19 Magnetoelectric Near Fields
549
Fig. 19.12 a A structure of microwave waveguide with a ferrite disk and small metallic ring.
b Circulating surface electric current on a metallic ring. c The electric current on the surface of a
metallic ring has the spin degree of freedom
located above a ferrite disk at a distance of 0.05 mm. The circulating current will give
an angular-momentum flux. The intensity of the flux is proportional to the gradient
of twisting angle, which plays the role of the phase gradient. A critical phase gradient
is required to enable the process. This occurs only at the MDM resonance. In this
case, the persistent charge current in the ring is correlated to the persistent magnetic
current in a ferrite disk. The electric current on the surface of the metallic ring has
the spin degree of freedom (see Fig. 19.12c). When using 2D models in our main
studies of MDM oscillations, we can conclude now that, generally, an analysis of ME
fields should be made based on the 3D model. At the MDM resonance, the current
induces on a test metal ring is a topological soliton structure which is quantized
simultaneously in poloidal and toroidal directions. This 3D continuous vector-field
structure—a hopfion (or Hopf soliton)—cannot be unknotted without cutting [90,
91]. It is also worth noting that the helicity properties of the 3D structure of the ME
549
Fig. 19.12 a A structure of microwave waveguide with a ferrite disk and small metallic ring.
b Circulating surface electric current on a metallic ring. c The electric current on the surface of a
metallic ring has the spin degree of freedom
located above a ferrite disk at a distance of 0.05 mm. The circulating current will give
an angular-momentum flux. The intensity of the flux is proportional to the gradient
of twisting angle, which plays the role of the phase gradient. A critical phase gradient
is required to enable the process. This occurs only at the MDM resonance. In this
case, the persistent charge current in the ring is correlated to the persistent magnetic
current in a ferrite disk. The electric current on the surface of the metallic ring has
the spin degree of freedom (see Fig. 19.12c). When using 2D models in our main
studies of MDM oscillations, we can conclude now that, generally, an analysis of ME
fields should be made based on the 3D model. At the MDM resonance, the current
induces on a test metal ring is a topological soliton structure which is quantized
simultaneously in poloidal and toroidal directions. This 3D continuous vector-field
structure—a hopfion (or Hopf soliton)—cannot be unknotted without cutting [90,
91]. It is also worth noting that the helicity properties of the 3D structure of the ME