Transmission Eigenvalues and the Bare Conductance in the Crossover to Anderson Localization

We measure the field transmission matrix $t$ for microwave radiation propagating through random waveguides in the crossover to Anderson localization. From these measurements, we determine the dimensionless conductance $g$ and the individual eigenvalues ${\tau }_n$ of the transmission matrix $t{t}^{†}$ whose sum equals $g$. In diffusive samples, the highest eigenvalue, ${\tau }_1$, is close to unity corresponding to a transmission of nearly 100%, while for localized waves, the average of ${\tau }_1$, is nearly equal to $g$. We find that the spacing between average values of $\ln {\tau }_n$ is constant and demonstrate that when surface interactions are taken into account it is equal to the inverse of the bare conductance.

Figure 1

(a) Schematic experiment setup. Source and detector wire antennas are mounted on a two-dimensional translation stage, so they could move freely on a two- dimensional grid. Measurements are made for two orthogonal linear polarizations by rotating the wire antennas. (b) Speckle patterns from two different source positions $a$ and $a^{\prime }$ at $a$ single frequency in one random realization for diffusive waves with $g=6.9$, (upper row) and localized waves with $g=0.37$, (lower row).

Figure 2

Spectra of the transmittance $T$ and transmission eigenvalues ${\tau }_n$ for (a) diffusive sample of $L=23\mathrm{cm}$ with $g=6.9$ and (b) localized sample of $L=40\mathrm{cm}$ with $g=0.37$. The black dashed line gives $T=\sum _{a,b=1}^N|t_{ab}{|}^2=\sum _{n=1}^N{\tau }_n$ and the solid lines are spectra of ${\tau }_n$.

Figure 3

“Crystallization” of transmission eigenvalues. (a) Probability density of $\ln {\tau }_n$ (lower curves) and the density of $\ln \tau$ (top dash curve), $P(\ln \tau )=\sum _{n}P(\ln {\tau }_n)$ for diffusive sample with $g=6.9$. (b) Probability density of $u=u_n\equiv \ln {\tau }_n-⟨\ln {\tau }_n⟩$ for $n$ between 2 and 10, for the same sample as in (a) compared with a Gaussian distribution. (c) Variation of $⟨\ln {\tau }_n⟩$ with channel index $n$ for sample lengths $L=23$ (circle), 40 (square) and 61 (triangle) cm for both diffusive (green open symbols) and localized (red solid symbols) waves fitted, respectively, with black dash lines.

Figure 4

Illustration of the photon diffusion model for the spatial variation of intensity, $I(z)$.

Figure 5

Identification of $g^{\prime \prime }$ with the bare conductance $g_0$. The constant products of $g^{\prime \prime }{L}_{\mathrm{eff}}$ for three different lengths for both diffusive and localized samples give the localization length $\xi$ in the two frequency ranges.