linear imaging approximations4 - cornell...

David Muller 2006 1

Linear Imaging Approximations

Notes to accompany the lectures delivered by David A. Muller at the SummerSchool on Electron Microscopy: Fundamental Limits and New Science heldat Cornell University, July 13-15, 2006.

Reading and References:Chapter 1-3 of “Advanced Computing in Electron Microscopy” by E. J. Kirkland

David Muller 2006 2Single atom

Sensitivity:

Electron Energy Loss Spectrometer

Annular Dark Field (ADF) detector

yx

200 kV IncidentElectron Beam

(∆E=1 eV)

Incr

easi

ngen

ergy

loss

1 atom wide (0.2 nm) beam is scannedacross the sample to form a 2-D image

Elastic Scattering ~ "Z contrast"

Scanning Transmission Electron Microscopy

0 0.5 1 1.

ADF Signal Er M4 Edge

Distance (nm)

3 Å

P. Voyles, D. Muller, J. Grazul, P. Citrin, H. Gossmann, Nature 416 826 (2002)U. Kaiser, D. Muller, J. Grazul, M. Kawasaki, Nature Materials, 1 102 (2002)

David Muller 2006 3

ReciprocityReciprocity (or STEM vs. CTEM)(or STEM vs. CTEM)

CTEM STEM

Reciprocity (for zero-loss images):A hollow-cone image in CTEM an annular-dark field image in STEM.

Specimen

Illuminationangle

α

Collectorangle

β

Specimen

Objective Aperture

Objective Aperture

ViewingScreen Gun

Gun Detectors

However: In STEM, energy losses in the sample do not contribute to chromatic aberrations (Strong advantage for STEM in thick specimens)

David Muller 2006 4

Reciprocity

From L. Reimer, Transmission Electron Microscopy

Reciprocity: Electron intensities and ray paths in the microscope remain the same if (i) the direction of rays is reversed, and (ii) the source and detector are interchanged.

Proof follows from time-reversal symmetry of the electron trajectories and elastic scattering (to all orders).

Reciprocity does not hold for inelastic scattering:Sample is after probe forming optics in STEM - energy losses in sample do not cause chromatic blurring in the image

Sample is before the imaging optics in TEM – energy losses in the sample do cause chromatic blurring in the image. Imaging thick samples in TEM can be improved by energy filtering (so on the zero-loss image is recorded). This is not needed for STEM.

David Muller 2006 5

Reciprocity

From L. Reimer, Transmission Electron Microscopy

Image recordedIn parallel

Image recorded serially by scanning the source

Condensoraperture(before sample)

Objective aperture

(after sample)

Controls coherence

Controls resolution

Collectoraperture

Condensoraperture

STEMTEM

David Muller 2006 6

Geometric Optics Geometric Optics –– A Simple LensA Simple Lens

x x

Objectplane

imageplane

Backfocalplane

frontfocalplane

Lensat z=0

α

Focusing: angular deflection of ray α distance from optic axis

David Muller 2006 7

Geometric Optics Geometric Optics –– A Simple LensA Simple Lens

x x

Objectplane

imageplane

Backfocalplane

frontfocalplane

Lensat z=0

α1

α1

Wavefronts in focal plane are the Fourier Transform of the Image/Object

α1

David Muller 2006 8

α

λ1

0 =kλαrr

≈k

{ } xdexfxfkF xki rrrr rr 2.2)()()( π∫∞

∞−==F

All plane waves at angle αpass through the same pointIn the focal plane.

If the component of the wavevectorin the focal plane is ,thena function in the back focal plane, F(k)is the Fourier transform of a function in the image plane f(x)

kr

Fourier Transforms

{ } sdekFkFxf xki rrrr rr.21 )()()( π−∞

∞−

− ∫==F

Forward Transform: (image- > diffraction)

Inverse Transform: (diffraction ->image)

Note: in optics, we define k=1/d, while in physics k=2π/d

David Muller 2006 9

FT of A Circular Aperture

⎪⎩

⎪⎨⎧

>

<=

max

max

,0

,1)(

kk

kkkA r

rr

FT

rkrkJrA

max

max1

2)2(2)(

πππ=

An ideal lens would have an aperture A(k)=1 for all k. However, there is a maximum angle that can be accepted by the lens, αmax, and so there is a cut-off spatial frequency kmax= k0 αmax,

Focal PlaneImage Plane

David Muller 2006 10

The Aperture Function in the Image Plane

0

1

2

3

-3 -2 -1 0 1 2 3

A(x)

x

rkrkJrA

max

max1

2)2(2)(

πππ=

First zero at 2πkmaxr = 0.61

0

4

8

-3 -2 -1 0 1 2 3|A

(x)|2

x

2)(rA

The image of single point becomes blurred to (why?) 2)(rA


Linear Imagingf(x) g(x)h(x)

imageobject Blurring function

( ) )()(')'()'( xhxfdxxxhxfxg ⊗≡−= ∫∞

∞−

Image = object convolved (symbol ) with the blurring function, h(x)⊗

(Kirkland chapter 3)

( ) )()( kHkFkG =In Fourier Space, convolution becomes multiplication (and visa-versa)

Point spread function

Contrast Transfer Function


Contrast Transfer FunctionThe contrast transfer function (CTF) is the Fourier Transform of the point spread function (PSF)

The CTF describes the response of the system to an input plane wave.By convention the CTF is normalized to the response at zero frequency (i.e. DC level)

( ))0()(~

HkHkH =

A low-pass filter

k

( )kH~

A high-pass filter

k

( )kH~

(division by k in Fourier Space->integration in real space)

(multiplication by k in Fourier Space->differentiation in real space)

smoothing Edge-enhancing


High-pass filter

Edge-enhancing

Original image

A high-pass filter

Final image

Fourier Transform

x

Inverse Transform

=

David Muller 2006 14smoothing

Original image

A low-pass filter

Final image

Fourier Transform

x

Inverse Transform

=

Low pass filter


The Aperture Function in the Image Plane

0

1

2

3

-3 -2 -1 0 1 2 3

A(x)

x

rkrkJrA

max

max1

2)2(2)(

πππ=

First zero at 2πkmaxr = 0.61, with kmax = (2π/λ)θmax

i.e. r = 0.61/(2πkmax)~ 0.61 λ/θmax

0

4

8

-3 -2 -1 0 1 2 3|A

(x)|2

x

2)(rA

The image of single point becomes blurred to (why?) 2)(rA


ε

From Wyant and Creath

Adding a central beam-stopto the aperture increases the tails on the probe but only slighty narrows the central peak.

Is this the best we can do?


Coherent vs. Incoherent Imaging(Kirkland, Chapter 3.3)

Lateral Coherence of the Electron Beamfor an angular spread βmax (Born&Wolf):

max

16.0β

λ≈∆ cohx

Image resolutionmaxαλ

≈d

Combining these 2 formula we get:

Coherent imaging:

Incoherent imaging:

maxmax 16.0 αβ <<

maxmax 16.0 αβ >> (usually ) maxmax 3αβ >

Wave Interference inside allows us to measure phase changes

as wavefunctions add:

α

β

cohx∆

No interference, phase shifts are not detected. Intensities add

**222abbababa ψψψψψψψψ +++=+

22ba ψψ +

obj


Coherent Imaging(Kirkland 3.1)

Convolve wavefunctions, measure intensities

( ) )()( xAxx objectimage ⊗=ψψ

( ) ( ) 2xxg imageψ=

( )xobjectψ

( )kimageψ

Lens has a PSF A(x)

We measure the intensity, not the wavefunction

( ) )()( kAkk objectimage ⋅=ψψ

( ) 2)()( xAxxg object ⊗= ψ

α

β~0


Incoherent Imaging (Kirkland 3.4)

Convolve intensities, measure intensities

( ) 222)()( xAxx objectimage ⊗= ψψ

( ) ( ) 2xxg imageψ=

( )xobjectψ

( )kimageψ

We measure the intensity, not the wavefunction

( ) [ ] [ ])()()()( ** kAkAkkk objectobjectimage ⊗⋅⊗= ψψψ

( ) 22)()( xAxxg object ⊗= ψ

α

β Lens has a PSF2)(xA

Lost phase information, only work with intensities

CTF


Coherent vs Incoherent Imaging

Phase Object

Amplitude Object

Contrast Transfer Function

We measure

Point Spread Function

IncoherentCoherent

( ) 2)()( xAxxg object ⊗= ψ ( ) 22

)()( xAxxg object ⊗= ψ

)(xA

[ ])(Im kA

2)(xA

2* )()( kAkA ⊗[ ])(Re kA


Convolutions(from Linear Imaging Notes, Braun)


Resolution Limits Imposed by the Diffraction Limit(Less diffraction with a large aperture – must be balanced against Cs)

Lens

000

61.0sin61.0

αλ

αλ

≈=n

d

Gaussian image plane

The image of a point transferred through a lens with a circular aperture of

semiangle α0 is an Airy disk of diameter

0d

(for electrons, n~1, and the angles are small)

α0

(0.61 for incoherent imaging e.g. ADF-STEM, 1.22 for coherent or phase contrast,. E.g TEM)


Electron Wavelength vs. Accelerating Voltage

0.00871890.813521 MeV

0.0196870.77653300 keV

0.0250780.69531200 keV

0.0370130.54822100 keV

0.122040.01919410 keV

0.387630.0624691 keV

1.22630.0062560100 V

12.2640.00197841 V

λ (Ǻ)v/cAccelerating Voltage

0

0.01

0.02

0.03

0.04

0.05

0 200 400 600 800 1000

RelativisticNon-relativistic

λ (A

ngst

rom

s)

Electron Kinetic Energy (keV)


Resolution Limits Imposed by Spherical Aberration, C3(Or why we can’t do subatomic imaging with a 100 keV electron)

Lens

33min 2

1 αCd =

Plane ofLeast Confusion


C3=0

C3>0

For Cs>0, rays far from the axis are bent too strongly and come to a crossover before the gaussian image plane.

For a lens with aperture angle α, the minimum blur is

mind

Typical TEM numbers: C3= 1 mm, α=10 mrad → dmin= 0.5 nm


Balancing Spherical Aberration against the Diffraction Limit(Less diffraction with a large aperture – must be balanced against Cs)

2303

2

0

220

2

2161.0

⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=+≈ α

αλ Cddd stot

For a rough estimate of the optimum aperture size, convolve blurring terms-If the point spreads were gaussian, we could add in quadrature:

1

10

100

1 10

Prob

e Si

ze (A

ngst

rom

s)

α (mrad)

ds

d0

Optimal apertureAnd minimum

Spot size

4/34/13min 66.0 λCd =


Balancing Spherical Aberration against the Diffraction Limit(Less diffraction with a large aperture – must be balanced against C3)

4/34/13min 43.0 λCd =

A more accurate wave-optical treatment, allowing less than λ/4 of phase shift across the lens gives

Minimum Spot size:

4/1

3

4⎟⎟⎠

⎞⎜⎜⎝

⎛=

CoptλαOptimal aperture:

At 200 kV, λ=0.0257 Ǻ, dmin = 1.53Ǻ and αopt = 10 mrad

At 1 kV, λ=0.38 Ǻ, dmin = 12 Ǻ and αopt = 20 mrad

We will now derive the wave-optical case


Spherical Aberration (C3) as a Phase Shift

Lens


C3 =0

C3 >0

Phase shift from lens aberrations:

∆s = path difference due to wave aberration

Wavefronts (lines of equal phase)

( ) ( )αλπαχ s∆=

2

α

For spherical aberration but there are other terms as well ( ) 434

1 αα Cs =∆

(remember wave exp[ i(2π/λ) x] has a 2π phase change every λ)

David Muller 2006 28J.C. Wyant, K. Creath, APPLIED OPTICS AND OPTICAL ENGINEERING, VOL. Xl

An Arbitrary Distortion to the Wavefront can be expanded in a power series(Either Zernike Polynomials or Seidel aberration coefficients)

=)',( θρχ

Zernike Polynomials


An Arbitrary Distortion to the Wavefront can be expanded in a power seriesSeidel Aberration Coefficients (Seidel 1856)

Or more generally

EM community notation is similar:

C5,0 and C7,0 are also important


An Arbitrary Distortion to the Wavefront can be expanded in a power seriesHere are some terms plotted out


Phase Shift in a Lens

( ) ( )αχαϕ ie=

Electron wavefunction in focal plane of the lens

⎟⎠⎞

⎜⎝⎛ ∆−= 24

3 21

412)( αα

λπαχ fC

Where the phase shift from the lens is

Keeping only spherical aberration and defocus

(Kirkland, Chapter 2.4)

( )

⎪⎩

⎪⎨⎧

>

<=

max

max

,0

,)(

kk

kkekA

ki

r

rr

χ


-2π

-π

0

π

2π

3π

4π

0 2 4 6 8 10 12

Phas

e Sh

ift (r

ad)

α (mrad)

α0

π/2

χ(α)

1/4C3α4

1/2∆fα2

Goal is to get the smallest phase shift over the largest range of anglesOptimizing defocus and aperture size for ADF

Step 1: Pick largest tolerable phase shift: in EM λ/4=π/2, in light optics λ/10Step 2: Use defocus to oppose the spherical aberration shift within the widest π/2 bandStep 3: Place aperture at upper end of the π/2 band


Goal is to get the smallest phase shift over the largest range of angles

Step 1: We assume a phase shift <λ/4=π/2 is small enough to be ignored

Step 2: Use defocus to oppose the spherical aberration shift within the widest π/2 band

4/34/13min 43.0 λCd =

Minimum Spot size:

4/1

3

4⎟⎟⎠

⎞⎜⎜⎝

⎛=


opt

dα

λ61.0min ≈

Step 3: Place aperture at upper end of the π/2 band & treat as diffraction limited

Optimal defocus: ( ) 21

λSopt Cf =∆

(The full derivation of this is given in appendix A of Weyland&Muller

Optimizing defocus and aperture size for ADF


(derivation is different, given in Kirkland 3.1)

4/34/13min 77.0 λCd =

Minimum Spot size:

4/1

3

6⎟⎟⎠

⎞⎜⎜⎝

⎛=


opt

dα

λ22.1min ≈

Optimal defocus: ( ) 21

5.0 λSopt Cf =∆

(The full derivation of this is given in appendix A of Weyland&Muller

Optimizing defocus and aperture size for TEM

Look for a uniform phase shift of ±π/2 across lens


Phase Shift in a Lens with an Aberration Corrector

( ) ( )αχαϕ ie=

Electron wavefunction in focal plane of the lens

⎟⎠⎞

⎜⎝⎛ +++∆−= K4

54

32

61

41

212)( ααα

λπαχ CCf

Where the phase shift from the lens is

O.L. Krivanek et al. Ultramicroscopy 96 (2003) 229

5th order spherical aberration


-2

-1

0

1

2

3

4

0 5 10 15 20 25 30

Phas

e Sh

ift (r

ad)

α (mrad)

α0

π/2

χ(α)

-1/4C3α4

1/2∆fα2

1/6C5α6


Optimizing Aperture size with a C3 Corrector

Step 1: Pick largest tolerable phase shift: in EM λ/4=π/2, in light optics λ/10Step 2: Use defocus and C3 (now negative) to balance C5 within the widest π/2 bandStep 3: Place aperture at upper end of the π/2 band



Optimizing defocus and aperture size

Step 1: We assume a phase shift <λ/4=π/2 is small enough to be ignored

Step 2: Use C3 to oppose the C5 shift within the widest π/2 band

6/56/15min 42.0 λCd =

Minimum Spot size:

6/1

5

6/1

5

47.1323

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

CCoptλλαOptimal aperture:

opt

dα

λ61.0min ≈

Step 3: Place aperture at upper end of the π/2 band & treat as diffraction limited

Optimal C3: ( ) 312

533 CC opt λ−=


Contrast Transfer Functions of a lens with Aberrations

Coherent Imaging CTF: [ ] ( )[ ]kSinkA χ=)(Im

( )

⎪⎩

⎪⎨⎧

>

<=

max

max

,0

,)(

kk

kkekA

ki

r

rr

χ

Aperture functionof a real lens

Generated with ctemtf

0 0.1 0.2 0.3 0.4 0.5 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Spatial Frequency (in 1/A)

MTF

E= 200keV, Cs= 1.2mm, df= 600A, Beta= 0mrad, ddf= 0A

coherent

partiallycoherent


Contrast Transfer Functions of a lens with Aberrations

Incoherent Imaging CTF:

( )

⎪⎩

⎪⎨⎧

>

<=

max

max

,0

,)(

kk

kkekA

ki

r

rr

χ

Aperture functionof a real lens

Generated with stemtf

2* )()( kAkA ⊗

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.2

0

0.2

0.4

0.6

0.8

1

1.2

k in inv. Angstroms

MTF

Cs= 1.2mm, df= 400A, E= 200keV, OA= 10mrad, d0=0A

550A

900A

0A

CTF for different defocii

Theorem:Aberrations will neverIncrease the MTFFor incoherent imaging

David Muller 2006

Phase vs. ADF Contrast

TEM: Bandpass filter:low frequencies removed = artificial sharpeningADF : Lowpass filter: 3 x less contrast at 0.3 nm than HRTEM

-1

-0.5

0

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

STEMHRTEM

Con

tras

t Tra

nsfe

r Fun

ctio

n

k (Inverse Angstroms)

(JEOL 2010F, Cs=1mm)

David Muller 2006

Phase vs. ADF ContrastRandom white noise

ADF CTFBF CTF

a-C support films look like this

David Muller 2006

Phase vs. ADF Contrast

David Muller, 2000

ADF : 40% narrower FWHM, smaller probe tails

(JEOL 2010F, Cs=1mm, Scherzer aperture and focus)

0

0.1

0.2

0.3

0.4STEM-CTEM

PSF


CTF

PSF

(200 kV, C3=1.2 mm)

Effect of defocus and aperture size on an ADF-STEM image

David Muller 2006

ADF of [110] Si at 13 mr, C3=1mm

Strong {111} fringes Strong {311} fringes

2 clicks overfocus

Best 111 and 311 fringes occur at different focus settingsIf the aperture is too large

What happens with a too-large aperture?


Aperture Size is Critical(200 kV C3=1mm)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

13 mrad10 mrad

CTF

k (1/A){111}

Si

{220}Si

30% increase in aperture size ~50% decrease in contrast for Si {111} fringes


Aperture Size is Critical

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1

10 mrad13 mrad

Bea

m C

urre

nt E

nclo

sed

(pA

)

Radius (nm)

r80%

= 0.25 nm

r80%

= 0.8 nm

r64%

= 0.1 nm

All the extra probe current falls into the tails of the probe – reduces SNR

(200 kV C3=1mm)


Finding the Aperture with the smallest probe tails

(Kirkland, Fig 3.11)

4/1

3

22.1 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

Coptλα

( ) 21

min 8.0 λSCf =∆


Depth of Field, Depth of Focus

200

061.0

tan αλ

α≈=

dD

For d=0.2 nm, α=10 mrad, D0= 20 nm For d= 2 nm, α=1 mrad, D0= 2000 nm!

For d= 0.05 nm, α= 50 mrad, D0= 1 nm!


Depth of Field in ADF-STEM: 3D Microscopy?

D=6 nm: θ0 = 25 mr

(300 kV, 25 mrad)

0

1

2

3

4

5

6

-300 -200 -100 0 100 200 300

25 mrad10 mrad

Peak

Inte

nsity

Depth (A)


Depth of Field in ADF-STEM: 3D Microscopy?

1

10

100

0 20 40 60 80 100

100 kV200 kV300 kV

Dep

th R

esol

utio

n (A

)

Probe Forming Semi-Angle (mrad)

D=1nm: θ0 > 50mr

Today: D~ 6-8 nm SuperSTEM: ~1 nm

Cs>0

Cs~0

C5~0


Does Channeling Destroy the 3D Resolution?(Multislice Simulation of [110] Si @ 200 kV, 50 mrad)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 5 10 15 20 25 30

Sign

al (F

ract

ion

of I 0)

Thickness (nm)

∆f=20 nm∆f=1 nm

NO! (at least in plane – relative intensities between different depths are still out)

Fringe contrast


Summary

Contrast Transfer Functions: Coherent:

Lower resolution, higher contrastEasy to get contrast reversals with defocusAperture size only affects cutoff in CTF

Incoherent:

Higher resolution, lower contrastHarder to get contrast reversals with defocusAperture size is critical – affects CTF at all frequencies

4/34/13min 43.0 λCd =

4/1

3

4⎟⎟⎠

⎞⎜⎜⎝

⎛=

Coptλα

4/34/13min 77.0 λCd =

4/1

3

6⎟⎟⎠

⎞⎜⎜⎝

⎛=

Coptλα

linear imaging approximations4 - cornell...

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