Schottky’s conjecture on multiplication of field enhancement factors

Abstract

Of great interest to high power microwave, millimeter wave to terahertz sources, x-ray tubes, electrons guns, etc., is the electric field enhancement obtained from sharp emitting structures fabricated by various microfabrication methods. In this paper, we use conformal mapping to investigate the field enhancement of several rectilinear geometries, including a single rectangular ridge, a trapezoidal ridge, and their superposition, i.e., one ridge on top of another. We show that the composite field enhancement factor of the double ridge with a microprotrusion on top of a macroprotrusion is dominated by the product of the individual protrusions’ field enhancement factors over a very wide range of geometric aspect ratios, as conjectured by Schottky. Simplified scaling laws are proposed. Significant deviation from Schottky’s product rule occurs almost exclusively when the half-width of the macroprotrusion is less than the height of the microprotrusion. Accurate expressions of the divergent electric field near the sharp edges are derived.