Supercell periodicity of all solutions at rational Bloch phases in one-dimensional photonic crystals

Bloch waves in one-dimensional periodic optical systems are typically regarded as quasiperiodic fields. When the Bloch phase is a rational multiple of π, the Floquet multiplier becomes a root of unity, and the corresponding Bloch-wave solutions are strictly periodic on a finite supercell. At such spectral points within allowed bands, the two linearly independent Bloch waves form a basis of the solution space. As a consequence, every solution of the underlying Hill equation is periodic on the same supercell (or on a divisor of it). While this property follows from Floquet theory, it is not usually stated explicitly in the context of photonic crystals. The result is illustrated for a one-dimensional binary photonic crystal, where the field distribution repeats exactly over a finite number of unit cells.