The anode proximity effect for generic smooth field emitters

The proximity of the anode to a curved field electron emitter alters the electric field at the apex and its neighbourhood. A formula for the apex field enhancement factor, $\gamma_a(D)$, for generic smooth emitters is derived using the line charge model when the anode is at a distance $D$ from the cathode plane. The resulting approximately modular form is such that the anode proximity contribution can be calculated separately (using geometric quantities such as the anode-cathode distance $D$, the emitter height $h$ and the emitter apex radius of curvature $R_a$) and plugged into the expression for $\gamma_a(\infty)$. It is also shown that the variation of the enhancement factor on the surface of the emitter close to the apex is unaffected by the presence of the anode and continues to obey the generalized cosine law. These results are verified numerically for various generic emitter shapes using COMSOL Multiphysics. Finally, the theory is applied to explain experimental observations on the scaling behavior of the $I-V$ field emission curve.