Transcript Technology for a better society

High Resolution
TEM
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Resolution of an optical system
Rayleigh criterion
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http://micro.magnet.fsu.edu/primer
The resolving power of an optical system is limited by the diffraction occurring at the optical
path every time there is an aperture/diaphragm/lens.
The aperture causes interference of the radiation (the path difference between the green
waves results in destructive interference while the path difference between the red waves
results in constructive interference).
An object such as point will be imaged as a disk surrounded by rings.
The image of a point source is called the Point Spread Function
Point spread function (real space)
1 point
2 points
resolved
2 points
unresolved
Aperture and resolution of an
optical system
Diffraction spot
on image plane
= Point Spread Function
Objective
Tube lens
Intermediate
image plane
Sample
Back focal plane aperture
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Aperture and resolution of an
optical system
Diffraction spot
on image plane
= Point Spread Function
Objective
Tube lens
Intermediate
image plane
Sample
Back focal plane aperture
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Aperture and resolution of an
optical system
Diffraction spot
on image plane
= Point Spread Function
Objective
Tube lens
Intermediate
image plane
Sample
Back focal plane aperture
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Aperture and resolution of an
optical system
Diffraction spot
on image plane
= Point Spread Function
Objective
Sample
Tube lens
Intermediate
image plane

Back focal plane aperture
The larger the aperture at the back focal plane (diffraction plane), the larger 
and higher the resolution (smaller disc in image plane)
NA = n sin()
where:
 = light gathering angle
n = refractive index of sample
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New concept:
Contrast Transfer Function (CTF)
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Optical Transfer Function (OTF)
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OTF(k)
Resolution limit
Image
contrast
K or g
(Spatial frequency,
periods/meter)
Observed
image
Object
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Kurt Thorn, University of California, San Francisco
Definitions of Resolution
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OTF(k)
As the OTF cutoff frequency
1/kmax
= 0.5  /NA
|k|
As the Full Width at Half Max
(FWHM) of the PSF
FWHM
≈ 0.353  /NA
As the diameter of the Airy disk
(first dark ring of the PSF)
= “Rayleigh criterion”
Airy disk diameter
≈ 0.61  /NA
Kurt Thorn, University of California, San Francisco
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Remember: reciprocal/frequency space
To describe a wave, specify:
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Frequency (how many periods/meter?)
Direction
Amplitude (how strong is it?)
Phase (where are the peaks & troughs?)
A wave can also be described
by a complex number at a point:
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Distance from origin
Direction from origin
Magnitude of value at the point
Phase of number
complex
ky
k = (kx , ky)
kx
Kurt Thorn, University of California, San Francisco
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Remember: frequency space
and the Fourier transform
ky
Fourier
Transform
ky
kx
kx
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The Transfer Function Lives in Frequency Space
Object
OTF(k)
Observed
image
|k|
ky
Observable
Region
kx
Kurt Thorn, University of California, San Francisco
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Remember: the Properties of the
Fourier Transform
F(k) 
2 ikr
f
(r)e
dr

Completeness:
The Fourier Transform contains all the information of the original image
Symmetry:
The Fourier Transform of the Fourier Transform is the original image
Fourier
transform
Kurt Thorn, University of California, San Francisco
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The OTF and Imaging
True
Object
convolution

Fourier
Transform
Observed
Image
PSF
?
=
OTF

Kurt Thorn, University of California, San Francisco
=
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Convolutions
(f  g)(r) = f(a) g(r-a) da
Why do we care?
• They are everywhere…
• The convolution theorem:
h(r) = (fg)(r),
If
then
h(k) = f(k) g(k)
A convolution in real space becomes
a product in frequency space & vice versa
Symmetry: g  f = f  g
So what is a convolution, intuitively?
• “Blurring”
• “Drag and stamp”
g
f

y
=
y
x

fg
y
=
x
Kurt Thorn, University of California, San Francisco
x
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Resolution in HRTEM
In optical microscopy, it is possible to define point resolution as the ability to resolve individual
point objects. This resolution can be expressed (using the criterion of Rayleigh) as a quantity
independent of the nature of the object.
The resolution of an electron microscope is more complex. Image "resolution" is a measure of
the spatial frequencies transferred from the image amplitude spectrum (exit-surface wavefunction) into the image intensity spectrum (the Fourier transform of the image intensity). This
transfer is affected by several factors:
• the phases of the diffracted beams exiting the sample surface,
• additional phase changes imposed by the objective lens defocus and spherical aberration,
• the physical objective aperture,
• coherence effects that can be characterized by the microscope spread-of-focus and
incident beam convergence.
For thicker crystals, the frequency-damping action of the coherence effects is complex but for a
thin crystal, i.e., one behaving as a weak-phase object (WPO), the damping action can best be
described by quasi-coherent imaging theory in terms of envelope functions imposed on the
usual phase-contrast transfer function.
The concept of HRTEM resolution is only meaningful for thin objects and, furthermore, one has
to
distinguish
between
point
resolution
and
information
limit.
O'Keefe, M.A., Ultramicroscopy, 47 (1992) 282-297
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Microscope:
Transforms each point on the specimen into an
extended region (at best, a circular disk) in the
final image.
Each point on the specimen may be different,
we describe the specimen by a specimen
function, f(x,y).
The extended region in the image which
corresponds to the point (x,y) in the specimen
is then described as g(x,y)
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Two nearby points, A and B, they will produce
two overlapping images gA and gB
Each point in the image has contributions from
many points in the specimen.
How much each point in
the specimen contributes
to each point in the
image
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h(r): How information in real space is transferred from the specimen to the
image
H(u): How information (or contrast) in u space is transferred to the image.
H(u) is the Fourier transform of h(r)
H(u) is the contrast transfer function.
Now these three Fourier transforms are related by
G(u) = H(u) F(u)
So a convolution in real space gives multiplication in reciprocal space
The factors contributing to H(u) include:
Apertures
Attenuation of the wave
Aberration of the lens
The aperture function A(u)
The envelope function E(u)
The aberration function B(u)
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We write H(u) as the product of these three
terms
H(u) = A(u) E(u) B(u)
The aperture function A(u):
The objective diaphragm cuts off all values of u (spatial frequencies) greater than
(higher than) some selected value governed by the radius of the aperture.
The envelope function E(u):
Has the same effect but is a property of the lens itself, and so may be either more or
less restricting than A(u).
The aberration function B(u):
Is usually expressed as
Spherical
coefficient
Defocus
Electron
ofaberration
the
wavelength
objective
lens
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High spatial frequencies correspond to large distances from the optic axis in
the DP. The rays which pass through the lens at these large distances are
bent through a larger angle by the objective lens. They are not focused at
the same point by the lens, because of spherical aberration, and thus cause
a spreading of the point in the image. The result is that the objective lens
magnifies the image but confuses the fine detail. The resolution we require
in HRTEM is limited by this ‘confusion’
 Each point in the specimen plane is transformed into an extended region
(or disk) in the final image.
 Each point in the final image has contributions from many points in the
specimen.
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Spherical aberration
Electrons with high spatial frequency (large distance from optical axis) are bent
The resolution we require in HRTEM is limited by this ‘confusion’
through a larger angle
• Each point in the specimen plane is transformed into an extended region (or disk)
in the final image.
• Each point in the final image has contributions from many points in the specimen.
Causes: Spreading of the point in the image
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Weak Phase-Object Approximation
For a very thin specimen, the amplitude of a transmitted
wave function will be linearly related to the projected
potential of the specimen.
The projected potential is taking account of variations in
the z-direction, and is thus very different for an electron
passing through the center of an atom compared to one
passing through its outer regions.
Assumptions:
- Sample very thin
- Amplitude A(x,y) = 1 (unity)
- Represent the specimen as
phase object
- Small absorptions
- Vt(x,y) <<1
WPOA fails for an electron wave passing through the
center of a single uranium atom! An atomic layer of U
would be too thick for the WPOA.
A model to represent the specimen:
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H(u) = A(u) E(u) B(u) = A(u) E(u) exp(iχ(u))
Only the imaginary part
Intensity transfer function:
(Objective lens transfer function)
T(u) = A(u) E(u) 2sin χ(u)
T(u) = A(u) exp (iχ) exp(-π2Δ2λ2u4/2) exp(-π2uc2q2)
Where: q = Cs λ3u3 + Δf λ u
Δ = Cs (𝜎 2 𝑈/𝑈 2 ) + (4𝜎 2 𝐼/𝐼 2 ) + (𝜎 2 𝐸/𝐸 2 )
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Transfer function T(u)
1. T(u), formulation applies to any specimen
2. T(u) is not the CTF of HRTEM
The image wave function is not an observable quantity! What we observe in an image is
contrast, or the equivalent in optical density, current readout, etc., and this is not linearly
related to the object wave function.
Fortunately, there is a linear relation involving observable quantities under the special
circumstances, where the specimen acts as a WPO
If we have WPO : T(u) called CTF
- No amplitude contribution
- Output of the transmission system is an observable quantity (image contrast)
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T(u) = A(u) E(u) 2sin χ(u)
Ignore E(u)
Phase distortion function
has the form of a phase shift expressed as 2π/λ
times the path difference traveled by those
waves affected by spherical aberration (Cs),
defocus (Δz), and astigmatism (Ca).
T(u) = 2 A(u) sin χ(u)
CTF is:
- Oscillatory
- There are bands of good transmission separated by gaps (zeros) where no transmission
occurs
- Maxima: when phase-distortion function assumes multiple odd values of ±π/2
- Zero contrast for χ(u) = multiple of ±π
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T(u)<0
Positive phase contrast:
Phase shift of –π/2 due to diffraction
Atoms appear dark against bright background
T(u)>0
Negative phase contrast:
Phase shift of +π/2 due to diffraction (adds amplitude: 'in phase')
Atoms appear bright against dark background
T(u)=0
No detail in the image
(assuming Cs>0)
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Contrast transfer function
sin χ(k) = sin(πλ∆fk2 +1/2πCsλ3k4)
k
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Contrast transfer function
sin χ(k)
sin χ(k) = sin(πλ∆fk2 +1/2πCsλ3k4)
k:
parameters: λ=0.0025 nm (200 kV), cs =1.1 mm, Δf=60 nm
The CTF oscillates between -1 (negative contrast transfer) and +1 (positive contrast transfer).
The exact locations of the zero crossings (where no contrast is transferred, and information is
lost) depends on the defocus.
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Point resolution
Point resolution: related to the finest detail that can be directly interpreted in terms of
the specimen structure. Since the CTF depends very sensitively on defocus, and in
general shows an oscillatory behavior as a function of k, the contribution of the different
scattered beams to the amplitude modulation varies. However, for particular underfocus
settings the instrument approaches a perfect phase contrast microscope for a range of k
before the first crossover, where the CTF remains at values close to –1. It can then be
considered that, to a first approximation, all the beams before the first crossover
contribute to the contrast with the same weight, and cause image details that are directly
interpretable in terms of the projected potential.
Optimisation of this behaviour through the balance of the effects of spherical aberration
vs. defocus leads to the generally accepted optimum defocus1 −1.2(Csλ)1/2. Designating
an optimum resolution involves a certain degree of arbitrariness. However, the point
where the CTF at optimum defocus reaches the value –0.7 for k = 1.49C−1/ 4λ−3/4 is
usually taken to give the optimum (point) resolution (0.67C1/4λ3/4). This means that the
considered passband extends over the spatial frequency region within which transfer is
greater than 70%. Beams with k larger than the first crossover are still linearly imaged,
but with reverse contrast. Images formed by beams transferred with opposite phases
cannot be intuitively interpreted.
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The Contrast Transfer Function
1
sin   sin( πfu 2  πCs3u 4 )
2
Plot: T(u) = sin χ
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The effect of different Cs and Δf on CTF
Important points to notice:
•
CTF is oscillatory: there are "passbands" where it is
NOT equal to zero (good "transmittance") and there
are "gaps" where it IS equal (or very close to) zero
(no "transmittance").
•
When it is negative, positive phase contrast occurs,
meaning that atoms will appear dark on a bright
background.
•
When it is positive, negative phase contrast occurs,
meaning that atoms will appear bright on a dark
background.
•
When it is equal to zero, there is no contrast
(information transfer) for this spatial frequency.
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Other important features:
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CTF starts at 0 and decreases, then
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CTF stays almost constant and close to -1 (providing a
broad band of good transmittance), then
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CTF starts to increase, and
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CTF crosses the u-axis, and then
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CTF repeatedly crosses the u-axis as u increases.
•
CTF can continue forever but, in reality, it is modified by
envelope functions and eventually dies off. Effect of the
envelope functions can be represented as
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Scherzer defocus
Every zero-crossing of the graph corresponds to a contrast
inversion in the image.
Up to the first zero-crossing k0 the contrast does not change
its sign.
The reciprocal value 1/k0 is called Point Resolution.
The defocus value which maximizes this point resolution is called the Scherzer
defocus.
Optimum defocus: At Scherzer defocus, one aims to counter the term in u4 with the
parabolic term Δfu2 of χ(u). Thus by choosing the right defocus value Δf one flattens
χ(u) and creates a wide band where low spatial frequencies k are transferred into
image intensity with a similar phase.
Working at Scherzer defocus ensures the transmission of a broad band of
spatial frequencies with constant contrast and allows an unambiguous
interpretation of the image.
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Defocus
The presence of zeros in CTF means that we have
gaps in the output signal. Obviously, the best CTF is
the one with the fewest zeros and with the broadest
band of good transmittance (where CTF is close to 1). Back in 1949 Scherzer suggested an optimum
defocus condition which occurs at:
∆𝑓 = − 𝐶𝑠 λ
this value is now called "1 scherzer".
Sometimes the value of 1.2 scherzer is called
"Scherzer defocus" (we are going to call the value of
1.2 scherzer as "extended scherzer" throughout this
manual). "Extended scherzer" provides even broader
band of transmittance with CTF still close enough to
-1.
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Scherzer defocus
Δ f = - (Csλ)1/2
Scherzer condition
Δ f = -1.2(Csλ)1/2
Extended Scherzer condition
http://www.maxsidorov.com/ctfexplorer/webhelp/effect_of_defocus.htm
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b/d
1) Optimum focus conditions:
a) Scherzer focus:
∆𝑓 = − 𝐶𝑠 λ
b) Scherzer resolution:
𝑥𝑚𝑖𝑛 = 0.71(𝐶𝑠1/4 λ3/4 )
c) Extended Scherzer focus:
∆𝑓 = −
d) Extended Scherzer resolution:
Optimum aperture size:
2) Optimum information limit:
Instrumental resolution
given by the envelope function
4/3
𝐶𝑠 λ = −1.2 𝐶𝑠 λ
/
𝑥𝑚𝑖𝑛 = 0.66(𝐶𝑠1 4λ3/4 )
(Point resolution)
𝑢𝑚𝑎𝑥 = 1.52(𝐶𝑠 −1/4 λ−3/4 )
𝑥𝑖 =
𝜋λ∆/2
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The Envelope function
The resolution is also limited by the spatial coherence of the
source and by chromatic effect:
Teff = T(u)EcEa
The envelope function imposes a “virtual aperture” in the back focal plane of the
objective lens
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• Information limit goes well beyond point resolution limit for FEG
microscopes (due to high spatial and temporal coherency).
• For the microscopes with thermionic electron sources (LaB6 and W), the
info limit usually coincides with the point resolution.
• Phase contrast images are directly interpretable only up to the point
resolution (Scherzer resolution limit).
• If the information limit is beyond the point resolution limit, one needs to
use image simulation software to interpret any detail beyond point
resolution limit.
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Information limit
Information limit: corresponds to the highest spatial frequency still appreciably transmitted to the
intensity spectrum. This resolution is related to the finest detail that can actually be seen in the image
(which however is only interpretable using image simulation). For a thin specimen, such limit is
determined by the cut-off of the transfer function due to spread of focus and beam convergence (usually
taken at 1/e2 or at zero).
These damping effects are represented by E or Etc a temporal coherency envelope (caused by chromatic
aberrations, focal and energy spread, instabilities in the high tension and objective lens current), and E
or Esc is the spatial coherency envelope (caused by the finite incident beam convergence, i.e., the beam is
not fully parallel).
The Information limit goes well beyond point resolution limit for FEG microscopes (due to high spatial
and temporal coherency). For the microscopes with thermionic electron sources (LaB6 and W), the info
limit usually coincides with the point resolution. \
The use of FEG sources minimises the loss of spatial coherence. This helps to increase the information
limit resolution in the case of lower voltage ( ≤ 200 kV) instruments, because in these cases the temporal
coherence does not usually play a critical role. However the point resolution is relatively poor due to the
oscillatory behavior of the CTF. On the other hand, with higher voltage instruments, due to the increased
brightness of the source, the damping effects are always dominated by the spread of focus and FEG
sources do not contribute to an increased information limit resolution.
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Damped contrast transfer function
Microscope examples
(Scherzer)
Thermoionic, 400 kV
FEG, 200 kV
Spatial
envelope
Point
resolution
Information
limit
Temporal
envelope
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Important points to notice
•
CTF is oscillatory: there are "passbands" where it is NOT equal to zero (good "transmittance") and
there are "gaps" where it IS equal (or very close to) zero (no "transmittance").
•
When it is negative, positive phase contrast occurs, meaning that atoms will appear dark on a
bright background.
•
When it is positive, negative phase contrast occurs, meaning that atoms will appear bright on a
dark background.
•
When it is equal to zero, there is no contrast (information transfer) for this spatial frequency.
•
At Scherzer defocus CTF starts at 0 and decreases, then
•
CTF stays almost constant and close to -1 (providing a broad band of good transmittance), then
•
CTF starts to increase, and
•
CTF crosses the u-axis, and then
•
CTF repeatedly crosses the u-axis as u increases.
•
CTF can continue forever but, in reality, it is modified by envelope functions and eventually dies
off.
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Defocus: 1 scherzer ("True" Scherzer defocus
Defocus: 1.2 scherzer ("Extended" Scherzer defocus).
In general, this is the best defocus to take HR-TEM
images.
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Defocus: 1.9 scherzer ("2nd Passband" defocus).
Produces a nice and broad passband which starts NOT at
zero. CTF is positive so that it produces a negative phase
contrast ("white atoms")
Defocus: 0.4 scherzer ("Minimum Contrast" defocus)
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Defocus: 0 (low contrast, not good). Note: Zero-defocus
images are NOT "Minimum Contrast" images. Minimum
contrast occurs at about 0.4 scherzer (shown above).
Defocus: 10 scherzer (large and generally not good for
HR-TEM)
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