Graef M. Introduction to conventional transmission electron microscopy

Подождите немного. Документ загружается.

648 Phase contrast microscopy

10.4.2 Fresnel fringes for non-magnetic objects

In the previous section, we saw that Fresnel (fringe) contrast occurs wherever the

phase of the electron wave has a non-vanishing second derivative (curvature). This

is valid for any phase, not just for phase of a magnetic origin, so in this section

we will explore brieﬂy the nature of Fresnel fringes for non-magnetic objects.

From the Aharonov–Bohm equation we see that a non-magnetic object will cause

a phase shift which is entirely due to the electrostatic lattice potential. At low

magniﬁcations, signiﬁcantly lower than those used for high-resolution observations,

the only important component of the electrostatic lattice potential is the mean inner

potential V

, so that the phase shift due to a foil of thickness t is given by φ

= σ V

where σ is the interaction constant. From here on we will drop the subscript 0 on

the mean inner potential and simply write V .

The ION routine

fresnel.pro can be used to display Fresnel images for objects

with a spatially varying mean inner potential. The routine provides the following

pre-deﬁned sample “shapes”: a strip of material, a foil with a circular hole, a disk-

shaped island, a square island, and a thin strip of material with a slightly lower mean

inner potential than the surrounding matrix. The routine calls a Fortran program

that is very similar to the

LorentzExample.f90 program discussed earlier. The only

difference is that the phase computation is much simpler than for a magnetic object:

if the geometry of the object is known, then the local thickness can be computed

for a given orientation, and hence the phase shift will be known.

Figure 10.35 shows four pairs of simulated images for a circular hole in a matrix.

The matrix is 20 nm thick and has a mean inner potential of 30 V; the normal

absorption length is taken to be 50 nm. The exit wave function is then equal to

φ(x, y) = e

−t/ξ



iσ Vt

for x

+ y

> r

with r the radius of the hole. The origin is taken at the center of the hole. The exit

wave is unity inside the hole. The image pair a–b of Fig. 10.35 shows Fresnel fringes

for the defocus values ±150

µm (same values for all images in this ﬁgure). The

beam divergence angle is θ

= 0.05 mrad, and several Fresnel fringes are visible

near the edge of the hole, which is indicated by a thin white circle in all simulated

images. The ﬁrst fringe inside the hole is bright for positive defocus and dark for

negative defocus. In the presence of astigmatism (C

= 100 µm and C

= 60

◦

the distance between the ﬁrst fringe and the edge of the hole is no longer constant

and varies around the hole, as shown in image pair c–d. Conversely, if the ﬁrst

fringe is not at the same distance from the edge for all points along the hole, then

astigmatism must be present. Correcting the astigmatism is then simply reduced to

making the Fresnel fringe continuous around the hole at a uniform distance from

the edge.

10.5 Exit wave reconstruction 649

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

150 nm

∆f = + 150 µm

∆f = - 150 µm

= 0.05 mrad

= 0.005 mrad

Fig. 10.35. Fresnel fringe simulations for a 300 nm diameter circular hole in a 20 nm thick

matrix with a mean inner potential of 30 V. The defocus is +150 µm for the top row, and

−150 µm for the bottom row. The four images on the left have θ

= 0.05 mrad, while the

others have θ

= 0.005 mrad. Image pairs c–d and g–h contain astigmatism (C

= 100 µm

and C

= 60

◦

). The images were computed using the ION routine fresnel, available from

the

website.

If the beam divergence angle is decreased to θ

= 0.005 mrad (this is possible for

a ﬁeld emission microscope), then a much larger number of Fresnel fringes can be

observed, as illustrated by image pair e–f in Fig. 10.35. The presence of astigmatism

(g–h) can be recognized easily from the asymmetry of the fringes. The number of

fringes and the defocus range for which they are visible are illustrated in Fig. 10.36.

The top image represents a through-focus series ( f from −150 to +150

µm) for

a solid–vacuum interface; the foil thickness is 20 nm, with a mean inner potential

of 30 V. The interface is indicated by a dashed line. The beam divergence angle

is θ

= 0.1 mrad, which represents a low beam coherence. The Fresnel fringes are

visible for small defocus magnitudes but disappear for larger values. For a beam

divergence of 0.001 mrad, many fringes are visible (bottom of Fig. 10.36) even

for large defocus values. The fringes inside the foil may not always be visible,

depending on the thickness and absorption length of the foil.

10.5 Exit wave reconstruction

10.5.1 What are we looking for?

In virtually every TEM experiment we are ultimately interested in the structure

and/or chemistry of a particular material or defect. This means that we would

650 Phase contrast microscopy

-150 µm

150 µm

∆f = 0 µm

vacuum

solid

100 nm

= 0.1 mrad

= 0.001 mrad

Fig. 10.36. Through-focus series of Fresnel fringes near a straight edge. The microscope

accelerating voltage is 200 kV, foil thickness 20 nm, mean inner potential 30 V, and two

different beam divergence angles are shown.

like to ﬁnd out where the individual atoms are located, or how the magnetization is

distributed throughout the thin foil, or where charges accumulate near an electrically

active boundary, and so on. For non-periodic features, such as most defects, we

may have a structure model that we can use to simulate images which can then be

compared with the observations. This is the standard approach which we will call

the forward approach:

atom positions → lattice potential → exit wave function → simulated images.

(10.80)

From the differences between observed and simulated images we can often derive

information to reﬁne the model, and we may be able to iterate the model parameters

until the simulated images converge to the observed ones.

10.5 Exit wave reconstruction 651

In the forward approach, we typically do not modify the observed intensities,

other than perhaps some noise ﬁltering and intensity and/or magniﬁcation scaling.

We use the observed images (they could also be diffraction patterns) as the reference

state against which to validate the model. We could also think about an opposite

approach, in which we attempt to derive the structure directly from the observed

images, with as few adjustable parameters as possible. This is known as the inverse

approach or, more commonly, as the inverse problem. While it may seem obvious

that this would be a possible approach, the inverse problem is actually a highly

non-trivial problem, since it requires that we reconstruct both the amplitude and

the phase of the electron wave at the exit plane of the sample. Phase information

is present in the images, but generally not in a form that can be extracted easily.

Ultimately, we want to ﬁnd out what the electrostatic lattice potential looks like,

since it contains information on both the atom types and locations. This means

that we must invert the Schr¨odinger equation, in addition to removing the transfer

function of the microscope. The inverse problem can then be stated schematically

as follows:

experimental images → exit wave function → lattice potential → atom positions.

(10.81)

The ﬁrst step, from image to wave function, is known as exit wave restoration

or reconstruction. It is a highly non-linear problem, since it essentially involves

the deconvolution of the microscope transfer function according to equation (2.56)

on page 107. We know from the ﬁrst section of this chapter that the microscope

transfer function has multiple zero-crossings, so that a simple deconvolution is

not possible. The second step requires inversion of the Schr¨odinger equation,

and this is again a non-trivial step. Both problems have received considerable at-

tention, even since the early days of TEM research. There are several solutions

available in the literature, but none of them is entirely satisfactory in the sense

that no single method can be applied to all possible situations. Exit wave recon-

struction remains a topic of signiﬁcant interest, in particular now that spherical

aberration can be removed from the image formation process using an aberration

corrector.

There is a signiﬁcant body of literature on wave function restoration. The main

ideas go back to the work by Gerchberg and Saxton [GS72], Schiske [Sch68],

and ultimately to Cowley and Moodie [CM57]. The phase of the image wave

can be found from combinations of multiple images acquired at different micro-

scope defocus values, sometimes combined with diffraction information. Schiske’s

original work introduces so-called ﬁlter functions, which are essentially weight-

ing functions used to combine images taken at multiple defocus values in such

a way that a least-squares best estimate of the exit wave function is obtained.

652 Phase contrast microscopy

This procedure is now known as a generalized Wiener ﬁlter, as described in

[HK94, Chapter 73].

Kirkland and coworkers [KSUF80, Kir82, KSUF82] have successfully imple-

mented a reconstruction scheme based on a minimum mean-square formalism

and applied it to what has effectively become a “standard sample”, chlorinated

Cu-phthalocyanine. Adaptive least-squares ﬁlters were introduced by Hawkes

[Haw74], and a review of various ﬁlter methods can be found in [Sch84]. The

Gerchberg–Saxton algorithm makes use of an image and a diffraction pattern taken

from the same region [GS72, GS73]; a multiple image algorithm, proposed by

Misell [Mis73], and elaborated by many others, can in principle be used to recover

the phase from two images at different microscope defocus, but no experimental

veriﬁcation of the method has been reported. Kirkland’s multiple input maximum a

posteriori method (MIMAP) [Kir84, KSUF85] again works in principle, but does

not function satisfactorily for realistic experimental conditions. Non-linear recon-

struction schemes based on the earlier work have also been proposed [CJOdBVD92]

and have been shown to work in selected experimental situations. For an in-depth

review of other reconstruction schemes we refer to Chapters 73 and 74 in [HK94].

A review of exit wave reconstruction using electron holography methods can be

found in [VAJ99].

In the next section, we will illustrate the process of exit wave reconstruction

for Lorentz imaging. The reconstruction process for Lorentz images is much more

straightforward than for conventional high-resolution images, since we can make

good use of the fact that the Lorentz deﬂection angle is small compared with the

Bragg angle. As we will see in the next section, this makes it possible to derive the

phase of the exit wave from just three experimental images.

10.5.2 Exit wave reconstruction for Lorentz microscopy

We start from equation (10.78) and write the in-focus intensity a

as I (r, 0); the out-

of-focus image intensity is then written as I (r,f ). Next, we derive an equation for

the phase φ; consider an under-focus and over-focus image, for the same defocus

magnitude | f |:

I (r, | f |) = I (r, 0) −

λ| f |

2π

∇·

(

I (r, 0)∇φ

)

(

| f |

)

2ln2



I (r, 0)∇



I (r, 0) − I (r, 0)(∇φ)

;

I (r, −| f |) = I (r, 0) +

λ| f |

2π

∇·

(

I (r, 0)∇φ

)

(

| f |

)

2ln2



I (r, 0)∇



I (r, 0) − I (r, 0)(∇φ)

10.5 Exit wave reconstruction 653

Subtracting the second equation from the ﬁrst, and rearranging terms we have:

−

2π

I (r, | f |) − I (r, −| f |)

2| f |

=∇·

[

I (r, 0)∇φ

]

. (10.82)

In the limit of vanishingly small defocus the left-hand side becomes a derivative and

we arrive at the so-called transport-of-intensity equation (TIE)

†

[Tea83, GRN95b,

GRN95a, GN96, PN98]:

∇·

[

I (r, 0)∇φ

]

=−

2π

∂ I (r, 0)

∂z

(10.83)

The TIE equation was also derived by Van Dyck and Coene in 1987 starting from

equation (5.35) with

V = 0 (i.e. the free-space propagation equation) [VDW87].

Note that the beam divergence term, which is even in the defocus, cancels out in

the difference between under-focus and over-focus images. This means that phase

reconstruction for Lorentz microscopy images does not require a highly coherent

electron beam.

10.5.2.1 Solving the TIE equation

Equation (10.83) can be solved numerically in a number of different ways. Perhaps

the most straightforward method relies on the use of fast Fourier transforms. The

Fortran source code for the algorithm can be found on the

website in the routine

TIE.f90. A formal solution to the TIE equation can be derived by introducing a new

variable , such that

∇(r, 0) ≡ I (r, 0)∇φ. (10.84)

If the phase is entirely determined by the magnetic structure of the thin foil,

then its gradient can be used to compute the in-plane magnetic induction

conﬁguration:

∇φ =

∇

I (r, 0)

=−

[

B(r) × n

]

t(r), (10.85)

where n is the unit normal in the beam direction and t is the local sample thickness.

Using the function  deﬁned in equation (10.84), the TIE becomes a Poisson-

type equation [PN98]:

∇

(r, 0) =−k

∂ I (r, 0)

∂z

. (10.86)

†

The name of this equation was ﬁrst coined by M.R.Teague [Tea83], who showed that this formalism could be

applied to phase retrieval.

654 Phase contrast microscopy

While a direct analytical solution is difﬁcult to obtain, it is possible to write down

a symbolic solution as follows:

(r, 0) =−k∇

−2

∂ I (r, z)

∂z

. (10.87)

The symbol ∇

−2

refers to the two-dimensional inverse Laplacian operator, and

we will describe shortly a simple numerical implementation. Taking the two-

dimensional gradient of this equation we obtain:

∇φ(r, 0) =−

I (r, 0)

∇∇

−2

∂ I (r, 0)

∂z

, (10.88)

where we have used equation (10.84). After applying the two-dimensional diver-

gence operator ∇·, and bringing the Laplacian operator to the right-hand side we

ﬁnd the symbolic solution for the phase [PN98]:

φ(r, 0) =−k∇

−2



∇·



I (r, 0)

∇

−2



∂ I (r, 0)

∂z

>

. (10.89)

It is straightforward to show that the same equation can be used to recover the phase

for an out-of-focus image; the argument (r, 0) can then be replaced by (r,f ).

While equation (10.89) looks rather menacing, it is actually not very difﬁcult

to implement an efﬁcient numerical procedure to solve for the phase. In fact,

we can take the formal solution one step further and note that (at least formally)

we have

∇

−2

∇=∇

−1

=∇∇

−2

, (10.90)

and we arrive at the formal solution

φ(r, 0) =−k∇

−1



I (r, 0)

∇

−1



∂ I (r, 0)

∂z

>

(10.91)

From equation (10.82) we see that we need to acquire two out-of-focus images,

I (r, − f ) and I (r, + f ). In addition, according to equation (10.89), we need

the in-focus image I (r, 0). The longitudinal derivative ∂ I /∂z can be computed

readily, provided both images are perfectly aligned and have precisely the same

magniﬁcation.

†

From equations (10.76) and (10.77) we can derive the following

relation:

∇

[···] =−4π

−1

F[···]|q|

, (10.92)

†

Recall that the magniﬁcation changes with defocus, in particular when a low-ﬁeld lens with a long focal length

is used.

10.5 Exit wave reconstruction 655

where [···] indicates a function of the coordinate variable r. We leave it to the

reader to show that

∇

−2

[···] =−

4π

−1



F[···]

|q|



. (10.93)

If we combine the inverse Laplacian with the gradient operator ∇, then we can

show that with respect to the Cartesian reference frame e

we have for every

function α(r)

∇

−1

α(r) =∇∇

−2

α(r) =∇

−2

∇α(r);

=−

4π

∂

∂x



−1



[α(r)]

|q|



;

=−

2π

α(r



)



|q|

2πiq·(r−r



)





;

2π

α(r − R

⊥

)



|q|

2πiq·R

⊥



⊥

;

≡

2π

α(r − R

⊥

)G(R

⊥

)dR

⊥

;

2π

G(r) ⊗ α(r).

The vector function G(r) is equal to the inverse Fourier transform of the vector

function q/|q|

, and is not deﬁned for q = 0. The formal solution (10.91) to the

TIE equation can then be expressed as a double convolution product:

φ(r,f

) =

64π

λ f

(G(r)⊗) ·



G(r) ⊗ ∂ I (r,f

)/∂z

I (r,f

)



. (10.94)

It is understood that the integrations are carried out on a grid with unit grid spacing.

In component notation we have

G(r) ⊗ [···] = e

⊗ [···] = e



|q|

2πiq·r



⊗ [···],

and for numerical implementation we can use

φ =−

64π

λ f



i=1

⊗



⊗ ∂ I /∂z



, (10.95)

where γ

are the components of the vector function G. The prefactor follows from

a simple dimensional analysis of the formal solution (10.91). This particular form

of the symbolic solution shows clearly that the accuracy of the magnitude of the

reconstructed phase depends directly on a careful calibration of the microscope

656 Phase contrast microscopy

Pseudo Code PC-23 Reconstruct exit wave phase from Lorentz–Fresnel images.

Input: Three or ﬁve images, δ,  f , λ {Microscope calibration!}

Output: Reconstructed exit wave phase

compute array of

|q|

vectors

compute longitudinal derivative

∇

verify conservation criterion

convolve with G(r) (using FFT operations)

divide by the in-focus image

convolve with G(r) (using FFT operations)

normalize with the prefactor

64π

λ f

magniﬁcation (through δ) and the defocus step size ( f ). Pseudo code for the

implementation of equation (10.95) can be found in

PC-23 . Since part of the

reconstruction involves a convolution product, one must take care of aliasing effects

and truncate the Fourier transforms at

. Since the inverse Laplacian is effectively

a low-pass ﬁlter, this does not cause any loss of information.

In addition to the anti-aliasing measures, we must also symmetrize the in-

put images, to ensure that the solution to the TIE equation is unique. The rea-

sons for this symmetrization are beyond the scope of this textbook and we re-

fer the interested reader to [VZDG02] for the details. In practice, we double the

size of the images by application of two orthogonal mirror planes, as shown in

Fig. 10.37. The algorithm is then applied to the symmetrized images. The main

reason for this symmetrization is the use of the fast Fourier transform, which re-

quires a periodic input image. An example computation is shown in the following

section.

Finally, the right-hand side derivative in the TIE must average to zero, indicating

that no electrons are lost from the image with varying defocus. This conservation

criterion guarantees that the solution to the TIE is unique [GRN95a].

While we have used the small-angle approximation to the transfer function, it

can be shown (e.g. [BPN00, Chapter 5] and [VDW87]) that the TIE equation is

valid for a much more general class of experimental conﬁgurations. It is possible

to apply the same formalism to high-resolution phase contrast images, and one can

then reconstruct the phase of the image wave from a through focus series consisting

of at least three images. When the phase of the image wave is known, a Fourier

transform propagates the wave back to the back focal plane of the objective lens,

where one can then deconvolve the transfer function. This phase reconstruction

method is rather promising when used in combination with a spherical aberration

10.5 Exit wave reconstruction 657

(a)

(b)

Fig. 10.37. Symmetrization of the images needed to solve the transport-of-intensity equa-

tion using a fast Fourier transform algorithm.

corrector, since the zero-crossings of the transfer function can then be avoided

entirely.

10.5.2.2 Example application of the TIE formalism

The TIE solution algorithm has been applied to the Fresnel images of Fig. 4.28.

The area indicated by a square was taken as the input image (160 × 160 pixels).

After aligning the three images using a cross-correlation algorithm, the algorithm in

pseudo code

PC-23 was applied. The resulting reconstructed phase is shown as a

grayscale image in Fig. 10.38(a) and a contour plot in (b). The x and y components

of the gradient of φ are proportional to the y and x components of the integrated

magnetic induction and are shown in Figs 10.38(c) and (d); bright areas indicate

positive components. A more familiar representation of the magnetic induction in

the foil is shown in Fig. 10.39: each vector represents the average induction over

a4× 4 pixel area. Vector lengths are scaled with respect to the longest vector

which has unit magnitude. The vortex and cross-tie wall are clearly visible in the

reconstructed map.

As a ﬁnal example we consider a 2 × 1

µm rectangular Permalloy island de-

posited on a Si

membrane, observed in Fresnel mode on a JEOL 4000EX

microscope equipped with a Gatan post-column energy ﬁlter (GIF). The under-

and over-focus images are shown in Figs 10.40(a) and (b), along with the numer-

ically computed derivative ∂ I /∂ z, shown in (c). The image consists of 300 × 300

pixels. The reconstructed phase is shown as a grayscale plot (d) and as cos φ (e).

The square region outlined in (c) is shown in (f) as a vector plot, obtained from the

x and y components of ∇

⊥

φ.