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The roughness of a photocathode could lead to an additional uncorrelated divergence of the emitted electrons and therefore to an increased thermal emittance. To calculate the emittance growth due to the cathode roughness, people usually choose a simple 2D sinusoidal surface model to avoid mathematical complexity. In this paper, we demonstrate an analytical method, which is inspired by the point spread function that has been widely used in radiation imaging field, to accurately evaluate the emittance growth due to the random roughness of a real-life cathode. Both analytical and numerical studies are performed. Our analytical formulas clearly reveal the relationship between the surface roughness and the emittance growth. Both analytical and numerical results surprisingly show that in the typical 3D random surface case, the influence of the surface roughness on the emittance growth is much smaller than the 2D sinusoidal case with typical roughness properties, however with roughness properties which are matched to the 3D case, the emittance growth conditions in these two cases are very similar. Even with applied electric field strength up to 120  MV/m, the total emittance growth is still below 10%. It implies that the large emittance growth (50%–100%) observed on metallic cathodes in some experiments, which is generally believed to be the result of the electric field on the rough surface, might be due to some other reasons

Analytical study on emittance growth caused by roughness of a metallic photocathode