Field at circular hole in an infinite flat plate.
(Taken from the footnotes of the test data files test2d15.dat and test3d13.dat).
Taking E1 and E2 as the asymptotic fields for z<0 and z>0 respectively, then for a hole of diameter R the exact expression for the potential along the axis is
V(0,z) = (E1-E2)*(z/pi)*(arctan(z/R)+(R/z)) + (E1+E2)*(z/2)
The algorithm for off-axis potential, derived from the equations given by P W Hawkes and E Kasper, Principles of Electron Optics, is:
Using reduced coordinates (that is, distances are divided by the radius of the hole), and taking the asymptotic field to be zero and unity at negative and positive z respectively,
d1 = sqrt((r - 1)^2 + z^2), d2 = sqrt((r + 1)^2 + z^2) (= distances from the point to the upper and lower edges of the hole)),
u = sign(z)*sqrt(((d1 + d2)/2)^2 - 1), v = sqrt(-((d1 - d2)/2)^2 + 1),
potl = v*(-u/2 - (1 + u*arctan(u))/pi)
To see the form of the potential and field near the edge of the aperture, put
r = 1 + a*sin(theta), z = a*cos(theta),
and expand the potential in powers of a. The term of lowest order in the potential is then
potl = -sqrt(a*u)/pi,
u = 1 - sin(theta)
(which varies from 0 when r = 1 + a and z = 0, to 2 when r = 1 - a and z = 0).
The radial and axial components of the field have the lowest order terms:
er = 0.5*sqrt(u/a)/pi, ez = -0.5*cos(theta)*sqrt(1/(u*a))/pi.,
and so both er and ez go to infinity as a goes to zero.
At large distances from the aperture the magnitude of the change in potential caused by the aperture is: