= Solution
\b[A <photonic crystal> has a spatially periodic <dielectric> response on the scale of the optical <wavelength>.] For a nonmagnetic, lossless, nondispersive <dielectric>, the source-free <Maxwell equations> can be written as the eigenproblem
$$
\nabla\times\left[\epsilon_r(\mathbf r)^{-1}\nabla\times\mathbf H\right]=\frac{\omega^2}{c^2}\mathbf H.
$$
The <Bloch theorem> gives $\mathbf H_{n\mathbf k}=e^{i\mathbf k\cdot\mathbf r}\mathbf u_{n\mathbf k}(\mathbf r)$ with periodic $\mathbf u$. The allowed frequencies form <photonic band structures>. Coupling of counterpropagating waves near a <Brillouin zone> boundary opens a stop band. A one-dimensional <dielectric> multilayer is a simple example; two-dimensional hole arrays and three-dimensional <dielectric> networks allow more extensive control.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2016/paper-1-photonic-bands.png]
{title=An exactly calculated one-dimensional dielectric bilayer has allowed bands separated by a stop band}
{source=paper-1-photonic-bands.py}
A \b[complete <photonic band gap>] excludes bulk propagating modes for every direction and <polarization> in a <frequency> interval. Periodicity alone does not guarantee such a gap: dimensionality, symmetry and refractive-index contrast matter. A missing or altered element can support a localized defect state in the gap, making a cavity; a line defect can guide light. Finite crystals have leakage and wavelength-dependent transmission rather than an infinitely sharp exclusion. Applications include selective mirrors, optical filters, small resonators and integrated waveguides.<Mathematical Methods in Nanophotonics, MIT 18.369, Spring 2016 by Steven G. Johnson>[Lecture 2: Maxwell eigenproblem; lectures 9–14: Bloch states, one-dimensional gaps and defect localization]{ref}
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