One dimensional Kronig-Penney model with positional disorder: Theory versus experiment

We study the effects of random positional disorder in the transmission of waves in the one-dimensional Kronig-Penny model formed by two alternating dielectric slabs. Numerical simulations and experimental data revealed that the so-called resonance bands survive even for relatively strong disorder and large number of cells, while the nonresonance bands disappear already for weak disorder. For weak disorder we derive an analytical expression for the localization length and relate it to the transmission coefficient for finite samples. The obtained results describe very well the experimental frequency dependence of the transmission in a microwave realization of the model. Our results can be applied both to photonic crystals and semiconductor superlattices.

Figure 1

Kronig-Penney model with Teflon bars of constant length $d_b\left(n\right)=d_b\left(n+1\right)=d_b$ and air spacings $d_a\left(n\right)$ defined by Eq. (2.1).

Figure 2

(Color online) The microwave guide. Left: overview of the brass waveguide of perimeter $P=234\text{cm}$ with height $h=1\text{cm}$ and width $w=2\text{cm}$. Right: an enlargement of the waveguide showing the two carbon pieces (black) and one of the electric dipole antennas. The other lies on a brass lid (not shown) inserted near the other carbon piece. The white pieces are the Teflon bars.

Figure 3

(Color online) Transmission spectrum for a 26-cell array. Top: $7.3<\nu <11\text{GHz}$. Bottom: $11<\nu <15\text{GHz}$. Curve (a): experimental data for the periodic array. Curve (b): $\left|S_{12}^N\right|$, multiplied by 1/2, for a slightly random $\left(ϵ=0.049\right)$ array. Curve (c): $\left|S_{12}^{\left(0\right)N}\right|$ for the perfectly periodic array $\left(d_a=d_b=4.078\text{cm}\right)$. Dashed vertical lines mark the position of the Teflon resonances.

Figure 4

(Color online) Transmission spectrum $\left|S_{12}^N\right|$ for 26-cell arrays: (a) weak $\left(ϵ=3.0\times {10}^{-2}\right)$, (b) medium $\left(ϵ=12.3\times {10}^{-2}\right)$, and (c) strong disorder $\left(ϵ=49.0\times {10}^{-2}\right)$. Transfer matrix calculations are shown by thin black solid curves and experimental measurements by thick red solid curves. Dashed curves correspond to analytical expression (5.18) for the inverse localization length. Perpendicular lines mark the position of the Teflon resonances.

Figure 5

(Color online) Transmission spectrum $\left|S_{12}^N\right|$ (solid curves) and $\text{exp}\left(-l_{\text{loc}}\right)$ (dashed curves) for $N=100$ and $N=400$: (a)–(c) $N=100$ cells; (a) weak, (b) medium, and (c) strong disorder. (d)–(f) $N=400$ cells; (d) weak, (e) medium, and (f) strong disorder.

Figure 6

(Color online) (a) Form factor $F\left(\nu ,d_a,d_b,n_a,n_b\right)$ for $d_a=d_b=4.078\text{cm}$, $n_a=1$, and $n_b=\sqrt{2.08}$; (b) inverse localization length $l_{\text{loc}}^{-1}$ for $N=400$ for weak (solid curve), medium (dashed curve), and strong (dotted curve) disorder.