaussian distribution for the defocus with standard deviation D is assumed   1 e2 : ð18Þ pðeÞ ¼ pffiffiffiffiffiffi exp  2pD 2D2 The defocus spread D is mainly determined by the fluctuations in the accelerating voltage, objective lens current, and the incident electron energy: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2 DI DV DE D ¼ Cc 4 þ þ ð19Þ I0 V0 E0 with Cc the chromatic aberration coefficient, DV and DI the standard deviations of the statistically independent fluctuations of the accelerating voltage V0 and objective lens current I0, respectively, and DE the intrinsic energy spread, that is, the standard deviation of the statistically independent fluctuations of the incident electron energy E0 of the electrons in the electron source. In order to incorporate the effect of partial temporal coherence, Eq. (14) needs to be integrated over the defocus including the weight factor pðeÞ. This results into Z DL ðgÞ ¼ 8pa2 expð4p2 a2 g 2 Þ expðipelg 2 ÞpðeÞ de ! 2 1 p2 D2 l g 4 ¼ 8pa2 expð4p2 a2 g 2 Þ pffiffiffiffiffiffi exp 2 2pD !   Z 2 e ipDlg 2 de exp  pffiffiffi þ pffiffiffi 2D 2 ! 2 p2 D2 l g 4 ¼ 8pa2 expð4p2 a2 g 2 Þexp : ð20Þ 2

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Normalizing Eq. (20) such that DL(0) ¼1, gives the object dependent damping envelope for linear imaging ! 2 p2 D2 l g 4 DL ðgÞ ¼ expð4p2 a2 g 2 Þexp : 2

ð21Þ

Note that for ideal point scatterers, for which a¼0, this expression leads to the damping envelope function due to partial temporal coherence for linear imaging. However, for realistic objects, for which a a 0, an object dependent correction is incorporated. Following [15,24], the linear structural information limit is defined as the inverse of gL for which the damping envelope drops to a value of 1/s with s the signal-to-noise ratio. From Eq. (21), it follows that gL can be derived by solving the following quadratic equation in g2L :

p2 D2 l2 2

gL4 þ4p2 a2 gL2 lns ¼ 0,

from which it follows that 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2 2 @ lns 2pa2 2pa2 A 2 þ : gL ¼  2 pDl Dl Dl

ð22Þ

ð23Þ

Typically, the value 1/s is taken equal to exp(  2) or 13.7%. From this equation, it is then clear that gL ¼ ð2=pDlÞ1=2 , or rL ¼ ðplD=2Þ1=2 , if a is chosen equal to 0 in addition. This is the well-known expression defining the information limit. If the signal-to-noise ratio s is higher than exp(2), the cut-off frequency gL will shift towards higher frequencies. The expression between brackets in Eq. (23) is a correction factor for realistic object functions, for which a a0. 4.2. Non-linear damping envelope From Eq. (12), it follows that the term describing the nonlinear contributions to the image intensity distribution is given by   Ez 2 INL ðrÞ ¼ 4c1s ð24Þ jf1s ðrÞ  tðrÞj2 sin2 2 2 with c1s jf1s ðrÞ  tðrÞj2 given by Eq. (16). The non-linear imaging contribution in reciprocal space is given by its Fourier transform, which is equal to ! !

8pa2 exp 2p2 a2 1 þ 

e2 l2 2 2 g : 4pa2

ð25Þ

In order to compute the non-linear structural information limit, Eq. (25) will be integrated over the defocus including the weight factor pðeÞ given by Eq. (18) ! ! Z l2 g 2 1 2 2 2 2 þ e2 pðeÞ de DNL ðgÞ ¼ 8pa expð2p a g Þ exp  8a2 2D2 16pa3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expð2p2 a2 g 2 Þ: 2 2 2 D l g þ 4a2

ð26Þ

Normalizing Eq. (26) such that DNL(0) ¼1, gives the object dependent damping envelope for non-linear imaging 1 DNL ðgÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expð2p2 a2 g 2 Þ: D2 l2 g 2 1þ 4a2

ð27Þ

In correspondence with the linear structural information limit, the non-linear structural information limit rNL is set by the inverse of the value gNL for which Eq. (27) drops to a value of 1/s.

This value can be obtained by solving the following equation: 1 1 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expð2p2 a2 gNL Þ¼ : s 2 2 2 D l gNL 1þ 4a2

ð28Þ

Note that in contrast to the derivation of an expression for gL, for which an expression in closed analytical form could be derived, given by Eq. (23), Eq. (28) has to be solved numerically. However, from Eq. (28) it is immediately clear that the cut-off value gNL will also shift towards higher frequencies if the signal-to-noise ratio s increases. Moreover, if the defocus spread D would be equal to 0, it follows from the comparison of the non-linear damping envelope with the linear damping pffiffiffi envelope that, independent of the value 1/s, gNL would be 2 times larger than gL, in agreement with what has already been mentioned pffiffiffi in Section 3.1. This implies an improvement by a factor of 2 for the non-linear structural information limit as compared to the linear structural information limit. However, in the presence of temporal coherence, this simple relation is no longer valid since partial temporal coherence has a different effect on the transfer of linear and nonlinear terms.

4.3. Damping envelope due to linear and non-linear imaging contributions In practice, however, the linear and non-linear damping envelopes cannot be measured independently. Both linear and non-linear imaging contributions will add simultaneously to a diffractogram or Young’s fringes experiment. In order to study the combined effect, the Fourier transform of the image intensity distribution should be calculated taking the integration over the focal spread into account. Apart from the constant background ‘1’, which leads to a dpeak at g ¼0, it follows from Eqs. (12), (14), (20), (25), and (26) that   Ez ðDNL ðgÞDL ðgÞÞ ð29Þ DL=NL ðgÞ ¼ 32pa2 sin2 2 with DL(g) and DNL(g) defined by Eqs. (21) and (27), respectively. In a diffractogram, which is given by the modulus square of Eq. (29), one will thus measure the difference between the linear and non-linear damping envelope. Perhaps surprisingly, this difference is equal to 0 at g ¼0. The reason for this is similar to what has been mentioned in Section 3.1, namely conservation of intensity. The intensity in a peak is compensated by a reduction of intensity around the peak. This is not only valid in case of aberration-free imaging but is more generally valid. Note that this effect is neglected when only linear imaging is considered. The linear damping envelope DL(g) itself is indeed different from zero at g ¼0. To conclude this section, it should be mentioned once more that the simplified channelling theory has been used to describe dynamical scattering. As a result of dynamical scattering, nonlinear imaging contributions can no longer be neglected. The existence of an analytical expression describing the exit wave has the advantage that the effect of temporal coherence can be described by means of damping envelopes in closed analytical form for the linear as well as for the non-linear imaging contributions. This could be studied by means of purely numerical calculations in Fourier space using the transmission crosscoefficient as well, probably leading to more accurate results. However, the existence of closed analytical expressions describing the damping envelopes is important in order to get a better understanding of the effect of temporal coherence.

S. Van Aert et al. / Ultramicroscopy 110 (2010) 1404–1410

5. Discussion In the previous section, expressions for the linear and nonlinear damping envelope functions have been derived resulting into Eqs. (21) and (27), respectively. Next, the linear and nonlinear structural information limit could be derived from the inverse of the value of g for which the damping envelope function reduces to a value of 1/s. An expression in closed analytical form has been found for the linear structural information limit, given by the inverse of Eq. (23), whereas the non-linear structural information limit follows from the numerical solution of Eq. (28). Although the non-linear structural information limit cannot be expressed in closed analytical form, it is clear from Eq. (28) that, in correspondence to the result for the linear structural information limit, the solution will depend on the product Dl and on a. Fig. 2 shows the damping envelope functions DL(g) for linear imaging (Eq. (21)), DNL(g) for non-linear imaging (Eq. (27)), and the combination (Eq. (29)), here presented as the absolute value of the difference between DNL(g) and DL(g), for different microscope ˚ corresponding settings. The value of a has been set equal to 0.2 A, to a typical value for the width of the S-state. From these figures, it is clear that the linear and non-linear structural information limit are different. The reason for this is that at high spatial frequencies, the linear damping envelope decreases exponentially 2 as expðp2 D2 l g 4 =2Þ, whereas the non-linear damping envelope only decreases as 1=ðDlgÞ as follows from Eqs. (21) and (27), respectively. This confirms the well-known result that the cut-off frequency in a diffractogram will usually be set by the non-linear imaging contributions [3,4] provided that the signal-to-noise ratio is sufficiently large. Moreover, this different behavior of the linear and non-linear damping causes that the relative discrepancy

between the linear and non-linear structural information limit will be more important if the acceleration voltage is lowered and the product Dl increases. Finally, the figures show that in case of a crystal consisting of identical columns, the combination of linear and non-linear imaging contributions may lead to additional circular bands in the diffractogram of an image, which are not due to coherent aberrations.

6. Conclusion In high-resolution transmission electron microscopy (HRTEM), the information limit is often used as criterion to define the attainable resolution. It corresponds to the inverse of the maximum spatial object frequency that is linearly transmitted with sufficient intensity from the exit plane of the object to the image plane and is limited due to partial temporal coherence. In practice, the information limit is often measured from a diffractogram or from Young’s fringes assuming a weak phase object scattering beyond the inverse of the information limit. However, for an aberration corrected electron microscope, with an information limit in the sub-angstrom range, such objects can no longer be considered as weak phase objects. If dynamical scattering becomes important, the non-linear and linear interaction may be equally important. Therefore, the non-linear interaction may set the experimental cut-off frequency observed in a diffractogram or in Young’s fringes. In this paper, the linear and the non-linear information transfer have been quantified in terms of closed form analytical expressions. Therefore, use has been made of the channelling theory, which provides an expression in closed analytical form for the exit wave. Whereas the cut-off E=300 kV, ΔE=1 eV, Δ I=0 A, Δ V=0 eV, 2 Cc = 1.4 mm, a = 0.2 A, Δλ = 0.9 A

E=80 kV, ΔE=1 eV, Δ I=0 A, Δ V=0 eV, 2 Cc=1.4 mm, a=0.2 A, Δλ=7.3 A

1 linear damping nonlinear damping combination

0.8

damping envelope

damping envelope

1

0.6 0.4 0.2

linear damping nonlinear damping combination

0.8 0.6 0.4 0.2 0

0 0

1

2

0

3

1

2

3

−1

−1

g (angstrom )

g (angstrom )

E=80 kV, ΔE = 0.227 eV, Δ I=0 A, Δ V=0 eV, 2 Cc = 1.4 mm, a = 0.2 A, Δλ = 1.6 A

E=300 kV, ΔE = 0.227 eV, Δ I=0 A, Δ V=0 eV, 2 Cc = 1.4 mm, a = 0.2 A, Δλ = 0.2 A

1

1 linear damping nonlinear damping combination

0.8

damping envelope

damping envelope

1409

0.6 0.4 0.2 0

linear damping nonlinear damping combination

0.8 0.6 0.4 0.2 0

0

1

2

g (angstrom−1)

3

0

1

2

3

g (angstrom−1)

Fig. 2. Damping envelope functions DL(g) for linear (Eq. (21)), DNL(g) non-linear imaging (Eq. (27)), and the combination (Eq. (29)), for different microscope setting. Here the combination is presented as the absolute value of the difference between DNL(g) and DL(g).

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S. Van Aert et al. / Ultramicroscopy 110 (2010) 1404–1410

frequency set by the linear transfer can be directly related with the attainable resolution, information from the non-linear transfer can only be extracted using quantitative, model-based methods. In contrast to the historic definition of the information limit depending on microscope parameters only, the expressions derived in this paper explicitly incorporate their dependence on the structure parameters as well. In order to emphasize this dependence and to distinguish from the usual information limit, the expressions derived for the inverse cut-off frequencies have been referred to as the linear and non-linear structural information limit. It has been shown that partial temporal coherence has different effects on the transfer of the linear and non-linear terms. The function describing the damping of the linear term is found to be identical to the damping envelope due to partial temporal coherence for linear imaging, although an object-dependent correction has now been included. Also an expression describing the damping of the non-linear term has been derived. This function decreases as 1=ðDlgÞ at high spatial frequencies, whereas 2 the linear damping envelope decreases as expðp2 D2 l g 4 =2Þ. These findings confirm the well-known result that the non-linear imaging contributions are damped less than the linear imaging contributions at high spatial frequencies causing a discrepancy between the linear and non-linear structural information limit. The discrepancy between the linear and non-linear structural information limit will be more important if the acceleration voltage is lowered provided that the signal-to-noise ratio is sufficiently large. Finally, it has been shown that in case of a crystal consisting of identical columns, the contribution of both linear and non-linear imaging contributions may lead to additional circular bands in the diffractogram of an image, which cannot solely be explained by coherent aberrations.

Acknowledgements S. Van Aert and D. Van Dyck gratefully acknowledge the financial support from the Fund for Scientific Research—Flanders (FWO) (Project no. G.0188.08). J.H. Chen is grateful to the National Basic Research (973) Program of China (No. 2009CB623704), the National Natural Science Foundation of China (No. 50771043), and the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province for the financial support.

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