On the theory of the modal expansion method for the EM fields inside a closed volume

The theory of the modal expansion method for the transient electromagnetic fields inside a closed region is reviewed. Although the concept of modal expansion is well known in the theory of electromagnetism, this formulation may make it more comprehensive where the scope of applicability, the effect of surface excitations, and the introduction of real modal fields to avoid complex notations in the time-domain formulation is concerned. It can be shown that, without resorting to the explicit expression for the quasi-static solution, which should be used to satisfy the inhomogeneous boundary conditions, the transient responses of EM fields inside a closed region could be expressed explicitly in terms of the natural modes of the system. This formulation assumes an arbitrarily inhomogeneous anisotropic material, volume and surface sources of arbitrary temporal and spatial dependence, and arbitrary initial electric and magnetic field distributions. For simplicity, the effects of losses are neglected in this formulation.