A Complex-Envelope FDTD Formulation Using Alternating In-Phase and Quadrature Field Variables

Real-valued partial differential equations (PDEs) are obtained by substituting the rectangular form of the complex envelope (CE) field and source quantities into the CE versions of Faraday´s and Ampere´s laws, and then separating each resulting complex PDE into real and imaginary parts. These real-valued PDEs result in a CE FDTD scheme that uses only real numbers and operations. As presented here, this CE FDTD scheme alternates at intervals of half a time step between solving for the real and imaginary portions of the CE fields, exactly as the Yee grid alternates spatial components of the fields at half spatial steps. The scheme is demonstrated here for the two-dimensional (2-D) transverse-magnetic case. A split-step method is used in the implicit formation such that only tridiagonal matrices have to be solved. FDTD results are presented for a 2-D cavity with an electric current source.