Flaws in classical proofs of vector Kirchhoff integral theorem and its new strict proof

The vector Kirchhoff integral theorem (VKI) is an important formula in electromagnetic (EM) theory, especially it is a basis of the optical diffraction theory. Recently, it has been found that there exist some flaws in the proofs presented in the literature. There are mainly two types of methods to prove the VKI. The first type of method is to employ the vector analysis to prove the VKI directly. Some flaws of this type of proof presented in the literature have been found and pointed out in this paper. The second type of method is to employ the scalar Kirchhoff Integral (SKI) to directly obtain the VKI. The SKI was first derived by Kirchhoff (1882). In spite of its mathematical inconsistency and its physical deficiencies, the SKI works remarkably well in the optical domain and has been the basis of most of the work on diffraction. However, the proofs for SKI usually need the scalar radiation conditions. The scalar radiation condition was first proposed by Sommerfeld to ensure the uniqueness of the solution of certain exterior boundary value problems in mathematical physics. But whether the scalar radiation conditions were suitable for the EM was not sure. In fact, for electromagnetic field, we have another vector radiation conditions which have been verified to be adaptable for all the radiation and scattering fields. It is difficult to obtain the scalar radiation conditions directly by just separating three Cartesian directions from the vector one, because the different scalar components are coupled together after the rotation and cross product operation. Actually, few strict proofs could be found to support the fact that EM satisfies the scalar radiation condition. So as the supplementary, the scalar radiation conditions will be derived in detail with far-field approximation method in this paper. To avoid using the scalar radiation condition which may bring some non-rigorousness, we perform a new strict proof for the VKI by using the vector analysis identities. The rest of this paper is organized as follows. In Section 2, the different proofs presented in the classical books will be analyzed in detail. The flaws existing in these proofs will be pointed out. After that, in Section 3, based on the Stratton-Chu formula, a new strict proof will be given with using the vector identities. In Section 4, a sensitivity analysis is numerically performed to confirm our demonstration. Finally, the conclusions are drawn from the present study in Section 5. The scalar radiation conditions will be discussed in the appendix.