GB2095028A - Image correction in electron microscopes - Google Patents

Abstract

In a transmission electron microscope the observable intensity of electrons passing through a specimen contains information about the amplitude of the wave function but not its phase. The invention provides this phase information by recording the intensity distribution with and without an apodizing filter 20 in the electron beam path and processing these two distributions to retrieve the phase. This filter e.g. of carbon, has a thickness which varies in a manner so as to produce an exponential factor of exp(-2 pi at) to the amplitude of the transmitted wave function and a linear factor to the phase of the wave function, e.g. as shown, it may be a linear wedge. <IMAGE>

Description

SPECIFICATION Phase retrieval processing in electron microscopy This invention relates to image processing in electron microscopy.

In a transmission electron microscope electrons pass through a thin specimen and thence through an objective lens, a small number of intermediate lenses and a projection lens before striking a phosphor screen where an image or a diffraction pattern is observed. All these lenses introduce unwanted aberrations into the electron beam, which results in a distortion of the observed image of diffraction pattern, however the distortion is normally dominated by the aberrations in the objective lens and the other lenses may, to a good approximation, be regarded as aberration free. In order to compensate for the effect of the aberrations and hence obtain information about the object, it is necessary to know both the amplitude and the phase of the wave function in the image or the diffraction pattern.The observable intensity is equal to the squared modulus of the wave function and so contains information about the amplitude of the wave function but not the phase. This may be termed the phase retrieval problem.

In mathematical terms, the observable intensity, I(x, w), is equal to the squared modulus of some possibly complex wave function, F(x, w), which is related to another possibly complex wave function, f(t, s), by a finite Fourier transform,

In the imaging case, I(x, w) is the observed intensity in the image plane, F(x, w) is the wave function in the image plane and f(t, s) is the wave function in one of the finite back focal or diffraction planes of the lens system. In the diffraction pattern case, I(x, w) is the observed intensity in the diffraction plane, F(x, w) is the wave function in the diffraction plane and f(t, s) is the wave function in one of the image planes of the lens system.

To reconstruct the possibly complex object structure requires a knowledge of F(x, w) or f(t, s), however the observable quantity is l(x, w)=IF(x, w)i2, over some finite measurement interval. The phase retrieval problem is to find the phase of F(x, w) from intensity measurements.

A number of methods for solving the phase retrieval problem have been proposed. Work up to 1 977 is reviewed in detail by Ferwerda, Topics in Current Physics Vol. 9 1 978. Recent work is briefly reviewed by Baltes, Topics in Current Physics Vol. 20 1 980. An overview is given by Ross, Fiddy and Moezzi, Optica Acta 27 1433 1980, in a paper which also proposes a novel method for the solution of the problem, involving the processing of two images of the same specimen; one in focus the other just out of focus.

According to this invention a method of determining the phase of the wave function associated with an observable intensity distribution comprises the steps of recording the intensity distribution, recording the intensity distribution obtained with an apodising filter of form approximating to exp(--27cat) located in a (t, s) plane then processing these two intensity distributions to retrieve the phase.

Apparatus for carrying out the method of this invention comprises a transmission electron microscope and an apodising filter of variable thickness material, such as carbon, capable of movement into and out of the electron beam in an appropriate plane.

The filter thickness may vary in a linear manner in one direction so as to introduce an exponential factor of exp(--27cat) to the amplitude of the transmitted wave function and a linear factor to the phase of the wavefunction.

The invention will now be described, by way of example only, with reference to the accompanying drawings of which: Figure 1 is schematic diagram of a transmission electron microscope operated in an imaging mode.

Figure 2 is a schematic diagram of a transmission electron microscope operated in a diffraction pattern mode.

Figure 3 is a flow diagram showing the method for retrieving the phase of the wave function.

Figure 4 is an alternative flow diagram.

As shown in figures 1 and 2 a transmission electron microscope comprises a vacuum chamber 1 containing an electron source 2 and a series of electron lenses 3-8 arranged on an axis 9. Of these lenses 3 and 4 are condenser lenses, 5 is the objective lens, 6 and 7 are intermediate lenses and 8 is the projector lens. The specimen 10 is mounted in a holder above the objective lens 5. Beneath the projector lens 8 is a phosphor coated screen 11 that glows when struck by electrons to provide an observable intensity distribution. The screen 11 can be moved aside to allow the electrons to strike a film plate 12 for permanent recording of the intensity distribution. A apodising filter 20 of varying thickness carbon is mounted for movement into any one of planes 13 to 19.Typically the filter thickness will be in the range .02 to .4 yrn. Although carbon is the obvious choice of material for the filter other uniform and preferably amorphous materials could be used.

In the imaging mode of operation (Fig. 1), a specimen is prepared and mounted in the usual manner An image is formed on the screen 11 and focal adjustments are made to focus the image. The image l(x, w) is recorded on film 1 2 or by some other means e.g. with a video camera. The exponential apodising filter 20 is inserted into the electron beam in one of the back focal or diffraction planes 1 3, 14, 1 5 or 16, and focal adjustments made to focus the modified image. The modified image l'(x, w) is recorded on film 12 or by some other means.

In the diffraction pattern mode of operation (Fig. 2), a specimen 10 is prepared and mounted in the usual manner. A diffraction pattern is formed on the screen 11 and focal adjustments are made to focus the diffraction pattern . The diffraction pattern l(x, w) is recorded on film 1 2 or by some other means. The exponential apodising filter 20 is inserted into the electron beam on one of the image planes 17, 1 8 or 19, and focal adjustments made to focus the modified diffraction pattern. The modified diffraction pattern l'(x, w) is recorded on film 12 or by some other means.

In both modes of operation the subsequent processing of the recorded intensity distributions I(x, w) and l'(x, w) to retrieve the phase of F(x, w) is the same. The mathematical basis for determining the phase of F(x, w) is as follows:- The analytic continuation of F(x, w) into the complex z=x+iy plane may be written,

Functions such as F(z, w), which may be written as a finite Fourier transform, are entire functions of exponential type. This property allows F(z, w) to be expanded in an infinite product known as a Hadamard product

where C1 and C2 are constants for any given wand zJ(w)=xj(w)+iyj(w) is the location of the jth complex zero for a given w.

From equation (4) it is clear that if all the zj(w) for a particular w can be located and l(x, w) is known then F(x, w) for that w may be constructed and hence by repetition for all other w values F(x, w) may be found for all x and w in the measurement interval. Unfortunately, from a measurement of l(x, w), the location. of the zeros cannot, uniquely, be found. A function which can be computed from l(x, w) isl(z,w), I(z, w)=F(z, w)(F(z*, w))* (5) This can be achieved in practice by applying the equations,

for a succession of y values.

Inspection of equation (5) shows that l(z, w) contains not only the zeros of F(z, w) but also those of (F(z*, w))*. The zeros of (F(z*, w))* lie in locations, which are the complex conjugates of the zeros of F(z, w); hence from a knowledge of l(x, w) one can search for the zeros of l(z, w) and determine xj(w) and Iyi(w)l but not the signs of the y.

To solve the phase retrieval problem the sign of every non-zero yj(w) in the measurement interval is determined as follows. Determination of the location of the complex zeros of F(z, w) is based on two measurements. The first is a measurement of l(x, w); from which, as described above, xj(w) and Iy(w)I may be determined for all zeros in the measurement interval. To determine the sign of each of the non-zero yj(w) a second measurement is made. This second measurement is denoted by I'(x, w), where, and, I'(x, w)=jF'(x, w)12 (8)

where a is a real positive constant and the lateral shift of l'(x, w), due to the linear phase factor introduced by the filter, is compensated for by an equal shift of the (x, w) plane origin.

The method for determining the sign of each non-zero yj(w) is as follows. The first step is to compute the analytic continuation l'(z, w) from l'(x, w) using the procedure described above (equations (6) and (7)). Now, I'(z, w)=F'(z, w)(F'(z*, w))*, (10) where,

so l'(z, w) contains the zeros of F'(z, w), denoted by z'j(w)=x'j(w)+iy'j(w), and the zeros of (F'(z*, w))*, which are in complex conjugate positions. Hence x'j(w) and ly:(w)l may be determined, for all zeros in the measurement interval.

Comparison of equations (2) and (11) shows that, F'(z, w)=F(z-ia, w) (1 2) from which it is clear that, x'j(w)=xj(w), (13) y'j(w)=y(w)+a. (14) From (14) it follows that, |(w)|=|yj(w)|+a; yj(w) positive, Iy'(w)I=Iy(w)I-a; yj(w) negative, gyJ(w)lEa, (15) tyti(w)i=aIyi(w)l; yj(w) negative [y(w) < a.

A knowledge of iy(w)i, IYi(w)I a and the rules given by equation (15) are sufficient to uniquely determine the sign of yj(w), for all non-zero yl(w) in the measurement interval. This may be achieved in practice simply, quickly and without ambiguity by an algorithm which decides which of the equalities (1 5) holds, to the best approximation, for each measured gyj(w)l and ty'j(w)l and the known value of a.

Having located the zeros, F(x, w) may be constructed using equation (3) with z=x; outside the measurement interval zeros should be placed on the real axis at positions zi(w)=j/b--a and the number of zeros used should be sufficiently large that the inclusion of more would not lead to significant changes. The values of the constants C, and C2 may be found by adjusting the amplitude of the F(x, w) constructed from the zero locations to conform with the measured form.

-Figure 3 shbws a flow chart of the computational steps of this zero location method for retrieving the phase of F(x, w) from measurements of l(x, w) and l'(x, w).

The zero location method of retrieving the phase of F(x, w) is not the only one which may be applied to data consisting of l(x, w) and l'(x, w). The arguments used in developing the zero location method may be regarded simply as a proof that measurements of I(x, w) and l'(x, w) are sufficient to uniquely determine the phase of F(x, w). An alternative method of retrieving the phase from l(x, w) and l'(x, w) would be an iterative algorithm similar to that first proposed by Gerchberg and Saxton, Optik 35 2371972.

Figure 4 shows a flow chart of the computational steps of one of a number of different iterative methods that could be used to retrieve the phase of F(x, w) from measurements of l(x, w) and l'(x, w).

Claims (filed on 23/2/82) 1. A method of determining the phase of the wave function associated with an observable intensity distribution comprising the steps of recording the intensity distribution, recording the intensity distribution obtained with an apodising filter of form approximating to exp(--2#at) located in a (t, s) plane then processing these two intensity distributions to retrieve the phase.

2. Apparatus for carrying out the method of claim 1 comprising a transmission electron

**WARNING** end of DESC field may overlap start of CLMS **.

Claims (7)

**WARNING** start of CLMS field may overlap end of DESC **. and, I'(x, w)=jF'(x, w)12 (8) where a is a real positive constant and the lateral shift of l'(x, w), due to the linear phase factor introduced by the filter, is compensated for by an equal shift of the (x, w) plane origin. The method for determining the sign of each non-zero yj(w) is as follows. The first step is to compute the analytic continuation l'(z, w) from l'(x, w) using the procedure described above (equations (6) and (7)). Now, I'(z, w)=F'(z, w)(F'(z*, w))*, (10) where, so l'(z, w) contains the zeros of F'(z, w), denoted by z'j(w)=x'j(w)+iy'j(w), and the zeros of (F'(z*, w))*, which are in complex conjugate positions. Hence x'j(w) and ly:(w)l may be determined, for all zeros in the measurement interval. Comparison of equations (2) and (11) shows that, F'(z, w)=F(z-ia, w) (1 2) from which it is clear that, x'j(w)=xj(w), (13) y'j(w)=y(w)+a. (14) From (14) it follows that, |(w)|=|yj(w)|+a; yj(w) positive, Iy'(w)I=Iy(w)I-a; yj(w) negative, gyJ(w)lEa, (15) tyti(w)i=aIyi(w)l; yj(w) negative [y(w) < a. A knowledge of iy(w)i, IYi(w)I a and the rules given by equation (15) are sufficient to uniquely determine the sign of yj(w), for all non-zero yl(w) in the measurement interval. This may be achieved in practice simply, quickly and without ambiguity by an algorithm which decides which of the equalities (1 5) holds, to the best approximation, for each measured gyj(w)l and ty'j(w)l and the known value of a. Having located the zeros, F(x, w) may be constructed using equation (3) with z=x; outside the measurement interval zeros should be placed on the real axis at positions zi(w)=j/b--a and the number of zeros used should be sufficiently large that the inclusion of more would not lead to significant changes. The values of the constants C, and C2 may be found by adjusting the amplitude of the F(x, w) constructed from the zero locations to conform with the measured form. -Figure 3 shbws a flow chart of the computational steps of this zero location method for retrieving the phase of F(x, w) from measurements of l(x, w) and l'(x, w). The zero location method of retrieving the phase of F(x, w) is not the only one which may be applied to data consisting of l(x, w) and l'(x, w). The arguments used in developing the zero location method may be regarded simply as a proof that measurements of I(x, w) and l'(x, w) are sufficient to uniquely determine the phase of F(x, w). An alternative method of retrieving the phase from l(x, w) and l'(x, w) would be an iterative algorithm similar to that first proposed by Gerchberg and Saxton, Optik 35 2371972. Figure 4 shows a flow chart of the computational steps of one of a number of different iterative methods that could be used to retrieve the phase of F(x, w) from measurements of l(x, w) and l'(x, w). Claims (filed on 23/2/82)

1. A method of determining the phase of the wave function associated with an observable intensity distribution comprising the steps of recording the intensity distribution, recording the intensity distribution obtained with an apodising filter of form approximating to exp(--2#at) located in a (t, s) plane then processing these two intensity distributions to retrieve the phase.

2. Apparatus for carrying out the method of claim 1 comprising a transmission electron

microscope characterised by an apodising filter of variable thickness, means for locating the filter in the electron beam, and means for recording the intensity distribution of electrons passing through a specimen with and without the filter in the electron beam path.

3. Apparatus according to claim 2 wherein the filter thickness varies in a linear manner in one direction so as to introduce an exponential factor approximating to exp(--27cat) to the amplitude of the transmitted wave function and a linear factor to the phase of the wave function.

4. Apparatus according to claim 3 wherein the filter thickness varies in a simple wedge shape.

5. Apparatus according to claim 3 wherein the filter thickness varies between 0.2 and 0.4 ym.

6. Apparatus according to claim 2 wherein the filter is of carbon material.

7. Apparatus for phase retrieval processing in an electron microscope constructed arranged and adapted to operate substantially as hereinbefore described with reference to the accompanying Figures 1 to 4. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~