Bi-hyperbolic isofrequency surface in a magnetic-semiconductor superlattice

Abstract

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 ...

Lossless and loss-induced topological transitions of isofrequency surfaces in a biaxial gyroelectromagnetic medium

“…The superlattice behaves as an anisotropic double-negative medium for the extraordinary waves, while for the ordinary waves it is a double-positive one. For the extraordinary waves the isofrequency surface appears to be in a form of bihyperboloid, 32 which is significantly different from those obtained above for Regions I and III ( Fig. 8(a)).…”

“…Such an unusual bi-hyperbolic topology is a result of simultaneous effects of both the structure periodicity and influence of an external magnetic field. 32 The isofrequency surface of the ordinary waves transits to an ellipsoid-like form, since the condition |µ 2 xy | < |µ xx µ yy | is satisfied. This ellipsoid lies inside of such a complicated bi-hyperbolic form.…”

“…In particular, in such a superlattice the topological transitions from a closed ellipsoid to open Type I and Type II hyperboloids as well as a bi-hyperboloid have been observed for the extraordinary waves. 32 The latter is a recently found form of topology of isofrequency surfaces which appears in the result of simultaneous tuning of both geometrical parameters of the superlattice and external static magnetic field influence.…”

Topologically Protected and Highly Localized Surface Waves in Gyro‐Electromagnetic Metamaterials

“…Therefore, z is a constant and independent of frequency. [36][37][38][39][40] We analyze topological phases in the electromagnetic duality metamaterial model, which can be expressed as = . [10,[41][42][43] The corresponding medium can be implemented by using a periodic multi-layered structure of magnetic and semiconductor layers.…”

Topological Transition Points in a Magnetic-Semiconductor Periodic Structure in an External Magnetic Field

“…Topological transition occurs at the frequency ω 4 , where ε xx = 0. The isofrequency surface for ordinary waves is closed ellipsoid, while for extraordinary waves it represents complex surface which is a combination of closed ellipsoid and open surfaces [18].…”