In this thesis, we investigate the wave propagation in inhomogeneous complex media, including Dirac materials, bi-isotropic metamaterials, and plasmas theoretically, using the invariant imbedding method (IIM). We pursue to find phenomena that are well-known in one's field but unconventional in the other's field, comparing two different kinds of waves that have similarities in the forms of the wave equation. In order to investigate the wave propagation in various materials, we generalize IIM.
At first, we investigate the surface wave excitation in the interface between a metal/a bi-isotropic medium, and bi-isotropic/bi-isotropic media. We also derive the dispersion relation of surface waves. The results obtained using IIM agree with those obtained from the dispersion relation perfectly. We generalize the concept of a conjugate matched pair to bi-isotropic media and obtain several conditions under which the omnidirectional total transmission, which we call the super-Klein tunneling, occurs through conjugate matched pairs consisting of Tellegen media and of chiral media. We find that these conditions are closely linked to those for the omnidirectional excitation of surface waves.
Secondly, we investigate the mode conversion of transverse electromagnetic waves
into longitudinal plasma oscillations in chiral media and magnetized plasma. We verify the conditions where mode conversion occurs and find the conditions where mode conversion enhanced. We explain the giant over-reflection as the inverse process of perfect absorption.
Lastly, we investigate the Anderson localization phenomena of electromagnetic waves in anisotropic media and matter waves in pseudospin-1/2, -1 Dirac materials. By applying the perturbation expansion method to the invariant imbedding equations, we derive concise analytical expressions for the localization length, which are extremely accurate in the weak and strong disorder regimes. From analytical considerations, we provide an interpretation of the delocalization of waves at a special incident angle, as a phenomenon arising when the wave impedance is effectively uniform. Similarly, the ordinary Brewster effect of EM waves, the Klein tunneling of massless pseudospin-N particles and the super Klein tunneling of massless pseudospin-1 particles which are the total transmission phenomena of waves at a condition, is interpreted as an impedance matching phenomenon.
