# Problems

## Maxwell's equations

FAME is used for solving three-dimensional source-free Maxwell's equations governed by the following equations

where ω represents the frequency, *E* and *H* are the electric and magnetic fields, respectively. The magnetic induction *B* and the electric displacement field *D* satisfy the constitutive relations

in which ε and μ represent the permittivity and the permeability, respectively, ζ and ξ are magnetoelectric parameters. If the waves travel in vacuum, then these parameters satisfy

where *ε _{0}* is the vacuum permittivity and

*µ*

_{0}is the vacuum permeability.

Three type of (complex) media are considered.

### Isotropic materials

"**Isotropic**" means that the physical and chemical properties of the material are not changed by the characteristics of the direction. In other words, the performance value measured in different direction are the same. Most materials, such as all the gas, liquid and amorphous, are isotropic, moreover, the electric field *E* and electric flux density *D* are parallel to one another, and the relationships between the fields can be expressed using constants. In isotropic material,

### Anisotropic materials

"**Anisotropic**" is opposite to isotropic, that is, these fields are not necessarily parallel for more complex materials, such as crystals and many metamaterials. All or part of the physical and chemical properties of the material are changed with different direction. In an-isotropic material, the permittivity tensor and permeability tensor are given as

### Bi-isotropic materials

In **bi-isotropic** materials, the electric and magnetic fields are coupled. That is, the coupling parameters ζ and ξ were nonzero. Furthermore, in a negative index material, these magnetoelectric parameters are described as

where χ and γ are reciprocity and chirality parameter, respectively. Here is a tabular of classification: