Anderson localization of partially incoherent light

We study Anderson localization and propagation of partially spatially incoherent wavepackets in linear disordered potentials, motivated by the insight that interference phenomena resulting from multiple scattering are affected by the coherence of the waves. We find that localization is delayed by incoherence: the more incoherent the waves are, the longer they diffusively spread while propagating in the medium. However, if all the eigenmodes of the system are exponentially localized (as in one- and two-dimensional disordered systems), any partially incoherent wavepacket eventually exhibits localization with exponentially decaying tails, after sufficiently long propagation distances. Interestingly, we find that the asymptotic behavior of the incoherent beam is similar to that of a single instantaneous coherent realization of the beam.

Figure 1

The disordered potential and its eigenmodes. (a) Small section of the dimensionless disordered potential $V(x)=-2\delta n(x)({\mathit{kx}}_0){}^2/n_0$, where $k=2\pi n_0/\lambda$, $\lambda =514$ nm, $x_0=1$ $\mu$m, $n_0=1.45$. (b, c) Typical eigenfunctions of the structure with a smaller (b) and larger (c) spatial extent, in a sample of width $L=4$ mm. (d) Logarithmic plot of $|u_n(x)|$ from (b) with exponentially decaying tails, which are the fingerprint of Anderson localization.

Figure 2

(a) Lyapunov exponents ${\gamma }_n$ extracted from the eigenfunctions $u_n$ (solid blue line) and via the transfer matrix approach (black squares). Shown are averages over 40 different realizations of the random potential; the standard deviation is smaller than the thickness of the line. (b) The Thouless (solid blue line) and the Heisenberg (dashed red line) times (propagation distances) vs. $n$ for the system with $L=4$ mm. Horizontal line depicts $Z=10$ cm. See text for details.

Figure 3

Intensities of the incoherent beam after $Z=10$ cm of propagation, for different system parameters, averaged over 40 realizations of the disorder. (a,b) Beam intensity for absorbing (a) and reflecting (b) boundary conditions for beams with ${\sigma }_I=10$ $\mu$m, and ${\sigma }_C=1$ $\mu$m (top black line), $3$ $\mu$m (middle blue line), and $5$ $\mu$m (bottom red line); $L=4$ mm. (c) Beam intensity for absorbing boundary conditions and three values of $L$: $2$, $4$, and $8$ mm (top to bottom) [${\sigma }_I=10$ $\mu$m, ${\sigma }_C=1$ $\mu$m]. (d) The intensity for one specific instantaneous initial realization of the incoherent field (lower blue line) and the time-averaged intensity (upper black line) [${\sigma }_I=10$ $\mu$m, ${\sigma }_C=1$ $\mu$m, $L=8$ mm, absorbing boundary conditions].

Figure 4

Spatial power spectrum of the incoherent beam after $Z=10$ cm of propagation in a disordered medium with absorbing (thin dashed blue line) and reflecting (solid red line) edges, for an initial beam (thick solid black line) defined by ${\sigma }_I=10$ $\mu$m, ${\sigma }_C=1$ $\mu$m. All plots are averages over 40 realizations of the disorder.