xmpl2d57, 57th 'example' data file for CPO2D

Einzel 3-tube lens with curved surfaces.


Prepared for paper 68 in the publications list,  Optimization of the shapes of electrodes of electrostatic lenses, by F H Read, Microscopy and Microanalysis 21 Suppl S4, 276-279 (2015)

See the paper for more detailed information.

The use of the curved surface gives a large improvement in the spherical aberration, compared with traditional designs with flat-faced cylinders.


See also xmpl2d56, which deals with aperture lenses.

To quantify the quality of the image we consider an incoming parallel beam of diameter db and a given set of constraints (see below).  The diameter of the disc of least confusion of the image can be shown to be

dl = Cs4/(16*f13)*db3

where Cs4 (referred to the object) is the limit of Csobj*M4 as M→0, as defined by E. Harting and F.H. Read, Electrostatic Lenses, Elsevier, Amsterdam, 1976 (which is avalable as part of the CPO package).  We ignore higher-order aberrations.  Also of interest is the relative potential penetration ∆V at the object and image positions (which ideally should be  field-free areas).  Using db = 0.5*D the corresponding values of dl/D and ∆V for the 2D and 3D lenses considered are:




2-aperture reference lens



2-aperture lens, optimised spacing



2-aperture lens, curved electrodes



3-tube einzel lens, reference lens



3-tube einzel lens, curved electrodes



Quoting from the paper:

Firstly, it is obvious that improvements in aberration properties can always be made when degrees of freedom are added to a design, but here we have found that the improvements can be surprisingly large, giving reductions in image sizes by factors of 6 and 8 for the present lenses.  

Secondly, the source of the improvements for both types of lens studied here is clearly connected to the reduction in the strengths of the fields inside the lenses, which is achieved by extending the overall lengths of the lenses.  In the case of the tube lens the shortness of the inner tube also contributes to this.

Thirdly, the most critical regions are those near to the entrance and exit positions, where the rate of change of field strength tends to be largest, and so it has been found that improvements are obtained by shielding these regions as much as possible.  This is achieved for the aperture lens by using rounded parts that are concave to the centre of the lens, and for the tube lens by giving the edges of the tubes in those regions a large radius of curvature.  

The constraints mentioned above:

As with all optimizations, the results are entirely dependent on the restraints imposed on the system.  Without any restraints a system can be improved almost without limit, except of course the limits imposed by any theoretical limits.  The following results are therefore arbitrary, in the sense that the restraints are arbitrary.

We shall be considering double-aperture accelerating lenses that have apertures of diameter D and 3-tube einzel lenses that have tubes of internal diameter D.  All the results that involve distances scaled as D, so therefore in general the unit length D will not be mentioned (except in the description of the restraints below). 

The restraints that we imposing in the present study are:

(1) the diameter of the outermost electrode (or containment electrode) is 5D,

(2) the working distance (from the last electrode to the image position), must be >= D,

(3) the image is real (so that the image size is not affected by putting an image electrode at the image position),

(4) the overall length S of the lens (first to last apertures) must be <= 6D,

(5) the electrode configuration is symmetrical about the center.

(6) only the 3rd.order spherical aberration is considered.

For the double aperture lenses we add the restraint:

(7) the accelerating ratio of energies is 10.

And for the 3-tube einzel lenses we add:

(8) the Gaussian focus must be at Q = 4D, measured from the physical center of the lens.

Of these constraints, number (5) is the least serious, and in fact is broken for the lens shown in Fig. 1c, but is otherwise retained for simplicity.