Surface electric fields and spatial derivatives of Green's function of layered media based on half-space extraction

We consider the case when both source and field points are located on the same interface separating two different dielectric media. The electric fields and spatial derivatives of an electric-field Green's function in the spatial domain are calculated numerically without using potentials. For the electric field and its spatial derivatives, the integrand of the Sommerfeld integral grows with k_ρ for large k_ρ. For the fourth derivative, the integrand grows as the k_ρ^11/2 power. To evaluate the integrals, we use a half-space extraction technique. After the half-space extraction, the integrand of the Sommerfeld integral of layered media decays exponentially, and the integral is calculated along the Sommerfeld integration path. The half-space extraction part is calculated by numerical integration along the vertical branch cuts. Numerical results of the surface electric fields and spatial derivatives up to the fourth order for multilayered media are illustrated for distances from 0.02 to 20 free-space wavelengths. Results are illustrated for various layer thicknesses.