Positive phase evolution of waves propagating along a photonic crystal with negative index of refraction

We analyze propagation of electromagnetic waves in a photonic crystal at frequencies at which it behaves as an effective medium with a negative index in terms of refraction at its interface with free space. We show that the phase evolution along the propagation direction is positive, despite the fact that the photonic crystal displays negative refraction following Snell's law, and explain it in terms of the Fourier components of the Bloch wave. Two distinct behaviors are found at frequencies far and close to the band edge of the negative-index photonic band. These findings contrast with the negative phase evolution that occurs in left-handed materials, so care has to be taken when applying the term left-handed to photonic crystals.