## Abstract

In this manuscript, we experimentally and numerically investigate the chaotic dynamics of the state-of-polarization in a nonlinear optical fiber due to the cross-interaction between an incident signal and its intense backward replica generated at the fiber-end through an amplified reflective delayed loop. Thanks to the cross-polarization interaction between the two-delayed counter-propagating waves, the output polarization exhibits fast temporal chaotic dynamics, which enable a powerful scrambling process with moving speeds up to 600-krad/s. The performance of this all-optical scrambler was then evaluated on a 10-Gbit/s On/Off Keying telecom signal achieving an error-free transmission. We also describe how these temporal and chaotic polarization fluctuations can be exploited as an all-optical random number generator. To this aim, a billion-bit sequence was experimentally generated and successfully confronted to the dieharder benchmarking statistic tools. Our experimental analysis are supported by numerical simulations based on the resolution of counter-propagating coupled nonlinear propagation equations that confirm the observed behaviors.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Nowadays, chaotic dynamics in optical systems find numerous applications, among which the most emblematic are cryptographic secure communications [1–3], reservoir computing [4,5] or random bit generation [6–15]. Similarly to other physical areas, the common feature of those systems is that the route to chaos is achieved by means of a time-delayed feedback. In this contribution, we particularly focus our attention on the chaotic dynamics of the polarization state occurring in optical fiber and driven by a counter-propagating feedback setup. Beyond its fundamental aspect, understanding and harvesting the randomness of the light state-of-polarization (SOP) can find several practical applications. For instance, polarization scrambler devices are mainly implemented in optical communication and exploit polarization randomness in order to ensure polarization diversity in transmission links so as to mitigate the impact of polarization mode dispersion [16-17]. There are also mandatory apparatus when testing the performances of polarization-sensitive fiber devices, integrated systems or optical components. While commercially available polarization scramblers are usually based on the cascade of fiber resonant coils, rotating wave-plates, or fiber squeezers and opto-electronic elements [18–23], we have recently introduced an alternative approach based on the nonlinear Kerr effect occurring in optical fibers [24,25]. Fundamentally, it relies on an additional operating mode, i.e. the chaotic regime, of a bistable device called Omnipolarizer, originally designed to operate as an all-optical polarization funnel [26,27]. The principle of operation consists in an incident signal with a fixed polarization-state which nonlinearly interacts through a cross-polarization process with its own backward replica generated at the fiber-end thanks to an amplified reflective loop. When a strong power imbalance between the two counter-propagating waves is applied, the system becomes unstable and chaotic temporal fluctuations of the output SOP are then observed, thus leading to an optical scrambling system [24]. In this new contribution, we go significantly beyond our previous works and show that the polarization instability and its route to chaos can be greatly improved by feeding this self-organized system with an optical delay [28]. In particular, the instability threshold and transient regimes leading to a chaotic regime have been found to be greatly reduced when a time-delayed feedback is incorporated in the system with a time scale larger than the characteristic nonlinear time. We then report on a series of experiments involving a 1-km long highly nonlinear fiber (HNLF) and show that scrambling speeds up to 600-krad/s can be achieved. Moreover, a 2.5-reduction in terms of power threshold compared to our previous results reported in [24] as well as a dramatic reduction of fluctuations in the scrambling speed and output degree-of-polarization (DOP) have been observed. Experimental measurements are well confirmed by numerical predictions based on the numerical resolution of a set of counter-propagating nonlinear propagation equations. The performance of this all-optical chaotic scrambler has then been evaluated for telecom applications on a 10-Gbit/s On/Off Keying (OOK) signal showing that an error-free transmission can be achieved in a fully chaotic scrambling regime.

The second practical application highlighted in this contribution relies on the possibility for a chaotic system to harvest its randomness properties in order to generate random numbers [6–13]. Indeed, a genuine random number generator (RNG) must produce unpredictable, unreproducible and unbiased sequences of numbers. For that specific reason, many true RNGs are based on the peculiar properties that characterize chaotic dynamics. Practically, the advantage of using an optical approach is that one can generate random numbers at high repetition rate directly in the physical layer rather than using classical algorithmic techniques. Examples of such all-optical RNG include optoelectronic devices such as chaotic oscillations of high-bandwidth lasers [6,7], polarization chaos from a VCSEL diode [8], supercontinuum generation [9], homodyne detection of vacuum states [10], cosmic photons [11], spontaneous emission [12], superluminescent diodes [13] or exploiting the randomness inherent to quantum mechanics effects [14,15]. In the last section of this manuscript, we propose to exploit the chaotic SOP dynamics induced by the time-delayed feedback loop to generate random binary sequences. In this proof-of-concept experiment, the evolution over time of the output Stokes parameters has been recorded and sampled according to a fixed threshold so as to compute a binary sequence of one billion of bits. The degree of randomness of the generated bit sequence has been evaluated using the standard statistical benchmark provided by the dieharder testing suite [29], which shows that polarization chaos in optical fiber can be an efficient source of randomness for the generation of random numbers.

## 2. Principle and modeling

The system under-study is depicted in Fig.
1. It basically consists in a nonlinear Kerr medium, here a segment
of randomly birefringent telecommunication fiber of length
*L*, in which an initial forward signal defined by its
Stokes vector ** S** = [

*S*] and characterized by a fixed input SOP interacts nonlinearly with its own counter-propagating replica

_{1}, S_{2}, S_{3}**= [**

*J**J*] through a cross-polarization interaction. The forward signal

_{1}, J_{2}, J_{3}**is generated at the fiber-end by means of an amplified reflective device incorporating a time-delay (in practice, this optical delay is made of a second segment of standard fiber**

*J**Ld*for which the propagation regime is supposed to be linear). The dynamics of the system, in particular the chaotic regime, is mainly driven by the amplification factor

*g*of the reflective apparatus defined as the power ratio between the backward and forward signals measured in

*z*=

*L*:

*g*$=\left|J\left(L,t\right)\right|/\left|S\left(L,t\right)\right|$. The normalized unitary vectors $s=S/\left|S\right|$ and $j=J/\left|J\right|$ indicate their corresponding SOPs. For such a system, the spatio-temporal dynamics of

**and**

*S***are described by the following set of coupled nonlinear propagation equations [26, 30]:**

*J**z*and

*t*denote the propagation distance and time coordinates, respectively,

*c*is the speed of light,

*γ*the nonlinear Kerr coefficient,

*D*the diagonal matrix $diag\left(-8/9,-8/9,8/9\right)$ and

*α*the fiber losses, while the symbol $^$ denotes the vectorial product. Since typical temporal fluctuations come in the microsecond scale, only continuous-waves are considered in simulations, thus the chromatic dispersion has been neglected.

In order to simulate such a system including the time-delayed feedback, we
implement the following numerical procedure. Equations (1) are numerically resolved along the whole
propagation distance *L* + *Ld* while the
boundary conditions and time-delay are taken into account in such a way
that $J\left(L+Ld,t\right)=$ *gR*$S\left(L+Ld,t\right)$, where *R* denotes the
rotation matrix of the reflective device, while the nonlinear Kerr
coefficient *γ* as well as fiber losses are taken to zero
along the *Ld* segment. In this way, the nonlinear coupling
between the counter-propagating waves is only effective for
*z* = [0, *L*].

Before describing the experimental and numerical results, we would like
here to briefly comment by means of simple qualitative arguments the
mechanism underlying the fast process of polarization scrambling. As will
be discussed below, polarization scrambling is characterized by a fast and
disordered motion of the Stokes vectors on the surface of the Poincaré
sphere. This scrambling process originates from the fact that even weak
polarization fluctuations present in the incident waves are magnified
through the nonlinear coupling that exists between the wave itself and its
counter-propagating amplified replica. In fact, this feedback avalanche
process prevents the Omnipolarizer to reach a stationary solution, leading
to large polarization temporal fluctuations at its output. Moreover, the
insertion of an optical delay within the feedback loop allows to achieve a
complete decorrelation between the incident and backward wave
fluctuations, which greatly helps the system to enter into its chaotic
operation regime. On the other hand, from a fundamental point of view, a
key property of Eqs. (1) is
that, at variance with usual nonlinear Schrödinger equations governing
light propagation in optical fibers, here the dynamics is dominated by the
counter-propagating configuration of the interaction, i.e., there are no
second-order dispersion effects in Eqs. (1). This property introduces an undetermined sign into the
expression of the kinetic energy of the waves (setting *α*
= 0 the Hamiltonian of Eqs.
(1) is not bounded from above or below [31]). As a consequence, the system can exhibit a fast
decoherence process by creating rapid spatial fluctuations in the motion
of ** S** and

**, because the local increase of kinetic energy due to such rapid fluctuations in the forward**

*J***component can be compensated by a corresponding negative reduction in the backward wave component**

*S***(the detailed microscopic fluctuations of the backward waves still being decorrelated) [31]. Moreover, in the limit of a conservative interaction of Eqs. (1), such a fast scrambling process was shown to be responsible for an unexpected process of “unconstrained thermalization”: At variance with standard thermalization to equilibrium, here the system can freely increase the amount of disorder, because such an increase is no longer constrained by energy conservation, and can thus occur much faster than a slow conventional thermalization process [31].**

*J*## 3. Experimental setup

In order to study the chaotic dynamics of that system, we have implemented
the experimental setup depicted in Fig.
2. A 1-km length of HNLF is used as nonlinear Kerr medium. The HNLF
is characterized by a nonlinear coefficient *γ* = 9
W^{–1}km^{–1} and fiber losses *α* of 0.7
dB/km. The HNLF is then encapsulated between two optical circulators. The
input signal is amplified using an Erbium doped fiber amplifier (EDFA-1),
then the first circulator allows to inject the incident fully-polarized
light into the fiber and reject the counter-propagating replica
simultaneously. At the opposite end of the HNLF, the time-delayed feedback
apparatus consists in a fiber loop made of the second circulator, a 90:10
tap coupler to extract the output signal, a kms long spool of standard
single mode fiber (SSMF) as additional delay line and a second amplifier
(EDFA-2). The gain of the EFDA-2 is carefully controlled to adjust the
amplification factor *g*. A polarization controller is also
inserted within the loop in order to control the polarization rotation R
of the reflected beam. For fundamental studies, the input signal consists
in a fully-polarized 100-GHz bandwidth incoherent wave centered at 1550
nm. This incident signal is generated from an Erbium-based amplified
spontaneous noise source (ASE) sliced into its spectrum domain thanks to
an optical filter followed by an inline polarizer. This large bandwidth
input signal is used to avoid any impairment due to the stimulated
Brillouin backscattering in the fiber under-test but this process is
compatible with any type of incident signal. The signal is then amplified
up to 14 dBm by means of the EDFA-1 before injection into the fiber. At
the output of the system, the SOP of the resulting signal is characterized
by means of a standard commercial polarimeter. Furthermore, for random bit
generation experiments, the output signal SOP is projected on an inline
polarizer in order to transfer the polarization chaos into intensity
fluctuations. The resulting random signal is then recorded by means of a
1-GHz photodiode and a fast oscilloscope before digitalization
process.

In a final step, in order to further assess the performance of this all-optical chaotic scrambler for telecom applications, the incoherent wave has been substituted by a 10-Gbit/s OOK signal at 1550 nm. This return-to-zero (RZ) optical signal is generated from a 10-GHz mode-locked fiber laser (MLFL) delivering 2.5-ps pulses at 1550 nm. This 10-GHz pulse train is then intensity modulated thanks to a LiNbO3 Mach-Zehnder modulator driven by a 10-Gbit/s pulse pattern generator (PPG). Note that the initial pulse train is also phase modulated at 100 MHz in order to prevent any deleterious effect from Brillouin backscattering.

## 4. Experimental results

Figure 3(a) displays a 3-dimensionnal operation diagram of our system, which is
recorded at the system output as a function of the amplification factor
*g*. More precisely, it corresponds to the projection of
the output SOP in the S_{2}-S_{3} plane. For that
measurements, the input power is fixed to 14 dBm while the optical
delay-line for the forward signal consists in a 5-km long spool of SSMF.
Three different regions for the *g* parameter can be
observed. First-of-all, for a moderate level of backward power
(*g* < 8, ~20 dBm), it can be clearly seen that the two
waves do not interact, consequently, the output SOP remains almost
constant. For higher values of *g*, typically (8 ≤
*g* < 20, ~24 dBm), the system becomes unstable and
starts to oscillate. In this transient regime, more or less complicate
close trajectories can be observed whose complexity and frequency of
appearance increase with the level of backward power. Unstable fixed
points can be also observed. Moreover, in this transient regime, the
dynamics of the system was found to be dependent on both the input SOP and
the rotation matrix *R*. In contrast, by increasing further
the *g* values beyond 20 allows the system to enter into
the chaotic regime. In this case large fluctuations and aperiodic chaotic
behaviors are observed independently of the input SOP and the rotation
matrix *R*, leading to a full scrambling of the output SOP.
In order to illustrate the scrambling and chaotic behavior of the output
SOP, we have compared in Figs.
3(b)-3(e) the output Poincaré sphere for different values of the
amplification factor. While the output signal is characterized by a fixed
SOP for a weak value of *g* = 1 (a single point on the
sphere in Fig. 3(b)), we can
clearly note in Fig. 3(c) that the
output SOPs describe close trajectories in the transient regime (here
*g* = 10) and then exhibit a more or less complex behavior
for larger values of the reflective coefficient (*g* = 17
in Fig. 3(d)), before covering
almost homogeneously the complete surface of the sphere for high values of
*g* (*g* = 53 in Fig. 3(e)). This demonstrates the scrambling potential
of the underlying process. To go deeper into the analysis, we have also
reported in Figs. 3(b)-3(e) the
corresponding RF spectra of the output *S*_{1}
Stokes component. As the reflection coefficient *g* is
increased, one can easily notice that the resulting spectrum evolves from
a DC component to well-defined set of discrete frequencies in the
transient regime, until reaching a broad continuum of frequencies without
any discrete lines (Fig. 3(e)),
which further evidences the aperiodic and random nature of SOP
fluctuations in the chaotic regime. Moreover, to further understand the
key role of the time-delay introduced into the reflective loop as well as
its amplification factor *g*, we have carried out a series
of experiments and numerical simulations in three different configurations
for a fixed feedback delay: 0 delay (Omnipolarizer configuration reported
in [24]), 1 km of SSMF included
into the reflective loop and then 5 km. Furthermore, we have carefully
recorded 100 realizations for each value of the parameter
*g*, each of them having a different polarization rotation
matrix *R* to ensure the chaotic regime is reached
independently of the SOP of the backward replica. The input power is still
kept constant to 14 dBm. The scrambling performances have been evaluated
by means of the degree-of-polarization (DOP) and scrambling speed
(${V}_{scr}$) defined as:

*g*, while Figs. 4(d)-4(f) report the corresponding scrambling speed

*V*. Red solid lines represent average values of DOP and speed, while the shaded areas display respective fluctuations (standard deviation in grey and maximum excursion in ochre).

_{scr}From Figs. 4, the influence of the
delay-line becomes evident. Indeed, for an increasing delay starting from
0 to 5 km, the threshold value of *g* required to enter
into a genuine chaotic scrambling regime and thus reach a DOP close to
zero (at least <5%) is found to be significantly reduced. More
precisely, compared to our previous works (here zero delay case in Fig. 4(a)), the insertion of a delay-line
within the reflective loop allows to achieve at least a 2.5-fold reduction
of the chaotic amplification factor threshold. Moreover, it is important
to stress that the strong system fluctuations, typically generated in the
transient regime, progressively vanish for *g* ≥ 40 when a
1-km long delay-line is inserted in the system and just above 30 (backward
power of 26 dBm) with a 5-km spool of delay, thus making the scrambling
performances more predictable and reliable. The same behavior can be
observed for the scrambling speed in Figs.
4(d)-4(f) for which at least a 2-fold increase of
*V _{scr}* is achieved for the same amplification
factor value when a 5-km long delay-line is inserted in the system. The
performances and repeatability of the device are also greatly improved
with a large reduction of fluctuations in the output scrambling speed.
These results underline the fact that the polarization instability and its
route to chaos can be significantly improved by feeding this
self-organized system with an optical time-delay. We attribute this
behavior to the fact that the polarization fluctuations of the backward
waves are mutually decorrelated from each other, which greatly helps the
system to enter into the chaotic regime. Finally, we have also reported in
Figs. 4 by means of black circles
the results obtained from numerical resolution of Eqs. (1) averaged over 24
realizations, each involving a different rotation matrix

*R*and including the exact experimental parameters, in particular taking into account for the limit response of our polarimeter (1Msa/s). We can observe an excellent agreement between our numerical predictions and the experimental data, thus validating our theoretical model and providing a reliable tool for designing this home-made chaotic polarization scrambler.

In order to highlight in more details the key role played by the
delay-line, we have carried out the following additional measurements. For
a fixed maximal amplification factor of *g* = 53, we have
measured the resulting scrambling speed and DOP at the output of the
Omnipolarizer as a function of the fiber length inserted into the
reflective loop. Moreover, to further characterize the chaotic nature of
our system, we have also calculated the corresponding Lyapunov coefficient
*L* following the procedure described in [32]. These measurements are summarized in
Figs. 5, and have been averaged over 100 realizations involving different
rotation matrix *R*.

We can fairly observe the strong impact of the feedback delay-line with a
clear threshold around 200 m which enables to reach the maximum scrambling
speed near 610 krad/s (blue circles) and very low values of DOP (red
triangles). This threshold behavior is consistent with the nonlinear
response of the system and can be explained by the fact that, to enter
into a fully chaotic regime, the decorrelation time-scale between both
counter-propagating waves is governed by the nonlinear length of the
system defined by $1/\gamma P$, leading to 180 m for *g* =
53. Note also the excellent agreement between our measurements and
numerical predictions (averaged over 24 runs), depicted in Fig. 5(a) by means of blue stars for the
scrambling speed and red crosses for DOP. Finally, the calculation of the
experimental Lyapunov coefficient *L*, depicted here in
Fig. 5(b), shows that thanks to the
inclusion of a delay-line in the system, *L* becomes
largely positive for a fiber length beyond 200 m when *g* =
53, confirming aforementioned conclusions and providing a clear signature
of the chaotic nature of the SOP at the output of the system. As already
highlighted in Fig. 4, we can also
notice the strong reduction of fluctuations in the system performance
owing to the feedback delay, thus making our device more reliable.
Finally, Fig. 5(c) displays the
typical temporal evolution of the *S _{1}* Stokes
parameter at maximum scrambling speed and for 5 km of delay, directly
recorded at the output of the fiber beyond a polarizer. We can clearly see
that typical temporal fluctuations come in the microsecond scale.

## 5. Polarization scrambling of a 10-Gbit/s optical signal

The chaotic all-optical scrambler under-study has been tested in a
telecommunication configuration. For this proof-of-principle, the
partially coherent wave (100-GHz) described in the first section has been
substituted with a 10-Gbit/s RZ signal centered at 1550 nm. The pulse
width has been chosen as short as 2.5 ps in order to evaluate the impact
of the scrambler for higher repetition rates of data or high-frequencies
analogic signals. The delay-line inserted into the reflective loop is
fixed to 5 km while the input power is kept constant to 14 dBm. To ensure
that our device operates in a genuine chaotic regime, the amplification
factor has been chosen to *g* = 53, corresponding to a
backward power of 28 dBm. Figure 5
summarizes our results. Firstly, Figs.
6(a) and 6(b) display respectively the input and output Poincaré sphere of the
10-Gbit/s signal. While the input SOP is totally fixed at the input of the
system, the output Poincaré sphere appears entirely covered, thus
confirming that an efficient scrambling process can be achieved, even with
high-repetition rate pulsed signals. Furthermore, Figs. 6(c) and 6(d) depict the output eye-diagrams when
the backward signal is OFF (Fig.
6(c)) and in scrambling configuration (Fig. 6(d), pump ON). We can observe that the shape of
the pulses is ideally preserved with a clearly opened output eye-diagram,
which validates the applicability of our polarization scrambler to RZ
telecom signals. Note however the presence of an additional amplitude
jitter into the scrambled eye-diagram in Fig. 6(d). This source of noise was attributed to the Rayleigh
backscattering imposed by the backward signal on the output beam.
Nevertheless, in a wavelength division multiplexing configuration, one
could exploit one isolated pump channel in order to filter out this
deleterious noise source, as already proposed in [24]. To further assess the quality of the transmitted
signal, we have performed bit-error-rate (BER) measurements as a function
of the average power incoming on the receiver. Figure 6(e) compares the back-to-back configuration
(blue circles) with the pump OFF (red dots) and ON (purple diamonds) cases
for two values of the backward power (29 dBm and 30 dBm). We can first
stress that an error-free transmission is achieved for the scrambled
signal, confirming that our scrambler is fully compatible with such
single-channel RZ telecom configuration. However, a slight power penalty
(0.5 dB) has been measured between the pump ON/OFF curves, which is
attributed to the Rayleigh backscattering induced by the intense backward
signal. A power penalty of 1.5 dB has been detected compared to the input
configuration, which is mainly attributed to the deleterious effects of
chromatic dispersion and Kerr effect on the ultra-short pulses used in our
experiments.

## 6. Random bit generation

The genuine chaotic nature of our all-optical scrambler provides a good
opportunity to exploit this optical system as RNG, a field that has
recently received much attention in photonics [6–13]. To this aim, in the last section of this
manuscript, we take advantage of the chaotic evolution of the output
Stokes parameters to experimentally generate random bit sequences. For
this proof-of-concept experiment, the evolution over time of a Stokes
parameter has been recorded and concatenated from a large number of
different realizations in order to construct a 10^{9} bit
sequence. Such long sequences are mandatory in order to perform standard
statistical Dieharder tests as described below [29]. The principle of operation of random bit generation
from polarization chaos is to convert a Stokes component of the output
field of the scrambler into either a 0 or 1 depending on its relative
value according to some specific threshold, here calculated from the
median value of the waveform. In this series of experiments, the
*S _{1}* component is recorded beyond an inline
polarizer by means of a photo-receiver connected to a 1-Gsa/s
oscilloscope, 10

^{6}slots of 100-ms each were then acquired, concatenated and sampled. For that purpose, the chaotic polarization scrambler uses a 5-km delay-line as well as an amplification factor

*g*= 53. The input signal corresponds to the 100-GHz partially coherent wave described in the first section of the paper. Figure 7(a) displays a part of a typical experimental realization of the raw data (blue solid-line) as well as the post-processing involved in the random numbers extraction. Basically, the waveform is first under-sampled at a clock rate whose frequency is chosen well below the typical correlation length of the signal under test so as to ensure a reliable randomization. In our practical case, the clock (in black) has been chosen to 10 kHz for a typical scrambling speed of 610 krad/s. The

*S*signal is then sampled (red points) at each rising edge of the clock. After suitable thresholding by calculating the median value of the sampled signal, the binary random sequence represented in Fig. 7(b) has been obtained. Note that 5 days of continuous recording involving more than 1.2 To of raw data have been necessary to construct the billion bit sequence required for the dieharder test. Finally, in order to improve the randomness of the sequence and successfully pass the benchmark tests, we also remove any residual correlation and bias associated with binary conversion using an exclusive-or (XOR) gate between the initial sequence and its time-delayed replica [6, 9, 12, 13]. A delay of 100 bits was here applied. The degree of randomness of the computed binary sequences are first tested through calculation of the autocorrelation trace (Fig. 7(c)), and the cross-correlation function between two different sequences (Fig. 7(d)). These first results reveal a vanishing cross-correlation for all of the generated random bit streams, indicating that all the sequences are different and that the proposed technique is a good candidate for RNG.

_{1}To further assess the degree of randomness of the generated billion bit sequence, we have implemented the standard statistical benchmark dieharder tests. Results are summarized in Fig. 8 [see also Dataset 1 [33] for complete results] and show that the generated sequences pass all the most commonly used statistical tests (p-value > 0.01), thus demonstrating that the polarization chaos induced by a counter-propagating time-delayed feedback is a suitable source of randomness to generate random binary sequences. It is important to note that the Dieharder suite also includes all the statistical tests developed by the National Institute for Standards and Technology (NIST), as well as some extra tests [29], which all successfully passed with resulting values between 0.04 and 1, thus demonstrating the genuine randomness of the generated sequence.

## 7. Conclusions

In summary, we have experimentally investigated the chaotic dynamics of the state-of-polarization in a nonlinear optical fiber fed by an intense time-delayed backward replica. The fundamental basis of this work was initially proposed in [24] as an additional working regime of the device called Omnipolarizer [26] and dedicated to polarization scrambling. This system relies on a nonlinear cross-polarization interaction occurring in an optical fiber between an incident signal and its own high-power counter-propagating replica, generated at the fiber-end through an amplified reflective loop apparatus. In this new contribution, we go significantly beyond these previous results and show that the polarization instability and its route to chaos can be significantly improved by feeding this self-organized system thanks to a time-delayed feedback. Indeed, the polarization fluctuations of the incident and reflected waves are completely decorrelated from each other, which significantly helps the system to enter into a chaotic regime. For that particular configuration, we have shown that the instability threshold and transient regimes can be substantially reduced when a delay-line is incorporated in the reflective loop, with a time scale longer than the nonlinear characteristic time of the system. We have then reported on a series of experiments involving a 1-km long HNLF-based polarization scrambler including different lengths of delay-line within the reflective loop, which revealed that scrambling speeds up to 600-krad/s can be achieved. A 2.5-fold reduction in terms of power threshold compared to our previous observations as well as a significant reduction of the system performance fluctuations in the scrambling speed and output DOP have been observed. The performance of this all-optical scrambler for telecom applications has been evaluated on a 10-Gbit/s OOK RZ signal with error-free transmission. Finally, we also described how the chaotic nature of polarization fluctuations can be exploited to generate ensembles of random bit sequences. More precisely, through a digitalization process of the Stokes parameters at the output of the system, we have experimentally generated a billion bit sequence at a repetition rate of 10 kHz whose randomness has successfully passed the benchmark dieharder test. The speed of our random bit generator is quite low compared to recent publications, which can achieved several of Gbit/s in [6, 7, 12, 13]. However, these results represent the first proof-of-principle demonstration of random bit generation through a counter-propagating cross-polarization interaction. Moreover, the speed of our system as well as its compactness could be improved by implementing ultra-high nonlinear materials or high-confinement waveguides such as soft-glass optical fibers or silicon waveguides integrated on a CMOS compatible chip in order to achieve the Gbit/s repetition rate. Finally, our experimental observations have been well confirmed by numerical simulations based on the resolution of counter-propagating coupled nonlinear propagation equations.

## Funding

European Research Council (ERC Starting Grant PETAL) (306633); Agence Nationale de la Recherche (ANR) (APOFIS); Labex Action (ANR-11-LABX-0001-01); FEDER; Conseil Régional de Bourgogne Franche Comté.

## Acknowledgments

We thank Dr. Daniel Brunner from Femto-st institute in Besancon, as well as Dr. Anthony Martin from the University of Geneva for fruitful discussions.

## References and links

**1. **A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at
high bit rates using commercial fibre-optic links,”
Nature **438**(7066),
343–346 (2005). [CrossRef] [PubMed]

**2. **L. Keuninckx, M. C. Soriano, I. Fischer, C. R. Mirasso, R. M. Nguimdo, and G. Van der Sande, “Encryption key distribution
via chaos synchronization,” Sci. Rep. **7**, 43428 (2017). [CrossRef] [PubMed]

**3. **L. Larger and J. M. Dudley, “Nonlinear dynamics:
Optoelectronic chaos,” Nature **465**(7294),
41–42 (2010). [CrossRef] [PubMed]

**4. **D. Sussillo and L. F. Abbott, “Generating coherent patterns
of activity from chaotic neural networks,”
Neuron **63**(4),
544–557 (2009). [CrossRef] [PubMed]

**5. **F. Duport, A. Smerieri, A. Akrout, M. Haelterman, and S. Massar, “Fully analogue photonic
reservoir computer,” Sci. Rep. **6**(1), 22381
(2016). [CrossRef] [PubMed]

**6. **A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit
generation with chaotic semiconductor lasers,”
Nat. Photonics **2**(12),
728–732 (2008). [CrossRef]

**7. **I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random
bit generator,” Nat. Photonics **4**(1),
58–61 (2010). [CrossRef]

**8. **M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization
chaos from a laser diode,” Nat.
Photonics **7**(1),
60–65 (2013). [CrossRef]

**9. **B. Wetzel, K. J. Blow, S. K. Turitsyn, G. Millot, L. Larger, and J. M. Dudley, “Random walks and random
numbers from supercontinuum generation,” Opt.
Express **20**(10),
11143–11152 (2012). [CrossRef] [PubMed]

**10. **C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique
quantum random numbers based on vacuum states,”
Nat. Photonics **4**(10),
711–715 (2010). [CrossRef]

**11. **C. Wu, B. Bai, Y. Liu, X. Zhang, M. Yang, Y. Cao, J. Wang, S. Zhang, H. Zhou, X. Shi, X. Ma, J. G. Ren, J. Zhang, C. Z. Peng, J. Fan, Q. Zhang, and J. W. Pan, “Random Number Generation with
Cosmic Photons,” Phys. Rev. Lett. **118**(14), 140402
(2017). [CrossRef] [PubMed]

**12. **C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number
generator using amplified spontaneous emission,”
Opt. Express **18**(23),
23584–23597 (2010). [CrossRef] [PubMed]

**13. **X. Li, A. B. Cohen, T. E. Murphy, and R. Roy, “Scalable parallel physical
random number generator based on a superluminescent
LED,” Opt. Lett. **36**(6),
1020–1022 (2011). [CrossRef] [PubMed]

**14. **B. Sanguinetti, A. Martin, H. Zbinden, and N. Gisin, “Quantum random number
generation on a mobile phone,” Phys. Rev.
X **4**(3), 031056
(2014). [CrossRef]

**15. **T. Lunghi, J. B. Brask, C. C. W. Lim, Q. Lavigne, J. Bowles, A. Martin, H. Zbinden, and N. Brunner, “Self-testing quantum random
number generator,” Phys. Rev. Lett. **114**(15), 150501
(2015). [CrossRef] [PubMed]

**16. **F. Bruyere, O. Audouin, V. Letellier, G. Bassier, and P. Marmier, “Demonstration of an optimal
polarization scrambler for long-haul optical amplifier
systems,” IEEE Photonics Technol.
Lett. **6**(9),
1153–1155 (1994). [CrossRef]

**17. **Z. Li, J. Mo, Y. Wang, and C. Lu, “Experimental evaluation of
the eﬀect of polarization scrambling speed on the performance of PMD
mitigation using FEC,” in Optical Fiber
Commun. Conference (2004).

**18. **R. Noé, B. Koch, D. Sandel, and V. Mirvoda, “100-krad/s endless
polarisation tracking with miniaturised module card,”
Electron. Lett. **47**(14),
813–814 (2011). [CrossRef]

**19. **W. H. J. Aarts and G. Khoe, “New endless polarization
control method using three fiber squeezers,”
J. Lightwave Technol. **7**(7),
1033–1043 (1989). [CrossRef]

**20. **P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK
transmission with automatic polarization
stabilization,” IEEE Photonics Technol.
Lett. **21**(11),
745–747 (2009). [CrossRef]

**21. **Y. K. Lizé, R. Gomma, R. Kashyap, L. Palmer, and A. E. Willner, “Fast all-fiber polarization
scrambling using re-entrant Lefèvre controller,”
Opt. Commun. **279**(1),
50–52 (2007). [CrossRef]

**22. **F. Heismann, “Compact electro-optic
polarization scramblers for optically amplified lightwave
systems,” J. Lightwave Technol. **14**(8),
1801–1814 (1996). [CrossRef]

**23. **J.-W. Kim, S.-H. C. W.-S. Park, W. S. Chu, and M. C. Oh, “Integrated-optic polarization
controllers incorporating polymer waveguide birefringence
modulators,” Opt. Express **20**(11),
12443–12448 (2012). [CrossRef] [PubMed]

**24. **M. Guasoni, P. Y. Bony, M. Gilles, A. Picozzi, and J. Fatome, “Fast and Chaotic Fiber-Based
Nonlinear Polarization Scrambler,” IEEE J.
Sel. Top. Quantum Electron. **22**(2), 88
(2016). [CrossRef]

**25. **P.-Y. Bony, M. Guasoni, P. Morin, D. Sugny, A. Picozzi, H. R. Jauslin, S. Pitois, and J. Fatome, “Temporal spying and
concealing process in fibre-optic data transmission systems through
polarization bypass,” Nat. Commun. **5**, 4678 (2014). [CrossRef] [PubMed]

**26. **J. Fatome, S. Pitois, P. Morin, E. Assémat, D. Sugny, A. Picozzi, H. R. Jauslin, G. Millot, V. V. Kozlov, and S. Wabnitz, “A universal optical all-fiber
Omnipolarizer,” Sci. Rep. **2**(1), 938
(2012). [CrossRef] [PubMed]

**27. **P.-Y. Bony, M. Guasoni, E. Assémat, S. Pitois, D. Sugny, A. Picozzi, H. R. Jauslin, and J. Fatome, “Optical flip-flop memory and
data packet switching operation based on polarization bistability in a
telecomunication optical fiber,” J. Opt. Soc.
Am. B **30**(8),
2318–2325 (2013). [CrossRef]

**28. **J. Morosi, A. Akrout, A.
Picozzi, M. Gilles, M. Guasoni, and J. Fatome, “Polarization Chaos in
Nonlinear Optical Fibers Induced by a Reflective Delayed Loop,” in
*Conference on Lasers and Electro-Optics*, OSA
Technical Digest (online) (Optical Society of America, 2017), paper
SM2M.3. [CrossRef]

**29. **http://www.phy.duke.edu/~rgb/General/dieharder.php

**30. **V. Kozlov, J. Nuño, and S. Wabnitz, “Theory of lossless
polarization attraction in telecommunication fibers,”
J. Opt. Soc. Am. B **28**(1),
100–108 (2011). [CrossRef]

**31. **M. Guasoni, J. Garnier, B. Rumpf, D. Sugny, J. Fatome, F. Amrani, G. Millot, and A. Picozzi, “Incoherent Fermi-Pasta-Ulam
recurrences and unconstrained thermalization mediated by strong
phase-correlations,” Phys. Rev. X **7**(1), 011025
(2017). [CrossRef]

**32. **M. T. Rosenstein, J. J. Collins, and C. J. A. De Luca, “Practical method for
calculating largest Lyapunov exponents from small data
sets,” Physica D **65**(1-2),
117–134 (1993). [CrossRef]

**33. **Dataset 1: Results of
the dieharder tests for the experimental 1-billion bit sequence
generated from polarization chaoshttps://doi.org/10.6084/m9.figshare.5725807.