(0) Question:

Can we define a proper refractive index of photonic crystals near the bandgap?

(1) Meaning of phase refractive index in periodic structures

Refractive index is generally meaningless in periodic structures ! (Fig. 1) (breakdown of Snell's law)

This is true for diffraction gratings and weakly-modulated photonic crystals.

Fig.1: Phase refractive index calculated by textbook formula for 1D photonic crystal and 1D empty lattice.

How about strongly-modulated photonic crystals?

(2) How about strongly-modulated photonic crystals?

Dispersion surface becomes rounded near the PBG (similar to Fermi surface in metals)(Fig. 2). Such dispersion surface resembles to that of homogeneous dielectric materials (Fig. 3).
Thus, Snell's law can be applied and effective refractive index can be defined.
However, the propagation direction can be negative because dw/dk can be negative in photonic crystals.

Fig. 2: dispersion surfaces of hexagonal PCs near the PBG

Fig. 3: dispersion surface plot for light incident. PC near the PBG (left) and dielectric (right).

,(3) Effective index states in photonic crystals:
Analogy to effective-mass states in semiconductor

Fig. 4. Effective-index model for photonic crystals and effective-mass model for electronic crystals in the vicinity of bandgaps.

In the vicinity of the PBG, (1) Snell's law holds (2) Effective index can be defined. (3) Light propagation direction can be negative. This leads to negative refraction phenomenon in the negative-index state

(4) Negative Refraction

(5) Numerical Simulation of Negative Refraction