Full Transmission and Reflection of Waves Propagating through a Maze of Disorder

Thirty years ago, theorists showed that a properly designed combination of incident waves could be fully transmitted through (or reflected by) a disordered medium, based on the existence of propagation channels which are essentially either closed or open (bimodal law). In this Letter, we study elastic waves in a disordered waveguide and present direct experimental evidence of the bimodal law. Full transmission and reflection are achieved. The wave field is monitored by laser interferometry and highlights the interference effects that take place within the scattering medium.

Figure 1

Experimental setup. The $\mathbf{S}$ matrix is measured in the time domain, between two arrays of points placed 5 mm away from each side of the disordered slab. The array pitch is 0.8 mm ($\sim \lambda /4$). Flexural waves are generated on each point by a pulsed laser via thermoelastic conversion over a focal spot of $1{\mathrm{mm}}^2$. The normal component of the plate vibration is measured with an interferometric optical probe. The laser source and the probe are both mounted on 2D translation stages.

Figure 2

(a) Real part of the $\mathbf{S}$ matrix measured at $f=0.36\mathrm{MHz}$. The black lines delimit transmission and reflection matrices as depicted in Eq. (1). (b) Eigenvalues $s_i$ (red dots) and ${\widehat{s}}_i$ (blue squares) of the measured and normalized scattering matrices, $\mathbf{S}$ and $\widehat{\mathbf{S}}$, respectively. The eigenvalues are displayed in the complex plane. The black continuous line denotes the unit circle. (c),(d) Transmission eigenvalue histograms, $\rho (T)$ and $\widehat{\rho }(\widehat{T})$, averaged over the frequency bandwidth. Both distributions are compared to the bimodal law ${\rho }_b$ [red continuous line, Eq. (3)].

Figure 3

Absolute value of the wave field in the scattering medium at $f=0.36\mathrm{MHz}$ associated with (a), an incident plane wave, (b), a closed eigenchannel, (c), an open eigenchannel deduced from the measured $\mathbf{S}$ matrix, and (d), an open eigenchannel deduced from the normalized $\widehat{\mathbf{S}}$ matrix. The corresponding intensities averaged over the wave guide section ($y$ axis) are shown versus depth $x$ in lower panels (e)–(h). They are all normalized by the intensity at the plane of sources ($x=0$).