Optimizing Light Storage in Scattering Media with the Dwell-Time Operator

We prove that optimal control of light energy storage in disordered media can be reached by wave front shaping. For this purpose, we build an operator for dwell times from the scattering matrix and characterize its full eigenvalue distribution both numerically and analytically in the diffusive regime, where the thickness $L$ of the medium is much larger than the mean free path $\ell$. We show that the distribution has a finite support with a maximal dwell time larger than the most likely value by a factor $(L/\ell {)}^2\gg 1$. This reveals that the highest dwell-time eigenstates deposit more energy than the open channels of the medium. Finally, we show that the dwell-time operator can be used to store energy in resonant targets buried in complex media, without any need for guide stars.

Figure 1

Eigenvalue distribution of the dwell-time operator $Q_d$, evaluated for a disordered slab (length $kL=300$) embedded in a multimode waveguide ($N=287$). Analytical predictions (solid lines) are compared with numerical results (dots) obtained from the solution of the wave equation for 128 realizations of the slab, with dielectric function $\epsilon (\mathbf{r})=n_1^2+\delta \epsilon (\mathbf{r})$; $n_1=1.5$ and $\delta \epsilon (\mathbf{r})$ is uniformly distributed in $[-a,a]$. Results for three values of $a$ are represented, corresponding to $k\ell =21.4$, 9.3, 5.8.

Figure 2

(a) Intensity profiles inside a disordered slab (integrated over the transverse dimension) resulting from the propagation of different input states ${\psi }^{\mathrm{in}}$: the first mode of the waveguide (similar to a plane wave), the most open channel, and the eigenstate ${\psi }^{\max }$ of $Q_d$ associated with the largest eigenvalue ${\tau }^{\max }$. (b) Dwell times $\tau =⟨{\psi }^{\mathrm{in}}|Q_d|{\psi }^{\mathrm{in}}⟩$, averaged over 128 configurations (dots), corresponding to the different states ${\psi }^{\mathrm{in}}$ shown in (a) versus the Thouless time ${\tau }_{\mathrm{Th}}=L^2/{\pi }^2{D}_B=2L^2/{\pi }^2\ell v$. All times are normalized by $⟨\tau ⟩=(\pi /2)L/v$. Solid lines correspond to analytical predictions (see text).

Figure 3

(a), (b) Typical intensity patterns $I(x,y)$ resulting from the propagation of the states associated with the maximal eigenvalue of the absorption operator (a) and the extremal eigenvalue of $Q_d^H$ (b), in a disordered medium ($kL=150$, $k\ell =21.4$) containing three absorbers placed at depth $kx=100$ (white squares). Only the second absorber is resonant ($\mathcal{F}={10}^{-4}$, ${\omega }_0/\mathrm{\Gamma }={10}^4$). (c) Slices $I(y)={\int }_{80/k}^{95/k}dxI(x,y)$ of the patterns (a) and (b), averaged over 32 configurations.