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

Transfer Function

H(x) is also called the “transfer function” of the system

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

Ideal CTF

0, 02, 0

uT u

u

A u E u 1Assume 2sinT u u

0, 0

2, 0u

uu

T(u)>0 positive phase contrast

Real CTF: (case 1: Cs=0)

2u f u

f < 0: underfocusingf > 0: overfocusing

Real CTF: (case 2: f=0) 3 41

2 su C u

Contributes to negative contrast

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

Phase At Scherzer defocus2.0 mm

125 KeVsC

E

CTF at Scherzer defocusE=125 keVCs=2.0 mm

93.4 nmschf

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:

PassbandsWe could make the phase stationary at higher u:

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:

Image of periodic specimen (III)

0g gF u F u g F u F u g

0g gG u G u g G u G u g

Three-beam case:

Example

1 sin 2F x iB gx

2 2 2 20e 1 e e e

2 2igx igx igx igx

g gB BF x F F F

1A u 0, 2

2, 2

u gu

g u

2 2e 1 e 1 sin 22 2

igx igxiB iBG x B gx

Given:

Write: