Magnetic-field effects on one-dimensional Anderson localization of light

Transport of coherent waves in multiple-scattering media may exhibit fundamental, nonintuitive phenomena such as halt of diffusion by disorder called Anderson localization. For electromagnetic waves, this phenomenon was observed only in one and two dimensions so far. However, none of these experiments studied the contribution of reciprocal paths nor their manipulation by external fields. In order to weaken the effect of reciprocity of coherent wave transport on Anderson localization in one dimension, we studied light propagation through stacks of parallel Faraday-active glass slides exposed to magnetic fields up to 18 T. Measurements of light transmission statistics are presented and compared to one-dimensional (1D) transfer-matrix simulations. The latter reveals a self-organization of the polarization states in this system leading to a saturation of the Faraday rotation-induced reciprocity breaking, an increase of the localization length, and a decrease of transmission fluctuations when reciprocity is broken. This is confirmed experimentally for samples containing small numbers of slides while for larger samples a crossover from a 1D to a quasi-1D transport regime is found.

Figure 1

(a) Localization through a 1D stack of glass slides, and direction of the external magnetic field. (b),(c) Transition from 1D to quasi-1D due to spreading of the multiply reflected laser beam.

Figure 2

Simulated polarization states in transmission for samples of $N=1,6,8$, or 30 plates represented on the Poincaré sphere (Mollweide projection) at $B=18\mathrm{T}$ for the Faraday active plates studied in the paper. The color encodes the logarithm of the total transmission ($T=T_x+T_y$). $\mathrm{X}$ (respectively $\mathrm{Y}$) represents linear polarization on the $x$ (respectively $y$) axis, D and A stand for diagonal and antidiagonal (linear polarization on the first and second bisectors), and R and L are the circular polarizations (right and left).

Figure 3

Average of the logarithm of $T_x$, $T_{xx}$, and $T_{xy}$ without and with (a) Faraday rotation (FR) or (b) optical activity (OA) for the same rotation angle of ${48}^{\circ }$ for a single passage through a single plate. This corresponds to a circular birefringence $\mathrm{\Delta }n=4.72\times {10}^{-5}$ for FR in SF57 at $B=18\mathrm{T}$ and at the average thickness of the plates.

Figure 4

Variance of the normalized transmitted intensity $\mathrm{Var}\left(s\right)$ as a function of the number of plates $N$. (a) Faraday rotation, and (b) optical activity. Similarly to Fig. 3, the birefringence and the average thickness of the plates are the same for the FR and OA simulations.

Figure 5

1D light transport setup: a laser beam is sent through an OD filter to control the incident intensity before being expanded to $\approx 0.5\mathrm{cm}$. This expanded beam illuminates the sample made of SF57 plates (varying from 0 to 50) placed inside the 18-T superconducting magnet. A linear polarizer can be placed in front of the complementary metal-oxide semiconductor (CMOS) camera to select one plane of polarization. The sample is placed inside a sample holder which is rotated constantly via a rubber hose by a motor.

Figure 6

Measured images for four increasing numbers of plates ($N=0$, 5, 15, 20). The measurement was performed at $B=0\text{T}$ without polarizer in front of the camera. Color bars show measured intensities in a.u.

Figure 7

Histograms of the normalized transmitted intensity $s=T/〈T〉$ without [upper row (a)] and with [lower row (b), $B=18\mathrm{T}$] external magnetic field for $N=6,8$, and 30 plates. Simulation data are traced in blue and experimental data in orange.

Figure 8

Experimental data of the mean logarithmic relative transmission as a function of the number of plates $N$ at $B=0$ and 18 T. Data are measured with a polarizer at the output. The light curves are the simulation results [same data as in Fig. 3].

Figure 9

Experimental data of the variance of the normalized transmitted intensity $\mathrm{Var}\left(s\right)$ through a stack of $N$ Faraday active glass slides with (orange) and without (blue) magnetic field. Data are measured with a polarizer at the output. The ellipse highlights the measurements where $\mathrm{Var}\left(s_{xx}\right)$ is lower with magnetic field than without in the 1D regime. The light curves are the simulation results [same data as in Fig. 4].

Figure 10

Direction of propagation of the light electromagnetic fields $E$ in front (1) and behind (2) any optical system (dielectrics, interface,...).

Figure 11

Simulated polarization states in reflection for samples of $N=1,2,5$, or 30 plates represented on the Poincaré sphere (Mollweide projection) at $B=18\mathrm{T}$ for the Faraday active plates studied in the paper. The color encodes the total reflection ($R=R_x+R_y$). X (respectively Y) represents linear polarization on the $x$ (respectively $y$) axis, D and A stand for diagonal and antidiagonal (linear polarization on the first and second bisectors), and R and L are the circular polarizations (right and left).

Figure 12

Simulated reflection coefficients in the case of the same circular birefringence plotted against the sample length, for Faraday rotation and for optical activity for the same rotation angle of ${48}^{\circ }$ for a single passage through a single plate. This corresponds to a circular birefringence $\mathrm{\Delta }n=4.72\times {10}^{-5}$ for FR in SF57 at $B=18\text{T}$ and at the average thickness of the plates.

Figure 13

Experimental data of the mean logarithmic relative transmission as a function of the magnetic field $B$ for $N=0$, 25, and 30 plates.

Figure 14

3D image of a measured speckle pattern before image processing (upper left). The intensity is plotted on the $z$ axis while $x$ and $y$ are the camera pixel values. A Gaussian was fitted with an offset and an elliptical region of interest was defined (upper right). The offset was then subtracted from the image (lower left) and the image was weighted by the Gaussian to remove the intensity differences caused by the incident Gaussian laser beam.