Brightness calculation formula for Kohler illumination beam

A two-lens optical system entails two beams: a crossover beam and a Kohler illumination beam. We observed that upon varying the excitation corresponding to the first lens from large to small values, the crossover beam changed to a Kohler illumination beam with increased brightness. We derived new calculation formulas for the Kohler illumination beam. Brightness B and beam current Ib can be expressed as B = Bco (ϕco/ϕ)2 and Ib = Bco (παϕco)2/4, where Bco, ϕco, ϕ, and α denote the crossover beam brightness, crossover beam size, Kohler illumination beam size, and beam semi-angle, respectively; additionally, ϕco and ϕ are obtained at the same first lens excitation. These two equations were experimentally validated. We obtained a brightness of 2.05 × 108 A/cm2 sr using a beam energy and an emission current of 20 keV and 54 µA, respectively. Notably, this value surpasses the Langmuir limit of 4.11 × 105 A/cm2 sr by 499 times.