Scalar-Wave Dispersion in Vectorial Photonic Crystals via Site-Adapted p Orbitals
Electromagnetic waves are intrinsically vectorial and require description via polarization, unlike scalar fields such as acoustic pressure or electronic wavefunctions. In three dimensions, the transversality constraint further prevents any globally smooth transverse-polarization frame at the $\Gamma$ point, which would apparently rule out a simple scalar band structure for three-dimensional (3D) photonic crystals. We show here that site-adapted $p$-orbitals can realize scalar-wave dispersion: the induced band representation is isomorphic to the scalar elementary band representation up to a one-dimensional character twist, so the symmetry-enforced degeneracies and compatibility relations are the same. We demonstrate this mechanism experimentally in 3D photonic meta-crystals, where the local $p$-orbital axes adapt from site to site according to symmetry. In contrast to a fixed-polarization reduction (e.g., in 2D), our construction preserves site-polarization textures while simultaneously supporting a scalar network with one amplitude per site. Thus, it offers a pathway from vectorial photonic degrees of freedom to scalar band engineering, keeping polarization as an active design knob.