Xmpl2d58, 58th 'example' data file for CPO2D

Calbick aperture lens.

Calbick aperture lens, with rounded edges.

The Calbick formula is f = 4*V/E, where eV is the nominal energy at the aperture, E is the change in field, and the aperture diameter is the unit of distance.

In the present case the electron is decelerated to 1eV in a field of strength 5, therefore f = 4.0.

Here the edge is rounded, with rad = 0.01, but this appears to have only a small effect on the results given below (for the range used here, 0 to 0.01).

Extrapolating zc versus alpha**2, where zc is the intercept of the ray with the axis and alpha is the angle, the focus is at zf = 4.11621 for alpha = 0, and the slope is 90.93.

Putting f = r/tan(alpha) we find by extrapolation that f = 4.12387 at alpha = 0, not the theoretical value 4.0.

The position zp of the principal plane is zf - f = -0.00766.

The radial position of a ray at the focal plane is C*alpha**3.

Therefore C = (zf - zc)/alpha**2. The value of this is 90.93, see above.

This value is near to f_theoretical**2/2*R (= 80.0) and nearer still to f_calculated**2/2*R (= 85.03), where R is the radius of the hole.

Varying R, the difference between the calculated and theoretical values of f seem to be proportinal to R. The dependence of C on R has not been studied.