The cosine law of field enhancement factor variation: Generic emitter shapes

The cosine law of field enhancement factor variation was recently derived for a hemi-ellipsoidal emitter and numerically established for other smooth emitter shapes (Biswas et al., Ultramicroscopy, 185, 1 (2018)). An analytical derivation is provided here for general smooth vertical emitter shapes aligned in the direction of the asymptotic electrostatic field. The law is found to hold in the neighbourhood of the emitter apex from where field emission pre-dominantly occurs.