Zero-mean circular Bessel statistics and Anderson localization

We demonstrate that a circular Bessel density function describes the electromagnetic field statistics in the Anderson localization regime using example numerical terahertz field data in strongly scattering media. This density function for localized fields provides a measure that allows identification and description in a manner akin to the Gaussian density function for weakly interacting scatterers, the mathematical framework to date for statistical optics. Our theory provides a framework for improved understanding of wave propagation in random media, random scattering media characterization, and imaging in and through randomly scattering media.

Figure 1

The random medium simulation geometry (not to scale) was composed of randomly distributed $50 \mu \mathrm{m}$ diameter cylindrical holes in an 8 mm long by 2 mm wide LN region $\left({\epsilon }_r=41.7\right)$: PML—perfectly matched layer; PEC—perfect electric conductor. A 0.75 THz plane wave with H in the $\widehat{\mathbf{z}}$-direction was incident from the left.

Figure 2

Probability density function of normalized total transmission power, $p\left(\widehat{T}\right)$, from numerical data (symbols) and Eq. (8) (curves) on linear (a) and semi-log (b) scales. The randomly located $50\mu \mathrm{m}$ diameter holes in LN $\left({\epsilon }_r=41.7\right)$ had dielectric constants of 20, 10, and 1, and a fill fraction of 0.42. The hole Gaussian density function had $\langle x\rangle =67\mu \mathrm{m}$, $\langle y\rangle =69\mu \mathrm{m}$ and ${\sigma }_{x,y}=7\mu \mathrm{m}$. As the hole dielectric constant decreases (increasing scatter), $g$ decreases as well.

Figure 3

Probability density function for the normalized intensity, $\widehat{I}$: negative exponential (green); theoretical result from Eq. (9) (red); $K$-distribution (3) (blue); numerical data (black dots). The scatterers were $50 \mu \mathrm{m}$ diameter dielectric holes in LN, where (a) ${\epsilon }_r=1$ with $g=0.32$ and $\alpha =0.12$, and (b) ${\epsilon }_r=20$ with $g=42$ and $\alpha =37$. In both figures, the Gaussian-distributed holes had $\langle x\rangle =78\mu \mathrm{m}$, $\langle y\rangle =60\mu \mathrm{m}$, and ${\sigma }_{x,y}=7\mu \mathrm{m}$.

Figure 4

Probability density function for the natural logarithm of normalized intensity, $\ln \widehat{I}$: fit to Eq. (10) with $\alpha =0.267$ (green dashed); fit to Gaussian with mean $\mu =-3.05$ and variance ${\sigma }^2=8.37$ (red); numerical data (black dots). The numerical data are the same as in Fig. 3.

Figure 5

Magnetic field statistics at the output plane, on both linear and semi-log scales for clarity: (a) and (b) are for strong scatter $\left({\epsilon }_r=1\right)$, where the theoretical line is calculated using Eq. (7); (c) and (d) show the weak scatter case $\left({\epsilon }_r=20\right)$, with the red line being the fit to a Gaussian density function. The numerical data are the same as in Fig. 3.

Figure 6

Statistics of the real part of the magnetic field taken at different depths inside the random medium: (a) strong scatter $\left({\epsilon }_r=1\right)$; (b) weak scatter $\left({\epsilon }_r=20\right)$. The red line shows the theoretical prediction using Eq. (7). The numerical field data are the same as that used to generate the results in Fig. 3.