Localization-delocalization transition of photons in one-dimensional random $n$-mer dielectric systems

In this work, we have both theoretically and experimentally investigated the photonic transmission in a one-dimensional random $n$-mer (RN) dielectric system. Due to the positional correlation in the RN structure, the localization-delocalization transitions of photons happen at expected frequencies of photons. Multiple resonant transmissions are found in the photonic band gap. At each resonant mode, zero-Lyapunov exponent and undecayed field distribution of electromagnetic waves have been found through the whole system. Furthermore, the channel is opened for photonic transport at the resonant frequency, and the density of states of photons increases step by step as frequency increases. The theoretical results are experimentally demonstrated in RN dielectric multilayer films of $\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ti}{\mathrm{O}}_2$ for visible and near infrared light.

Figure 1

(a) The photonic transmission spectrum and (b) the dispersion relation in RD multilayer of $\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ti}{\mathrm{O}}_2$. (c) The photonic transmission spectrum and (d) the dispersion relation in a random multilayer of $\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ti}{\mathrm{O}}_2$. The dielectric materials $A$ and $B$ are $\mathrm{Si}{\mathrm{O}}_2$ and $\mathrm{Ti}{\mathrm{O}}_2$, respectively. The layer thicknesses $d_A$ and $d_B$ in these two multilayer films are chosen to satisfy $n_A{d}_A=n_B{d}_B\equiv a$, and the total thickness of the multilayer is $d$. The total number of layers in each film is $N=365$.

Figure 2

The electric field distributions along the layer growth in the RD multilayer of $\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ti}{\mathrm{O}}_2$. (a) and (b), the extended distributions around the resonant frequency ${\omega }_{\mathit{RD}}=\pi c∕2a$, i.e., ${\lambda }_{\mathit{RD}}=700\mathrm{nm}$; (c) and (d), the amplitude distributions between the extended and localized states away from the resonant mode ${\lambda }_{\mathit{RD}}$; and (e) and (f), the localized distributions away from the resonant mode ${\lambda }_{\mathit{RD}}$. Here, the layer thicknesses $d_A$ and $d_B$ are chosen to satisfy $a=n_A{d}_A=n_B{d}_B=175\mathrm{nm}$. The total number of layers is $N=365$, and $d$ is the total thickness of the film.

Figure 3

The transmission coefficient and the density of states $g\left(\omega \right)$ as a function of the frequency in several RN multilayers. (a) and (f), a random-dimer multilayer with $N=365$; (b) and (g), a random-trimer multilayer with $N=480$; (c) and (h), a random-tetramer multilayer with $N=595$; (d) and (i), a random 5-mer multilayer with $N=710$; and (e) and (j), a random 6-mer multilayer with $N=825$. Other parameters are the same as in Fig. 1.

Figure 4

The Lyapunov coefficient $\Gamma$ as a function of the frequency in several RN multilayers. (a) A random-dimer multilayer with $N=365$, where ${\omega }_{\mathit{RD}}=(1∕2)(\pi c∕a)$; (b) a random-trimer multilayer with $N=480$, where ${\omega }_R=(1∕3)(\pi c∕a)$, $(2∕3)(\pi c∕a)$, respectively; (c) a random-tetramer multilayer with $N=595$, where ${\omega }_R=(1∕4)(\pi c∕a)$, $(1∕2)(\pi c∕a)$, $(3∕4)(\pi c∕a)$, respectively; (d) a random 5-mer multilayer with $N=710$, where ${\omega }_R=(1∕5)(\pi c∕a)$, $(2∕5)(\pi c∕a)$, $(3∕5)(\pi c∕a)$, $(4∕5)(\pi c∕a)$, respectively; and (e) a random 6-mer multilayer with $N=825$, where ${\omega }_R=(1∕6)(\pi c∕a)$, $(1∕3)(\pi c∕a)$, $(1∕2)(\pi c∕a)$, $(2∕3)(\pi c∕a)$, $(5∕6)(\pi c∕a)$, respectively. Other parameters are the same as in Fig. 1.

Figure 5

Cross-sectional SEM image of the RD $\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ti}{\mathrm{O}}_2$ multilayer film with $N=30$. The RD sequence is marked, where dielectric materials $A$ and $B$ are $\mathrm{Si}{\mathrm{O}}_2$ and $\mathrm{Ti}{\mathrm{O}}_2$, respectively. Note that the cross section of the sample has not been polished, and in order to increase the conductivity, the sample is covered by a gold film with a thickness of $5\mathrm{nm}$ or so.

Figure 6

The measured and calculated transmission coefficient $T$ as a function of the wave number ${\lambda }^{-1}$ in the RN multilayer films of $\mathrm{Si}{\mathrm{O}}_2∕\mathrm{Ti}{\mathrm{O}}_2$. (a) A random-dimer multilayer with $N=30$, and (b) a random-trimer multilayer with $N=40$, (c) a random-tetramer multilayer with $N=50$, and (d) a random 5-mer multilayer with $N=60$. Note that in each figure, the solid curve is for the measured spectrum, while the dashed curve is for the calculated one. Here absorption of dielectric materials has been included.