Bloch impedance in negative index photonic crystals

The condition of both relative permittivity and permeability equal to $\ensuremath{-}1$ in negative index photonic crystals was addressed by the retrieval of the so-called Bloch impedance. The dispersion characteristics of a photonic crystal made of an array of air holes in an InGaAsP semiconductor layer were first compared (i) by solving the eigenvalue problem by plane wave expansion and (ii) by calculating the complex transmission and reflection coefficients for a finite slab. From the latter, the effective refractive index and the Bloch impedance are deduced by using a transfer matrix technique. The criterion of optical index $n=\ensuremath{-}1$, which is the key condition for the same focus for all angle incidence, is shown to be satisfied at a frequency where ${\ensuremath{\epsilon}}_{\mathit{eff}}\ensuremath{\approx}\ensuremath{-}5$ and ${\ensuremath{\mu}}_{\mathit{eff}}\ensuremath{\approx}\ensuremath{-}0.2$ and thus far from the condition for impedance matching. A dielectriclike mode of the negative index branch pointed out by field mapping explains the impossibility of effective permittivity and permeability matching to $\ensuremath{-}1$.