A generalized geometrical representation of coupled mode theory

The evolution of a general two-level system under arbitrary operators is cast in a three-dimensional form. An earlier geometrical representation is generalized to include contradirectional, parametric, and skew-Hermitian coupling as well as systems with loss or gain. A formal connection to the theory of rigid body dynamics is made and explicit linearity and transformation properties are rigorously established in a coordinate-free form. The connection with mechanics is shown to permit transformations to rotating coordinate systems, a useful technique in analyzing typical guided wave systems.