Perfect transmission and self-similar optical transmission spectra in symmetric Fibonacci-class multilayers

We study the transmission properties of light through the symmetric Fibonacci-class $\left[\mathrm{SFC}\mathrm{}\right(n\left)\mathrm{}\right]$ quasiperiodic dielectric multilayers, which possess a mirror symmetry. For a normal incidence of light, many perfect transmission peaks (the transmission coefficients are unity) are numerically obtained. The transmission coefficient exhibits a two-cycle feature in a family of the $\mathrm{SFC}\mathrm{}\left(n\right)\mathrm{}$ with an odd n, while a three-cycle feature in another family with an even n. The scaling factors $f\left(n\right),$ which give a description of the self-similar behaviors of transmission spectra, are analytical obtained. Let $m_{\mathrm{ij}}^{\mathrm{}\left(n\right)\mathrm{}}\mathrm{}\left(k\right)\mathrm{}$ $\mathrm{}(i,j=1,2\mathrm{})\mathrm{}$ be the elements of the total transfer matrix of the $k\mathrm{th}$ generation of $\mathrm{SFC}\mathrm{}\left(n\right);$ it is proven that the positions (wavelength) of the perfect transmission peaks can be uniquely determined by $m_{12}^{\mathrm{}\left(n\right)\mathrm{}}{\mathrm{}\left(k\right)+m}_{21}^{\mathrm{}\left(n\right)\mathrm{}}\mathrm{}\left(k\right)=0.\mathrm{}$ The analytical results are very well confirmed by the numerical calculations.