Magnetostatic microwaves in circular dielectric waveguide with anisotropic uniaxial μ-negative media

The quasi-magnetostatic (briefly, magnetostatic) microwaves propagation in a long, circular cross-section waveguide made of an anisotropic uniaxial mu-negative medium (here, a metamaterial consisting of split-ring resonators) is studied by using the quasi-static Maxwell's equations. Such waves do not exist in a circular dielectric waveguide made of an isotropic mu-negative medium. A Laplace equation for the magnetostatic potential of the system is obtained and solved. We derive the dispersion relation for the modes and present numerical solutions. Also, general expressions are derived and illustrated graphically for the power flow and the energy density of the different magnetostatic modes. Physically, these magnetostatic waves owe their existence to the anisotropic property of the system, where in the absence of this property, they disappear.