Hanbury Brown and Twiss correlations of Anderson localized waves

When light waves propagate through disordered photonic lattices, they can eventually become localized due to multiple scattering effects. Here we show experimentally that while the evolution and localization of the photon density distribution is similar in the two cases of diagonal and off-diagonal disorder, the density-density correlation carries a distinct signature of the type of disorder. We show that these differences reflect a symmetry in the spectrum and eigenmodes that exists in off-diagonally disordered lattices but is absent in lattices with diagonal disorder.

Figure 1

Observation of Anderson localization in lattices with diagonal and off-diagonal disorders. Top panels show simulation of the average wave-packet expansion in lattices with (a) diagonal and (b) off-diagonal disorder, when excited at a single site, showing similar dynamics. (c, d) Experimental measurements of the output distributions for the two types of disorder (blue, solid lines) as compared to the same distribution in a periodic lattice (green, dashed lines).

Figure 2

Measured density-density correlations ${\Gamma }_{r,q}=\langle I_r{I}_q\rangle /\langle I_r\rangle \langle I_q\rangle$ for localized wave packets in (a) lattices with diagonal disorder. The strong diagonal feature reflects finite coherence length of the waves (see text). (b) The extracted correlation function. (c) Density correlations in lattices with off-diagonal disorder, showing checker-like correlations. (d) The extracted correlation function, showing oscillating correlations.

Figure 3

Spectral symmetry in disordered lattices. (a) The spectrum (band) of eigenvalues for lattices with off-diagonal disorder. Note the symmetry in the eigenvalues with respect to zero, for every eigenvalue $\lambda$ there exists a counterpart eigenvalue $\tilde{\lambda }=(-\lambda )$ (see text). (b) In lattices with diagonal disorder no exact symmetry exists in the spectrum of eigenvalues. (c) An example of the spatial distribution of a pair of eigenmodes in the off-diagonal case that have symmetric eigenvalues [in this example the extreme eigenvalues in the system ${\lambda }_{\max }$ (solid line) and ${\lambda }_{\text{min}}$ (dashed)]. The two eigenmodes occupy the same region of the lattice and have the same spatial distribution, but vary in phase structure. A similar result can be observed for all other symmetric pairs. (d) In this diagonal disorder case, the eigenmodes associated with opposite eigenvalues occupy different regions of the lattice. (e) Experimental measurement of two spectrally symmetric eigenmodes in a lattice with off-diagonal disorder, showing their spatial similarity. (f) Measurement showing that in the diagonal disorder case the same location in the lattice cannot host two spectrally symmetric localized modes (see text).