Optical characteristics of a two-cylinder electrostatic lens

Abstract

In this paper formulae are developed for the first and second focal lengths, and the positions of the first and second principal planes of a type of electrostatic lens which has been the subject of study (mostly experimental) in several previous papers. The lens, which is commonly used in electron optical devices, lends itself to a theoretical study, although this does not appear to have been attempted before. It consists of two equal semi-infinite cylinders placed end to end so that their axes coincide and the ends are separated by a small gap. If the cylinders are at potentials V1 and V2 and we write σ = V2/V1, the system behaves as an electron lens when σ > 0 and as an electron mirror when σ < 0. In the latter case some experimental results have been given by Nicoll(1) who also studied the focusing action in the case σ > 0 and, in particular, the formation of intermediate images when σ ≪ 1 and when σ ≫ 1. But for the precise formulation of the relationship between σ and the number of cross-overs a theoretical study, based on the paraxial equation, would be necessary. The problem will be indicated below. An experimental determination of the lens characteristics for values of σ from about 2 to 15 and for several gap widths has been made by Spangenberg(2), whose results will be compared with those obtained in the present paper. The two-cylinder lens has also been studied by Klemperer and Wright(3) using an experimental and a numerical (trigonometrical) method, and some crude analytical results have been given by Gray(4).

A note on the Petzval field curvature in electron-optical systems

“…Using the formula (see 7 4. We now calculate the value of & in the case of the two-cylinder electrostatic lens, which was studied in the preceding paper (9). If the potentials of the two cylinders are Hence E -^V 2 f 2~X , 2 + X Now, writing A = (!+%)* and /i = (1 -# ) * , it is easy to show that…”

“…In the case of a non-zero separation e between the cylinders, we derive the general (integral) formula for the Petzval curvature and then confine ourselves to the case e<^ 1. The potential <fi(z) is given [see (9) ( l -£ t a n h w e ) 2 1 --log{(l -£,tanhwe)coshwe} J _i . |_ we J This is t h e exact formula, b u t in order to effect the integration we replace the term containing the logarithm b y (a + fc^ + cf, 2 ) 1 , where v , , vtanhwe , l v t a n h 2 w e a = \--coshwe, 6 =^ and c = .…”