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Ultramicroscopy
Volume 107, Issue 8, August 2007, Pages 635-643
 
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doi:10.1016/j.ultramic.2006.12.005    How to Cite or Link Using DOI (Opens New Window)
Copyright © 2007 Elsevier B.V. All rights reserved.

Astigmatic intensity equation for electron microscopy based phase retrieval

Tim C. Petersena, Corresponding Author Contact Information, E-mail The Corresponding Author and Vicki J. Keastb

aAustralian Key Centre for Microscopy and Microanalysis, The University of Sydney, Sydney, NSW 2006, Australia bSchool of Mathematical and Physical Sciences, The University of Newcastle, Callaghan NSW 2308, Australia

Received 15 October 2006; 
revised 5 December 2006; 
accepted 19 December 2006. 
Available online 8 January 2007.

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Abstract

Phase retrieval, in principle, can be performed in a transmission electron microscope (TEM) using arbitrary aberrations of electron waves; provided that the aberrations are well-characterised and known. For example, the transport of intensity equation (TIE) can be used to infer the phase from a through-focus series of images. In this work an “astigmatic intensity equation” (AIE) is considered, which relates phase gradients to intensity variations caused by TEM objective lens focus and astigmatism variations. Within the paraxial approximation, it is shown that an exact solution of the AIE for the phase can be obtained using efficient Fourier transform methods. Experimental requirements for using the AIE are the measurement of a through-focus derivative and another intensity derivative, which is taken with respect to objective lens astigmatism variation. Two quasi-experimental investigations are conducted to test the validity of the solution.

Keywords: Astigmatism; Phase retrieval; Transmission electron microscopy

PACS classification codes: 42.30.Rx; 41.85.−p; 68.37.−d

Article Outline

1. Introduction
2. Theory
3. Simulation and experiment
3.1. Astigmatic intensity equation test using a low resolution phase object
3.2. Astigmatic intensity equation test at high resolution
4. Conclusion
Acknowledgements
Appendix A. Appendix
A.1. Derivation of Eq. (5)
A.2. Fourier inversion of the astigmatic intensity equation
A.3. Digital micrograph script
References








Ultramicroscopy
Volume 107, Issue 8, August 2007, Pages 635-643
 
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