Abstract: Theory of lenslet interactions in a fly’s eye lens

Abstract

In an electron-optical fly’s eye lens, comprised of several parallel plates with arrays of aligned holes, the field at each lenslet is distorted by the fields of the surrounding lenslets. This interaction sets a limit on how closely packed the array can be (or how large the lenslet holes can be for a given spacing) and therefore has a direct bearing on performance. In this paper a method is developed for calculating the effect of the interaction for arbitrary lens geometry. First, from the symmetry of the environment of each lenslet, Neumann boundary conditions are derived on the bounding polygon between lenslets. These boundary conditions are expanded in a Fourier-perturbation series about the circle which is the zeroth harmonic of the polygon. Laplace’s equation is then solved within the bounding circle for each harmonic to first-order in the perturbation theory, for example by numerical relaxation. Second- and higher-order solutions can be similarly obtained if necessary. From the zeroth Fourier component of the resulting axial potential, Gaussian properties of the lenslet are obtained, along with the spherical aberration coefficient. The next nonvanishing harmonic (fourth for a square array of lenslets, sixth for an hexagonal array) determines a corresponding aberration coefficient, which describes the major effect of lenslet interactions on the focused spot. Higher-order aberration coefficients are calculated from higher-harmonic potentials, if needed. Finally, as an example, the method is applied to a three-element einzel lens with a square array of lenslets. Fourfold aberrations, which are the dominant interaction effect, are found to be significant if the hole diameter in the center element is greater than about 70% of the lenslet spacing, in rough agreement with experimental results. At this diameter such aberrations amount to roughly one tenth of the spherical aberration.