Symplectic analysis for periodical electro-magnetic waveguides

Symplectic analysis is introduced into electro-magnetic waveguide theory, by using Hamiltonian system theory in which the transverse electric and magnetic field vectors are the dual vectors. The method can accommodate arbitrary anisotropic material and includes the interface conditions between adjacent segments of the waveguide. An electro-magnetic stiffness matrix is introduced which relates to the two ends of each segment of the waveguide. Both the pass- and stop-band stiffness matrices for plane waveguides with constant cross-section are given analytically and also a transformation matrix is given to permit abrupt changes of cross-section to occur. The variational principle is applied to obtain the segment combination algorithm needed to generate the electro-magnetic stiffness matrix related to the two ends of the fundamental periodical segment. Then the Wittrick–Williams algorithm is used to extract the eigenvalues. Thereafter, an energy band analysis is performed for a periodical waveguide, e.g., a grating, by using the symplectic eigensolutions.