Universality of eigenchannel structures in dimensional crossover

The propagation of waves through transmission eigenchannels in complex media is emerging as a new frontier of condensed matter and wave physics. A crucial step towards constructing a complete theory of eigenchannels is to demonstrate their spatial structure in any dimension and their wave-coherence nature. Here we show a surprising result in this direction. Specifically, we find that as the width of diffusive samples increases transforming from quasi-one-dimensional (1D) to two-dimensional (2D) geometry, notwithstanding the dramatic changes in the transverse (with respect to the direction of propagation) intensity distribution of waves propagating in such channels, the dependence of intensity on the longitudinal coordinate does not change and is given by the same analytical expression as that for quasi-1D. Furthermore, with a minimal modification, the expression describes also the spatial structures of localized resonances in strictly 1D random systems. It is thus suggested that the key ingredients of eigenchannels are not only universal with respect to the disorder ensemble and the dimension, but also of 1D nature and closely related to the resonances. Our findings open up a way to tailor the spatial energy density distribution in opaque materials.

Figure 1

Simulations show that as the width $W$ increases, so that a quasi-1D $\left(N=5\right)$ waveguide turns into a 2D slab $\left(N=800\right)$, the eigenchannel structure $|E_{\tau }{(x,y)|}^2$ in a single disordered medium undergoes dramatic changes in the transverse $y$ direction. For $N=5$, the transverse structure is always declocalized (a). For $N=800$, the transverse structure exhibits very rich localization behaviors for fixed disorder configuration: for example, the structure exhibits one, two, and three localization peaks in (b), (c), and (d), respectively. Moreover, the transverse structures of eigenchannels are qualitatively the same as the structures of $|v_n{\left(y\right)|}^2$. For all panels, the eigenvalue $\tau \approx 0.6$ and $L=50$.

Figure 2

Example of a transmittance spectrum $\mathcal{T}\left(\mathrm{\Omega }\right)$ (top) and the resonance structure ${\tilde{I}}_{{\mathcal{T}}_n}\left(x\right)$ corresponding to the resonant frequency ${\mathrm{\Omega }}_n$ (bottom).

Figure 3

Simulation results of the $x$-dependent participation ratio ${\xi }_n\left(x\right)$ for different values of $N$.

Figure 4

A numerical example shows that $v_n\left(y\right)$ (high transmission) is localized approximately in the region of $9\le y/L\le 12$ (a) and (b). This localization of $v_n\left(y\right)$ at the input edge leads to the localization of eigenchannel structure in the $y$ direction in the interior of the medium (d). When the phase of $v_n\left(y\right)$ in the region of $9.0\le y/L\le 9.1$ is twisted by $\pi$ [(c), dotted-dashed line], the eigenchannel structure is unaffected (e). Whereas if the $\pi$-phase twist is introduced in the region of $10.3\le y/L\le 10.4$ [(c), dashed line], the eigenchannel structure is significantly changed (f). $N=200$.

Figure 5

A numerical example shows that $v_n\left(y\right)$ (low transmission, ${\tau }_n=0.009)$ exhibits two localization peaks, which are located approximately in the region of $3\le y/L\le 5$ and $9\le y/L\le 11$, respectively (a). This necklace statelike structure of $v_n\left(y\right)$ at the input edge leads to two localization peaks of eigenchannel structure in the $y$ direction in the interior of the medium (b). When the phase of $v_n\left(y\right)$ in the region of $9\le y/L\le 10$ is twisted by $\pi$, the localization peak of the eigenchannel structure corresponding to the localization region $9\le y/L\le 11$ of $v_n\left(y\right)$ is significantly changed, whereas the other peak corresponding to the localization region $3\le y/L\le 5$ of $v_n\left(y\right)$ is unaffected (c). $N=200$.

Figure 6

Simulations of the resonance structures in 1D show that the $h^{*}$ function is universal with respect to both the resonant transmission $\mathcal{T}$ and the disorder strength $s$. This property is similar to the universality of the $h$ function that determines the eigenchannel structures in 2D and quasi-1D.

Figure 7

Simulations show that for 2D slabs with different values of $N$, the ensemble averaged depth profile $W_{\tau }\left(x\right)$ (symbols) are well described by the analytic expression given by Eqs. (12, 13, 14) for quasi-1D waveguides (black solid lines). Note that at a given $\tau$ and $x$, all symbols for distinct $N$ overlap. $L$ is fixed to be 50 and five ratios of $W/L$ are considered, which are 3, 3.6, 4.8, 7.2, and 12, corresponding to $N=50$, 60, 80, 120, and 200, respectively.

Figure 8

The ensemble-averaged resonance structure $I_{\mathcal{T}}\left(x\right)$ for two different disorder strengths: $s=0.1$ (top) and $s=0.3$ (bottom), whose corresponding localization-to-sample length ratios are 12 and 3, respectively.

Figure 9

Simulations show that by shaping the input field ${\mathbf{\psi }}_{\text{in}}$, one can realize different profiles of energy density inside the medium. (a) The input field ${\mathbf{\psi }}_{\text{in}}=\frac{1}{\sqrt{2}}({\mathit{v}}_h+{\mathit{v}}_l)$, with ${\mathit{v}}_{h\left(l\right)}$ corresponding to the right-singular vector of certain highly (low) transmitting eigenchannel. (b) The phase of ${\mathit{v}}_l$ is twisted by $\pi$ in the region $0.3\le y/L\le 0.4$. (c) The phase of ${\mathit{v}}_h$ is twisted by $\pi$ in the region $10.3\le y/L\le 10.4$.