# Background

FAME is aimed at calculating the band structure of **three-dimensional** **linear nondispersive** photonic crystals and metamaterials. The governing equation considered here is the source-free Maxwell's equations in the frequency domain, i.e.

where ω is the frequency of the electromagnetic waves. The description of the media consists in the material parameters of the constitutive relations

in which ε and μ are the permittivity and the permeability of the media, respectively, ξ is the magnetoelectric coupling parameter of the media and ξ* means the complex conjugate of ξ . More importantly, in our scenario, these parameters ε, μ and ξ are periodic in three directions, i.e.

where **a_1, a_2** and **a_3** are lattice translation vectors.

Then due to the famous Bloch theorem, electromagnetic waves have to satisify

where F refers to E, B, D or H in Maxwell's equations and **k **is the wavevector .

Depending on the specific form of ε, μ and ξ, up to now, three types of (complex) media can be routinely dealt with by FAME.

### i) Isotropic photonic crystals

"**Isotropic**" simply means that the properties of the media are the same in all directions. Specifically, in isotropic photonic crystals, ξ vanishes identically, and

### ii) Anisotropic photonic crystals

"**Anisotropic**", as opposed to "isotropic", means the properties of the media vary with directions. In anisotropic photonic crystals, ξ also vanishes identically, and the permittivity and permeability are in general 3-by-3 Hermitian positive-definite dyadics as follows

### iii) Bi-isotropic complex media

In **bi-isotropic** complex media, the permittivity and the permeability is the same as the isotropic case, and the magnetoelectric coupling parameter ξ is not zero but can be further described by

where χ and γ are reciprocity and chirality parameter, respectively. Depending on whether χ and/or γ are zero, the **bi-isotropic** complex media can be classified into the following four types, including the isotropic one:

So far, only the Pasteur media can be dealt with by FAME.

Any three-dimensional crystal can be identified as one of the 7 lattice systems and the 14 **Bravais lattices**. Practical knowledge of all lattice systems and Bravais lattices can be found, for example, in https://en.wikipedia.org/wiki/Bravais_lattice . For a specific three-dimensional crystal, the lengths of lattice translation vectors **a_1, a_2** and **a_3** and the angle between any two of them are required for our simulation.

It is worth noting that it is the factor exp(i **k \cdot a_l), **l=1,2,3** **that will play an important role in the solution to Maxwell's equations, not **k **itself. There is no need for the wavevector **k **to sweep the whole three-dimensional space**. **In fact, we can define reciprocal lattice vectors to be wavevectors **h** such that the following equality holds for given lattice translation vectors** ****a_1, a_2** and **a_3**

The solution **h** is just integer multiples of primitive translation vectors **b_1, b_2** and **b_3,** as defined below, of the reciprocal lattice,

Then the wavevector k can be spanned by **b_1, b_2** and **b_3** with coefficients lying between 0 and 1, i.e.

In other words, it is sufficient that the wavevector k varies within the first Brillouin zone which is just the parallelepiped formed by **b_1, b_2** and **b_3**. Furthermore, thanks to** **the symmetries in the point group of the lattice, the first Brillouin zone can be generated from the **irreducible Brillouin zone.** Particularly, in practical band structure calculations, we only concern about **k**'s which are along the segments connecting any two corner points of the **irreducible Brillouin zone**. The irreducible Brillouin zone of various lattices and their transformation relation have documented detailed in the reference https://doi.org/10.1016/j.commatsci.2010.05.010 .