Faustmann's formula

In physics, engineering and materials science, bi-isotropic materials have the special optical property that they can rotate the polarization of light in either refraction or transmission. This does not mean all materials with twist effect fall in the bi-isotropic class. The twist effect of the class of bi-isotropic materials is caused by the chirality and non-reciprocity of the structure of the media, in which the electric and magnetic field of an electromagnetic wave (or simply, light) interact in an unusual way.

Definition

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

$D=\epsilon E+\xi H$

$B=\mu H+\zeta E$

D, E, B, H, ε and μ are corresponding to usual electromagnetic qualities. ξ and ζ are the coupling constants, which is the intrinsic constant of each media.

This can be generalized to the case where ε, μ, ξ and ζ 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

$\chi -i\kappa =\frac{\xi }{\sqrt{\epsilon \mu }}$

$\chi +i\kappa =\frac{\zeta }{\sqrt{\epsilon \mu }}$

after substitute the above equations into the constitutive relations, gives

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

$B=\mu H+(\chi +i\kappa )\sqrt{\epsilon \mu }E$

Classification

	non-chiral $\kappa =0$	chiral $\kappa \ne 0$
reciprocal $\chi =0$	simple isotropic medium	Pasteur Medium
non-reciprocal $\chi \ne 0$	Tellegen Medium	General bi-isotropic medium

Examples

Pasteur media can be made by mixing metal helices of one handedness into a resin. Care has been 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 is a sort of inductor. The magnetic component of an EM wave will induces a current on the wire and further influence the electric component of the same EM wave.

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

$D=\epsilon E-i\kappa \sqrt{\epsilon \mu }H$

the D field was delayed the respond from the H-field by a phase i.

Tellegen media is an 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 are turned by the electric part of an EM wave, the magnets will also be changed, due to they bounded together. The change of magnets direction will change the magnetic component of the same EM wave.

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

$B=\mu H+\chi \sqrt{\epsilon \mu }E$

The D field was responded immediately from the H-field.

See also

• Anisotropy
• Chirality (electromagnetism)
• Metamaterial
• Reciprocity (electromagnetism)
• Maxwell's_equations#Constitutive_relations