- Email: [email protected]
Measurement of the phase contrast transfer function and the cross-correlation peak using young interference fringes
Ultramicroscopy 5 (1980) 139-145 © North-Holland Publishing Company
MEASUREMENT OF THE PHASE CONTRAST TRANSFER FUNCTION AND THE CROSSCORRELATION PEAK USING YOUNG INTERFERENCE FRINGES F. ZEMLIN and P. SCHISKE Fritz-Haber-lnstitut der Max-Planck-Gesellschafl, Teilinstitut fiir Elektronenmikroskopie, Faradayweg4-6, Berlin (West)33/Dahlem Received 6 December 1979 Cross-correlation of micrographs taken under identical optical conditions allows measurement of (a) the optical transfer function free from electron and photographic noise and (b) object change, especially that due to radiation damage. Light optical methods based on superposition diffraetograms are discussed. A subtraction procedure is described for measuring the optical transfer function, and a two-step diffraction process is presented for measuring object change. As an example, experiments with radiation damage in carbon foil are described. Here increased stability regarding radiation-induced change was found with increasing dose, making possible measurement of deviations from the "zero dose image".
1. Introduction
2. Theory of superposition diffractograms
The phase contrast transfer function T(k) of an electron microscope can be measured by producing light optical diffractograms from micrographs of objects whose distance distributions can be considered random [1]. Carbon foil is usually considered adequate. Nevertheless, the zeroes of the transfer function which are expected theoretically show up only poorly in the photometer registrations of the diffractograms. The transferred intensities are superimposed on a noisy background which decays slowly towards liJgher spatial frequencies. This background impairs the measurements; the minima are shifted relative to the zeroes of the T(k) and the calculated envelope [2,3] can be recognized only qualitatively. For practical microscopy, it would be important to have a simple routine method which could achieve a correction readily *. Superposition diffractograrns (SDs) [4-7] carry the necessary information. In this paper, two simple methods of evaluating the SDs both of which achieve the separation of noise and signal - are discussed and tested experimentally.
Superposition diffractograms are generated from double-exposed micrographs or by inserting two single. exposed micrographs simultaneously in the beam path of the diffractometer. In both cases a lateral shift A (the "Sprungdrift" of ref. [4]) is applied. In the second case angular misorientation is avoided by making tentative small rotational adjustments while keeping A ~ const. When the maximum correlation (i.e., maximum contrast of the interference fringes mentioned below) - especially for high spatial frequencies - has been found, the micrographs are ftxed mechanically [8]. With the usual assumptions of weak phase object and linear photographic response, the optical diffractogram is determined as follows. We consider an absolute stable object f(x) which is exposed two times with exactly the same optical parameters (defocus, exposure time .... ) except for the relative shift A. (Note: two-dimensional vectors x, A, y ..... k are used. In most expressions the vectors have been suppressed. Integrations are two-dimensional and extended over the whole plane, except when stated explicitly otherwise. Lower case function letters refer to functions in direct space: f(x), i(x) ..... Fourier transformation leads to functions in reciprocal
* See Editorial, Ultramicroscopy 4 (1979) 393. 139
F. Zemlin, P. Schiske / Phase contrast transfer function
140
space: F(k), l(k) ..... which are denoted by capital letters. Averaging is expressed by a crossbar, complex conjugation by * (* is used as a sign of convolution). In consequence of the high magnification, for our purposes the specimen can be considered infinite. To this infinite extension correspond the distributions f(x), F(k) (object), r(x), R(k) (noise), and i(x), l(k) (image). The intensity i(x) on the final screen of the microscope,
i(x) = f
I(k)eikx dk,
(1)
which can be seen as the definition of the Wiener spectrum E(k) of i(x). It is convenient to consider an ensemble of objects, all prepared in the same way, generating an ensemble of image functions in (x). The Wiener spectrum E(k) will be characteristic for the whole ensemble: En (k) = E(k). Averaging over the ensemble eq. (5) can be condensed formally to
l(k) I* (l) = E(k) 6 (k - l).
(6)
Here the 5-function expresses the statistical homogeneity of i(x). E(k), which is characteristic of the microscope and the type of preparation of the specimen, is the quantity which is of primary interest to the microscopist. Da(k) merely reflects the properties of a sample taken at random which is used to infer E(k). From eq. (5) it is seen that ((21r)2/a)Da(k) for increasing area approximates E(k), at least with regard to a fixed diffraction resolution Ak. For fixed area, using eqs. (2) and (6) one has
is recorded as a small variatfon in the light optical transmission of the micrograph. It is convenient to put [(x) = 0 (suppression of a constant which carries no information). Fraunhofer diffraction of i(x) generates an amplitude distribution in the screen of the light optical diffractometer which corresponds to the Fourier transform of the original intensity distribution. Since the detector of the diffractometer responds to the square of the amplitude and since, contrary to the mathematical assumptions used for the definition of I(k), the area of investigation, a, is of finite extent, the final result D(k) is not simply a ° signal proportional to l(k), but the square of a convolution of I(K) with a shape function Ba (k), which accounts for the finite extent of the area illuminated in the diffractometer:
Da(k) = f(k) * IB(k)l 2 .
D(k) = Da(k) = II(k) * Ba(k)l 2 •
The image intensity i(x), and hence also I(k), relate to the object function f(x), F(k) by
(2)
Here * denotes convolution with regard to k, and the shape function Ba(k) can be found from the window function ba(x) (= 1 inside, = 0 outside the area, a, which is illuminated):
(7)
B(k) is a peak-shaped function with width 5k ~ 21r/L, where L ~ ~/a gives the size of the window b(x) (i.e., the part of the image actually used). This describes the limits of the simplified relation (2~')2 - -
a
D(k)
~ E(k)
.
I(k) = F(k) T(k) " [1 + exp(iAk)] + R(k),
(8)
(9)
To suppress the speckle, D(k) = Da(k) has to be evaluated integrating over finite patches Ak (photometer aperture):
where T(k) is the (phase) contrast transfer function of the microscope and R(k) represents noise (electron noise and photographic granularity). The sum [1 + exp(iAk)] corresponds to the shifted single image (equal exposure). Introducing the Wiener spectra, I¢(lkl) of the specimen f(x) and N(Ikl) of the noise r(x) (electron and photographic granularity), one has
f Da(k) dk.
E(k) = 2WOkl)(l + cos Ak) l T(k)[ 2 + N(Ikl).
ba(x) = fBa(k) eik'x d k .
(3)
(4)
ak
Here the statistical independence of signal and noise,
Taking conceptually the limit a ~ oo, one has lim (2rr)---~2 f ea(k) dk = f E(k) dk, a --..~.~
(10)
a Ak
Ak
f(x) n(x)= O, (5)
has been used. W is related to the auto-correlation
(11)
F. Zemlin, P. Schiske /Phase contrast transfer function
141
function of the object
fw(Ikl) eiky dk = c(lYl)=f(x)f(x +y).
(12)
This can be inferred from the relation
F(k)F*(l) = W(Ikl) ~i(i~ - / ) ,
(13)
which is analogous to eq. (6). Statistical isotropy is taken care of by using Ikl, lyl (not k, y) as arguments of W and c.
3. Subtraction method [9 ] 1
The factor (1 + cos Ak) describes the modulation of the superposition diffractogram (Young interference fringes - YIF). Scanning along the lines 1 + cos Ak = 0, i.e., the minima of the YIF, N(k) can be measured for all values Ikl > rr/IAI, using the theorem of Pythagoras (fig. 1). Subtraction of N from eq. (10) gives the signal
W(k)" (1 + cos A k ) " IT(k)l 2.
(14)
If the object f(x) has a white spectrum, that means l¢(Ikl) = I, we are measuring the pure instrument function IT(k)l. In fact, the spectrum of carbon foil is not white. But if either W(Iki) or IT(k)l is known the other factor can be found by this method. Actually in scanning along the..,minima of the YIF a finite aperture is used. Thus a small part of the signal, eq. (14), is included in the measurements of the
'
I (3,h) ~
Fig. 2. Evaluation of the superposition diffractogram of fig. 1: (a) density scan parallel to the maxima of the YIFs (smoothed); (b) density scan parallel to the minima of the YIFs (smoothed); (c) difference of curves (a) and (b); (d) fitting this theoretical phase contrast transfer function to the corrected curve (c) requires a defocus of -3(Csh) 1/2 .
noise N. It seems that an aperture diameter of approximately ~ of the fringe distance is tolerable. In our experiments (fig. 2) ~o of the fringe distance was used.
3.1. Frequency filtering of superposition diffractograms Another approach consists of frequency fftltering. For example, scanning along concentric circles and using analog electronics, Tonar [ 10] has determined l#(Ikl) • cos(Ak) • IT(k)l 2 ,
/
Ik12=Is12. I--~Iz b Fig. 1. (a) Superposition diffractogram with Young interference fringes (YIF). (b) Geometry of YIFs, schematic. With k, spatial frequency, A, spacing of the YIF; s, the scanned distance.
(15)
by measuring the amplitude of an alternating current. This method, however, is only practicable when even small deviations from rotational symmetry (axial astigmatism in the electron microscope) are avoided. Both difficulties - the error introduced by finite size of photometer aperture and the sensitivity to axial astigmatism - can be overcome by a second Fourier transform,
d2 (x) =
[fD(k) e - i ~
dk[ 2 ,
(16)
14 2
F. Zemlin, P. Schiske /Phase contrast transfer function
using the light optical diffractometer again as analog computer. This procedure does not need any additional electronic equipment and was found to present no experimental difficulty. Since the diffractograms (spectra) are Fourier transforms of autocorrelation functions, the second diffraction leads back to correlation functions. This can be considered an advantage of this method since measurement of correlation peaks is becoming increasingly important in quantitative electron microscopy, particularly for the investigation of radiation-sensitive specimens [11-15]. It was found that especially the height of correlation peaks can be measured with great ease by this method. The second Fourier transform is not applied to the spectrum W([k[)but to W([kl) • IT(k)[ 2. Multiplication by IT(k)[ 2 is replaced by convolution with the spread function,
s(x)--
f,r(k), 2 e
dk,
(17)
s(x) depicts the finite resolution of the electron microscope. The resulting second diffractogram d: (x) is represented by d2 (x) = e 2 (x)" h 2 (x).
(18')
Using * for the convolution with regard to x one has e(x)
=
cOxl) * s(x) * (28(x)
+/i(x - A) + 5(x + A)) + n(Ixl)
(19)
with n(Ixl) = f N ( I k l ) e ikx dk.
(20)
To a broad noise spectrum N(Ikl) there corresponds a sharp peaked n(Ixl). Eq. (19) describes three peaks, centered at x = 0 and x = +A. The factor h (x) merely expresses the finite extension of the part of the image actually used for analysis: h(x) = b(x) * b(x) .
(21)
In practice this region is so big that the relative variation of h(x) may be neglected in the region of interest which is given by the sharp peaks of c(Ixl) resp. c o x + AJ) (fig. 3). To isolate the contribution of c(jx - Aj), thereby eliminating also n(ixl), photometric measurements are made near x = A from which
Fig. 3. The optical diffractogram (Fourier transform) of the superposition diffractograms shows three (correlation) peaks due to the cosine-modulation of the YIF system.
the values found at other places of the annulus ixl [A[ are subtracted. Thus the background due to the limiting value c([x[) ~ f-2 for large x is eliminated. In this way one determines the height and shape of the peak of c(ix - A[) * s(x).
4. Unstable objects Equal exposure times can be achieved experimentally without much effort. Basic relations for unequal exposure times are worked out and tested in ref. [4]. Specimens which are not stable due to electron irradiation have to be described by two specimen func. tions f l (x):/:f2 (x) corresponding to the two exposures. The calculations for this case can be seen as a generalization of those necessary for unequal exposure times. In place of the auto-correlation function c(Ixl) now a cross-correlation function c12 (Ixl) = c21 (ix[) has to be used alongside the auto-correlation functions cll (Ixl), c22(Ixl). Eq. (12) must be generalized to fW~t3(ikl )
e iky
dk = Cet(J(lY I)
= f~(x) f~(x +y) = c0a(lyl),
% g = 1, 2.
(22)
Eq. (10) is replaced by D(k) = [Wl~(Ikt) + W22(Ikl) -
2W12(Ikl)] • IT(k)l 2 +N(IkJ)
+ 2(1 + cos A k ) . W12(Jkl) • IT(k)l 2.
(23)
F. Zemlin, P. Schiske /Phase contrast transfer function
From eq. (23) it follows that for severe axial astigmatism the subtraction procedure [9] has to be modified. Making l A I big enough, the values of T(k) taken along the lines (a)
A. k = +zr/2,
(b) A. k =0,
can be made nearly equal (compared for equal values of ik[) even if the imaging in the electron microscope is not rotationally symmetric. By subtraction of D(k) taken along (a) from D(k) taken along (b) (both for equal Ikl), W12(Ik,) • IT(k)h2 can be found in good approximation for (b). Since a given pair of micrographs may be shifted with arbitrary orientation of A, this modification of the subtraction method has, in principle, no limitation. No essential modification is needed for the method of second diffraction. Eq. (19)is replaced by e(x) = [ c l l ([xi) + c22([xl) + c12(Ix - AI) + cl2(Ix + Ai)] * s(x) + n(Ixl).
(24)
The cross-correlation peak c12(Ix - AI) * s(lxl) can be isolated in the same way as the auto-correlation peak, eq. (19), in the case of the stable object.
1 -2
1 -3
143
5. Application In the following the application of this measuring procedure is demonstrated. The specimen was a thin (about 20 A) carbon foil which had not yet been exposed to electron irradiation. In order to measure the radiation damage at minimum dose, the mini-exposure technique was applied [16], i.e., the adjustment was made at an adjacent region. Immediately before the first exposure the irradiated region was shifted by switching off a deflection system. From the central region, in rapid succession, a sequence of exposures (1, 2, ..., n) was recorded at 2 × 10 s magnification. Fig. 4 shows a sequence of superposition diffractograms of these images: beginning from the upper left, the image pair (1,2), next to it (I, 3) etc. up to the lower right (1, 7) As can be seen, the modulation of the YIF decreases slowly. The decrease is stronger at high spatial frequencies than at low frequencies. This decrease of the YIFs is a measure of the radiation damage. Moreover, fig. 4 (right) shows the superposition diffractogram of the image pair (7, 8). As can be seen, the YIFs are very strongly modulated there, more strongly than at
1 -4
'!!!I!ttlit! 7-8
1 - 5
1 -6
1 - 7
(3.~) -~ I
I
Fig. 4. Sequence of superposition diffractograms of a dose series, demonstrating radiation damage.
F. Zernlin, P. $chiske /Phase contrast transfer function
144
XCF
XCF
1.0
10 .7
.7
.5
.5 .3 .2
.3 0
560
1060 1500 20'00
, =
O/~
Fig. 5. Cross-correlation peaks versus dose of the dose series of fig. 4. The extrapolation to zero dose is controlled independently (see text).
the initial image pair (1, 2). This observation was confirmed by many measurement sequences; it demonstrates the known phenomenon that the carbon foil stabilizes at irradiation. After a high pre-irradiation the structure changes only slightly with further irradiation - compared to the initial state. This sequence of diffractograms was evaluated with the procedure of section 3. In fig. 5 the correlation peaks are plotted versus the dose. From the curve it can be seen that: (1) the correlation peaks decrease exponentially in the beginning; (2) with increased pre-irradiation the decrease of the correlation peaks becomes weaker, i.e., the foil becomes stable. It was of interest to learn how strongly the first image of the series deviates from the "zero dose image". Therefore, the curve was extrapolated to dose zero. Any extrapolation is afflicted with. uncertainty. Hence an attempt was made to obtain this value in another way. The intersection with the ordinate axis which belongs to an image pair of a stable specimen corresponds to the correlation peak 1. Since the carbon foil stabilizes itself, i.e., its radiation sensitivity decreases, we plotted the measured crosscorrelation peak of the image pair (7, 8) directly on the ordinate - i.e., we did ascribe the dose zero to this cross-correlation peak. It turns out that the crosscorrelation peak of image pair (1, 2) is smaller by 25% than the "auto-correlation peak of the zero dose image". This point coincides, within the uncertainty of the measurement, with the point found by extrapolation. The knowledge of this fact makes it possible
.2
-
0
500
1000
15'00
Fig. 6. Cross-correlation peaks of another dose series: (×) optical analog, (*) digital evaluation.
to normalize all values by putting this peak = 1. It is clear that this procedure rests on the following conditions: (1) the specimen stabilizes, and (2) all imaging conditions in the microscope remain constant. In order to check the analog calculation, the correlation functions were also calculated digitally using the photometric data from the micrographs. The results of both methods are compared in fig. 6 and are in good agreement. (For technical reasons the digital data from the micrograph series of fig. 4 could not be obtained and hence are missing in fig. 5; a second set of micrographs was used for this important comparison.)
6. Conclusie" Two procedures using only light optical analog computations have been described in detail because they can be helpful in quantitative electron microscopy, If a medium-size digital computer is available, the microscopist can, after scanning the micrographs or the light optical diffraetograms, calculate the T(k) and the correlation function ci2 with considerable ease [17]. On the other hand, the repertoire of light optical methods'is not exhausted by the two methods discussed here. For instance~ elimination of the unmodulated part of the SD by coherent optical filtering seems promising. Another possibility is the use of optical (fine-grained photographic negative) or electronic computer [5] (storage unit [18])subtraction
F. Zemlin, P. Schiske /Phase contrast transfer function of the images g l , g2 (gd = gl -- ~2hifted) • By comparison of the SDs generated by gd [18] and by gs = gl + ~hifted, the background noise can be eliminated.
Acknowledgements We are indebted to Professor E. Zeitler for encouragement and to Dr. E. Reuber, Dr. G. Lehmpfuhl and K. Weiss for experimental help and discussions.
References [1] [2] [3] [4] [5]
F. Then, Z. Naturforsch. 20a (1965) 154. K.H. Hanssen and L. Trepte, Optik 33 (1971) 182. J. Frank, Optik 38 (1973) 519. J. Frank, Optik 30 (1969) 171. W. Hoppe, R. Langer, J. Frank and A. Feltynowski, Naturwissenschaften 56 (1969) 267. [6] J. Frank, P.H. Bussler, R. Langer and W. Hoppe, Bet. Bunsenges. Physik. Chem. 74 (1970) 1105.
145
[7] R. Langer, J. Frank, A. Feltynowski and W. Hoppe, Bet. Bunsenges. Physik. Chem. 74 (1970) 1120. [8] W.A. Saxton, in: Prec. 8th Intern. Congr. on Electron Microscopy, Canberra, 1974, Vol. 1, Eds. J.V. Sanders and D.J. Goodchild (Australian Academy of Science, Canberra) pp. 314-315. [9] P. Schiske and F. Zemlin, Mikroskopie 32 (1976) 208. [10] K. Tonar, Mikroskopie 32 (1976) 236. [11] J. Frank and A1Ali, Nature 256 (1975) 376. [12] R. Hegerl, A. Feltynowski and B. Grill, in: Prec. 9th Intern. Congr. on Electron Microscopy, T9ronto, 1978, Vol. 1, Ed. J.M. Sturgess (Microscopical Society of Canada, Toronto) pp. 214-215. [13] W. Hoppe, R. Hegerl and R. Guckenberger, Z..Naturforsch. 33a (1978) 857. [14] N.R. Arnot and W.O. Saxton, Optik 53 (1978) 271. [ 15] J. Gassmann, in: Advances in Structural Research by Diffraction Methods, Vol. 7, Eds. W. Hoppe and R. Mason, pp. 121-136. [16] R.C. Williamsand H.WIFisher, J. Mol. Biol. 52 (1970) 121. [17] J. Frank, in: Prec. Electron Microscopy Soc. of America, 34th Ann. Meeting, 1976, pp. 478-479. [18] K.-H. Herrmann, D. Krahl and H.-P. Rust, Ultramicroscopy 3 (1978) 227.
MEASUREMENT OF THE PHASE CONTRAST TRANSFER FUNCTION AND THE CROSSCORRELATION PEAK USING YOUNG INTERFERENCE FRINGES F. ZEMLIN and P. SCHISKE Fritz-Haber-lnstitut der Max-Planck-Gesellschafl, Teilinstitut fiir Elektronenmikroskopie, Faradayweg4-6, Berlin (West)33/Dahlem Received 6 December 1979 Cross-correlation of micrographs taken under identical optical conditions allows measurement of (a) the optical transfer function free from electron and photographic noise and (b) object change, especially that due to radiation damage. Light optical methods based on superposition diffraetograms are discussed. A subtraction procedure is described for measuring the optical transfer function, and a two-step diffraction process is presented for measuring object change. As an example, experiments with radiation damage in carbon foil are described. Here increased stability regarding radiation-induced change was found with increasing dose, making possible measurement of deviations from the "zero dose image".
1. Introduction
2. Theory of superposition diffractograms
The phase contrast transfer function T(k) of an electron microscope can be measured by producing light optical diffractograms from micrographs of objects whose distance distributions can be considered random [1]. Carbon foil is usually considered adequate. Nevertheless, the zeroes of the transfer function which are expected theoretically show up only poorly in the photometer registrations of the diffractograms. The transferred intensities are superimposed on a noisy background which decays slowly towards liJgher spatial frequencies. This background impairs the measurements; the minima are shifted relative to the zeroes of the T(k) and the calculated envelope [2,3] can be recognized only qualitatively. For practical microscopy, it would be important to have a simple routine method which could achieve a correction readily *. Superposition diffractograrns (SDs) [4-7] carry the necessary information. In this paper, two simple methods of evaluating the SDs both of which achieve the separation of noise and signal - are discussed and tested experimentally.
Superposition diffractograms are generated from double-exposed micrographs or by inserting two single. exposed micrographs simultaneously in the beam path of the diffractometer. In both cases a lateral shift A (the "Sprungdrift" of ref. [4]) is applied. In the second case angular misorientation is avoided by making tentative small rotational adjustments while keeping A ~ const. When the maximum correlation (i.e., maximum contrast of the interference fringes mentioned below) - especially for high spatial frequencies - has been found, the micrographs are ftxed mechanically [8]. With the usual assumptions of weak phase object and linear photographic response, the optical diffractogram is determined as follows. We consider an absolute stable object f(x) which is exposed two times with exactly the same optical parameters (defocus, exposure time .... ) except for the relative shift A. (Note: two-dimensional vectors x, A, y ..... k are used. In most expressions the vectors have been suppressed. Integrations are two-dimensional and extended over the whole plane, except when stated explicitly otherwise. Lower case function letters refer to functions in direct space: f(x), i(x) ..... Fourier transformation leads to functions in reciprocal
* See Editorial, Ultramicroscopy 4 (1979) 393. 139
F. Zemlin, P. Schiske / Phase contrast transfer function
140
space: F(k), l(k) ..... which are denoted by capital letters. Averaging is expressed by a crossbar, complex conjugation by * (* is used as a sign of convolution). In consequence of the high magnification, for our purposes the specimen can be considered infinite. To this infinite extension correspond the distributions f(x), F(k) (object), r(x), R(k) (noise), and i(x), l(k) (image). The intensity i(x) on the final screen of the microscope,
i(x) = f
I(k)eikx dk,
(1)
which can be seen as the definition of the Wiener spectrum E(k) of i(x). It is convenient to consider an ensemble of objects, all prepared in the same way, generating an ensemble of image functions in (x). The Wiener spectrum E(k) will be characteristic for the whole ensemble: En (k) = E(k). Averaging over the ensemble eq. (5) can be condensed formally to
l(k) I* (l) = E(k) 6 (k - l).
(6)
Here the 5-function expresses the statistical homogeneity of i(x). E(k), which is characteristic of the microscope and the type of preparation of the specimen, is the quantity which is of primary interest to the microscopist. Da(k) merely reflects the properties of a sample taken at random which is used to infer E(k). From eq. (5) it is seen that ((21r)2/a)Da(k) for increasing area approximates E(k), at least with regard to a fixed diffraction resolution Ak. For fixed area, using eqs. (2) and (6) one has
is recorded as a small variatfon in the light optical transmission of the micrograph. It is convenient to put [(x) = 0 (suppression of a constant which carries no information). Fraunhofer diffraction of i(x) generates an amplitude distribution in the screen of the light optical diffractometer which corresponds to the Fourier transform of the original intensity distribution. Since the detector of the diffractometer responds to the square of the amplitude and since, contrary to the mathematical assumptions used for the definition of I(k), the area of investigation, a, is of finite extent, the final result D(k) is not simply a ° signal proportional to l(k), but the square of a convolution of I(K) with a shape function Ba (k), which accounts for the finite extent of the area illuminated in the diffractometer:
Da(k) = f(k) * IB(k)l 2 .
D(k) = Da(k) = II(k) * Ba(k)l 2 •
The image intensity i(x), and hence also I(k), relate to the object function f(x), F(k) by
(2)
Here * denotes convolution with regard to k, and the shape function Ba(k) can be found from the window function ba(x) (= 1 inside, = 0 outside the area, a, which is illuminated):
(7)
B(k) is a peak-shaped function with width 5k ~ 21r/L, where L ~ ~/a gives the size of the window b(x) (i.e., the part of the image actually used). This describes the limits of the simplified relation (2~')2 - -
a
D(k)
~ E(k)
.
I(k) = F(k) T(k) " [1 + exp(iAk)] + R(k),
(8)
(9)
To suppress the speckle, D(k) = Da(k) has to be evaluated integrating over finite patches Ak (photometer aperture):
where T(k) is the (phase) contrast transfer function of the microscope and R(k) represents noise (electron noise and photographic granularity). The sum [1 + exp(iAk)] corresponds to the shifted single image (equal exposure). Introducing the Wiener spectra, I¢(lkl) of the specimen f(x) and N(Ikl) of the noise r(x) (electron and photographic granularity), one has
f Da(k) dk.
E(k) = 2WOkl)(l + cos Ak) l T(k)[ 2 + N(Ikl).
ba(x) = fBa(k) eik'x d k .
(3)
(4)
ak
Here the statistical independence of signal and noise,
Taking conceptually the limit a ~ oo, one has lim (2rr)---~2 f ea(k) dk = f E(k) dk, a --..~.~
(10)
a Ak
Ak
f(x) n(x)= O, (5)
has been used. W is related to the auto-correlation
(11)
F. Zemlin, P. Schiske /Phase contrast transfer function
141
function of the object
fw(Ikl) eiky dk = c(lYl)=f(x)f(x +y).
(12)
This can be inferred from the relation
F(k)F*(l) = W(Ikl) ~i(i~ - / ) ,
(13)
which is analogous to eq. (6). Statistical isotropy is taken care of by using Ikl, lyl (not k, y) as arguments of W and c.
3. Subtraction method [9 ] 1
The factor (1 + cos Ak) describes the modulation of the superposition diffractogram (Young interference fringes - YIF). Scanning along the lines 1 + cos Ak = 0, i.e., the minima of the YIF, N(k) can be measured for all values Ikl > rr/IAI, using the theorem of Pythagoras (fig. 1). Subtraction of N from eq. (10) gives the signal
W(k)" (1 + cos A k ) " IT(k)l 2.
(14)
If the object f(x) has a white spectrum, that means l¢(Ikl) = I, we are measuring the pure instrument function IT(k)l. In fact, the spectrum of carbon foil is not white. But if either W(Iki) or IT(k)l is known the other factor can be found by this method. Actually in scanning along the..,minima of the YIF a finite aperture is used. Thus a small part of the signal, eq. (14), is included in the measurements of the
'
I (3,h) ~
Fig. 2. Evaluation of the superposition diffractogram of fig. 1: (a) density scan parallel to the maxima of the YIFs (smoothed); (b) density scan parallel to the minima of the YIFs (smoothed); (c) difference of curves (a) and (b); (d) fitting this theoretical phase contrast transfer function to the corrected curve (c) requires a defocus of -3(Csh) 1/2 .
noise N. It seems that an aperture diameter of approximately ~ of the fringe distance is tolerable. In our experiments (fig. 2) ~o of the fringe distance was used.
3.1. Frequency filtering of superposition diffractograms Another approach consists of frequency fftltering. For example, scanning along concentric circles and using analog electronics, Tonar [ 10] has determined l#(Ikl) • cos(Ak) • IT(k)l 2 ,
/
Ik12=Is12. I--~Iz b Fig. 1. (a) Superposition diffractogram with Young interference fringes (YIF). (b) Geometry of YIFs, schematic. With k, spatial frequency, A, spacing of the YIF; s, the scanned distance.
(15)
by measuring the amplitude of an alternating current. This method, however, is only practicable when even small deviations from rotational symmetry (axial astigmatism in the electron microscope) are avoided. Both difficulties - the error introduced by finite size of photometer aperture and the sensitivity to axial astigmatism - can be overcome by a second Fourier transform,
d2 (x) =
[fD(k) e - i ~
dk[ 2 ,
(16)
14 2
F. Zemlin, P. Schiske /Phase contrast transfer function
using the light optical diffractometer again as analog computer. This procedure does not need any additional electronic equipment and was found to present no experimental difficulty. Since the diffractograms (spectra) are Fourier transforms of autocorrelation functions, the second diffraction leads back to correlation functions. This can be considered an advantage of this method since measurement of correlation peaks is becoming increasingly important in quantitative electron microscopy, particularly for the investigation of radiation-sensitive specimens [11-15]. It was found that especially the height of correlation peaks can be measured with great ease by this method. The second Fourier transform is not applied to the spectrum W([k[)but to W([kl) • IT(k)[ 2. Multiplication by IT(k)[ 2 is replaced by convolution with the spread function,
s(x)--
f,r(k), 2 e
dk,
(17)
s(x) depicts the finite resolution of the electron microscope. The resulting second diffractogram d: (x) is represented by d2 (x) = e 2 (x)" h 2 (x).
(18')
Using * for the convolution with regard to x one has e(x)
=
cOxl) * s(x) * (28(x)
+/i(x - A) + 5(x + A)) + n(Ixl)
(19)
with n(Ixl) = f N ( I k l ) e ikx dk.
(20)
To a broad noise spectrum N(Ikl) there corresponds a sharp peaked n(Ixl). Eq. (19) describes three peaks, centered at x = 0 and x = +A. The factor h (x) merely expresses the finite extension of the part of the image actually used for analysis: h(x) = b(x) * b(x) .
(21)
In practice this region is so big that the relative variation of h(x) may be neglected in the region of interest which is given by the sharp peaks of c(Ixl) resp. c o x + AJ) (fig. 3). To isolate the contribution of c(jx - Aj), thereby eliminating also n(ixl), photometric measurements are made near x = A from which
Fig. 3. The optical diffractogram (Fourier transform) of the superposition diffractograms shows three (correlation) peaks due to the cosine-modulation of the YIF system.
the values found at other places of the annulus ixl [A[ are subtracted. Thus the background due to the limiting value c([x[) ~ f-2 for large x is eliminated. In this way one determines the height and shape of the peak of c(ix - A[) * s(x).
4. Unstable objects Equal exposure times can be achieved experimentally without much effort. Basic relations for unequal exposure times are worked out and tested in ref. [4]. Specimens which are not stable due to electron irradiation have to be described by two specimen func. tions f l (x):/:f2 (x) corresponding to the two exposures. The calculations for this case can be seen as a generalization of those necessary for unequal exposure times. In place of the auto-correlation function c(Ixl) now a cross-correlation function c12 (Ixl) = c21 (ix[) has to be used alongside the auto-correlation functions cll (Ixl), c22(Ixl). Eq. (12) must be generalized to fW~t3(ikl )
e iky
dk = Cet(J(lY I)
= f~(x) f~(x +y) = c0a(lyl),
% g = 1, 2.
(22)
Eq. (10) is replaced by D(k) = [Wl~(Ikt) + W22(Ikl) -
2W12(Ikl)] • IT(k)l 2 +N(IkJ)
+ 2(1 + cos A k ) . W12(Jkl) • IT(k)l 2.
(23)
F. Zemlin, P. Schiske /Phase contrast transfer function
From eq. (23) it follows that for severe axial astigmatism the subtraction procedure [9] has to be modified. Making l A I big enough, the values of T(k) taken along the lines (a)
A. k = +zr/2,
(b) A. k =0,
can be made nearly equal (compared for equal values of ik[) even if the imaging in the electron microscope is not rotationally symmetric. By subtraction of D(k) taken along (a) from D(k) taken along (b) (both for equal Ikl), W12(Ik,) • IT(k)h2 can be found in good approximation for (b). Since a given pair of micrographs may be shifted with arbitrary orientation of A, this modification of the subtraction method has, in principle, no limitation. No essential modification is needed for the method of second diffraction. Eq. (19)is replaced by e(x) = [ c l l ([xi) + c22([xl) + c12(Ix - AI) + cl2(Ix + Ai)] * s(x) + n(Ixl).
(24)
The cross-correlation peak c12(Ix - AI) * s(lxl) can be isolated in the same way as the auto-correlation peak, eq. (19), in the case of the stable object.
1 -2
1 -3
143
5. Application In the following the application of this measuring procedure is demonstrated. The specimen was a thin (about 20 A) carbon foil which had not yet been exposed to electron irradiation. In order to measure the radiation damage at minimum dose, the mini-exposure technique was applied [16], i.e., the adjustment was made at an adjacent region. Immediately before the first exposure the irradiated region was shifted by switching off a deflection system. From the central region, in rapid succession, a sequence of exposures (1, 2, ..., n) was recorded at 2 × 10 s magnification. Fig. 4 shows a sequence of superposition diffractograms of these images: beginning from the upper left, the image pair (1,2), next to it (I, 3) etc. up to the lower right (1, 7) As can be seen, the modulation of the YIF decreases slowly. The decrease is stronger at high spatial frequencies than at low frequencies. This decrease of the YIFs is a measure of the radiation damage. Moreover, fig. 4 (right) shows the superposition diffractogram of the image pair (7, 8). As can be seen, the YIFs are very strongly modulated there, more strongly than at
1 -4
'!!!I!ttlit! 7-8
1 - 5
1 -6
1 - 7
(3.~) -~ I
I
Fig. 4. Sequence of superposition diffractograms of a dose series, demonstrating radiation damage.
F. Zernlin, P. $chiske /Phase contrast transfer function
144
XCF
XCF
1.0
10 .7
.7
.5
.5 .3 .2
.3 0
560
1060 1500 20'00
, =
O/~
Fig. 5. Cross-correlation peaks versus dose of the dose series of fig. 4. The extrapolation to zero dose is controlled independently (see text).
the initial image pair (1, 2). This observation was confirmed by many measurement sequences; it demonstrates the known phenomenon that the carbon foil stabilizes at irradiation. After a high pre-irradiation the structure changes only slightly with further irradiation - compared to the initial state. This sequence of diffractograms was evaluated with the procedure of section 3. In fig. 5 the correlation peaks are plotted versus the dose. From the curve it can be seen that: (1) the correlation peaks decrease exponentially in the beginning; (2) with increased pre-irradiation the decrease of the correlation peaks becomes weaker, i.e., the foil becomes stable. It was of interest to learn how strongly the first image of the series deviates from the "zero dose image". Therefore, the curve was extrapolated to dose zero. Any extrapolation is afflicted with. uncertainty. Hence an attempt was made to obtain this value in another way. The intersection with the ordinate axis which belongs to an image pair of a stable specimen corresponds to the correlation peak 1. Since the carbon foil stabilizes itself, i.e., its radiation sensitivity decreases, we plotted the measured crosscorrelation peak of the image pair (7, 8) directly on the ordinate - i.e., we did ascribe the dose zero to this cross-correlation peak. It turns out that the crosscorrelation peak of image pair (1, 2) is smaller by 25% than the "auto-correlation peak of the zero dose image". This point coincides, within the uncertainty of the measurement, with the point found by extrapolation. The knowledge of this fact makes it possible
.2
-
0
500
1000
15'00
Fig. 6. Cross-correlation peaks of another dose series: (×) optical analog, (*) digital evaluation.
to normalize all values by putting this peak = 1. It is clear that this procedure rests on the following conditions: (1) the specimen stabilizes, and (2) all imaging conditions in the microscope remain constant. In order to check the analog calculation, the correlation functions were also calculated digitally using the photometric data from the micrographs. The results of both methods are compared in fig. 6 and are in good agreement. (For technical reasons the digital data from the micrograph series of fig. 4 could not be obtained and hence are missing in fig. 5; a second set of micrographs was used for this important comparison.)
6. Conclusie" Two procedures using only light optical analog computations have been described in detail because they can be helpful in quantitative electron microscopy, If a medium-size digital computer is available, the microscopist can, after scanning the micrographs or the light optical diffraetograms, calculate the T(k) and the correlation function ci2 with considerable ease [17]. On the other hand, the repertoire of light optical methods'is not exhausted by the two methods discussed here. For instance~ elimination of the unmodulated part of the SD by coherent optical filtering seems promising. Another possibility is the use of optical (fine-grained photographic negative) or electronic computer [5] (storage unit [18])subtraction
F. Zemlin, P. Schiske /Phase contrast transfer function of the images g l , g2 (gd = gl -- ~2hifted) • By comparison of the SDs generated by gd [18] and by gs = gl + ~hifted, the background noise can be eliminated.
Acknowledgements We are indebted to Professor E. Zeitler for encouragement and to Dr. E. Reuber, Dr. G. Lehmpfuhl and K. Weiss for experimental help and discussions.
References [1] [2] [3] [4] [5]
F. Then, Z. Naturforsch. 20a (1965) 154. K.H. Hanssen and L. Trepte, Optik 33 (1971) 182. J. Frank, Optik 38 (1973) 519. J. Frank, Optik 30 (1969) 171. W. Hoppe, R. Langer, J. Frank and A. Feltynowski, Naturwissenschaften 56 (1969) 267. [6] J. Frank, P.H. Bussler, R. Langer and W. Hoppe, Bet. Bunsenges. Physik. Chem. 74 (1970) 1105.
145
[7] R. Langer, J. Frank, A. Feltynowski and W. Hoppe, Bet. Bunsenges. Physik. Chem. 74 (1970) 1120. [8] W.A. Saxton, in: Prec. 8th Intern. Congr. on Electron Microscopy, Canberra, 1974, Vol. 1, Eds. J.V. Sanders and D.J. Goodchild (Australian Academy of Science, Canberra) pp. 314-315. [9] P. Schiske and F. Zemlin, Mikroskopie 32 (1976) 208. [10] K. Tonar, Mikroskopie 32 (1976) 236. [11] J. Frank and A1Ali, Nature 256 (1975) 376. [12] R. Hegerl, A. Feltynowski and B. Grill, in: Prec. 9th Intern. Congr. on Electron Microscopy, T9ronto, 1978, Vol. 1, Ed. J.M. Sturgess (Microscopical Society of Canada, Toronto) pp. 214-215. [13] W. Hoppe, R. Hegerl and R. Guckenberger, Z..Naturforsch. 33a (1978) 857. [14] N.R. Arnot and W.O. Saxton, Optik 53 (1978) 271. [ 15] J. Gassmann, in: Advances in Structural Research by Diffraction Methods, Vol. 7, Eds. W. Hoppe and R. Mason, pp. 121-136. [16] R.C. Williamsand H.WIFisher, J. Mol. Biol. 52 (1970) 121. [17] J. Frank, in: Prec. Electron Microscopy Soc. of America, 34th Ann. Meeting, 1976, pp. 478-479. [18] K.-H. Herrmann, D. Krahl and H.-P. Rust, Ultramicroscopy 3 (1978) 227.