Bi-isotropic material

Definition

For most materials, the electric field E and electric displacement field D (as well as the magnetic field B and inductive magnetic field H) are parallel to one another. These simple mediums are called isotropic, and the relationships between the fields can be expressed using constants. For more complex materials, such as crystals and many metamaterials, these fields are not necessarily parallel. When one set of the fields are parallel, and one set are not, the material is called anisotropic. Crystals typically have D fields which are not aligned with the E fields, while the B and H fields remain related by a constant. Materials where either pair of fields is not parallel are called anisotropic.

In bi-isotropic media, the electric and magnetic fields are coupled. The constitutive relations are

:D = \varepsilon E + \xi H,

:B = \mu H + \zeta E,

D, E, B, H, \varepsilon and \mu are corresponding to usual electromagnetic qualities. \xi and \zeta are the coupling constants, which is the intrinsic constant of each media.

This can be generalized to the case where \varepsilon , \mu, \xi and \zeta are tensors (i.e. they depend on the direction within the material), in which case the media is referred to as bi-anisotropic.

Coupling constant

ξ and ζ can be further related to the Tellegen (referred to as reciprocity) χ and chirality κ parameter

:\chi - i \kappa = \frac{\xi }{\sqrt{\varepsilon \mu}}

:\chi + i \kappa = \frac{\zeta }{\sqrt{\varepsilon \mu}}

after substitution of the above equations into the constitutive relations, gives

:D = \varepsilon E+ (\chi - i \kappa) \sqrt{\varepsilon \mu} H

:B = \mu H + (\chi + i \kappa) \sqrt{\varepsilon \mu} E

Frequency dependence of the chirality parameter can be described by Condon model.

Classification

Examples

Pasteur media can be made by mixing metal helices of one handedness into a resin. Care must be exercised to secure isotropy: the helices must be randomly oriented so that there is no special direction.

The magnetoelectric effect can be understood from the helix as it is exposed to the electromagnetic field. The helix geometry can be considered as an inductor. For such a structure the magnetic component of an EM wave induces a current on the wire and further influences the electric component of the same EM wave.

From the constitutive relations, for Pasteur media, χ = 0,

:D = \varepsilon E - i \kappa \sqrt{\varepsilon \mu} H.

Hence, the D field is delayed by a phase i due to the response from the H field.

Tellegen media is the opposite of Pasteur media, which is electromagnetic: the electric component will cause the magnetic component to change. Such a medium is not as straightforward as the concept of handedness. Electric dipoles bonded with magnets belong to this kind of media. When the dipoles align themselves to the electric field component of the EM wave, the magnets will also respond, as they are bounded together. The change in direction of the magnets will therefore change the magnetic component of the EM wave, and so on.

From the constitutive relations, for Tellegen media, κ = 0,

:B = \mu H + \chi \sqrt{\varepsilon \mu} E

This implies that the B field responds in phase with the H field.

References

References

1. (2017). "Origins of Willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization". Physical Review B.
2. (2010). "Electromagnetic Anisotropy and Bianisotropy: A Field Guide". World Scientific.
3. Lakhtakia, Akhlesh. (1994). "Beltrami Fields in Chiral Media". World Scientific.
4. (1994). "Electromagnetic Waves in Chiral and Bi-isotropic Media". Artech House.
5. (2024). "Optical Tellegen metamaterial with spontaneous magnetization". [[Nature Communications]].