Measurement of the Probability Distribution of Optical Transmittance on the Crossover to Anderson localization

We report measurements of spectra of the field transmission matrix $t$ for microwave radiation propagating through waveguide filled with randomly positioned dielectric scattering spheres in the Anderson localization transition. Diagonalizing the matrix product \textit{tt}$^{\dag }$ gives the transmission eigenvalues \textit{$\tau $}$_{n}$, which yields the optical transmittance, $T=\sum\nolimits_{a,b=1}^N {\left| {t_{ba} } \right|^2} =\sum\nolimits_{n=1}^N {\tau _n } $. The ensemble average of the transmittance is equal to the dimensionless conductance, $g=$. We show the probability distribution of transmittance $P(T)$ changes from Gaussian to log-normal as the value of $g$ decreases. The distribution $P(T)$ is analyzed in terms of the underlying transmission eigenvalues \textit{$\tau $}$_{n}$. For random samples with $g\sim $3.9, we found $P(T)$ follows a Gaussian distribution. For $g\sim $0.37, we observe a highly asymmetric distribution for --ln$T. $The sharp drop for high values of $T$ is attributed to the restriction that \textit{$\tau $}$_{n}<$1 and the repulsion between transmission eigenvalues even for localized samples. For $g\sim $0.04, the distribution of transmittance is nearly log-normal. The variance of -ln$T$, \textit{$\sigma $}$^{2}$, scales linearly with $<$-ln$T>$ as predicted by single parameter scaling even for weakly localized waves.