Rigorous expressions of Huygens' principle in scalar theory

The Huygens' principle is thoroughly investigated under scalar theory. The rigorous expressions of Huygens' principle must be independent of ∂u/∂n, and their boundaries can only be taken as either spherical or flat; thus, three cases can be concluded. An extended version of Huygens' principle is proposed to cover these cases, whose rigorous expressions are shown in this paper. Specifically, when the radius of the spherical boundary approaches infinity, the corresponding expressions become the form corresponding to the flat boundary. Expressions with spherical boundary can change the area and average intensity of small angle diffraction pattern proportionally, thus providing a promising mathematical tool for the design of curved imaging systems.