Photonic crystal

Definition

Let $\Lambda \subset \R^3$ be the set of integer combinations of $d \in \set{1, 2, 3}$ linearly independent vectors, a lattice in their span. A photonic crystal is a non-magnetic linear medium in $\R^3$ whose relative permittivity $\varepsilon(\mathbf{x})$ is a real, measurable function that satisfies:

1. $\varepsilon(\mathbf{x} + \mathbf{R}) = \varepsilon(\mathbf{x})$ for all $\mathbf{R} \in \Lambda$ and $\mathbf{x} \in \R^3$;
2. $\varepsilon_{\min} \le \varepsilon(\mathbf{x}) \le \varepsilon_{\max}$ for constants $0 < \varepsilon_{\min} \le \varepsilon_{\max}$;
3. $\varepsilon$ is not constant, and it does not depend on the directions orthogonal to $\Lambda$.

Its time-harmonic modes are the solutions of the source-free Maxwell's equations, which reduce to $$\nabla \times \left( \frac{1}{\varepsilon(\mathbf{x})} \nabla \times \mathbf{H} \right) = \frac{\omega^2}{c^2} \mathbf{H}, \qquad \nabla \cdot \mathbf{H} = 0.$$ A photonic band gap is an open interval $I$ of frequencies $\omega > 0$ such that $\omega^2/c^2$ lies outside the spectrum of this operator for every $\omega \in I$.