A new fundamental equation for the band spectra of dielectric layer films

摘要

This paper derives and demonstrates a new fundamental equation for the frequency spectra of one-dimensional photonic crystals. Commonly, the frequency spectra.(q) as a function of Brillouin wavevector q for waves propagating through a one-dimensional photonic crystal are calculated from the trace of the transfer matrix. This paper develops a novel factored expression for band spectra calculations, tan(2) qd/2 = tan(k(N)a(N) -alpha(N)) x tan(k(N)a(N) -beta N), where N is the number of layers per period, d is the unit cell width and k(i) = n(i)omega/c is the local wavevector in the ith layer of width 2a(i) and refractive index n(i). Angles (alpha(N), beta(N)) depend on the parameters of all N layers but are independent of a(N); in particular, in the limit of two layers, (alpha(2), beta(2)) correspond to the even/odd parity solutions at the center and the edge of the Brillouin zone. The derived spectral expression for the first time provides separate eigenvalue conditions for consecutive band edges at the center and the edge of the Brillouin zone for any N, which is especially useful in separating nearly-degenerate band edges. Besides being applicable everywhere the transfer matrix formalism is used, such as in finding the Bloch phase that is necessary in finite crystal calculations, the formalism is especially convenient for tailoring bandgaps and for calculating impurity modes in dielectric stacks. Overall, the present results provide an alternate analytic structure for discussing and designing one-dimensional photonic crystals.