Master’s Thesis · Condensed Matter Theory, TU Dortmund · 2026

Why Blaze2D Exists: Photonic Moiré Crystals

The research Blaze was built to make possible: a two-scale theory of twisted photonic crystals.

Blaze2D is the computational engine of a master’s thesis, Photonic Band Theory of Moiré Crystals: A Two-Scale Approach. This page is a short tour of that research, what it built, and why it forced a new solver into existence. (A dedicated thesis site follows; this is the bridge.)

Photonic moiré crystals

Overlay two photonic crystals with a small relative twist and a new, much larger periodic pattern emerges: a moiré superlattice, the optical cousin of the patterns behind twisted bilayer graphene. The physics is richest at small angles, which is exactly where it becomes impossible to compute directly:

At generic twist angles the bilayer is incommensurate: there is no repeating unit cell at all, so a conventional solver has nothing to operate on.
At the special commensurate angles where periodicity is restored, the unit cell grows as roughly the inverse of the twist angle, so a brute-force “supercell” calculation explodes precisely where the interesting physics lives.

Photonic moiré lattices formed by twisting two crystals — Twisting two photonic crystals produces an emergent moiré superlattice whose period far exceeds that of the layers it is built from.

A two-scale envelope theory

The thesis sidesteps the supercell entirely. Locally, every patch of the twisted bilayer looks like an ordinary crystal with a two-atom basis; only the inter-layer registry drifts slowly from patch to patch. A single dimensionless quality parameter decides when this picture holds: the twist angle $\theta$ measured against the stop-band width $\gamma$ ,

\beta := \frac{\theta}{\gamma}.

When $\beta \ll 1$ the local crystal barely changes over the distance light needs to “feel” the lattice, and a two-scale separation is justified. A systematic expansion in the scale ratio $\eta = a/L$ (monolayer period over moiré period) then projects the full Maxwell operator onto a few retained bands, producing an effective Hamiltonian defined on the registry domain and independent of the twist angle:

\mathcal{H}_{\text{eff}} = \Lambda + \eta\!\left[\tfrac{1}{2}\!\left(v^{(i)}\Pi_i + \Pi_i v^{(i)}\right) + U^{(1)}_{\text{rem}}\right] + \eta^{2}\!\left[\tfrac{1}{2}\,\Pi_i\,M^{-1}_{ij}\,\Pi_j + U^{(2)}_{\text{rem}}\right] + \mathcal{O}(\eta^{3}).

Every ingredient is a physical field on the registry: the local band energies $\Lambda$ , the group velocities $v^{(i)}$ , the inverse-mass tensor $M^{-1}$ , the covariant momentum $\Pi_i$ carrying the Berry connection, and the Born–Huang leakage potentials $U_{\text{rem}}$ . The twist angle re-enters only through the single scalar $\eta$ .

Moiré miniband dispersion — Moiré minibands: the slow envelope modulation reorganizes the spectrum into flat, closely spaced bands.

Quadratic bandwidth scaling with twist angle — A θ² bandwidth scaling, predicted by the envelope theory and independently reproduced by the reference solver.

The headline quantitative result is this quadratic ( $\theta^2$ ) bandwidth scaling of the lowest minibands, predicted by the envelope theory and independently confirmed by a finite-difference (FDFD) reference that is itself checked against MPB.

The data MPB cannot give

The framework’s real payoff is geometric. Because it works from full Bloch eigenvectors, it produces spatially resolved fields (Berry connection and curvature, the Born–Huang potential, velocity and effective-mass maps) that decompose the moiré physics in a way no conventional supercell eigenvalue calculation can.

Berry curvature and Born–Huang geometric fields on the registry domain — Geometric fields on the registry domain: Berry curvature and the Born–Huang metric. These are the quantities the envelope Hamiltonian is built from, and the reason the solver had to expose eigenvectors, not just eigenvalues.

Why this required a new solver

The envelope pipeline asks for the full Bloch eigenvector at every point of a dense registry sweep: thousands of local eigenproblems per configuration. That specific combination of demands is what existing tools could not meet together:

Throughput. Thousands of solves per sweep make solver speed the binding constraint.
Data transparency. The theory consumes eigenvectors, group velocities, and Berry-phase data directly: operator-level quantities most solvers never expose.
Control and understanding. Owning the solver meant owning every numerical choice the geometric data depends on: band tracking through avoided crossings, gauge conventions, dielectric smoothing.

Blaze2D was written to fill exactly this gap: fast enough for the dense sweep, transparent enough to hand the envelope framework everything it needs, and validated against the same FDFD/MPB references used throughout the thesis. The speed and convenience were necessary; the data extraction was the part that made the research possible at all.

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