Autofocus and Astigmatism Correctionin Scanning Transmission Electron Microscopy

M. Rudnaya, W. Van den Broek, R. DoornbosJ. Maubach, R. Mattheij

CASA-day28.10.2009

Outline• Electron microscopy• Condor project

• Defocus and astigmatism• CASA-day 2008

• Variance as an image quality measure • Variance for STEM• 3-parameter optimization

• STEM Application

Outline• Electron microscopy

- Electrons vs. light- Astigmatism aberration

• Condor project

• Defocus and astigmatism• CASA-day 2008

• Variance as an image quality measure • Will the variance work for STEM?• 3-parameter optimization

• STEM Application

Light Microscopy: Resolution 0.0002 mm = 200 nm - red light: λ ≈ 800 nm - blue light λ ≈ 400 nm

Electron Microscopy

100 kV 300 kV 1 MV 3 MV

λ (pm) 3.70 1.97 0.87 0.36

• h is Planck’s constant,• E is the accelerating voltage• e and m are the electron’s

charge and mass

Electron microscopy: Resolution 0.002 nm

Electron microscopy: Atomic resolution

FEI’s TitanScanning TransmissionElectron Microscope (STEM)

Pictures from the PhD-thesis “Advanced Focus Methods in Electron Microscopy: Tomographic Reconstruction of the EELS Data Cube Autofocus of HAADF-STEM Images”by W. van den Broek

Astigmatism

Pictures from “A Robust Focusing and Astigmatism Correction Method for the Scanning Electron Microscope” by K.H. ONG, J.C.H. PHANG, J.T.L. THONG

Outline• Electron microscopy• Condor project

• Defocus and astigmatism• CASA-day 2008

• Variance as an image quality measure • Will the variance work for STEM?• 3-parameter optimization

• STEM Application

Condor Project

AcknowledgmensThis work has been carried out as a part of the Condor project at FEI company under the responsibilities of the Embedded Systems Institute (ESI). This project is partially supported by the Dutch Ministry of Economic Affairs under the BSIK program.

Outline• Electron microscopy• Condor project

• Defocus and astigmatism- Linear image formation model- Gaussian point spread function- Defocus and astigmatism in STEM

• CASA-day 2008

• Variance as an image quality measure • Will the variance work for STEM?• 3-parameter optimization

• STEM Application

Linear image formation model

Gaussian PSF

=

=

=

*

*

*

Defocus and Astigmatism

Autofocus and astigmatism correction

Outline• Electron microscopy• Condor project

• Defocus and astigmatism• CASA-day 2008

- Our approach- New challenges

• Variance as an image quality measure • Will the variance work for STEM?• 3-parameter optimization

• STEM Application

CASA-day 2008

ChallengeSome real sample’s Fourier transforms are difficult to analyze

Outline• Electron microscopy• Condor project

• Defocus and astigmatism• CASA-day 2008

• Variance as an image quality measure - Definition- Why would it work?

• Will the variance work for STEM?• 3-parameter optimization

• STEM Application

Variance as an image quality measure

Variance definition

Variance as an image quality measure

Variance definitionWhy will it work?

Assumptions

Parseval’s theorem

(A.1)

(A.2) (A.3)

(A.4)

(4)

(5)

(1)

(2)

(3)

(6)

(7)

Outline• Electron microscopy• Condor project

• Defocus and astigmatism• CASA-day 2008

• Variance as an image quality measure• Variance for STEM

- STEM point spread function- Numerical simulation for typical objects

• 3-parameter optimization

• STEM Application

STEM PSF

STEM defocus and astigmatism

d=d0=-45 nm d=-22.5 nm d=0 nmd=-67.5nmd=-90 nm

Ca=-10 nm

Ca=0 nm

Ca=10 nm

Numerical experiment - amorphous

Numerical experiment - crystalline

Outline• Electron microscopy• Condor project

• Defocus and astigmatism• CASA-day 2008

• Variance as an image quality measure • Will the variance work for STEM?• 3-parameter optimization

- Optimization challenges- Nelder-Mead simplex method

• STEM Application

3-parameter optimization• Destructive and time consuming function evaluations• No derivative information available

• (*) Local optima due to the noise• (*) Deviation in the function values for the same parameters

Nelder-Mead simplex method

Image from “Derivative-free optimization” by Andrew Conn

Outline• Electron microscopy• Condor project

• Defocus and astigmatism• CASA-day 2008

• Variance as an image quality measure • Will the variance work for STEM?• 3-parameter optimization

• STEM Application- Cross-section- Golden particle

Application

Evaluations and iterations

Experiment with the gold particle

Concluding remarksA new method for STEM autofocus and astigmatism correction is proposed. It is based on a 3-parameter optimization of the variance image quality measure. Nelder-Mead simplex method is used as optimization technique. The method works for both numerical simulation and a real STEM application.

Plans for the future- Noise robustness- Reduce the number of function evaluations

Questions

?

Window function

Focus Series (Gold-on-Carbon)

Astigmatic Imagesdefocus

AstY

0

1

-1 0 1

Spherical aberration

Lens

Image plane

Modified varianceModel

Real world:

Modified variance