Relativistic Considerations on Electron Optical Brightness

On the basis of Liouville's theorem, the relativistically corrected expressions are derived for the differential brightness in the orthogonal curvilinear coordinate system, both in terms of the total and normal momentum representations and of the total and normal velocity representations. The result is specialized to the case of the axial brightness for thermionic and field emission. It is shown that the relativistically corrected axial brightness is given by R*=j0e\varPhi*/πkT for thermionic emission, and is given by R*=j0e\varPhi*/πd for field emission, where \varPhi*=\varPhi(1+e\varPhi/2m0c2) is the relativistically corrected accelerating voltage. It is also demonstrated that the differential brightness is a fundamental quantity characterizing the quality of the electron gun.