Behavior of the field enhancement due to mutual depolarization on a pair of triangular emitters at short and large separations

The Schwarz-Christoffel transformation is used to analytically evaluate the Field Enhancement Factor (FEF) in the vicinity of the apex of two isosceles triangular emitters close to each other. It is shown that the fractional reduction between the apex-FEF of a single triangular emitter and this same FEF evaluated when another identical emitter is placed close to it, $−δ$⁠, may be well described by an exponential or a power-law behavior involving the ratio between the distance and the height of the emitters and/or their aspect-ratios at different regimes. This way the analytical model presented here intends to investigate the existence of universal depolarization laws for non-cylindrical emitters used in different scientific and technological applications involving large area field emitters. The results obtained here show a quadratic power-law decay of the fractional reduction with respect to the ratio between the distance and the height of the emitters for asymptotically large distances between them. This differs from the cubic decay frequently verified in the literature for three-dimensional structures but the robustness of this exponent, which remains independent of the aspect-ratio of the emitters, is also reinforced.