GBS765
Hybrid methods
Lecture 3
Contrast and image formation
10/20/14 4:37 PM
The lens ray diagram
Magnification M = A/a = v/u
and
1/u + 1/v = 1/f
where f is the focal length
The lens ray diagram
•  So we know how to magnify the
object in the object plane
•  But what is being magnified?
– i.e. what generates contrast?
•  The contrast depends on the
interaction between the object
and the incident electron wave
How (glass) lenses work
Electron lenses
Schematic diagram
of magnetic lens
Ray diagram for a lens
(electron or optical)
Path of an electron through a
magnetic field
Magnification vs. Resolution
• What is magnification?
Definition: Magnification is the ratio between the apparent size of an object and its
actual size.
•  What is resolution?
Definition: Resolution is the smallest detail that can be distinguished in an image
– is usually given in units of distance, d (nm, Å)
– higher resolution means smaller d
• Resolving power is the theoretical ability of a particular instrument to resolve fine
detail in a specimen
• What is the resolving power of the human eye?
The naked eye can see objects of about 0.1-0.2 mm
• A typical CCD camera has a pixel size of 15µm
• Photographic film can resolve about 1-2µm
• How much do you have to magnify an atom/a virus to see it with the naked eye/a CCD camera?
•  No point magnifying beyond the resolution limit!
What is contrast?
• Contrast is what allows us to see the object that we are looking
for"
" "(i.e. doesn t matter how high your resolving power is if you can t see the
specimen)"
"
• "Percent contrast = 100 x |Ii – Ib| / Ib "
• "However, the ability to distinguish the signal from the
background depends on the Signal-to-noise ratio:"
" "SNRrms = [Var(Signal) / Var(noise)]1/2"
"
• "How is contrast generated? "
"
" "– in light microscopy, photons are absorbed by the sample, giving rise to a
pattern of light and dark"
"
" "– in EM there is no absorption, but electrons are scattered (scattering contrast)"
"
" "– in addition, interference between scattered waves can lead to contrast
(=phase contrast) "
"
What is contrast?
•
Contrast is the difference in intensity of the object of interest and the
background
•
Contrast needs to be >5-10% to be detectable by eye
– computers can detect arbitrarily small differences in contrast
•
Whether the signal is detectable depends on the noise
Interaction
of electrons
Electron
scattering
and
diffraction
with the specimen
Scattering and diffraction
Scattering of electrons
e-
Incident beam
+
Secondary
electrons
Backscattered
electrons
• Electrons interact with the combined
electric field of the nucleus and the electrons
Diffraction of
Electrons
asX-rays:
waves:
2ry electrons
Auger
electrons
X-rays
Incident wave
Secondary scattered
(=diffracted) wave
•Wavelength
X-rays interactofonly
with the(nm)
electrons
electrons
≈ 1.22/√E
• Note
λthat
also electrons can be described
where
E iswith
acceleration
as a wave
wavelength voltage
! ≈ 1.22/ (eV)
√E
and also undergo diffraction phenomena.
Inelastically
scattered
Unscattered beam
Elastically scattered
Electron
scattering
and
diffraction
Scattering and diffraction
Scattering and diffraction
Scattering of electrons
e+
Incident wave
• Electrons interact with the combined
electric field of the nucleus and the electrons
Scattered wave
Diffraction of
Electrons
asX-rays:
waves:
Interference of
scattered waves:
diffraction
Incident wave
Secondary scattered
(=diffracted) wave
•Wavelength
X-rays interactofonly
with the(nm)
electrons
electrons
≈ 1.22/√E
• Note
λthat
also electrons can be described
where
E iswith
acceleration
as a wave
wavelength voltage
! ≈ 1.22/ (eV)
√E
and also undergo diffraction phenomena.
Types of contrast in TEM
•
The TEM image is produced mainly by the elastically scattered electrons.
•
These give rise to contrast by two main mechanisms:
–  Scattering contrast (mass-thickness contrast)
– Due to elastic interactions with atomic potential field
– Dependent on mass thickness differences in specimen.
(captured by the aperture)
– Generated by the loss of scattered electrons
– Mass-thickness differences are introduced/enhanced by the addition of heavy atom
stains
–  Phase contrast
– Generated by the interaction of coherently scattered and unscattered waves
• Same principle as diffraction
– Also depends on mass thickness differences
– Most important form of contrast in unstained
•
biological specimens
Note: Inelastic electrons also give rise to scattering contrast, but are focused in a different plane
due to different energy.
Scattering cross-section
• simulated paths of electrons"
through 100nm Cu and Au"
• Note: more scattering for "
higher Z (Au)"
• Mean free path as "
Fuinction of accelerating"
Voltage"
• Higher voltage: better"
penetration"
Scattering depends on:
– sample thickness
– atom number (Z)
– acceleration voltage
See simulation at
http://www.matter.org.uk/tem/electron_scattering.htm
Mass-thickness contrast
• Areas of greater thickness or higher density scatter more electrons
TEM mass-thickness contrast"
• Areas of higher mass thickness scatter
electrons more than others"
"
• Some of these scattered electrons are
captured by the aperture and lost from the
beam path"
"
• Areas of higher mass thickness will
therefore appear dark in the image"
"
• This is known as mass thickness
contrast, scattering contrast, aperure
contrast or amplitude contrast! "
Mass-thickness contrast
Biological samples and contrast
•  Biological material consists mostly of low Z atoms (C, N, O) that
scatter electrons to a similar extent
•  Thickness and density differences (which give rise to massthickness/scattering contrast) are also quite small
•  Therefore, scattering contrast in biological specimens is very low
•  Solutions:
1.  Introduce more highly scattering (high Z) atoms, such as Au, U or W
2.  Use phase contrast
•  Most common method for particulate samples: negative staining
–  Sample is embedded in a heavy atom staining solution and air dried.
•  Results in a heavy atom staining pattern that reflects object structure
Staining of biological specimens
• Staining increases the image constrast by introducing heavy atoms
(higher Z) that scatter electrons more strongly
• Negative staining results in a heavy atom staining pattern that is
excluded by and reflects the object structure
A"
B"
A: Positive staining
Specimen will appear dark against light background
B: Negative staining
Specimen will appear light against dark background
C: Incomplete negative staining
D: Flattening due to dehydration
C"
D"
Negative staining and scattering contrast
•
•
Negative staining results in a heavy
atom staining pattern that reflects the
object structure
Heavy atoms give rise to scattering
contrast
–  due to loss of highly scattered electrons
outside aperture
–  phase contrast is also important at high
resolution
•
Solves many problems:
1.  Water: gone
2.  Radiation damage: stain is radiation
resistant
3.  Contrast: heavy atoms introduce
scattering contrast
A"
B"
C"
D"
lens
BFP
image
Negative staining of bacteriophages
virions
procapsids
P2
(stained with 1%
uranyl acetate)
P4
Chang et al 2008 Virology 370, 352-361
Effect of aperture
No Ap.
40µm
• Smaller aperture gives higher contrast because more electrons
are removed from the beam.
• Lower kV also gives higher contrast
Types of contrast in TEM
•
The TEM image is produced mainly by the elastically scattered electrons.
•
These give rise to contrast by two main mechanisms:
–  Scattering contrast (mass-thickness contrast)
– Due to elastic interactions with atomic potential field
– Dependent on mass thickness differences in specimen.
(captured by the aperture)
– Generated by the loss of scattered electrons
– Mass-thickness differences are introduced/enhanced by the addition of heavy atom
stains
–  Phase contrast
– Generated by the interaction of coherently scattered and unscattered waves
• Same principle as diffraction
– Also depends on mass thickness differences
– Most important form of contrast in unstained
•
biological specimens
Note: Inelastic electrons also give rise to scattering contrast, but are focused in a different plane
due to different energy.
Electron
scattering
and
diffraction
Scattering and diffraction
• When the incident wavefront hits an atom it
sets up a secondary radially scattered wave.
• The radially scattered wave represents a
small perturbation of the incident wave/
Incident wave
Scattered wave
Interference of
scattered waves:
diffraction
• The resultant wave can be seen as a
combination of the incident wave and a π/2
phase shifted wave
• Scattered waves from several atoms may
interact constructively or destructively to form
an amplitude difference (=diffraction pattern).
This is too weak to observe except for crystals.
• The phase distribution pattern also carries
information about the specimen; however, it is
not directly observable.
See simulation at
http://www.falstad.com/ripple/
Coherence and monochromaticity
• Monochromatic illumination = all radiation has the same wavelength"
"– Important because electrons at different
wavelength are focused in different points
"
• Coherence (spatial coherence) means that the phase
of a wave is the same at different points, i.e. the wavefront arrives at the same time at different points in the x-y plane"
"– Lasers are coherent"
"– Use very small point source"
• Phase contrast depends on coherent illumination"
• For pure scattering contrast imaging, coherence is not important."
Condensor lenses and coherence"
Tungsten filament
FEG tip
• The source (electron gun) is not a perfect point, but has a finite size = incoherent"
"
• By using a highly excited (strong) C1 lens and selecting a small cone, coherence
is improved"
"
• The FEG source is much smaller and more intense than the tungsten or LaB6
sources = more coherent"
Diffraction
of light
from
a double slit
Diffraction
from
a periodic
specimen
(double slit)
Phase difference 0
Waves reinforce
Phase difference !/2
Waves cancel out
Angles θ at which waves reinforce
are given by Bragg’s law:
Phase difference !
Waves reinforce
Phase difference 3!/2
Waves cancel out
nλ = 2d sin θ
See simulation at
http://www.falstad.com/ripple/
Scattering angle and spatial frequency
Diffraction of light from a double slit
•  Any periodic function can be mathematically
Phasedescribed
difference 0
as a sum of sine waves
Waves reinforce
•  Each wave has a spatial frequency
(=resolution) that corresponds to a particular
spacing (ν = 1/d) Phase difference !/2
Waves cancel out
NOTE: do not confuse this wave with the
electron wave (with wavelength λ)
λ
•  Each spatial frequency ν (=spacing d) gives
rise to a wave scattered at a specific angle θ:
Phase difference !
Waves reinforce
sin θ ≈ θ = λ / d = λν
θ
d
Phase difference 3!/2
Waves cancel out
•  This is equivalent to a Fourier transform of the
object function F(θ) = FT {f(x)}
φ=0
φ=90°
F
F=|F|eiφ = |F|(cosφ + isinφ)
φ=180
How do we generate phase contrast?
Scattering and diffraction
•
•
•
•
•
•
Biological specimens scatter electrons weakly, thus
amplitude differences are very small
The specimen imposes a mass-thickness dependent
phase shift on the incident beam, which also carries
information about the specimen
This phase shift cannot be observed directly
In the light microscope, a phaseplate is inserted in
the beam path, which adds an extra phase shift to
the direct beam
In EM, phase plates are difficult to make
Instead an angle-dependent phase shift is imposed
on the beam in the focal plane by imperfections of
the lens (Cs) and defocus
–  Path differences in the scattered and unscattered beam leads
to amplitude reinforcement at specific angles
Incident wave
Scattered wave
Interference of
scattered waves:
diffraction
Generating phase contrast
• Scattered waves have a 90° phase shift
• The phase shift is not directly
observable
• The resultant amplitude change is very
small
• By adding an extra 90° phase shift to
the scattered waves, the phase shift is
converted to an observable amplitude
difference
• Extra phase shift is applied by
defocusing the objective lens
How do we generate phase contrast?
•
•
•
•
•
•
Biological specimens scatter electrons weakly, thus
amplitude differences are very small
The specimen imposes a mass-thickness dependent
phase shift on the incicent beam
The phase shift cannot be observed directly
In the light microscope, a phaseplate is inserted in the
beam path, which adds an extra phase shift to the direct
beam
In EM, phase plates are difficult to make
Instead an angle-dependent phase shift is imposed on
the beam in the focal plane by imperfections of the lens
(Cs) and defocus
Lenses and lens aberrations
•
•
•
A perfect lens recombines all waves scattered from a single point into a
single point
Lens aberrations lead to a reduction of the attainable resolution because
all the scattered electrons are not focused into the same point
But it also results in a relative phase shift that gives rise to phase contrast!
•  Spherical aberration Cs
– electrons scattered at different
angles are focused in different
points
• Cs limits the attainable resolution from
d ≈ 0.61 x λ / β
(Rayleigh criterion)
to
d ≈ 0.91 x λ3/4 x Cs1/4 !
known as the point-to-point resolution (at
Scherzer focus = optimal defocus)
– Typical value for Cs = 1–4 mm
• Computationally, information can be recovered
beyond this limit
Phase contrast
• The lens applies a phase shift χ, dependent on scattering angle θ
2$ 1
" (# ) =
( 2 &f# 2 + 14 Cs# 4 )
%
– where Cs is the spherical aberration and Δf is the defocus. λ is the electron wavelength
– NOTE that Δf<0 means defocus (under focus), Δf>0 is over focus
• The angle θ is simply related to the spatial frequency
so that
!
!
2
1
2
2
4
" (# ) = $%(&f# + Cs% # )
– note that ν = 1/d, where d is a distance measure (e.g. nm)
– at low resolution (e.g. >1 nm), Cs has little effect
!
θ ≈ λν
• How does this phase shift affect contrast?
Phase contrast
2
1
2
2
4
" (# ) = $%(&f# + Cs% # )
• The phase shift χ(ν) leads to contrast enhancements at specific
resolutions ν, commonly given by sin(χ(ν)), known as the Phase Contrast
Transfer Function (CTF)
!
– an additional amplitude component is given by cos(χ(θ)), the Amplitude
Contrast Transfer Function
– The CTF is often modeled as
2
2
! cos( " (# )) + (1! ! ) sin( " (# ))
(or something similar) in which α is the fraction of amplitude contrast
(typically around 5-10% for unstained cryo-EM samples)
• What does this function look like?
Generating phase contrast
• Scattered waves have a 90° phase shift
• The phase shift is not directly
observable
• The resultant amplitude change is very
small
• By adding an extra 90° phase shift to
the scattered waves, the phase shift is
converted to an observable amplitude
difference
• Extra phase shift is applied by
defocusing the objective lens
View of CTF curve at the aperture
Curve shows the degree of contrast formation at different scattering
angles θ at Scherzer focus ( optimal defocus)"
"
"sin χ represents phase contrast"
"cos χ represents amplitude contrast"
""
NOTE: the angle θ corresponds to a frequency ν (nm-1) "
and a corresponding spacing d (nm)
" θ ≈ λν = λ/d
Spence 2003
Contrast transfer functions
ACTF for in-focus,
perfect lens
PCTF for in-focus,
perfect lens
ACTF in- focus,
for lens with Cs =
4mm
PCTF defocus=67.5nm,
Cs=1mm (Scherzer focus)
PCTF in-focus,
Cs=4mm
PCTF defocus 500nm,
Cs=1.3mm
From Slayter and Slayter
Effect of focus
over
Scherzer
focus
under
Effect of defocus
Unstained cryo-EM sample of HCRSV (plant virus)
defocus 2.4 µm
defocus 0.9 µm
Fourier theory of imaging
Incident beam ψ0
ψ0 = 1
Specimen = ρxyz
φxy = ∫ρxyzdz
ψs ≈ 1 – iφ(x,y)
Specimen plane ψs
BFP =
diffraction plane
ψf = F{ψs} = δ(0) – iΦ(u,v)
ψf
CTF = exp[iχ(u,v)]
= cosχ(u,v) + isinχ(u,v)
ψf = ψf x CTF x A(u,v) x E(u,v)
≈ δ(0) – iΦ(u,v)sinχ(u,v)
Image plane
ψi
• Strictly speaking, cosχ(u,v) also comes in here
• Also, let s forget about A and E for the time being…
ψi = F{ψf } = 1 – iφ(x,y)*F{sinχ(u,v)}
I(x,y) = ψi2 = [1 – iφ(x,y)*F{sinχ(u,v)}]2 ≈ 1 – 2φ(x,y)*F{sinχ(u,v)}
F(u,v) = F{I(x,y)} = δ(0) – 2Φ(u,v) x sinχ(u,v)
CTF correction
!
F(u,v) = F{I(x,y)} = δ(0) – 2Φ(u,v) x sinχ(u,v)
• This equation tells us that – to an approximation – the FT of the image F(u,v) is
equal to the FT of the specimen projection Φ(u,v) multiplied by sinχ(u,v),
where sinχ(u,v) is the CTF.
• Therefore, if we know the CTF, all we need to determine the specimen projection
structure is to divide F(u,v) by sinχ(u,v) to get Φ(u,v). Then apply the inverse
FT to Φ(u,v) to get φ(x,y), the specimen function.
— Right?
Problem 1: first we have to figure out what sinχ(u,v) looks like
Problem 2: Division by zero!
Problem 3: Exponential falloff E(u,v)
Envelope functions
Instrument-dependent Resolution limitations:
1.  CTF, i.e. sinχ(ν) and cosχ(ν)
2.  Aperture A(ν)
3.  Exponential falloff E(ν) caused by
– Beam divergence
– Inelastic and multiple scattering
Exponential falloff E(ν) = exp[–ε(ν)],
where ε(ν) is an absurdly complex function of beam
divergence α, Cc, Cs, Δf and instrument instabilities
A simple version is
Envelope functions
Beam divergence angle
( $C %2" 3 # $&f" 2 ' 2 +
( s
) E (" ) = exp*#
*
ln2
,
)
( 0
1/ 2 3 2 +
2
2
2
0
3
* 2 $%" 2C / (V0 ) + 4/ ( I0 ) + / ( E 0 ) 5 c2
2
2
2 5
* 2
V
I
E
1 0
4 50
0
.exp*#2
5
4 ln2
* 2
55* 2
4,
) 1
Normally this decay function is modeled empirically, e.g. as
E(ν) = exp[–Bν2]
!
where B is the B factor (temperature factor) (typically B ≈ 100-500 Å-2)
The exponential decay gives rise to the information limit of the
instrument (which is different from the point-to-point resolution )
CTF without any falloff at high res (sin χ)
envelope function exp[-Bv2]
Information limit
First zero = point-to-point resolution
• At more defocus, the envelope function also falls off
more rapidly
• This is especially important in biological TEM
because a high defocus is used to produce phase
contrast
• An FEG has better coherence, i.e. less beam
divergence, so the envelope function is better at the
same defocus
Fourier transform of noise
Measuring the contrast transfer function
• The CTF can be observed by looking
at the FT of the image, especially in the
amorphous background"
"
• The FT is essentially equivalent to the
diffraction pattern of the object multipled
by the CTF"
Astigmatism
• Astigmatism is due to
imperfections in the imaging
system, causing the focus to
vary in different directions, and
resulting in an image that
cannot be focused"
"
• Astigmatism can be corrected
for by magnetic or electrostatic
deflectors in the microscope"
Resolution limitations in EM
•
•
•
•
•
Electron wavelength λ = 0.04Å
Lens imperfections Cs, Cc
Beam divergence
Defocus Δ
Contrast transfer function
Image = [Specimen projection] * PSF
ℑ{image} = ℑ{specimen projection} x CTF
CTF = [m cosχ(α) + (1-m)sinχ(α)] e-ε
χ(α) = πλ (Δ α2 – Cs λ2α4/2)
ε is a function of Δ, Cs, Cc and beam divergence. α is the spatial
frequency
Amplitude contrast m < 0.1
•
•
•
•
Drift
Detector imperfections
Signal-to-noise ratio
The specimen
Summary: contrast formation in EM
•
•
•
Contrast is what makes the specimen visible by microscopy
Contrast is formed primarily by the elastic interactions of the electrons with
the atoms in the sample
Two main mechanisms:
–  Scattering contrast – usually generated by heavy atom staining
–  Phase contrast – main mechanism in unstained biological samples (>90%)
•
•
•
Scattering contrast is formed by the loss of highly scattered electrons
outside the objective aperture
Phase contrast is formed by the constructive interaction of the scattered
wave and the unscattered wave and is generated mainly by defocusing (Δf)
the objective lens
Phase contrast is normally given as sin( ! (" ))
where
" (# ) = $%(&f# 2 + 1 C %2# 4 )
2
•
!
s
The envelope function represents an exponential falloff resulting from beam
divergence, current and voltage instabilities and lens imperfections and
defines the ultimate resolution limit (information limit)