Alternative expression of the Bloch wave group velocity in loss-less periodic media using the electromagnetic field energy

In periodic optical media, the group velocity is defined as the gradient with respect to wave-vector of the corresponding Bloch mode frequency dispersion curve, forming the photonic band structure. Instead of deducing it from the numerically computed photonic crystal band structure, the group velocity can be calculated directly from the integral of the Poynting vector over the crystal unit cell, the physical meaning of which is immediately perceivable. The related formula, which can be regarded as the application of Hellmann–Feynman theorem to electromagnetism, has been reported previously though without proof. We provide hereafter a full derivation of that formula starting from Maxwell’s equations and we discuss its usefulness in photonics.