The topology of isofrequency surfaces of a magnetic-semiconductor superlattice influenced by an external static magnetic field is studied. In particular, in the given structure, topology transitions from standard closed forms of spheres and ellipsoids to open ones of Type I and Type II hyperboloids as well as bi-hyperboloids were revealed and analyzed. In the latter case, it is found out that a complex of an ellipsoid and bi-hyperboloid in isofrequency surfaces appears as a simultaneous effect of both the ratio between magnetic and semiconductor filling factors and magnetic field influence. It is proposed to consider the obtained bi-hyperbolic isofrequency surface as a new topology class of the wave dispersion.A topology of photonic systems is related to global behaviors of a wave function accounting constitutive and structural parameters in the entire dispersion band [1]. It means that in the phase space of a propagating electromagnetic wave at a constant frequency the topology appears in the form of an isofrequency surface (also known as Fresnel wave surface or surface of wave vectors), which governs the wave propagation conditions along an arbitrary direction inside the corresponding optical material (in fact it expresses the relationships between the directions of the wave vector and the vector of group or phase velocity of the wave; for the first reading on construction of isofrequency surfaces according to the laws of geometrical optics we refer to methodological notes in [2]). By definition, each isofrequency surface belongs to a class of quadrics [3], from a large variety of which three particular nondegenerated forms are well known in optics-sphere, ellipsoid and hyperboloid.Thus, in an isotropic medium isofrequency surfaces appear in the closed form of a sphere, whereas in a uniaxial optical crystal they transit to a complex of a sphere and spheroid, which characterize propagation conditions of ordinary and extraordinary waves, respectively [4]. These isofrequency surfaces can intersect each other in some sections at certain singular points. In a biaxial crystal the complex consists of a sphere and ellipsoid. In other natural anisotropic media including acoustic crystals, plasmas and magnetically ordered (gyrotropic) media isofrequency surfaces can acquire both closed and open forms. In the latter case they resemble the form of a hyperboloid (see Fig. 8.3.2 in [5] for a taxonomy of isofrequency surfaces in anisotropic media). In this way, different forms of topology of the wave dispersion express the kind of anisotropy, namely the relations be- * volodymyr.i.fesenko@gmail.com tween components of permittivity and/or permeability tensors characterizing the medium.Indeed, in an anisotropic crystal when all principal values of its permittivity tensor are positive (i.e., ε > 0 and ε ⊥ > 0), the isofrequency surfaces have closed forms [ Fig. 1(a)]. Contrariwise, when one or two corresponding tensor's components are negative (i.e., the medium is "extremely" anisotropic), the topology appears in the open form ...

It is demonstrated that the effect of coexistence of bulk and surface polaritons within the same frequency band and wavevector space can be achieved in a magnetic-semiconductor superlattice providing a conscious choice of characteristic resonant frequencies and material fractions of the structure's underlying components as well as geometry of the external static magnetic field. The study is based on the effective medium theory which is involved to calculate dispersion characteristics of the long-wavelength electromagnetic modes of ordinary and extraordinary bulk polaritons and hybrid EH and HE surface polaritons derived via averaged expressions with respect to the effective constitutive parameters of the superlattice.

Crossing and anti-crossing effects in dispersion characteristics of both bulk and surface polaritons in a magnetic-semiconductor superlattice influenced by an external static magnetic field being in the Faraday geometry are discussed. The bulk polaritons are classified as eigenwaves with right-handed and left-handed elliptically polarized states, whereas the surface polaritons are considered as hybrid modes having a predominant effect of either magnetic or semiconductor subsystem, and distinctions in dispersion characteristics of such polaritons are revealed involving the concept of critical points.

Extraordinary dispersion features of both bulk and surface polaritons in a finely-stratified magnetic-semiconductor structure which is under an action of an external static magnetic field in the Voigt geometry are discussed in this letter. It is shown that the conditions for total overlapping dispersion regions of simultaneous existence of bulk and surface polaritons can be reached providing a conscious choice of the constitutive parameters and material fractions for both magnetic and semiconductor subsystems.The polariton is introduced as a quasi-particle which characterizes a coupling between electromagnetic waves (photons) and a diverse variety of dipole-active elementary excitations inherent to a matter such as phonons, plasmons, excitons, etc. Although the concept of quasiparticles is related to quantum mechanics, the polariton can be considered as a macroscopic phenomena concerning on interaction of electromagnetic waves with macroscopic normal modes (eigenwaves) of a matter assuming their wavelength are long enough so that the medium can be treated as a continuous one. In such a way the theory of polaritons is developed without specifying which kind of dipole excitation is coupled to electromagnetic waves, because the specific nature of this excitation is completely defined only by the dielectric function (e.g. permittivity) of a medium [1]. This approach implies a particular consideration of polaritons in a bulk material (bulk polariton) and on its surface (surface polariton).Although the nature of these two particular modes is the same and is related to the medium polarizability, surface polaritons are distinguished from bulk polaritons by the fact that their amplitudes decay exponentially away from the surface in the direction normal to it [2]. This means that the normal component of the wavevector of a surface polariton is purely imaginary and consequently it cannot propagate away from surface. Therefore, these surface modes do not couple linearly with bulk polaritons either inside or outside the surface. In particular, this fact manifests itself in the different areas of existence of bulk and surface polaritons on their dispersion characteristics.As already mentioned, from the macroscopic viewpoint, the dispersion conditions for bulk and surface polaritons are defined by the dielectric function of a * v.i.fesenko@ieee.org † tvr@rian.kharkov.ua ‡ fedorin.ilya@gmail.com medium, and in the case of surface modes the difference in signs of the dielectric functions of patterning materials is required (i.e. the real parts of the permittivity scalars of two patterning materials must have opposite signs). Nevertheless, if at least one of two patterning materials is anisotropic (due to its crystalline nature or as a result of an external action, such as application of the static magnetic field) the permittivity appears as a tensor quantity, and the dispersion characteristics of polaritons become to be more complicated [3,4], e.g. the propagation of surface polaritons can appear to be permissible, even though t...

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