Extended legality of curved boundary integral method

The angular spectrum method is an efficient approach for synthesizing electromagnetic beams from planar electric field distributions. The electric field definition is restricted to a plane, which can introduce inaccuracy when applying the synthesized beam to curved surface features. The angular spectrum method can also be interpreted as a pure source method defining the field symmetrically with respect to the creation plane. Recently, we generalized that symmetric field method to arbitrary source distributions, which are valid at any point on compact, regular surface Ω in R ³. We call this approach the Curved Boundary Integral method. The electromagnetic fields synthesized with this method satisfy the Helmholtz equation and are adjusted via amplitude and phase at the desired surface. The fields are obtained as a relatively simple integral. However, restrictions on where in space the synthesized field is valid were included in the mathematical proof length to avoid obscuring the main points. These restrictions can be significant depending on the shape and degree of curvature of surface Ω. In this article, we remove these restrictions so that the integral representation of the electromagnetic beam becomes valid at all points r∈R ³∖Ω, with a minor restriction. Its modification can work even on Ω. We demonstrate the importance of this extended legality with a source field parametrized into the torus surface. The electromagnetic radiation of this structure would not be valid at any point in space without this extension. Finally, we show that by changing the order of integration, the field singularity at each source point is eliminated.