Evolution of transient electromagnetic fields in spherical coordinate system

Transient electromagnetic fields in infinite space filled with inhomogeneous in radial direction transient medium without losses are presented by spherical modes. The initial nonstationary three-dimensional electrodynamic problem is transformed into the problem for one-dimensional evolutionary equations by means of the construction of modal basis for electromagnetic field with arbitrary time dependence. The completeness of the basis is proved by means of Weyl's Theorem about orthogonal splitting of Hilbert space. The transient electromagnetic field can be found directly without Fourier transform. The expansion coefficients of transient electromagnetic field are found from the set of evolutionary equations which describe independent propagation of the modes. Using Laplace transform and wave splitting the operator of transient field transformation is obtained for the case of homogeneous stationary medium.