Note on the asymptotic approximation of a double integral with an angular-spectrum representation

In this note, we are concerned with the asymptotic approximation of a class of double integrals which can be represented as an angular-spectrum superposition. These double integrals typically appear in electromagnetic scattering problems. Based on the synthetic manipulation of the method of steepest descent path, approximate expressions of the double integrals are derived in terms of the leading term of the contribution to the asymptotic expansions.