Optical Necklace States in Anderson Localized 1D Systems

We report on the observation of nonlocalized modes or necklace states of light waves in disordered systems in the Anderson localized regime. The samples consist of positional-disordered binary multilayer systems. Anderson localized modes manifest themselves as narrow high-transmission peaks in the transmission spectrum, whereas the average of the logarithm of the transmission coefficient decreases linearly with thickness. Optical necklace states are observed as modes with a characteristic multiresonance time response and relatively fast decay time.

Figure 1

Measured transmission spectra for three values of the sample thickness $L$. The spectra contain narrow high-transmission peaks that, on average, become narrower upon increasing $L$.

Figure 2

Spectral average of the logarithm of the measured transmission versus sample thickness. The error bars are obtained by repeating the measurement in various spots on the sample and therefore reflect lateral sample inhomogeneities.

Figure 3

Top panel: calculated intensity distribution inside the sample. The internal resonances are localized modes. In rare occasions multiple resonances couple and lead to extended necklace modes. A 4-resonance necklace state can be observed around $\lambda =1855\mathrm{nm}$. Bottom panel: Transfer-matrix calculation of the transmission of a short laser pulse. Dashed curves: localized modes with high (red) and low (black) $Q$ factor. Solid lines: necklace states, with two (green), respectively, four (blue) resonances. The refractive indices and pulse duration were taken equal to the experimental values.

Figure 4

Time-resolved transmission data. In (a) and (b) a localized mode is probed, with high, respectively, low $Q$ factor. In (c) a nearly symmetric pulse shape is observed that exhibits a fast decay time and a relatively large delay, typical for a multiple-resonance necklace state. Sample thickness: 250 layers. Gray curves: instrumental response (cross correlation between probe and gate), corrected for the trivial delay introduced by the effective refractive index $n_{\mathrm{eff}}=(d_A{n}_A+d_B{n}_B)/(d_A+d_B)$ of the sample.