Extended legality of curved boundary integral method.

Area of Science:

• Electromagnetism and Optics
• Computational Electromagnetics
• Mathematical Physics

Background:

• The angular spectrum method (ASM) is effective for synthesizing electromagnetic beams from planar sources.
• ASM's planar restriction limits accuracy for curved surfaces, treating fields symmetrically around the source plane.
• Existing methods struggle with accurate field synthesis on arbitrary, non-planar surfaces in 3D space.

Purpose of the Study:

• To generalize the symmetric field method for electromagnetic beam synthesis on arbitrary curved surfaces.
• To extend the validity of the synthesized electromagnetic field to all points in space, removing prior restrictions.
• To demonstrate the method's capability on complex geometries like a torus and eliminate field singularities.

Main Methods:

• Developed the Curved Boundary Integral (CBI) method, generalizing symmetric field approaches to arbitrary surfaces Ω in R³.
• Derived an integral representation for electromagnetic fields satisfying the Helmholtz equation, adjustable via amplitude and phase.
• Mathematically removed restrictions on the validity domain of the synthesized field, proving it holds for all r∈R³∖Ω.

Main Results:

• The CBI method provides an integral solution for electromagnetic fields on curved surfaces.
• The extended validity ensures the synthesized field is accurate across all space, crucial for complex source shapes.
• Demonstrated validity on a torus surface, where previous methods would fail; field singularities were eliminated by changing integration order.

Conclusions:

• The extended Curved Boundary Integral method accurately synthesizes electromagnetic beams for complex, curved surfaces.
• This approach overcomes the limitations of planar methods, offering a universally valid solution in 3D space.
• The method is robust, handling field singularities and enabling accurate electromagnetic radiation modeling for intricate structures.