A complete linear theory in four dimensional space-time is constructed for propagation and coupling of electromagnetic waves in the most general case, i.e. media where all three space directions are in homogeneous and where temporal changes, anisotropy and absorbtion are also included. The only condition is that the properties of the medium should be slowly varying, Although it is assumed that absorption may be strong. The differential equations for the waves are analytically solved for both individual propagation of the waves and coupling of two waves.

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