Linear Systems
x
G x F x H x x dx F x H x
-
Describes the output of a linear system
: object
: image
F x
G x
If the microscope is a linear system:
G x S F x
x
H x x H x x dx
-
Impulse response function
inputoutput
Microscope as a Linear System
G x F x H x
F u F x
G u G x
H u H x
G u F u H u
Convolution in direct space:
Fourier Transforms:
→Multiplication in reciprocal space G x
H x
F x
Convolution Theorem:
Contributions to H(u)
e
: aperture function
: envelope (damping) function
: aberration (phase) function
i uH u A u E u
A u
E u
u
Most important for phase-contrast imaging:
We need to find (u):
Path Length Correction due to Lens
2 2 2 2
2 2
2
2
0
2 2
1 12
2
r
r
s p r q r s r
r rp q s rp q
rp q s rp q
rs p q s rf
0rs s p q 2
02rs rf
We expect the same net path length for all focused rays:
0
1 1 1p q f
Ideal lens:
Snell’s Law
1 1 1
2 2 2
v f c nv f c n
1 2
1 2
sin sin
1 2
1 2
sin sincf
1 1 2 2sin sinn n
The wave crests must be continuous across the interface:
This applies to any type of wave(not only light).
refractive indices
Thin Optical Lens (I)1 1sin sinn
2 2sin sinn 1 2 2
2
1
0
tan rf
0
2 tan 21
dt rdr n f
Thickness change:
2 2
2 1 2 p
r rt rn f f
1 1
2 2
nn
2 12 2 1n n n
02 1
rn f
2
00 01 2 1
r
r
r dr rt r T T T t rn f n f
0 1pf
fn
Optical focal Length:
parabola focal length
air
glass
air
Snell's law:
Small angles:
Thin Optical Lens (II)Wavelength:
Phase difference:
Optical path-length difference:
0
0
vacuum:
medium:
c fc n
nf
0
0
0
1 12
12
2
r T t r
n t r
s rr
off-axis rays have a greater distance to travel
lens thinner at edges
01d t r
dr n f
1s r n t r
0
d s rdr f
2
00 02
r
r
r dr rs rf f
Uniform-Field Electron Lens
off-axis rays have a greater distance to travel
less momentum, longer wavelength
Path Length Correction due to LensA path-length correction for each trajectory through the lens should be included.
2
200 02
r
r
r f rrs r drf f
0f r f f r
0 0 0
1d s f rr r
dr f f r f f
0
d s r rdr f f r
Generalize:
The derivative of the path-length difference determines the focal length.
Non-ideal lens:
Influence of Spherical Aberration
22
200 0 0
42
0 0
2
12 4
r
sr
s
r r rs r C drf f f
r rCf f
2
0s
rf r Cf
From Chap. 6:
normal focusing aberration
Spherical aberration (wave optics)
3
40
43
4 400 04
s
rs s
r
C rds dr drf
C C rs r r drf f
3
40
sC rf
dsdr
When object is at focal point, if no aberration,all rays on image side (back) of lens will be parallel to lens
With aberration, off-axis rays will be tend toward the optic axis
Path-length difference: general case
Off-axis Ray:
2 2 2 2
2 2
0
2 2
r
r net
s z r q r s r
r rz q s rz q
s s s r
0s z q Axial Ray:
Assume object is on-axis,but not in focus.
42
0
1 1 1 12 4net sr rs r C
z q f f
42
0
12 4 sr rs r Cf f
Lens:
0
1 1 1p q f
Combine defocus and sample heightp z z
0f f f
0
0
1 1 1 1 1 1
1 1 1
z q f p z q f f
p q f
2 20
2 20
20
1 1 1
z fp f
z fp fz f
z q f f
0p f
only the difference matters
0 00 0
1 1q q qM f qp f f
Optical path-length difference
Net Effect:
42
0 0
1 12 4net s
r rs z f Cf f
2 41 1θ θ2 4 ss z f C
r z
42
2 4
0 0
1 1θ θ2 4 s
z zs z f Cf f
z z 0
0 0
1 11
f qf f q
0z f
Phase correction
2 4122 s
s f z C
R uL d
2 3 4
2 su f z u C u Frequency representation:
Phase shift:
Scattering angle:
Image function for weak-phase object
*
1 t
G x F x H x
G x i V x H x
A weak phase object produces an image function:
10
1
u
H x u H u
u H u H u
1 tG x i V x H x
The overall phase doesn't matter, so we might as well pick
0 1uH u
First term:
Simple situation
Intensity:
This gives no contrast to first order in
H x x
We might expect the ideal transfer function to be:
2 21 tI x G x V x 1
1 * 1t tG x i V x H x i V x
We will need a relative phase shift between the incident and scattered beams to see phase contrast
2
1 1
1
1 2 Im
1
t t
t
t
t
I x G x
i V x H x i V x H x
i V x H x H x
V x H x
I x V x T x
Intensity:
2ImT x H x
Contrast transfer function
CTF in reciprocal space
*
2 * 2 2
2 * 2
*
*
2 Im
e e e
e e
iu x iu x iux
x u
i u u x i u u x
u x x
u
T u T x
H x
i H x H x
i dx du H u H u
i du H u dx H u dx
i du H u u u H u u u
T u i H u H u
1 tI x V x T x
1 t tI u V x T x u V u T u
Special Case: H(u) is an even function
* *
H u H u
H u H u
Ifthen
*
*
2 Im
T u i H u H u
i H u H u
T u H u
Using the CTF
2Im 2 sinT u H u A u E u u
1
2 sin
t
t
t
I u V x T x
u V u T u
I u u V u A u E u u
expH u A u E u i u
H u H u Assume:
1, 1e
, 1i u u b
H ui u b
0, 1
, 12
u bu
u b
0, 1
2Im2, 1
u bT u H u
u b
10u 0u
H u
Ideal transfer function for phase object (I)
Ideal transfer function for phase object (II)
Re H u
Im H u
H u
Re H x
Im H x
H x
2 1 sinc 2H x i x i x b
causes relative phase change for
1, 1e
, 1i u u b
H ui u b
2 b 2b
1u b
Real CTF: (general)
Underfocus gives a region with constant phase, positive contrast
2 3 412 su f u C u
Scherzer Defocus 3 32 2 s
d uf u C u
du
Stationary Phase:
umin 23
12Cs
3umin4
umin 4
3Cs3
1 4
fsch 43
Cs
1 2
1.2 Cs 1 2
We need to pick what phase we want stationary:
This choice gives a relatively constant CTF.
min
2 2min0 sch s
u u
d uf f C u
du
min min max32sin 3
2T u u T min
3sin2
u
Resolution at Scherzer defocus
dsch 1
usch
0.66 Cs3 1 4
sin usch 0 usch 0
0 fsch usch2
12
Cs3usch
4
fsch 43
Cs
1 2
usch 2 3Cs3 1 4
1.51 Cs3 1 4
we already know:
Damping due to temporal incoherence
2 2 2 4exp 2cE u u
2 2
cE ICE I
EE
: Variation in energy
II
: Variation in objective-lens current
chromatic aberration coefficient
Beam convergence: one-lens condenser (I)
overfocusunderfocus
A range of illumination angles may be incident on each sample point.The illumination semiangle s is not the same as the beam convergence angle .
Beam convergence: one-lens condenser (II)
1s rr Hz z
r
rp
max
max
,
, otherwise
r r ss
s
r rz z
Semi-angle of illumination at a point on the specimen:
Illumination semi-angle in lens plane:
max as
RH
maxs s Max. illumination angle limited by aperture:
a ar
a
R Rq q
Semi-angle of convergence:
11 1a
a
qr
q R p
The limiting rays cross the axis at:
Beam convergence: one-lens condenser (III)
maxs
Large overfocus is usually better than underfocus.For underfocus, s is never less than r .
r
location of probe image: radius of probe image: convergence semi-angles:
CL excitation
crossoverparallel
abovebelow
Beam convergence: two-lens condenser (I)
2
1 21 1
rs
rL Hzq q
1r r p
max
2
1 1 1a
sR
H LL f H
max
2 2
max
,
, otherwise
r ss
s
r rz z
Semi-angle of illumination at a point on the specimen:
Beam convergence: two-lens condenser (II)Two-lens condenser system:
Allows small, parallel beam on sample.Parallel beam formed when probe image formed by CL is at front focal point of CM.Highly overfocused CL gives small illumination semi-angle on a sample point.
nearly parallelCL slightlyoverfocused
underfocus(CL off)
nearly optimal(CL highly overfocused)
Beam convergence: two-lens condenser (III)
location of probe images: radii of probe images: convergence semi-angles:
CL excitation
crossover parallel
abovebelow
Beyond crossover, s decreases with increasing CL3 strength ("Brightness").Parallel beam formed with CL3 overfocus to specific value; good for diffraction.Probe size increases with increasing CL3 strength below crossover.
Damping due to beam convergence (I)
0 0 0G u F u H u
2
1 e s
uk
s
S uk
For a non-parallel source, we have to integrate over u:
Gaussian spot profile:
There are two distinct ways to combine the incidence angles: coherently and incoherently
0u
I u du I u u S u
0u
G u du G u u S u
*0 0 0u
I u du G u u G u
Assuming a parallel beam:
Intensity:
coherent:
incoherent:
Damping due to beam convergence (II)
2
1 e e s
uki u u
us
G u du F u uk
0 e i uG u F u Assume only effect is a phase shift :
Let's assume the different incidence angles combine coherently.
2
1 e e s
uki u u
us
H u duk
1F x x F u H x S x
If the input is a delta function, the output is the transfer function:
Damping due to spatial incoherence (II)
2
exp2
ss
kE uu
2
2
2
1e e e
1e cos e
e exp e2
s
s
uki u iC u u
us
uki u
us
i u i uss
H u duk
du C u uk
kH u C u E u
Do the integral:
Spatial incoherence is represented by a damping function:
u u
uu u u u u u C uu
Assume ks is small.
Combine incoherence effects
c sE u E u E u
We can combine temporal and spatial incoherence functions:
Uniform-Field Electron Lens (I)m q p v A
ˆ ˆzB B B A ρ z
0 0
2 2B BB z a z a
0zB B u z a u z a
0 ˆ2B u z a u z a
A
//canonical momentum
1 1ˆˆ ˆz zA A AA A Az z
A z
//magnetic vector potential
//magnetic field in axially symmetric lens
//uniform-field model
ˆˆ ˆz v ρ z
0
0
0 0
ˆsinˆcos
ˆ
ˆcos2
z
L
z
m q m em k v k z a
m k z am ve B k z a
p v A v Aρ
z
L 0 cos k z a
0 sinzk v k z a
0
0
ˆsinˆcos
ˆ
z
L
z
k v k z a
k z av
v ρ
z
zz v
Uniform-Field Electron Lens (II)
L
z
kv
0
2LeB
m
0 ˆ ˆsinzm v k k z a p ρ z
220z z Lv v
0 z a Assume:
Uniform-Field Electron Lens (III)
ˆˆ ˆd d d dz r ρ z
e
ˆ
ˆ=
2= 2
1
f
i
f
i
i
hi
hh
d nh
n dh
r
r
r r
r
r r
r
p
p r rr p
r p r r
r p r
0 ˆ ˆsinzm v k k z a p ρ z
0 sinzd m v k k z a d dz p r
//electron wave function
//momentum operator
//momentum proportional to gradient of phase
// phase change
//momentum in uniform-field lens
//needed to find phase change
//change in number of wave fronts
Electron Lenses (III)
0
2 2, 0 sin 1
zz
z a
m vn z k k z a dzh
0 sind k k z a dz
2 20 sin 1zd m v k k z a dz p r
hm v
zz
hm v
21z zz
h um v
0u k 0 0L z zk v u v 21z zv v u
0
2 2, 0 sin 1
z
zz a
n z k k z a dz
//differential change
//# of wave fronts along z
Uniform-Field Electron Lens (IV)
0
0
2 2, 0
20
2 20 0
,
1 sin 1
1 sin 22 4
1 1 sin 2 12 4
k zz
x k ak z
x k a
z
n z k x k a dxa k a
x k a x k ak xk a
k kz zn z k aa a k a a
sin 2
22
0 2
tan ii k
1 tantan i
ik
: radius at i z a : angle w.r.t. optic axis at i z a
x k z
Uniform-Field Electron Lens
0i 0i
2 1u
2
1 sin 2 14
zu
z u zn z kaa a ka a
Expand to lowest order in u:
2
, 1 sin 2 1 sin 24
zu
z u zn z kaa a ka a
In this limit, the lens has no spherical aberration
Image of periodic specimen (I)
2e igxg
gF x F
2e igxg
g
gg
F u F xF
F u F u g
e i uH u A u
e i ug
gG u F u g A u
2
2
2
lime
lime e
e e
K iux
u K
K i u iuxg
u Kg
i g igxg
g
G x G u duK
F u g A u duK
G x F A g
Object function:
Transfer function (no attenuation):
Image function:
Image of periodic specimen (II)
2
2 2
2 2
2
lime
lime e e
lime e e
lime e
e
L iux
x L
L i g igx iuxg
x Lg
L i g igx iuxg
x Lg
Li g i g u xg
x Lg
i gg
g
gg
G u G x dxL
F A g dxL
F A g dxL
F A g dxL
F A g u g
G u G u g
e i g
g gG F A g
2e igxg
gG x G
FT of image function: