Properties of the dispersion relation in finite one-dimensional photonic crystals

Abstract

Using the transfer-matrix method we have expressed the dispersion relation of a finite, N-period, one-dimensional photonic crystal in terms of a frequency dependent function g(ω) determining important features of the band structure. We have then investigated the similarities and differences between the dispersion relation of a N-period crystal and that of an infinite one for finite and large values of the number N of unit cells. It is shown that in the frequency range where the infinite crystal has a bandgap, the dispersion relation of the finite crystal exhibits a bandgap of zero width for any value of N. The frequency ωC at which the null gap occurs corresponds to a zero of g(ω) and is independent of N. Around ωC and for sufficiently large values of N, the group velocity attains superluminal values. These results are general enough and were used to investigate the effects of finite crystal size on the properties of the zero-n− gap in photonic crystals consisting of alternating layers of right- and left-handed materials. In this case, the frequency at which the null gap occurs is insensitive to geometrical scaling of the structure.