Microwave magnetoelectric fields: helicities and reactive power flows

Symmetry principles play an important role with respect to the laws of nature. To put into a symmetrical shape the equations, coupling together the electric and magnetic fields, Maxwell added an electric displacement current. Such an additive, introduced for reasons of symmetry, resulted in appearing a unified-field structure: the electromagnetic field. The electric displacement current in Maxwell equations allows correct prediction of magnetic fields in regions where no free current flows and prediction of wave propagation of electromagnetic fields. The dual symmetry between electric and magnetic fields underlies the conservation of energy and momentum for electromagnetic fields [1]. It can be connected also with conservation of polarization of the electromagnetic field. In particular, this symmetry underlies the conservation of optical (electromagnetic) helicity [2–4]. As it is stated in Ref. [4], the dual electromagnetic theory inherently contains straightforward and physically meaningful descriptions of the helicity, spin and orbital characteristics of light. What kind of the source-free time-varying field structure one can expect to see when an electric displacement current is neglected and so the electromagnetic-field symmetry (dual symmetry) of Maxwell equations is broken? As an example of such a symmetry breaking, we can refer to the field structures studied in non-conductive artificial electromagnetic materials that exhibit zero (or near-zero) permittivity [5, 6]. In such materials, one has low-wave-number (index near zero) propagation of electromagnetic waves. So, the effective material parameters can be characterized by zero (or near-zero) permittivity. This clarifies the physical meaning of the term “zero permittivity.” For these metamaterials, no Maxwell correction (no electric displacement Abstract The dual symmetry between the electric and magnetic fields underlies Maxwell’s electrodynamics. Due to this symmetry, one can describe topological properties of an electromagnetic field in free space and obtain the conservation law of optical (electromagnetic) helicity. What kind of the field helicity one can expect to see when the electromagnetic-field symmetry is broken? The near fields originated from small ferrite particles with magneticdipolar-mode oscillations are the fields with the electric and magnetic components, but with broken dual (electric–magnetic) symmetry. These fields—called magnetoelectric (ME) fields—have topological properties different from such properties of electromagnetic fields. The helicity states of ME fields are topologically protected quantumlike states. In this paper, we study the helicity properties of ME fields. We analyze conservation laws of the MEfield helicity and show that the helicity density is related to an imaginary part of the complex power-flow density. We show also that the helicity of ME fields can be a complex value. The shown topological properties of the ME fields can be useful for novel nearand far-field microwave applications. The obtained results can find application for development of novel microwave metamaterials. Strongly localized ME fields, having both the real and imaginary helicity parameters, open unique perspective for sensitive microwave probing of structural characteristics of chemical and biological objects.