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Schottky Conjecture is analytically proved for multi-stage field emittersconsisting on the superposition of rectangular or trapezoidal protrusions on aline under some specific limit. The case in which a triangular protrusion ispresent on the top of each emitter is also considered as an extension of themodel. The results presented here are obtained via Schwarz-Christoffelconformal mapping and reinforce the validity of Schottky Conjecture when eachprotrusion is much larger than the ones above it, even when an arbitrary numberof stages is considered. Moreover, it is showed that it is not necessary torequire self-similarity between each of the stages in order to ensure thevalidity of the conjecture under the appropriate limits.

Analytical proof of Schottky Conjecture for multi-stage field emitters