Abstract: Refraction of electromagnetic or optical waves on the interface of LHMs and RHMs is studied by the uniform approach on the basis of the high frequency ray theory through the application of Fermat’s principle. And then the matrix optics is generalized through the combination of the refraction law gained and sympletic coordinates with phase items. The relevant expression for ray transmission can maintain the essence of the symplectic formulation and have the application in the RHMs and LHMs mixing optical systems.

Key words: matrix optics, LHMs, Fermat's principle, refraction, symplectic group