Band edge states of the ⟨n⟩=0 gap of Fibonacci photonic lattices

The stationary and normally incident electromagnetic modes in Fibonacci lattices with generating layers of positive and negative indices of refraction are calculated by a transfer-matrix technique. It is shown that the condition for constructive interference of reflected waves is fulfilled when the ratio of optical paths in positive and negative media are given by the golden ratio. Furthermore, in the long-wavelength limit, it is demonstrated that the edges of the < n > = 0 gap are the frequencies satisfying the conditions <epsilon > = 0 and <mu > = 0.