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

Lukas Schertel  Fachbereich Physik, Universität Konstanz, 78457 Konstanz,
Germany  Physik-Institut, Universität Zürich, 8057 Zürich, Switzerland  Oliver
Irtenkauf  Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany
Christof M. Aegerter  Physik-Institut, Universität Zürich, 8057 Zürich,
Switzerland  Georg Maret  Fachbereich Physik, Universität Konstanz, 78457
Konstanz, Germany  Geoffroy J. Aubry<EMAIL_ADDRESS>Fachbereich
Physik, Universität Konstanz, 78457 Konstanz, Germany  Departement de
Physique, Université de Fribourg, 1700 Fribourg, Switzerland

(August 27, 2024)

###### Abstract

Transport of coherent waves in multiple-scattering media may exhibit
fundamental, non intuitive 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 (1D), we
studied light propagation through stacks of parallel Faraday-active glass
slides exposed to magnetic fields up to 18 Tesla. Measurements of light
transmission statistics are presented and compared to 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.

Anderson localization, low dimensional coherent transport, magneto-optical
Faraday effect, broken reciprocity

## Introduction

Anderson localization (AL) is a coherent wave transport phenomenon which
results in the halt of diffusion due to interferences in multiple-scattering
media. Even though coherent light transport in highly multiple-scattering
media has been studied for years Wiersma (2013), experiments which mainly
focused on static or dynamic transmission properties still fail to show signs
of this supposedly universal phenomenon for light in three-dimensional media
Sperling  _et al._ (2016); Skipetrov and Page (2016). While it is believed
that this long-standing quest could be successfully achieved by further tuning
the sample properties by optimization of the scattering behavior Aubry  _et
al._ (2017); Schertel  _et al._ (2019); Escalante and Skipetrov (2017), signs
of localization could be hidden behind other perturbing signals such as
absorption or fluorescence Sperling  _et al._ (2016). In other words, one may
wonder whether the right quantities were measured up to now. The quest for
unequivocal signatures of AL is therefore of paramount importance Smolka  _et
al._ (2011); Ghosh  _et al._ (2017); Lopez-Bezanilla  _et al._ (2018); Prat
_et al._ (2019).

The observation of the coherent backscattering cone Wolf and Maret (1985);
*Albada1985 or weak localization, the precursor of AL, illustrated the
importance of interferences in the description of randomly scattered wave
phenomena. Robust interferences between reciprocal multiple-scattering paths
—which are at the heart of AL— survive even in strongly disordered materials
Bromberg  _et al._ (2016). Since coherent phenomena are very sensitive to the
phase of the wave, the magneto-optical Faraday effect which introduces
circular birefringence between counter propagating waves Rayleigh (1885) can
be used to manipulate coherent wave transport Golubentsev (1984); MacKintosh
and John (1988).

This effect breaks reciprocity and therefore suppresses weak localization of
light Erbacher  _et al._ (1993); Lenke and Maret (2000); Schertel  _et al._
(2017). Moreover, a dephasing between counter-propagating paths was used to
show the cancellinearly a fast step change in the refractive index of a random
photonic medium Muskens  _et al._ (2012), and the suppression and revival of
coherent backscattering of ultra-cold atoms Müller  _et al._ (2015) (as
proposed by Ref. Micklitz  _et al._ (2015)). These latter authors further
suggested to break time-reversal symmetry to study AL of cold atoms. This was
done very recently by Hainaut  _et al._ (2018) who used an artificial gauge
field on cold atoms trapped in 1D to induce a parity-time symmetry breaking
such that the system transits from the orthogonal to the unitary class.
Another mechanism of controlled breaking of reciprocity of optical paths by
the Doppler effect in hot atomic vapor was discussed in Cherroret  _et al._
(2019). External magnetic fields also affect weak localization in electronic
systems Bergmann (1984). In such systems, universal localization length
changes have been predicted in quasi-1D (Q1D) systems when breaking
reciprocity Pichard  _et al._ (1990); Thaha  _et al._ (1993), and non-
universal changes in dimensions above 2 Lerner and Imry (1995). Magnetic-
field-induced delocalization was also reported in solid-state systems
Katsumoto  _et al._ (1987); Lerner and Imry (1995), and several reviews
Beenakker (1997); Evers and Mirlin (2008) address the topic of the impact of
the external magnetic field on AL in electronic systems.

Here we study the effect of magneto-optical dephasing on light localization in
order to check its usefulness as an alternative probe of AL of light. We test
this idea in a system where light is known to localize, a 1D disordered
system, as in 1D every infinitesimal amount of disorder is sufficient to
localize the wave Abrahams  _et al._ (1979); Berry and Klein (1997). Note that
this idea should not be confused with the magnetic field localization
transition predicted in random ensemble of point scatterers Skipetrov and
Sokolov (2015); *Skipetrov2018c: in the latter system, the magnetic field does
not act on reciprocity but on the interatomic dipole-dipole interaction that
inhibits localization in such 3D systems without field Skipetrov and Sokolov
(2014); *Bellando2014. One-dimensional or quasi-one-dimensional
electromagnetic wave localization can be realized either in confined
geometries such as microwaves in a waveguide Kuhl  _et al._ (2008) or optical
fibers Karbasi  _et al._ (2012), or in open systems such as stacks of glass
slides Berry and Klein (1997); Lu  _et al._ (2010); Zhang  _et al._ (2008);
Park  _et al._ (2010), see Fig. 1.

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.

Bromberg  _et al._ (2016) showed the effect of the manipulation of reciprocity
on the coherent backscattering (weak localization) in a Q1D experiment using
an optical fiber setup. Here we use the Faraday effect to manipulate
reciprocity in the strong localization regime of stacks of Faraday active
glass slides.

Berry and Klein (1997) observed and explained why a stack of $N\gg 1$
transparent parallel plates of varying thickness acts essentially like a
mirror: due to random interferences between waves multiply reflected backward
and forward by the plates surfaces, the transmission $T$ of the intensity of
the normal incident light decreases exponentially as a function of $N$,

$\left<\ln T\right>=-2N\ln\left(\frac{1}{\tau}\right),$ (1)

and light is mostly reflected. This formula assumes that the wavelength of the
light is much smaller than the deviations around the mean value of the plates
thicknesses. Here, $\tau=\frac{4n}{(n+1)^{2}}$ is the transmission of one
glass$\rightarrow$air or air$\rightarrow$glass interface ($n$ is the
refractive index of the glass) at normal incidence calculated by the Fresnel
formulas, and the average $\left<\cdots\right>$ is done over the
configurations. Note that this is different from the situation of incoherently
added multiply reflected intensities which gives $T\propto 1/N$ (see, e.g.,
Ref. Berry and Klein (1997) and references there). As the above expression
includes reciprocal interferences, this situation was associated with
localization by Berry and Klein Berry and Klein (1997). The localization
length $\xi$, corresponding to the inverse of the slope of Eq. (1), is given
by

$\frac{1}{\xi}=2\ln\left(\frac{1}{\tau}\right)$ (2)

in units of plate number. For stacks of glass plates in air values of $\xi$
are typically $\leq 10$. Note that the exponential decay of Eq. (1) starts
even for samples sizes $N$ shorter than the localization length $\xi$ Berry
and Klein (1997).

The strong attenuation of the log-averaged transmission in the localized
regime is also connected to a narrowing of the modes allowed to be
transmitted. This leads to enhanced fluctuations of the transmission. Chabanov
_et al._ (2000) noted that the exponential decay of the static transmission
with the sample thickness is not clearly distinguishable from absorption
(similar to the 3D case) and that absorption may even suppress localization.
Moreover, the large fluctuations associated with localization imply that the
mean transmission is not the natural scaling parameter. Rather, the full
distribution or a parameter describing this distribution correctly is a better
choice. Thus, Genack and Chabanov Genack and Chabanov (2005) proposed to
quantify the transmission fluctuations by using the _variance_ of the
transmission $T$ _relative_ to its disorder ensemble averaged value $\langle
T\rangle$ to observe localization in their samples, even in the presence of
absorption. Zhang  _et al._ (2008) and Park  _et al._ (2010) then studied
localization via thickness dependent transmission and transmission statistics,
respectively, in a system showing a crossover from a 1D to a higher
dimensional (>1D) geometry by sending visible light through a stack of not
perfectly flat and not perfectly parallel glass slides. Even though in this
kind of system the transverse spreading of light is not bounded by reflecting
boundaries, they showed that for a large number of slides it was described
surprisingly well by the statistics of diffusive transport through long narrow
tubes (Q1D) Kogan and Kaveh (1995); Nieuwenhuizen and van Rossum (1995). This
is remarkable since the light transport is rather different in both cases: In
the narrow tube geometry, the mean free path is isotropic and shorter than (or
comparable to) the tube diameter, while in the slide stack geometry, the mean
free path is strongly anisotropic with a transverse value much larger than the
beam diameter. Therefore, mode correlation effects are much stronger in the
tube case. Nevertheless, for a small number of slides, one expects mode
correlations in the stack system as well which implies that its transmission
statistics should feature characteristics similar to perfect 1D localizing
systems. While such stacked systems are experimentally simple to realize, the
Q1D geometry might weaken or even suppress localization for a large number of
slides.

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 $B=18$ T for the Faraday active plates studied in the paper.
The color encodes the logarithm of the total transmission ($T=T_{x}+T_{y}$). X
(respectively Y) represents linear polarization on the $x$ (respectively $y$)
axis, D and A stand for diagonal and anti-diagonal (linear polarization on the
first and second bisectors), and R/L are the circular polarizations
(right/left).

In this paper, we study such a stack of slides made out of a Faraday active
glass. The interferences of the multiple reflections that lead to the
exponential decay in transmission are studied by manipulating the phases of
the waves via the Faraday effect in an external magnetic field applied
parallel to the direction of propagation of light (see Fig. 1(a)). We compare
our measurements with the results of transfer-matrix calculations from a
perfect 1D stack of glasses with the same parameters. The transfer-matrix
calculations show an increase of the localization length and a decrease of the
fluctuations in transmission when reciprocity is broken. They further show an
attraction of the polarization states towards the circular left and right
polarizations, thus leading to a saturation of the reciprocity breaking
mechanism in this 1D system. In the experiments, the probability distribution
of the normalized transmitted intensity allows us to identify the crossover
range from 1D to Q1D due to imperfections in flatness and parallelism of the
plates as sketched in Figs. 1(b) and (c). For samples smaller than this
crossover length, the experiments show reasonable agreement with the 1D
simulations and represent the first experimental manipulation of optical AL
via external fields. For large sample thicknesses, high transmission
fluctuations and AL are suppressed by the Q1D nature of the experimental
sample.

The paper is organized as follows: First, in § I we present the numerical
results obtained within the transfer-matrix formalism for the propagation of
an electromagnetic plane wave through an ideal stack of $N$ parallel Faraday
active glass slides. Next, in § II the experimental concept is presented.
Finally, in § III, we compare the numerical 1D results with our experimental
data.

##  I Numerical 1D results

In Appendix A, we present our extension of the transfer-matrix formalism in 1D
systems in the presence of circular birefringence mechanisms such as Faraday
rotation or optical activity. The transfer matrices were calculated using Eq.
(12), and the transmitted field was extracted using Eq. (13). The parameters
used for the simulations were chosen to be as close as possible to realistic
experimental parameters. We therefore choose to work with $\lambda_{0}=532$ nm
(wavelength in vacuum), $n_{0}=1$ (refractive index of the air gaps), $n=1.8$
(refractive index of the Faraday active material at $B=0$ T, Schott SF57
Goodman (2015)), $V=31$ rad/Tm (Verdet constant of the Faraday active
material, Schott SF57 Goodman (2015)), $d$ uniformly distributed on a span
$2\Delta d=10\,\mu$m around $\left<d\right>=1.5$ mm (thickness of both the
Faraday active slabs and of the air gaps in between). No absorption was put in
the simulations. We checked that the precision of the calculation was high
enough to compute the coefficients of the large transfer matrices up to $125$
plates. For each number of plates and for magnetic fields up to 25 T, $3\times
10^{4}$ different samples were calculated, and the whole statistics of all
transmission and reflection coefficients was recorded. With $a$ and $b$ being
either $x$ or $y$, the transmission (respectively reflection) coefficients are
defined as follows:
$T_{ab}=\left|E_{N}^{b\rightarrow}/E_{0}^{a\rightarrow}\right|^{2}$
(respectively
$R_{ab}=\left|E_{0}^{b\leftarrow}/E_{0}^{a\rightarrow}\right|^{2}$) are the
transmitted (respectively reflected) intensities with linear polarization $b$
from incident linear polarization $a$, $T_{a}=T_{ax}+T_{ay}$ is the total
transmitted intensity for incident linear polarization $a$. At $B=0$ T, the
two polarization channels are uncoupled, and therefore $T_{xy}=T_{yx}=0$.
Moreover for symmetry reasons, at any field, $T_{xx}=T_{yy}$, $T_{xy}=T_{yx}$,
and therefore $T_{x}=T_{y}$.

In our 1D model, one major difference between optical activity and Faraday
rotation is that in the first case, the rotation angle of the polarization
plane of each photon is proportional to the _position in the sample_ , while
in the latter it is proportional to the _path length of the photon in the
sample_. We therefore expect all reflected photons in the optical activity
case to come back with the same polarization as the incident photons, and to
mix the polarizations in the Faraday rotation case. This is indeed observed in
the numerical simulations (see Appendix B.1.1). Looking at how the reflected
intensities distribute over the $x$ and $y$ polarization channels for both
optical activity and Faraday rotation as a function of the sample length (see
Appendix B.1.2) further illustrates the fact that Faraday rotation breaks
reciprocity, while optical activity does not.

In transmission we also observe a spreading of the polarization states as
shown in Fig. 2. The polarization states for 1000 different samples consisting
of 1, 6, 8 or 30 plates are represented on the Poincaré sphere which
represents the Stockes parameter of polarized light. The important
polarization states are described in the legend of Fig. 2. Without magnetic
field, all the points are on the X point: light stays linearly polarized.
Applying a magnetic field induces a rotation of the points on the equator
(linearly polarized light) but also a spreading out of the equator
(elliptically polarized light). When the number of plates and the external
field are high enough the points are attracted by the poles (as observed for
$N=30$ plates in Fig. 2). The poles are the circularly polarized states. Those
circularly polarized states are not sensitive to the field induced circular
birefringence and therefore do not allow for magnetic-field-induced
reciprocity breaking: While Faraday rotation changes the phase of a path with
a given circular polarization, this phase change is identical to the one of
the reciprocal path with the same polarization, such that there is no field
dependent relative phase shift between both paths. Therefore, as soon as
points on the Poincaré sphere are getting close to the poles, they diffuse
less: the effect of the external magnetic field decreases. Note that this
system therefore shows a manifestation of self-organization of the
polarization state in an optical system Guasoni  _et al._ (2016); Vorontsov
and Miller (1995).

In Fig. 3(a), we plot the different transmission coefficients for the
intensity at 0 and 18 T when Faraday rotation is present.

Figure 3: Average of the logarithm of $T_{x}$, $T_{xx}$ and $T_{xy}$ without
and with (a) Faraday rotation or (b) optical activity for the same rotation
angle of 48∘ for a single passage through a single plate. This corresponds to
a circular birefringence $\Delta n=4.72\times 10^{-5}$ for FR in SF57 at
$B=18$ T and at the average thickness of the plates.

At $B=0$ T, $\left<\ln(T_{x})\right>=\left<\ln(T_{xx})\right>$ decreases
linearly with the number of plates as expected (see Eq. (1)) Berry and Klein
(1997). The extracted localization length is $\xi=5.85$ plates, which is very
comparable to the 5.87 plates value obtained from Eq. (2). This clearly shows
that 1D Anderson localization occurs in this system. However, the curves are
qualitatively different when the external magnetic field is turned on. At
$B=18$ T, oscillations are observed in both $\left<\ln(T_{xx})\right>$ and
$\left<\ln(T_{xy})\right>$ up to about 30 plates. These oscillations come from
the fact that for a small number of plates, the proportion of light
transmitted without reflection is high, such that the mean polarization state
is governed by the Faraday rotation of the light through the stack in one
direction. Looking for instance only at $\left<\ln(T_{xx})\right>$ leads to a
diminished transmission for 2, 6, … plates where the mean polarization is
perpendicular to the incident polarization. Calculating the Faraday rotation
of the directly transmitted light as a function of the total length of Faraday
material in the sample, one expects a perpendicular polarization for 1.9
plates, 5.6 plates, etc., consistent with the latter observation. At these
specific numbers, mainly multiply reflected light (which has undergone some
more or less stochastic polarization changes as shown in Fig. 2) is detected,
which gives rise to an increased observation of localized light and thus a
decreased $\left<\ln(T_{xx})\right>$. For an increasing number of plates, more
multiply reflected photons govern the mean transmission, such that the
distribution of polarization gets further broadened. For $N\gtrsim 30$,
$T_{xx}=T_{xy}$, and therefore
$\left<\ln(T_{x})\right>=\left<\ln(T_{xx})\right>+\ln 2$. In this regime, the
exponential decrease of the mean transmission is slowed down with magnetic
field. The ratio of the slopes of the linear fits to the $B=0$ T and $B=18$ T
data is found to be $r=1.1130\pm 0.0009$. This suggests that the localization
length in the system is increased by a factor $r$ when the magnetic field is
on. The subsistence of an exponential decay with magnetic field may suggest
that the magnetic field only disturbs some of the localized modes. In fact,
however, for sufficiently high values of $N$ (at high enough $B$) the mixed
elliptical polarization states progressively evolve into equal amounts of the
two circular polarized states (as illustrated above) which cannot evolve
further because the latter do not couple to FR. Therefore AL modes cannot be
manipulated further beyond this point. Increasing the product of the magnetic
field times the Verdet constant decreases the quasi-period of the oscillations
and moves the onset of the circular polarized regime to a smaller number of
plates.

We also ran the calculation for different values of the refractive index $n$
of the Faraday active material without any external magnetic field. Without
magnetic field, we verified that the slope of $\left<\ln(T_{xx})\right>$ as a
function of $N$ is equal to $1/\ln\tau$ (Eq. (1), Berry and Klein (1997)).
Even with this dependency, in the range explored ($1.4<n<2.0$, $B=9,18,27$ T),
we find that $r$ neither depends significantly on $n$ nor on $B$, once $B$ is
high enough ($r=1.113\pm 0.003$, mean and standard deviation on all the
calculated data). This result seems similar to the universal relation between
the localization length with and without time-reversal symmetry found for Q1D
electronic Pichard  _et al._ (1990) and quantum chaotic systems Thaha  _et
al._ (1993) with a large number of transverse modes, whereas the case studied
here is purely one-dimensional, that means with only one single transverse
mode. However, the relation found for the latter systems (removing completely
time-reversal symmetry doubles the localization length) does not seem to hold
in our case. One reason could be that time-reversal symmetry is not completely
broken in our case: even though increasing the magnetic field does not alter
the results anymore, the attraction of the polarization states toward the
circular polarized states for long samples (see Fig. 2) cancels time symmetry
breaking. Another reason could also be the fact that our system has two
polarization modes that are either not coupled (without field), or field-
dependent mixed with field. This is different from the $\mathcal{N}$ perfectly
mixed modes of the Q1D literature. In Ref. Beenakker (1997), Eq. (220) states
that for $\mathcal{N}=1$ mode (pure 1D system), the localization length should
be independent of the magnetic field ($r=1$), whereas for $\mathcal{N}=2$
modes (two coupled polarization states), $r=4/3$. Our results are consistent
with neither predictions, but the $\sim 10\%$ change is clearly established in
our numerical results. To illustrate this clear change, the same analysis is
done for optical activity (OA) in Fig. 3(b). In this case, no change in the
localization length is observed. Note that for the OA case, oscillations with
increasing number of plates are preserved in contrast to the Faraday rotation
case. For OA, the polarization state in transmission only depends on the
sample size in which case relative fluctuations from sample to sample vanish
for large $N$.

In 1D systems, the variance of the normalized transmitted intensity
$\mathrm{Var}\left(s\right)$, with $s=T/\left<T\right>$ is expected to
increase exponentially with number of plates $N$ indicating a strong increase
of the fluctuations in transmission in the localized regime Chabanov  _et al._
(2000). The blue solid line in Fig. 4(a) shows such an increase without
magnetic field (no FR).

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.

With a magnetic field ($B=18\leavevmode\nobreak\ $T) such an exponential
increase is still present for all normalized transmissions
$s_{x}=T_{x}/\left<T_{x}\right>$, $s_{xx}=T_{xx}/\left<T_{xx}\right>$ and
$s_{xy}=T_{xy}/\left<T_{xy}\right>$, but on a lower scale. This indicates that
high transmission fluctuations present in the localized regime are partly
suppressed by the Faraday effect. This observation is consistent with the
observation of an increased localization length in Fig. 3(a). Again
oscillations are observed in both $\mathrm{Var}\left(s_{xx}\right)$ and
$\mathrm{Var}\left(s_{xy}\right)$ for a small number of plates (see inset of
Fig. 4(a)). The selective detection of multiply reflected light for a certain
number of plates by polarization selection increases the transmission
fluctuations and leads to enhanced signs of localization similar to Fig. 3(a).
For $N>30$ the distribution of polarization states is further broadened such
that
$\mathrm{Var}\left(s_{x}\right)=\mathrm{Var}\left(s_{xx}\right)=\mathrm{Var}\left(s_{xy}\right)$.

In Fig. 4(b), the same analysis is performed for optical activity (with equal
rotation of the polarization) instead of Faraday rotation. In this case, no
change in the statistics is observed.

##  II Experimental concept

We describe now our experiment to study effects of reciprocity on Anderson
localization of light. We measured transmission statistics through a stack of
Faraday active slides as a function of an external applied DC magnetic field.
We designed the experimental setup (see Fig. 5) such that the linear incoming
polarization was Faraday-rotated by about $45^{\circ}$ for propagation through
one plate under the maximum available magnetic field (18 T) which was produced
by a superconducting solenoid with 41-mm-diameter vertical room-temperature
bore (Oxford Instruments). This FR was chosen to completely randomize the
phases of the waves on their way through the stack. We therefore used round
slides (2.5 cm diameter, $\left<d\right>\approx 1.5$ mm) of Schott SF57 glass,
a dense flint glass (high refractive index $n=1.8$ caused by the high lead
oxide concentration in the material Schott Advanced Optics (2015)) with strong
Faraday rotation Weber (2003). The Verdet constant was measured to be
$V\approx 31.0\pm 0.1\leavevmode\nobreak\ \mathrm{rad}/\mathrm{Tm}$ (in good
agreement with Weber (2003)) which corresponds to a rotation angle of
$\theta\approx 48^{\circ}$ for one plate at $\lambda_{0}=532$ nm and $B=18$ T.

Putting the plates on top of each other leads to somewhat irregular spacings
between the plates and slight relative inclinations of their normals (see Fig.
1(c) and Refs. Zhang  _et al._ (2008); Park  _et al._ (2010)). This is because
the plates were not specially treated to be optically flat so that they have a
surface roughness comparable to the wavelength of the incident light. In
addition, their thickness may not be perfectly equal nor perfectly homogeneous
and their normals rather slightly tilted off the direction of the incident
beam. This leads to varying spacings and thicknesses at different transverse
positions of the stack. The idea of this setup is to send a relatively wide
($\approx 0.5$ cm) parallel coherent light beam through this disordered stack
of $N$ glass plates ($0<N<50$) and to monitor the transmission patterns in the
far field by a camera. On progression through the stack the phase of the
incident plane wave obeys different shifts at different lateral positions and
the wave fronts become more and more distorted with increasing $N$ as further
discussed in the following section.

The sample holder was continuously rotated about an axis normal to the stack
via a rubber hose by a motor.

Figure 5: 1D light transport setup: a laser beam is sent through an OD filter
to control the incident intensity before being expanded to $\approx 0.5$ 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 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.

The motor turns the sample by one revolution per minute which corresponds to
the measurement time for each sample. During this time the camera (a 12-bit
CMOS camera with 1900$\times$1200 pixels from The Imaging Source, DMK 23UX174)
took 300 images (5 per second). The exposure time was chosen to be small
enough ($10^{-4}$ s) to minimize the averaging which occurs by turning the
sample. The linearly polarized coherent laser beam comes from a continuous
laser (Torus, Laser Quantum, coherence length $>100$ m). The output power can
be tuned between $P=0.5$ W and 0.75 W. An optical density (OD) filter is used
to further control the incident intensity to avoid camera saturation. The beam
is expanded using two lenses with $f_{1}=200$ mm and $f_{2}=1000$ mm in a
confocal setting, and then led through the sample in the magnet. The beam was
adjusted to pass parallel through the relatively long (1174 mm) bore by
aligning the back reflection of the first plate back on the incident beam. The
measurements involving $T_{xx}$ were done with a linear polarizer in front of
the camera to capture only the photons with the same polarization as the
incident light, and $T_{x}$ was measured without a polarizer.

##  III Experimental results vs 1D numerical simulations

Typical snapshots of the evolution of the transmission patterns with
increasing number $N$ of slides in the stack are shown in Fig. 6.

Figure 6: Measured images for four increasing numbers of plates ($N=0$, $5$,
$15$, $20$). The measurement was performed at $B=0T$ without polarizer in
front of the camera. Color bars show measured intensities in a.u. Figure 7:
Histograms of the normalized transmitted intensity $s=T/\left<T\right>$
without (upper row (a)) and with (lower row (b), $B=18$ T) external magnetic
field for $N=6,8$ and 30 plates. Simulation data are traced in blue and
experimental data in orange.

For $N=0$ (no sample) the intensity profile across the image surface is not
perfectly Gaussian as would be expected for a Gaussian incident beam but some
low contrast interference fringes can be noticed. They are due to the entrance
window of the CMOS camera which is somewhat inclined with respect to the
optical axis. With increasing $N$, an increasing number of similar fringe
patterns of slightly inclined sample slides coherently superimpose in the
transmitted pattern which, therefore, becomes more fine grained with
increasing length of the stack. For $N=8$ plates some residual fringes can
still be seen mixed with a more irregular pattern. The image with 20 plates
shows that an almost random pattern modulates the beam profile and the
intensity pattern becomes close to the familiar random interference pattern
(speckle) characteristic for fully randomly distributed phases Goodman (2015).
This is because with increasing $N$ the wavefront distortions rapidly become
larger than the wavelength and the patterns become more irregular because of
the increasing contributions of long multiply reflected waves. These
observations are qualitatively similar to those reported by Zhang  _et al._
(2008).

###  III.1 Zero-field transmission statistics

The non ideal alignment of the glass plates and their surface roughness thus
result in a crossover from a situation of interfering plane waves at small $N$
to a Q1D scattering geometry for a large $N$ similar to the experimental
realization by Park  _et al._ (2010). In the following section we use the
comparison between experimental and numerical intensity statistics to check
how close our experimental system comes to the simulated ideal 1D case, with
an emphasis on experimental signatures of 1D AL. We anticipate that the Q1D
case at high $N$ can not be captured by the numerical model.

Figure 7 displays the probability distributions of the transmission $p(s)$,
with $s=T/\langle T\rangle$, for 6, 8 and 30 plates, the upper (a) and lower
(b) row corresponding to zero and maximum field, respectively. Experimental
data are shown in orange while the 1D simulations are plotted in blue.
Overall, we find reasonable agreement between experimental and simulated
intensity distributions for low $N$, in particular at high magnetic fields,
while differences become progressively more pronounced with increasing $N$.

We consider first the zero-field case (a) in more detail. For small $N$ (6 or
8), both curves show a similar functional form at small and intermediate $s$
with a low value for $p(s)$ at small $s$ followed by a maximum. This compares
well with the 1D simulations of Park  _et al._ (2010) and shows that our
experimental results are close to the 1D situation at low $N$. Experimental
$p(s)$ deviate, however, substantially from our simulations at high
intensities in lacking the algebraic high intensity tail (see the $N=30$
plates case in Fig. 7(a)). We attribute this essentially to field correlations
visible by the stripe interferences described above which particularly spoil
the brightest localized modes requiring constructive interferences of the
longest multiple reflected paths. In addition, we believe that in our
experimental setting rotating the sample does not necessarily provide a good
ensemble average over a sufficient number of independent disordered
configurations as the fringes are seen to rotate with the sample. For
increasing $N$, the simulations exhibit a more and more pronounced algebraic
shape with increasing contributions at low and high intensities, which
highlights the anomalously high transmission fluctuations characteristic for
localization. The experimental distributions, however, are severely affected
by the crossover from the experimental 1D fringe situation to the Q1D speckle
situation Park  _et al._ (2010) and therefore tend towards an exponential
decay typical for a classical polarized speckle (see the $N=30$ plates case in
Fig. 7(a)). Note that, consistently, by comparing data taken with and without
an analyzer we verified that the polarization stays linear up the highest $N$
for the case $B=0$ T.

###  III.2 High-field transmission statistics

The lower row (b) of Fig. 7 shows that with a magnetic field ($B=18$ T), the
probability distributions of the normalized transmission $s$ obey dramatic
changes for both simulations and experiments. For small $N$ (here we show data
for $N=6$ and $N=8$) we find excellent agreement between simulations and
experiments. In the case of no polarization selected detection ($p(s_{x})$)
the distributions are relatively narrow and similar to those of a
superposition of two orthogonal polarized speckle intensities Goodman (2015).
This is due to the strong and widely distributed Faraday rotations which
result in a strong depolarization already after a few plates as illustrated in
Fig. 2 and in Appendix B.2. The polarization selected measurements ($s_{xx}$)
perfectly agree with the simulations for $N=6$ and $N=8$. This is because the
strong field induced path-length dependent depolarization generates enhanced
phase fluctuations for any given polarization which leads to a lower fringe
contrast and to a much better configuration averaging as in the zero-field
case. At high $N$ the experimental $p(s)$ tend towards the distributions of a
classical polarized ($p(s_{xx})$) and fully depolarized ($p(s_{x})$) speckle,
respectively, as expected for Q1D (see the $N=30$ plates case in Fig. 7(a)).
Therefore, the deviations between experiment and simulations at high $N$ are
due to the 1D to Q1D crossover in the experiment which spoils signatures of
localization.

The difference between the functional forms of $p(s_{xx})$ for $N=6$ and $N=8$
also reflects the oscillations in $\langle\ln\left(T_{xx}\right)\rangle$ in
the simulations shown in Fig. 3(a). Such oscillations are also present in the
experiments: in Fig. 8, we plot $\left<\ln T_{xx}\right>$ as a function of the
number of plates without and with magnetic field.

Figure 8: Experimental data of the mean logarithmic relative transmission as a
function of the number of plates $N$ at $B=0$ T 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(a)).

Without external magnetic field (blue open circles in Fig. 8), $\left<\ln
T_{xx}\right>$ is expected to decrease linearly Berry and Klein (1997) with
$N$ as also observed in the transfer-matrix calculations (blue line in Fig.
3(a) or light blue line in Fig. 8). This is well verified for low numbers of
plates ($N<10$) and deviations to this linear behavior due to the Q1D
situations are then observed. When the magnetic field is turned on, the
oscillations of $\left<\ln T_{xx}\right>$ (orange solid circles in Fig. 8) due
to the contribution of the non-reflected light are seen with the same period
as in the 1D simulations (dark orange dashed line in Fig. 3(a)). In the
transfer-matrix calculations the slope of the decay with $N$ is smaller for
the measurement with magnetic field than without. This was connected to an
increased localization length $\xi$. However, in the experiment no such change
of the slope is observed. This is explained by the fact that in the regime
where this change is expected to occur ($N>20$), signs of localization are
already affected by the Q1D nature of the scattering system.

The change of the probability distribution of the normalized transmission with
magnetic field (6 plates and 8 plates in Fig. 7) as well as the oscillations
in $\langle\ln T_{xx}\rangle$ with magnetic field, both in the experiments and
in the simulations, are signs of manipulation of AL via magnetic fields in a
perfect 1D geometry where localization exists from the first plate on. Even
though the samples are short compared to the mean-free path, 1D systems are
always localized —even for very small samples— as the exponential decay starts
from the first plate on (see Fig. 3(a)). A further probe on manipulation of
reciprocity and thus AL by magnetic fields via the Faraday effect will be done
in the following by analyzing the variance of the normalized transmitted
intensity $\mathrm{Var}\left(s\right)$.

###  III.3 Effect of magnetic field on Anderson localization

The main purpose of this paper is to study the effect of breaking reciprocity
on Anderson localization via Faraday rotation. Figure 9 shows the experimental
data for $\mathrm{Var}\left(s\right)$ versus $N$ without (blue open circles)
and with (orange filled circles) magnetic field. For a small number of plates
($N<8$), a similar increase as in the simulations is observed. In this regime,
oscillations are again observed in the magnetic-field measurements. Similar to
the simulations (inset in Fig. 4(a)) the minima of these oscillations fall
below the $B=0$ T measurement (see orange ellipse in Fig. 9). This indicates
the early stage of a slower increase of $\mathrm{Var}\left(s_{xx}\right)$ and
thus confirms the increase of the localization length with magnetic field
found in the simulations.

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(a)).

At $N>20$ plates for all measurements with and without magnetic field a
plateau occurs. The occurrence of such a plateau in the variance of the
transmitted intensity was interpreted by Park  _et al._ (2010) as a signature
of the crossover from 1D to Q1D. High intensity fluctuations are suppressed by
mode coupling in Q1D geometry as observed in Fig. 7 for $N=30$. The crossover
range is between $N=10$ and $N=20$, consistent with the probability
distributions of Fig. 7. This happens to be close to the expected localization
length $\xi\approx 6$ plates, such that no difference in the plateau value is
observed without and with magnetic field $B$ for $s_{xx}$. The relatively
thick plates (necessary to obtain a large rotation angle of Faraday rotation)
leads to an earlier crossover from 1D to Q1D (Park  _et al._ (2010) used
plates of average thickness 150 $\mu$m) and suppress localization for large
$N$. Note that on the other hand the high refractive index of SF57 ($n=1.84$
at $\lambda_{0}=633$ nm Weber (2003)) lowers the localization length $\xi$ by
increasing the scattering (i.e., reflection) compared to Park et al.
experiments($n=1.5$ Park  _et al._ (2010)).

## Conclusion

We used transfer-matrix simulations to show the effect of breaking reciprocity
on 1D Anderson localization of light via the magneto-optic Faraday effect. The
simulations show a magnetic-field-induced increase of the localization length
by a factor which neither seems to depend on the refractive index of the
plates nor on the magnetic field once Faraday rotation is high enough. This is
associated with a decrease of the transmission fluctuations which was also
observed in the simulations and confirmed in experiments. The non-trivial
change in the localization length is related to the fact that for high $N$ and
$B$, the polarization states are spread, but as soon as they come close to the
circular ones, they do not couple anymore to the external magnetic field
because reciprocity is not broken for circular polarization in a Faraday
active medium. The experimental data are a clear observation of manipulation
of AL of light by external magnetic fields. For a small number of plates, the
experimental data showed a similar behavior as predicted by the numerical
simulations. For stacks between 10 and 20 plates a crossover from 1D to Q1D is
observed. To be able to observe localization in 1D experiments even for large
$N$ the crossover length from 1D to Q1D would need to be increased. This might
be achievable by a better parallel alignment of the glass plates in the
stacks, e.g., by using piezo elements between plates, or by thinner glass
plates keeping the FR high enough to randomize the polarization state. The
first solution leads to a very sophisticated experimental sample, which might
be hard to realize. The second solution may be easier to realize.

In conclusion, we show in this paper that Anderson localization of light is
indeed affected by reciprocity breaking, even though the 1D configuration
studied here does not allow it to be fully broken. This paves the way towards
a new probe for Anderson localization of light in higher dimensional systems.

###### Acknowledgements.

We thank Luis S. Froufe-Pérez for fruitful discussions. We acknowledge support
by the Center for Applied Photonics (CAP, project CAP05), University of
Konstanz, and the Schweizerischer Nationalfonds (SNF, Grant No.
200020M_162846).

##  Appendix A Transfer matrix

###  A.1 1D Transfer-matrix formalism and Faraday rotation

In this section, we introduce the transfer-matrix formalism in the presence of
Faraday rotation. To make our paper self-contained, we start from the usual
definition of the transfer matrices.

Figure 10: Direction of propagation of the light electromagnetic fields $E$ in
front (1) and behind (2) any optical system (dielectrics, interface, …).

Given any optical system (see Fig. 10 for the geometry), the transfer-matrix
formalism decomposes the field in plane waves propagating in the forward $+z$
($\rightarrow$) and backward $-z$ ($\leftarrow$) directions, and relates the
field behind the system to the field in front of the system through the
transfer matrix $\mathcal{M}$,

$\begin{pmatrix}E_{2}^{\rightarrow}\\\
E_{2}^{\leftarrow}\end{pmatrix}=\mathcal{M}\begin{pmatrix}E_{1}^{\rightarrow}\\\
E_{1}^{\leftarrow}\end{pmatrix}.$ (3)

For instance, if light (with a wavelength $\lambda_{0}$ in vacuum) propagates
a distance $d$ in a homogeneous dielectric having a refractive index $n$, the
transfer matrix is

$M_{d}=\begin{pmatrix}\exp{-\frac{2\mathrm{i}\pi nd}{\lambda_{0}}}&0\\\
0&\exp{\frac{2\mathrm{i}\pi nd}{\lambda_{0}}}\end{pmatrix}.$ (4)

If the system is an interface between two dielectrics $n_{1}$ and $n_{2}$
perpendicular to the wave vector, the Fresnel formulae allow us to write the
transfer matrix of the interface Berry and Klein (1997)

$N_{n_{1}\rightarrow
n_{2}}=\frac{1}{2n_{2}}\begin{pmatrix}n_{2}+n_{1}&n_{2}-n_{1}\\\
n_{2}-n_{1}&n_{2}+n_{1}\end{pmatrix}.$ (5)

The transfer matrix of a dielectric slab of refractive index $n$ and thickness
$d$ placed in air ($n_{0}$) is then written as the matrix product
$N_{n\rightarrow n_{0}}\times M_{d}\times N_{n_{0}\rightarrow n}$.

Faraday rotation, which is a kind of circular birefringence, imposes to split
the field in the two circular polarization components
$E^{\rightarrow/\leftarrow}=\begin{pmatrix}E^{\mathrm{R}\rightarrow/\leftarrow},E^{\mathrm{L}\rightarrow/\leftarrow}\end{pmatrix}$.
In a Faraday active material, these components are related to each other by a
$4\times 4$ matrix $M_{\mathrm{FR}}$,

$\begin{pmatrix}E_{d}^{\mathrm{R}\rightarrow}\\\
E_{d}^{\mathrm{L}\rightarrow}\\\ E_{d}^{\mathrm{R}\leftarrow}\\\
E_{d}^{\mathrm{L}\leftarrow}\\\ \end{pmatrix}=M_{\mathrm{FR}}(d,\Delta
n)\begin{pmatrix}E_{0}^{\mathrm{R}\rightarrow}\\\
E_{0}^{\mathrm{L}\rightarrow}\\\ E_{0}^{\mathrm{R}\leftarrow}\\\
E_{0}^{\mathrm{L}\leftarrow}\\\ \end{pmatrix},$ (6)
$M_{\mathrm{FR}}=\begin{pmatrix}\exp{-\mathrm{i}n_{\mathrm{+}}\varphi}&0&0&0\\\
0&\exp{-\mathrm{i}n_{\mathrm{-}}\varphi}&0&0\\\
0&0&\exp{\mathrm{i}n_{\mathrm{+}}\varphi}&0\\\
0&0&0&\exp{\mathrm{i}n_{\mathrm{-}}\varphi}\end{pmatrix},$ (7)

with $n_{\pm}=n\pm\Delta n$ and $\varphi=\frac{2\pi d}{\lambda_{0}}$. Here
$2\Delta n$ is the refractive index difference for right and left circular
polarized waves, and is related to the Verdet constant $V$ of the Faraday
active material and to the applied magnetic field $B$ via $\Delta
n=\frac{\lambda_{0}BV}{2\pi}$. Note that a very similar matrix
$M_{\mathrm{OA}}$ can be written for optical activity (which does not break
reciprocity),

$M_{\mathrm{OA}}=\begin{pmatrix}\exp{-\mathrm{i}n_{+}\varphi}&0&0&0\\\
0&\exp{-\mathrm{i}n_{-}\varphi}&0&0\\\ 0&0&\exp{\mathrm{i}n_{-}\varphi}&0\\\
0&0&0&\exp{\mathrm{i}n_{+}\varphi}\end{pmatrix}.$ (8)

Using the Fresnel formulae between a dielectric and a Faraday active material
(see Appendix A.2, Table 2), we can write in this (circular) basis the
transfer matrix of an interface between air (of refractive index $n_{0}$) and
a Faraday active material,

$N_{n_{0}\rightarrow
n_{+},n_{-}}=\frac{1}{2}\begin{pmatrix}\frac{n_{0}+n_{+}}{n_{+}}&0&0&\frac{n_{+}-n_{0}}{n_{+}}\\\
0&\frac{n_{0}+n_{-}}{n_{-}}&\frac{n_{-}-n_{0}}{n_{-}}&0\\\
0&\frac{n_{-}-n_{0}}{n_{-}}&\frac{n_{0}+n_{-}}{n_{-}}&0\\\
\frac{n_{+}-n_{0}}{n_{+}}&0&0&\frac{n_{0}+n_{+}}{n_{+}}\end{pmatrix},$ (9)

and the transfer matrix of an interface between a Faraday active material and
air,

$N_{n_{+},n_{-}\rightarrow n_{0}}=\frac{1}{2n_{0}}\begin{pmatrix}{\scriptstyle
n_{0}+n_{+}}&0&0&{\scriptstyle n_{0}-n_{+}}\\\ 0&{\scriptstyle
n_{0}+n_{-}}&{\scriptstyle n_{0}-n_{-}}&0\\\ 0&{\scriptstyle
n_{0}-n_{-}}&{\scriptstyle n_{0}+n_{-}}&0\\\ {\scriptstyle
n_{0}-n_{+}}&0&0&{\scriptstyle n_{0}+n_{+}}\end{pmatrix}.$ (10)

In the experiments we worked with a linearly polarized incoming field
$E_{0}^{\rightarrow}=\begin{pmatrix}E_{0}^{x\rightarrow}\\\
E_{0}^{y\rightarrow}\end{pmatrix}$ such that we start these calculations in
the linear basis. Changing from the circular to the linear basis can be done
using the matrices for the basis change $P_{\mathrm{circ\rightarrow lin}}$ and
$P_{\mathrm{lin\rightarrow circ}}$ (Eqs. (18) and (19)). Without any loss of
generality let us consider an $x$-polarized incident wave. This implies
$E_{0}^{y\rightarrow}=0$. Having just a light source before the system, the
field after the system is connected via the transfer matrix $\mathcal{M}$ to
the field before the system,

$\displaystyle\begin{pmatrix}E_{N}^{x\rightarrow}\\\ E_{N}^{y\rightarrow}\\\
0\\\ 0\\\ \end{pmatrix}=\mathcal{M}\begin{pmatrix}E_{0}^{x\rightarrow}\\\ 0\\\
E_{0}^{x\leftarrow}\\\ E_{0}^{y\leftarrow}\\\ \end{pmatrix},$ (11)

with $E_{0}^{\leftarrow}=\begin{pmatrix}E_{0}^{x\leftarrow}\\\
E_{0}^{y\leftarrow}\end{pmatrix}$ the field reflected by the system. Note that
there is no field in the $-z$ direction ($\leftarrow$) after the system and
therefore $E_{N}^{x\leftarrow}=E_{N}^{y\leftarrow}=0$. The overall transfer
matrix $\mathcal{M}$ that connects the incoming and the outgoing fields
through a stack of $N\geq 2$ Faraday active slides of thicknesses
$\left\\{d_{i}\right\\}$ separated by $N-1$ air gaps of thicknesses
$\left\\{e_{i}\right\\}$ in the linear basis is then expressed by

$\displaystyle\mathcal{M}=$ $\displaystyle P_{\mathrm{circ\rightarrow
lin}}\times\prod_{i=N}^{2}\left[N_{n_{+},n_{-}\rightarrow
n_{0}}M_{\mathrm{FR}}(d_{i},\Delta n)N_{n_{0}\rightarrow
n_{+},n_{-}}M_{0}(e_{i})\right]\times N_{n_{+},n_{-}\rightarrow
n_{0}}M_{\mathrm{FR}}(d_{1},\Delta n)N_{n_{0}\rightarrow n_{+},n_{-}}\times
P_{\mathrm{lin\rightarrow circ}},$ (12)

$M_{0}(e_{i})$ being the transfer matrix of an air gap.

Knowing $\mathcal{M}$, the system (11) has four equations and four unknowns
($E_{0}^{x\leftarrow}$, $E_{0}^{y\leftarrow}$, $E_{N}^{x\rightarrow}$ and
$E_{N}^{y\rightarrow}$) and can therefore be solved analytically. Writing
$\mathcal{M}=\left(m_{ij}\right)_{0\leq i,j<4}$ using its matrix elements, the
solution is:

$\displaystyle\begin{cases}E_{0}^{x\leftarrow}&=E_{0}^{x\rightarrow}\cdot\frac{-m_{20}m_{33}+m_{23}m_{30}}{m_{22}m_{33}-m_{23}m_{32}}\\\
E_{0}^{y\leftarrow}&=E_{0}^{x\rightarrow}\cdot\frac{m_{20}m_{32}-m_{22}m_{30}}{m_{22}m_{33}-m_{23}m_{32}}\\\
E_{N}^{x\rightarrow}&=E_{0}^{x\rightarrow}\cdot\left(m_{00}+\frac{-m_{20}(m_{02}m_{33}-m_{03}m_{32})+m_{30}(m_{02}m_{23}-m_{03}m_{22})}{m_{22}m_{33}-m_{23}m_{32}}\right)\\\
E_{N}^{y\rightarrow}&=E_{0}^{x\rightarrow}\cdot\left(m_{10}+\frac{-m_{20}(m_{12}m_{33}-m_{13}m_{32})+m_{30}(m_{12}m_{23}-m_{13}m_{22})}{m_{22}m_{33}-m_{23}m_{32}}\right).\end{cases}$
(13)

This transfer-matrix method is used to calculate the transmission through $N$
slides with and without Faraday rotation. Different realizations of disorder
were realized by randomly varying the thicknesses $d_{i}$ and $e_{i}$ of both
the plates and the spaces between the plates uniformly over $\Delta d$ and
$\Delta e\gg\lambda_{0}$.

###  A.2 Fresnel formulae at the boundary of a circular birefringent medium

We derive the Fresnel formulae at normal incidence between air (refractive
index $n_{0}$) and a Faraday or optical active (FA or OA) medium described by
two refractive indices $n_{+}$ and $n_{-}$ (see Table 1 for the definitions).

Propagation direction | Polarization | FA | OA
---|---|---|---
forward $\rightarrow$ | RHC | $n_{+}$ | $n_{+}$
forward $\rightarrow$ | LHC | $n_{-}$ | $n_{-}$
backward $\leftarrow$ | RHC | $n_{+}$ | $n_{-}$
backward $\leftarrow$ | LHC | $n_{-}$ | $n_{+}$
Table 1: Refractive indices seen by the forward and backward propagating
circular right (RHC) and left (LHC) polarizations.

$\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$ and $\hat{\mathbf{z}}$ are unit vectors
defined in Fig. 10: $(\hat{\mathbf{x}},\hat{\mathbf{y}})$ is the polarization
plane, and $\hat{\mathbf{z}}$ is perpendicular to the interface. We define the
right ($\mathbf{R}$) and left ($\mathbf{L}$) basis vectors for the forward
($\rightarrow$) and backward ($\leftarrow$) propagation directions,

$\displaystyle\begin{cases}\mathbf{R}^{\rightarrow}=\frac{1}{\sqrt{2}}\left(\hat{\mathbf{x}}+i\hat{\mathbf{y}}\right)\\\
\mathbf{L}^{\rightarrow}=\frac{1}{\sqrt{2}}\left(\hat{\mathbf{x}}-i\hat{\mathbf{y}}\right)\\\
\mathbf{R}^{\leftarrow}=\frac{1}{\sqrt{2}}\left(\hat{\mathbf{x}}-i\hat{\mathbf{y}}\right)\\\
\mathbf{L}^{\leftarrow}=\frac{1}{\sqrt{2}}\left(\hat{\mathbf{x}}+i\hat{\mathbf{y}}\right).\end{cases}$
(14)

As an example, let us now assume that the incident light is circular right
polarized and propagates in the forward direction
$\mathbf{E}_{i}=\mathbf{E}_{i}^{\mathrm{R}}=E_{i}^{\mathrm{R}}\mathbf{R}^{\rightarrow}$.
We define the reflection $r$ and transmission coefficients $t$ for circular
right R and left L by

$\displaystyle\begin{cases}\mathbf{E}_{r}&=E_{i}^{\mathrm{R}}\left(r_{\mathrm{R}}\mathbf{R}^{\leftarrow}+r_{\mathrm{L}}\mathbf{L}^{\leftarrow}\right)\\\
\mathbf{E}_{t}&=E_{i}^{\mathrm{R}}\left(t_{\mathrm{R}}\mathbf{R}^{\rightarrow}+t_{\mathrm{L}}\mathbf{L}^{\rightarrow}\right).\end{cases}$
(15)

We call
$\mathbf{E}_{t}^{\mathrm{R}}=E_{i}^{\mathrm{R}}t_{\mathrm{R}}\mathbf{R}^{\rightarrow}$
and
$\mathbf{E}_{t}^{\mathrm{L}}=E_{i}^{\mathrm{R}}t_{\mathrm{L}}\mathbf{L}^{\rightarrow}$,
and write both continuity equations for the electric and the magnetic fields,

$\displaystyle\begin{cases}\hat{\mathbf{z}}\times\left(\mathbf{E}_{i}+\mathbf{E}_{r}\right)&=\hat{\mathbf{z}}\times\mathbf{E}_{t}\\\
\hat{\mathbf{z}}\times\left(\mathbf{k}_{0}\times\mathbf{E}_{i}-\mathbf{k}_{0}\times\mathbf{E}_{r}\right)&=\hat{\mathbf{z}}\times\left(\mathbf{k}_{+}\times\mathbf{E}_{t}^{R}+\mathbf{k}_{-}\times\mathbf{E}_{t}^{L}\right),\end{cases}$
(16)

with $\mathbf{k}_{0}$ the wave vector in the dielectric $n_{0}$, and
$\mathbf{k}_{+/-}$ the wave vectors for the components seeing the $n_{+/-}$
indices (all wave vectors are oriented in the $\rightarrow$ direction).
Solving this system at normal incidence gives

$\displaystyle\begin{cases}r_{\mathrm{R}}&=0\\\
r_{\mathrm{L}}&=\frac{n_{0}-n_{+}}{n_{0}+n_{+}}\\\
t_{\mathrm{R}}&=\frac{2n_{0}}{n_{0}+n_{+}}\\\ t_{\mathrm{L}}&=0.\end{cases}$
(17)

This procedure can now be done for all propagation directions, circular
polarizations, and for both FA and OA materials. The resulting Fresnel
formulae are summarized in Table 2.

Direction | Incident pol. | $r_{\mathrm{R}}$ | $r_{\mathrm{L}}$ | $t_{\mathrm{R}}$ | $t_{\mathrm{L}}$
---|---|---|---|---|---
Diel $\rightarrow$ FA, OA | R | 0 | $\frac{n_{0}-n_{+}}{n_{0}+n_{+}}$ | $\frac{2n_{0}}{n_{0}+n_{+}}$ | 0
Diel $\rightarrow$ FA, OA | L | $\frac{n_{0}-n_{-}}{n_{0}+n_{-}}$ | 0 | 0 | $\frac{2n_{0}}{n_{0}+n_{-}}$
Diel $\leftarrow$ FA | R | 0 | $\frac{n_{-}-n_{0}}{n_{0}+n_{-}}$ | $\frac{2n_{-}}{n_{0}+n_{-}}$ | 0
Diel $\leftarrow$ FA | L | $\frac{n_{+}-n_{0}}{n_{0}+n_{+}}$ | 0 | 0 | $\frac{2n_{+}}{n_{0}+n_{+}}$
FA $\rightarrow$ Diel | R | 0 | $\frac{n_{+}-n_{0}}{n_{0}+n_{+}}$ | $\frac{2n_{+}}{n_{0}+n_{+}}$ | 0
FA $\rightarrow$ Diel | L | $\frac{n_{-}-n_{0}}{n_{0}+n_{-}}$ | 0 | 0 | $\frac{2n_{-}}{n_{0}+n_{-}}$
FA $\leftarrow$ Diel | R | 0 | $\frac{n_{0}-n_{-}}{n_{0}+n_{-}}$ | $\frac{2n_{0}}{n_{0}+n_{-}}$ | 0
FA $\leftarrow$ Diel | L | $\frac{n_{0}-n_{+}}{n_{0}+n_{+}}$ | 0 | 0 | $\frac{2n_{0}}{n_{0}+n_{+}}$
Diel $\leftarrow$ OA | R | 0 | $\frac{n_{+}-n_{0}}{n_{0}+n_{-}}$ | $\frac{n_{+}+n_{-}}{n_{0}+n_{-}}$ | 0
Diel $\leftarrow$ OA | L | $\frac{n_{-}-n_{0}}{n_{0}+n_{+}}$ | 0 | 0 | $\frac{n_{+}+n_{-}}{n_{0}+n_{+}}$
OA $\rightarrow$ Diel | R | 0 | $\frac{n_{+}-n_{0}}{n_{0}+n_{-}}$ | $\frac{n_{+}+n_{-}}{n_{0}+n_{-}}$ | 0
OA $\rightarrow$ Diel | L | $\frac{n_{-}-n_{0}}{n_{0}+n_{+}}$ | 0 | 0 | $\frac{n_{+}+n_{-}}{n_{0}+n_{+}}$
OA $\leftarrow$ Diel | R | 0 | $\frac{n_{0}-n_{+}}{n_{0}+n_{+}}$ | $\frac{2n_{0}}{n_{0}+n_{+}}$ | 0
OA $\leftarrow$ Diel | L | $\frac{n_{0}-n_{-}}{n_{0}+n_{-}}$ | 0 | 0 | $\frac{2n_{0}}{n_{0}+n_{-}}$
Table 2: Fresnel coefficients calculated with the refractive indices
convention of Table 1 for light propagating in the forward $\rightarrow$ or
backward $\leftarrow$ direction. 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 $B=18$ 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 anti-diagonal
(linear polarization on the first and second bisectors), and R/L are the
circular polarizations (right/left).

###  A.3 Transfer matrix and linear/circular polarization

To include Faraday rotation in these calculations it is useful to switch from
a linear basis $\begin{pmatrix}x\\\ y\end{pmatrix}$ to a circular basis
$\begin{pmatrix}\mathrm{R}\\\ \mathrm{L}\end{pmatrix}$, as Faraday rotation
can also be seen as circular birefringence. For propagation in the $+z$
direction ($\rightarrow$) and respectively for the $-z$ direction
($\leftarrow$), the fields in the circular basis can be expressed as

$\begin{cases}E_{R}^{+}=\frac{1}{\sqrt{2}}\left(E_{x}^{+}+iE_{y}^{+}\right)\\\
E_{L}^{+}=\frac{1}{\sqrt{2}}\left(E_{x}^{+}-iE_{y}^{+}\right)\\\
E_{R}^{-}=\frac{1}{\sqrt{2}}\left(E_{x}^{-}-iE_{y}^{-}\right)\\\
E_{L}^{-}=\frac{1}{\sqrt{2}}\left(E_{x}^{-}+iE_{y}^{-}\right)\\\ \end{cases}\
.$

With this, the matrix for the basis change between linear
$\begin{pmatrix}E_{x}^{+}\\\ E_{y}^{+}\\\ E_{x}^{-}\\\ E_{y}^{-}\\\
\end{pmatrix}$ and circular $\begin{pmatrix}E_{\mathrm{R}}^{+}\\\
E_{\mathrm{L}}^{+}\\\ E_{\mathrm{R}}^{-}\\\ E_{\mathrm{L}}^{-}\\\
\end{pmatrix}$ polarized fields can be calculated:

$\displaystyle P_{\mathrm{circ\rightarrow
lin}}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1&0&0\\\ i&-i&0&0\\\ 0&0&1&1\\\
0&0&-i&i\end{pmatrix},$ (18)

$\displaystyle P_{\mathrm{lin\rightarrow
circ}}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&-i&0&0\\\ 1&i&0&0\\\ 0&0&1&i\\\
0&0&1&-i\end{pmatrix}.$ (19)

##  Appendix B Magneto-optical spreading of polarization

###  B.1 Transfer-matrix simulations in reflection

####  B.1.1 Polarization states

Figure 11 shows the simulated polarization states in reflection for different
realizations of disorder for samples consisting of $N=1,2,5$ or 30 plates when
an external magnetic field of 18 T is applied on a Faraday active sample made
of SF57. The incident light is linearly polarized along the $x$ axis. Without
magnetic field, the reflected light stays linearly polarized along the $x$
axis. With magnetic field (18 T, V=31 rad/Tm), one can observe that the
polarization states get indeed randomized over the whole Poincaré sphere for
intermediate plate number, before being attracted on the equator of the
sphere. This means that for large plate numbers, the reflected light is always
linearly polarized, but with a random polarization direction. For optical
activity no such change of polarization changes is observed: all the points
are exactly on the X point (figure not shown), meaning that for any sample at
any optical activity strength, the reflected photons are always linearly
polarized in the incident polarization direction.

####  B.1.2 Refection coefficients in the case of Faraday rotation and optical
activity

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∘ for a single passage
through a single plate. This corresponds to a circular birefringence $\Delta
n=4.72\times 10^{-5}$ for FR in SF57 at $B=18T$ and at the average thickness
of the plates.

We plot in Fig. 12 the ensemble average over disorder of the reflection
coefficients $R_{xx}$ and $R_{xy}$ as a function of the length of the sample,
for an external magnetic field of 18 Tesla and compare it to the case were
$M_{\mathrm{FR}}$ (Eq. (8)) is replaced by $M_{\mathrm{OA}}$ (Eq. (10)).
Without circular birefringence, $\left<R_{xy}\right>=0$ because the two
channels are uncoupled. When circular birefringence is turned on, for a large
enough sample, optical activity implies $\left<R_{xx}\right>=1$ and
$\left<R_{xy}\right>=0$ (the reflected photons leave the sample with the same
polarization as the incident light), while Faraday rotation implies
$\left<R_{xx}\right>=\left<R_{xy}\right>=0.5$ (the polarization of the
reflected photons is randomized). These observations illustrate the fact that
Faraday rotation breaks reciprocity, while optical activity does not.

###  B.2 Experimental observation in transmission

The spreading of the polarization states in transmission is discussed in Fig.
2 for the transfer-matrix simulations. On the experimental side, the strong
average Faraday rotation and its progressive spreading (depolarization) is
also directly seen in the polarized average transmission. Figure 13 shows
$\langle\ln T\rangle$ plotted as a function of the external magnetic field $B$
for different fixed numbers of 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.

The $N=0$ measurement (black points) is very stable and used as reference. The
oscillations of $\langle\ln T_{xx}\rangle$ due to the non reflected part of
the light beam are again observed (red and violet data in Fig. 13), and they
smear out for large field when the distribution of the polarization states has
become almost symmetric around the polar axis of the Poincaré sphere. The
pseudo-period of the oscillations decreases with the number of plates as there
is more average Faraday rotation as well as higher variance of FR in longer
samples (compare $N=25$ plates (red) and $N=30$ plates (violet) data of Fig.
13).

Note that a field-dependent oscillation behavior of $\left<\ln T\right>$ such
as seen in Fig. 13 is not found in genuine-Q1D (i.e., with reflecting walls
Chabanov  _et al._ (2000)) or 3D multiple-scattering situations where
polarization memory is lost on the scale of the transport mean free path
$\ell^{\star}$ and, as a consequence, the overall average FR remains zero in
transmission Lenke and Maret (2000). The oscillations highlight the fact that
light propagation in 1D or Q1D stacks of slides is essentially parallel or
antiparallel to $B$, while it is random in situations of diffuse multiple-
scattering (when $\ell^{\star}$ is smaller that the transverse scale of
confinement).

##  Appendix C Data analysis

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.

The measured images (see Fig. 6) were processed to obtain the mean
transmission $\left<T\right>$ (and/or the mean of the logarithm of the
transmission $\langle\ln T\rangle$) and the variance of the relative
transmitted intensity $s=T/\left<T\right>$. Figure 14 shows the different
steps of the image processing for one measured image of a transmission
speckle.

The intensity variations caused by the incident Gaussian beam need to be
removed to obtain the correct intensity statistics. A 3D plot of an original
image is shown in the upper left. A speckle pattern overlaying the Gaussian
beam can be observed. The intensity data were fitted by a 2D Gaussian with
widths $\sigma_{x}$ and $\sigma_{y}$ and an offset $b$. The offset is
necessary to account for noise and was subtracted from the images (lower
left). Moreover, an elliptical region of interest (ROI) was chosen with widths
2$\sigma_{x}$ and 2$\sigma_{y}$ respectively (see cutoff in the upper right
image). The elliptical form is necessary since for a large number of plates
the transmitted beam becomes elliptical due to non-perfect alignment of the
sample regarding the beam direction. This ellipse rotates with the rotating
sample such that the ROI was further narrowed by taking into account only the
overlap of the ellipses of all 300 images. The ROI was divided by the Gaussian
fit to obtain the final images as shown in the lower right in Fig. 14. For an
increasing number of glass slides increasing laser powers were used as the
transmission decreases with the number of plates. Therefore, the transmitted
images were normalized by the incident laser power.

## References

  * Wiersma (2013) D. S. Wiersma, Disordered photonics, Nat. Photon. 7, 188 (2013).
  * Sperling  _et al._ (2016) T. Sperling, L. Schertel, M. Ackermann, G. J. Aubry, C. M. Aegerter, and G. Maret, Can 3D light localization be reached in ‘white paint’?, New Journal of Physics 18, 013039 (2016).
  * Skipetrov and Page (2016) S. E. Skipetrov and J. H. Page, Red light for anderson localization, New Journal of Physics 18, 021001 (2016).
  * Aubry  _et al._ (2017) G. J. Aubry, L. Schertel, M. Chen, H. Weyer, C. M. Aegerter, S. Polarz, H. Cölfen, and G. Maret, Resonant transport and near-field effects in photonic glasses, Phys. Rev. A 96, 043871 (2017).
  * Schertel  _et al._ (2019) L. Schertel, I. Wimmer, P. Besirske, C. M. Aegerter, G. Maret, S. Polarz, and G. J. Aubry, Tunable high-index photonic glasses, Phys. Rev. Materials 3, 015203 (2019), arXiv:1810.04403 [physics.optics] .
  * Escalante and Skipetrov (2017) J. M. Escalante and S. E. Skipetrov, Longitudinal optical fields in light scattering from dielectric spheres and anderson localization of light, Annalen der Physik 529, 1700039 (2017).
  * Smolka  _et al._ (2011) S. Smolka, H. Thyrrestrup, L. Sapienza, T. B. Lehmann, K. R. Rix, L. S. Froufe-Pérez, P. D. García, and P. Lodahl, Probing the statistical properties of anderson localization with quantum emitters, New Journal of Physics 13, 063044 (2011).
  * Ghosh  _et al._ (2017) S. Ghosh, C. Miniatura, N. Cherroret, and D. Delande, Coherent forward scattering as a signature of anderson metal-insulator transitions, Phys. Rev. A 95, 041602(R) (2017).
  * Lopez-Bezanilla  _et al._ (2018) A. Lopez-Bezanilla, L. S. Froufe-Pérez, S. Roche, and J. J. Sáenz, Unequivocal signatures of the crossover to anderson localization in realistic models of disordered quasi-one-dimensional materials, Phys. Rev. B 98, 235423 (2018).
  * Prat  _et al._ (2019) T. Prat, D. Delande, and N. Cherroret, Quantum boomeranglike effect of wave packets in random media, Phys. Rev. A 99, 023629 (2019).
  * Wolf and Maret (1985) P. E. Wolf and G. Maret, Weak localization and coherent backscattering of photons in disordered media, Phys. Rev. Lett. 55, 2696 (1985).
  * van Albada and Lagendijk (1985) M. P. van Albada and A. Lagendijk, Observation of weak localization of light in a random medium, Phys. Rev. Lett. 55, 2692 (1985).
  * Bromberg  _et al._ (2016) Y. Bromberg, B. Redding, S. M. Popoff, and H. Cao, Control of coherent backscattering by breaking optical reciprocity, Phys. Rev. A 93, 023826 (2016).
  * Rayleigh (1885) L. Rayleigh, On the constant of magnetic rotation of light in bisulphide of carbon, Philosophical Transactions of the Royal Society of London 176, 343 (1885).
  * Golubentsev (1984) A. Golubentsev, The suppression of interference effects in the multiple scattering of light, Sov. Phys. JETP 59, 26 (1984), original version in Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki 86 (1), pp.47-59 (1984).
  * MacKintosh and John (1988) F. C. MacKintosh and S. John, Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media, Phys. Rev. B 37, 1884 (1988).
  * Erbacher  _et al._ (1993) F. A. Erbacher, R. Lenke, and G. Maret, Multiple Light Scattering in Magneto-optically Active Media, EPL (Europhysics Letters) 21, 551 (1993).
  * Lenke and Maret (2000) R. Lenke and G. Maret, Magnetic field effects on coherent backscattering of light, The European Physical Journal B - Condensed Matter and Complex Systems 17, 171 (2000).
  * Schertel  _et al._ (2017) L. Schertel, G. J. Aubry, C. M. Aegerter, and G. Maret, Coherent multiple light scattering in faraday active materials, The European Physical Journal Special Topics 226, 1409 (2017).
  * Muskens  _et al._ (2012) O. L. Muskens, P. Venn, T. van der Beek, and T. Wellens, Partial Nonlinear Reciprocity Breaking through Ultrafast Dynamics in a Random Photonic Medium, Phys. Rev. Lett. 108, 223906 (2012).
  * Müller  _et al._ (2015) K. Müller, J. Richard, V. V. Volchkov, V. Denechaud, P. Bouyer, A. Aspect, and V. Josse, Suppression and revival of weak localization through control of time-reversal symmetry, Phys. Rev. Lett. 114, 205301 (2015).
  * Micklitz  _et al._ (2015) T. Micklitz, C. A. Müller, and A. Altland, Echo spectroscopy of anderson localization, Phys. Rev. B 91, 064203 (2015).
  * Hainaut  _et al._ (2018) C. Hainaut, I. Manai, J.-F. Clément, J. C. Garreau, P. Szriftgiser, G. Lemarié, N. Cherroret, D. Delande, and R. Chicireanu, Controlling symmetry and localization with an artificial gauge field in a disordered quantum system, Nature Communications 9, 1382 (2018), arXiv:1709.02632 [quant-ph] .
  * Cherroret  _et al._ (2019) N. Cherroret, M. Hemmerling, V. Nador, J. T. M. Walraven, and R. Kaiser, Robust coherent transport of light in multilevel hot atomic vapors, Phys. Rev. Lett. 122, 183203 (2019), arXiv:1812.08651 [physics.atom-ph] .
  * Bergmann (1984) G. Bergmann, Weak localization in thin films: a time-of-flight experiment with conduction electrons, Physics Reports 107, 1 (1984).
  * Pichard  _et al._ (1990) J.-L. Pichard, M. Sanquer, K. Slevin, and P. Debray, Broken symmetries and localization lengths in anderson insulators: Theory and experiment, Phys. Rev. Lett. 65, 1812 (1990).
  * Thaha  _et al._ (1993) M. Thaha, R. Blümel, and U. Smilansky, Symmetry breaking and localization in quantum chaotic systems, Phys. Rev. E 48, 1764 (1993).
  * Lerner and Imry (1995) I. V. Lerner and Y. Imry, Magnetic-field dependence of the localization length in anderson insulators, Europhysics Letters (EPL) 29, 49 (1995).
  * Katsumoto  _et al._ (1987) S. Katsumoto, F. Komori, N. Sano, and S.-i. Kobayashi, Fine tuning of metal-insulator transition in Al0.3Ga0.7As using persistent photoconductivity, Journal of the Physical Society of Japan 56, 2259 (1987).
  * Beenakker (1997) C. W. J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69, 731 (1997).
  * Evers and Mirlin (2008) F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  * Abrahams  _et al._ (1979) E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979).
  * Berry and Klein (1997) M. V. Berry and S. Klein, Transparent mirrors: rays, waves and localization, European Journal of Physics 18, 222 (1997).
  * Skipetrov and Sokolov (2015) S. E. Skipetrov and I. M. Sokolov, Magnetic-field-driven localization of light in a cold-atom gas, Phys. Rev. Lett. 114, 053902 (2015).
  * Skipetrov (2018) S. E. Skipetrov, Localization transition for light scattering by cold atoms in an external magnetic field, Phys. Rev. Lett. 121, 093601 (2018).
  * Skipetrov and Sokolov (2014) S. E. Skipetrov and I. M. Sokolov, Absence of Anderson Localization of Light in a Random Ensemble of Point Scatterers, Phys. Rev. Lett. 112, 023905 (2014).
  * Bellando  _et al._ (2014) L. Bellando, A. Gero, E. Akkermans, and R. Kaiser, Cooperative effects and disorder: A scaling analysis of the spectrum of the effective atomic hamiltonian, Phys. Rev. A 90, 063822 (2014).
  * Kuhl  _et al._ (2008) U. Kuhl, F. M. Izrailev, and A. A. Krokhin, Enhancement of localization in one-dimensional random potentials with long-range correlations, Phys. Rev. Lett. 100, 126402 (2008).
  * Karbasi  _et al._ (2012) S. Karbasi, T. Hawkins, J. B., K. W. Koch, and A. Mafi, Transverse anderson localization in a disordered glass optical fiber, Opt. Mater. Express 2, 1496 (2012).
  * Lu  _et al._ (2010) Y. Lu, C. Miniatura, and B.-G. Englert, Average transmission probability of a random stack, European Journal of Physics 31, 47 (2010).
  * Zhang  _et al._ (2008) S. Zhang, J. Park, V. Milner, and A. Z. Genack, Photon delocalization transition in dimensional crossover in layered media, Phys. Rev. Lett. 101, 183901 (2008).
  * Park  _et al._ (2010) J. Park, S. Zhang, and A. Z. Genack, Intensity statistics and photon localization beyond one dimension, Phys. Rev. E 82, 045101(R) (2010).
  * Chabanov  _et al._ (2000) A. Chabanov, M. Stoytchev, and A. Genack, Statistical signatures of photon localization, Nature 404, 850 (2000).
  * Genack and Chabanov (2005) A. Z. Genack and A. A. Chabanov, Signatures of photon localization, Journal of Physics A: Mathematical and General 38, 10465 (2005).
  * Kogan and Kaveh (1995) E. Kogan and M. Kaveh, Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium, Phys. Rev. B 52, R3813 (1995).
  * Nieuwenhuizen and van Rossum (1995) T. M. Nieuwenhuizen and M. C. W. van Rossum, Intensity distributions of waves transmitted through a multiple scattering medium, Phys. Rev. Lett. 74, 2674 (1995).
  * Goodman (2015) J. W. Goodman,  _Statistical optics_ , edited by G. Boreman, Wiley Series in Pure and Applied Optics (John Wiley & Sons, 2015).
  * Guasoni  _et al._ (2016) M. Guasoni, P. Morin, P.-Y. Bony, S. Wabnitz, and J. Fatome, Self-induced polarization tracking, tunneling effect and modal attraction in optical fiber, Optics & Laser Technology 80, 247 (2016).
  * Vorontsov and Miller (1995) M. A. Vorontsov and W. B. Miller, eds.,  _Self-Organization in Optical Systems and Applications in Information Technology_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 1995).
  * Schott Advanced Optics (2015) Schott Advanced Optics, Technical safety information SF57 (2015), accessed: 2019-02-09.
  * Weber (2003) M. J. Weber,  _Handbook of Optical Materials_ , Laser & Optical Science & Technology (CRC Press, Boca Raton, 2003).