Lenses, Apertures, and Resolution

  • David B. Williams
  • C. Barry Carter


Electron lenses are the TEM’s equivalent of the glass lenses in a visible light microscope (VLM) and, to a large extent, we can draw comparisons between the two. For example, the behavior of all the lenses in a standard TEM can be approximated to the action of a convex (converging) glass lens on monochromatic light. The lens is basically used to do two things ■ Take all the rays emanating from a point in an object and recreate a point in an image ■ Focus parallel rays to a point in the focal plane of the lens


Electron Lens Object Plane Spherical Aberration Chromatic Aberration Gaussian Image 
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Some History

  1. Busch, H 1927 Über die Wirkungsweise der Konzentrierungsspule beider braunschen Röhre Arch. Elektrotech. 18 583–594. The original paper on focusing electron beams.Google Scholar
  2. Hawkes, PW (Ed.) 1997 Advances in Imaging & Electron Physics Vol. 100: Partial Cumulative Index Academic Press New York (now published by Elsevier). Essential reference for the historically minded.Google Scholar
  3. Hawkes, PW 2004 Recent Advances in Electron Optics and Electron Microscopy Ann. Fond. Louis de Broglie 29 837–855. An overarching yet concise review of recent advances in electron optics and microscopy, with a great collection of references, both historical and recent.Google Scholar

Lenses and Electron Trajectories

  1. Reimer gives a summary of lens defects and more on the derivation of equation 6.11.Google Scholar
  2. Grivet, P 1972 Electron Optics Pergamon Press New York.Google Scholar
  3. Hawkes, PW 1972 Electron Optics and Electron Microscopy Taylor & Francis Ltd. London. This account is particularly clear if you have an interest in the physics of electron lenses. An important discussion of how to take account of many aberrations when giving a figure of merit. In ‘Confusion in the Definitions of Resolution,’ we follow Hawkes’ clear reasoning regarding the plane of least confusion.Google Scholar
  4. Hawkes, PW (Ed.) 1982 Magnetic Electron Lenses Springer New York. A collection of review articles in true Peter Hawkes style; thorough, sound, erudite, and informative.Google Scholar
  5. Hawkes, PW and Kasper, E 1989, 1994 Principles of Electron Optics 13 Academic Press New York. Comprehensive but advanced. Volume 3 includes imaging in the TEM. If by now you’re getting the idea that Hawkes is the source of electron optical information, then you are right.Google Scholar
  6. Klemperer, O and Barnett, ME 1971 Electron Optics Cambridge University Press New York.Google Scholar
  7. Munro, E 1997 Electron and Ion Optical Design Software for Integrated Circuit Manufacturing Equipment J. Vac. Sci. Technol. B 15 2692–2701. More on electrons moving through the lens.Google Scholar
  8. Rempfer, GF 1993 Electrostatic Electron Optics in the 1940s and Today MSA Bull. 23 153–158. By an expert in the use of electrostatic lenses.Google Scholar

Aberration Correction

  1. The companion text goes into this in much more detail. In particular, you’ll find there that C s is actually better written as C3. There are many more ‘C s’ terms. These references give an introduction.Google Scholar
  2. Chang, LY, Kirkland, AI and Titchmarsh, JM 2006 On the Importance of Fifth-Order Spherical Aberration for a Fully Corrected Electron Microscope Ultramicroscopy 106 301–306.Google Scholar
  3. Krivanek, OL, Delby, N and Lupini, AR 1999 Towards Sub-Å Electron Beams Ultramicroscopy 78 1–11. Used in the Nioen STEM.Google Scholar
  4. Urban, K, Kabius, B, Haider, M and Rose, H 1999 A Way to Higher Resolution: Spherical-Aberration Correction in a 200 kV Transmission Electron Microscope J. Electr. Microsc. 48 821–826.Google Scholar


  1. All texts on TEM will include a discussion of resolution. Particularly useful are those in Reimer 1997, Edington 1976, Fultz and Howe 2002 and Hirsch et al. 1977.Google Scholar
  2. □ Points to be wary of when reading about definitions of Cs-limited resolution: (see references in Chapter 1)Google Scholar
  3. Sawyer and Grubb (2008) and Egerton 2005 use the Gaussian image radius referred back at the object plane, just as we do; i.e., r sph = C sβ3. Reimer 1997 and Fultz and Howe 2001 use the diameter of the disk in the plane of least confusion; i.e., d sph = 0.5C sβ3 although both also describe the radius at the Gaussian image plane as we do. Beware: Edington 1976 implies, and Hirsch et al. 1977 state, that C sβ3 is the radius of the disk in the plane of least confusion, which it is not, since by definition it must be less than the Gaussian-image radius (see Figure 6.11).Google Scholar
  4. Sawyer, LC, Grubb, DT and Meyers, DT 2008 Polymer Microscopy 3rd Ed. Springer New York. Rule of thumb for polymers.Google Scholar
  5. □ Points to be wary of when reading about depth of field and depth of focusGoogle Scholar
  6. Bradbury et al. 1989 give a particularly clear discussion of the topic. Reimer 1997 uses the term depth of focus for the depth of field and uses depth of image for depth of focus; a rare inconsistency! The terms are used interchangeably in SEM because there is no lens between the object and the image.Google Scholar
  7. Bradbury, S, Evennett, PJ, Haselmann, H and Piller, H 1989 Dictionary of Light Microscopy Royal Microscopical Society Handbook #15 Oxford University Press New York. For the VLM comparison.Google Scholar

Special Techniques

  1. Borisevich, AY, Lupini, AR, Travaglini, S and Pennycook, SJ 2006 Depth Sectioning of Aligned Crystals with the Aberration-Corrected Scanning Transmission Electron Microscope J. Electr. Microsc. 55 7–12. Moving to confocal imaging in the TEM.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.The University of Alabama in HuntsvilleHuntsvilleUSA
  2. 2.University of ConnecticutStorrsUSA

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