Scaling laws for the transmission of random binary dielectric multilayered structures

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.

Figure 1

Transmission spectra of (a) periodic and (b) random stratified binary media with $N={10}^2$ layers. For the random structure, we performed an average over ${10}^2$ distinct realizations of disorder, which plays quite distinct roles in the range of frequencies corresponding to transmitting and nontransmitting bands. The inset shows in detail the disorder induced transmission peak at the center of the stop band.

Figure 2

(a) Spectral average of the transmission versus the number of layers $N$. (b) Localization length in the vicinity of the half-wavelength mode. The average transmission decays with $1∕N^{1∕2}$, and the localization length diverges quadratically as one approaches the resonance mode. The estimate of the localization length was performed considering a finite structure with $N={10}^4$ layers, and the averages were taken over ${10}^3$ distinct random sequences. The saturation of the localization length very close to the resonance frequency is a finite-size effect.

Figure 3

Size dependence of the geometric $\left(\exp ⟨\ln T⟩\right)$, harmonic $\left({⟨1∕T⟩}^{-1}\right)$, and arithmetic $\left(⟨T⟩\right)$ average values of the transmission at the quarter-wavelength resonance frequency. Both the geometric and harmonic averages display an stretched exponential scaling, while the arithmetic average exhibits a slower power-law decay. Here, we averaged over ${10}^4$ distinct random sequences.

Figure 4

Probability distribution function of the transmission logarithm at the quarter-wavelength resonance frequency. The numerically obtained distribution is well fitted by a Gaussian (dashed line), thus corroborating the central limit theorem prediction. Data were the same as those used in Fig. 3 for $N={10}^4$ layers.

Figure 5

(a) Spectral average of the transmittance in a frequency range around the quarter-wavelength mode as a function of the number of layers. It exhibits an asymptotic stretched exponential decay as the number of layers is increased. (b) The disorder averaged localization length in the vicinity of the quarter-wavelength resonance showing its slow logarithmic divergence. Here, we averaged over ${10}^4$ random sequences. The localization length in (b) was estimated from structures having $5\times {10}^3$ layers.

Figure 6

Spectral average of the transmission around (a) the half- and (b) the quarter-wavelength resonances as a function of the disorder strength. Disorder plays opposite trends in the transmission behavior near each resonance. Around the half-wavelength resonance, Anderson localization is predominant and the transmission decreases with increasing disorder. On the other hand, near the quarter-wavelength resonance, disorder induces the emergence of states within the stop band and thus promotes a small transmittance in this frequency range. Data were obtained from $5\times {10}^3$ random sequences with ${10}^2$ layers each.