Scaling laws for the transmission of random binary dielectric multilayered structures

Abstract

We investigate several scaling aspects of the transmission spectrum of disordered one-dimensional dielectric structures. We consider a binary stratified medium composed of a random sequence of $N$ slabs with refraction indices satisfying the Bragg condition. The mode for which the optical thickness corresponds to half wavelength is insensitive to disorder and fully transparent. The average transmission in a frequency range around this resonance decays as $1∕{N}^{1∕2}$, and the localization length diverges quadratically as this resonance mode is approached. In the vicinity of the quarter-wavelength mode, the localization length diverges logarithmically and the frequency averaged transmission exhibits an stretched exponential dependence on the total thickness. At the quarter-wavelength resonance, the Lyapunov exponent for different realizations of disorder has a Gaussian distribution leading to distinct scaling laws for the geometric and arithmetic averages of the transmission. The scaling laws for the half- and quarter-wavelength modes are analogous to those found in electronic one-dimensional Anderson models with random dimers and pure off-diagonal disorder, respectively, which are known to display similar violations of the usual exponential Anderson localization.