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Polarization control in spun and telecommunication
optical bers
E. Assemat, D. Dargent, Antonio Picozzi, Hans-Rudolf Jauslin, Dominique
Sugny
To cite this version:
E. Assemat, D. Dargent, Antonio Picozzi, Hans-Rudolf Jauslin, Dominique Sugny. Polarization control
in spun and telecommunication optical bers. Optics Letters, Optical Society of America - OSA
Publishing, 2011, 36, pp.4038. �hal-00699910�
Polarization control in spun and telecommunication
optical fibers
Elie Assémat, Damien Dargent, Antonio Picozzi, Hans-Rudolf Jauslin, and Dominique Sugny*
Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 5209 CNRS-Université de Bourgogne, Dijon, France
*Corresponding author: dominique.sugny@u‐bourgogne.fr
Received June 22, 2011; revised September 6, 2011; accepted September 15, 2011;
posted September 15, 2011 (Doc. ID 149754); published October 7, 2011
We consider the counterpropagating interaction of a signal and a pump beam in a spun fiber and in a randomly
birefringent fiber, the latter being relevant to optical telecommunication systems. On the basis of a geometrical anal-
ysis of the Hamiltonian singularities of the system, we provide a complete understanding of the phenomenon of po-
larization attraction in these two systems, which allows to achieve a control of the polarization state of the signal beam
by adjusting the polarization of the pump. In spun fibers, all polarization states of the signal beam are attracted toward
a specific line of polarization states on the Poincaré sphere, whose characteristics are determined by the polarization
state of the injected backward pump. In randomly birefringent telecommunication fibers, we show that an unpolar-
ized signal beam can be repolarized into any particular polarization state, without loss of energy. © 2011 Optical
Society of America
OCIS codes: 190.4370, 060.4370, 190.0190.
Conventional polarizers are known to achieve light polar-
ization by wasting 50% of the energy of the beam. More
recently, by exploiting the nonlinearity of certain materi-
als, “universal polarizers” performing polarization of un-
polarized light with 100% efficiency have been proposed
[
1–3]. This phenomenon may be called “polarization at-
traction,” in the sense that all input polarization config-
urations are transformed into a well-defined polar ization
state, without any loss of energy of the beam. Polariza-
tion attraction has been demonstrated, in particular, in
an optical fiber system pumped at both ends by two
counterpropagating beams [
4]. Considering an isotropic
optical fiber, it was shown that, for a given state of po-
larization (SOP) of the backward pump, all polarization
states of the forward signal beam can be attracted toward
a particular polarization state, which is determined by
the SOP of the pump [
2,5,6]. Recent studies revealed that
an efficient polarization attraction may also occur in dif-
ferent types of optica l fibers, like, e.g., highly birefringent
spun fibers (HBSFs) [
7] and in randomly birefringent
fibers (RBFs) [
8,9].
From a different perspective, we recently showed
that the phenomenon of polarization attraction can be
explained through the analysis of the singularities of the
stationary Hamiltonian dynamics of the system [
6,10,11].
This theoretical approach revealed that the stationary so-
lutions of interest belong to a particular two-dimensional
object of the corresponding phase space representation,
the so-called singular torus, whose singular topology is at
the origin of the process of attraction. These recently de-
veloped mathematical techniques [
12,13] proved remark-
ably efficient for the understanding of the phenomenon of
polarization attraction in isotropic fibers [
6,10,11].
Our aim in this Letter is to provide a generalized com-
prehension of the phenomenon of polarization attraction
by analyzing the concrete examples of HBSFs and RBFs
recently considered in Refs. [
7–9]. Making use of the
mathematical techniques exploited in [
6,10,11], we show
that HBSFs exhibit a novel phenomenon of polarization
attraction. As opposed to the conventional attraction pro-
cesses discussed so far, in HBSFs, all SOPs of the signal
beam are not attracted to a single point but toward a
specific line of polarization states of the Poincaré
sphere. In the particular case of a vanishing ellipticity
of the spun fiber, the stationary states of HBSFs become
of the same type as the one of RBFs. The theory reveals
in this case that an unpolarized signal beam can be po-
larized into any desired polarization state, which is deter-
mined by the injected SOP of the backward pump.
Besides its fundamental interest, polarization attractio n
in RBFs is of great importance in optical telecommunica-
tion systems in order to achieve a repolarization of opti-
cal transmission lines without loss of energy [
8,9].
The evolution of the counterpropagating waves in
HBSFs and RBFs is ruled by [7]
(
∂
~
S
∂t
þ
∂
~
S
∂z
¼
~
S∧ðI
~
SÞþ
~
S∧ðJ
~
JÞ
∂
~
J
∂t
−
∂
~
J
∂z
¼
~
J∧ðI
~
JÞþ
~
J∧ðJ
~
SÞ
; ð1Þ
where
~
S and
~
J are the Stokes vectors for the forward and
backward waves, respectively, and ∧ denotes the vector
product. The radii of the forward and backward spheres,
S
0
and J
0
, which correspond to the signal and pump
powers, are assumed to be identical in the follow-
ing (S
0
¼ J
0
).
Polarization attraction in HBSFs. In this case, the di-
agonal matrices I and J are given by I ¼ Diag
ð0; 0; 2 sin
2
ϕ − cos
2
ϕÞ and J ¼ cos
2
ϕ × Diagð 1; −1; −2Þ,
where ϕ is the ellipticity of the fiber [
7]. Because of
the counterpropagating nature of the interaction, station-
ary boundary conditions
~
Sðz ¼ 0Þ and
~
Jðz ¼ LÞ are
imposed at the two ends of the fiber of length L. We nor-
malized the problem with respect to the nonlinear inter-
action time τ
0
¼ 1=ð γS
0
Þ and length Λ
0
¼ vτ
0
, where γ is
the nonlinear Kerr coefficient and v the group velocity of
the waves.
As in the case of conventional isotropic fibers [
6,10,11],
numerical simulations of Eq. (
1) reveal that, irrespective
of the initial conditions, the spatiotemporal dynamics ex-
hibit a relaxation toward a stationary state. Our approach
is based on the study of the singularities of the stationary
Hamiltonian trajectories. In particular, we showed in
Refs. [
6,10,11] that the stationary states selected by the
4038 OPTICS LETTERS / Vol. 36, No. 20 / October 15, 2011
0146-9592/11/204038-03$15.00/0 © 2011 Optical Society of America
space–time dynamics lie on the surface of a singular
torus, which can be viewed as a two-dimensional exten-
sion of the concept of separatrix, well known for systems
with 1 degree of freedom. The singular torus thus plays
the role of an attractor for the wave system [
6,10,11]. We
refer the reader to Ref. [
14] for a detailed presentation of
the singular reduction theory applied to spun and tele-
communication optical fibers. Here, this approach is only
briefly explained. The Hamiltonian structure of the
stationary Eq. (
1) is defined by
H ¼ cos
2
ϕðS
2
J
2
− S
1
J
1
þ 2S
3
J
3
Þ
−
2 sin
2
ϕ − cos
2
ϕ
2
ðS
2
3
þ J
2
3
Þ: ð2Þ
The stationary system admits another constant of mo-
tion K ¼ S
3
þ J
3
, which makes the system Liouville-
integrable [
12,13]. The corresponding energy-momentum
diagram for the HBSF is reported in Fig.
1(a) for ϕ ¼ π=4.
Besides the usual regular tori, the diagram ðH;KÞ reveals
the presence of a line of singular bitori for ϕ ∈0; π=2½.A
bitorus is the union of two tori glued along a circle (see
Fig.
1). The equation of the line of singular bitori in the
diagram ðH;KÞ is
H ¼ cos
2
ϕðK
2
− 1Þ−
sin
2
ϕ
2
K
2
: ð3Þ
This continuous line of singu larities reveals the exis-
tence of a new phenomenon of polarization attraction.
Indeed, for an isotropic optical fiber the diagram was
characterized by an isolated singular torus, which in turn
was shown to lead to a polarization attraction toward a
unique (or a discrete set of) polarization state(s) of the
signal [
10,11]. Here, the presence of the line of singula-
rities will be shown to lead to a polarization attraction
toward a continuous line of polarization states on the
Poincaré sphere. As explained below, some restrictions
have to be added to this effect, which also depends on the
value of J
3
. We investigated this attraction by performing
numerical simulations of the space–time Eq. (
1). We con-
sidered 64 different initial SOPs of the signal
~
Sðz ¼ 0Þ
uniformly distributed over the surface of the Poincaré
sphere (green points in Fig.
2), while the SOP of the pump
was kept fixed (yellow point). As illustrated in Fig.
2,all
individual signal SOPs are attracted, at z ¼ L, toward a
specific continuous line on the surface of the Poincaré
sphere (red points).
The line of polarization attraction can be calculated
analytically from our approach. For a given pump SOP
~
Jðz ¼ LÞ, and using S
1
ðLÞ
2
þ S
2
ðLÞ
2
þ S
3
ðLÞ
2
¼ 1 and
K ¼ S
3
ðLÞþJ
3
ðLÞ, we can express the Hamiltonian (2)
in terms of K and of S
1
ðLÞ [or alternatively S
2
ðLÞ]. Then,
the variable H can be eliminated by using Eq. (
3), which
thus gives a quadratic equation for the unknown S
1
ðLÞ[or
S
2
ðLÞ], whose coefficients depend on K, ϕ, and
~
Jðz ¼ LÞ.
This system can be solved analytically, which provides
the solution S
1;2
ðLÞ parameterized by the variable K.
Let us illustrate our results with some concrete exam-
ples. If J
1
ðLÞ¼1 [or J
2
ðLÞ¼1], the line of polariza-
tion attraction draws an eight-shaped figure on the
Poincaré sphere [see Fig.
2(a)]. In the particular case ϕ ¼
π=4 and J
2
ðLÞ¼ε, with ε ¼1, a straightforward
computation leads to the parameterized equation
8
<
:
S
1
¼jKj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 − K
2
p
S
2
¼ −εð1 − K
2
Þ
S
3
¼ K
; ð4Þ
where the first equation implies that jKj ≤ 1. The two dif-
ferent signs for S
1
draw the corresponding two parts of
the eight-shaped figure. For an arbitrary SOP of the in-
jected pump, the eight curve exhibits a complex deforma-
tion, as illustrated in Fig.
2(b). Note that the numerical
simulations show that only half of the closed curve com-
puted from our theory can be reached by the spatiotem-
poral dynamics (see Fig.
2). Indeed, the stationary
solutions that lie on the other half-curve exhibit an oscil-
latory behavior and are unstable [
10]. It is also im portant
Fig. 1. (Color online) Energy momentum diagram ðH; KÞ for
the HBSF with (a) ϕ ¼ π=4 and (b) ϕ ¼ 0. The (a) singular (red)
line and the (b) singular (red) point, play the role of attractors
for the space–time dynamics (
1). The dark crosses locate the
positions of the stationary states obtained by solving numeri-
cally the space–time Eq. (
1) for different initial conditions of
the signal (L ¼ 5,
~
JðLÞ¼ð0; 1; 0Þ). Note that, according to
Eq. (
4), the positions of the crosses satisfy jKj ≤ 1. Each point
of the singular red line in (a) is associated to a bitorus, while the
singular red point at K ¼ 0 in (b) corresponds to a sphere of
singular points (see the text).
Fig. 2. (Color online) Polarization attraction toward a contin-
uous line of polarization states for the HSBF: Numerical simu-
lations of the spatiotemporal Eq. (
1) on the Poincaré sphere
[ϕ ¼ π=4 for (a) and ϕ ¼ π=5 for (b)]. The green and red dots
denote respectively the initial (
~
Sð0Þ) and final (
~
SðLÞ) SOPs of
the signal. The yellow dot denotes the fixed pump SOP:
(a)
~
JðLÞ¼ð0; 1; 0Þ, (b)
~
JðLÞ¼ð0:7; 0:7; 0Þ for L ¼ 5. The blue
line is calculated analytically from our method.
October 15, 2011 / Vol. 36, No. 20 / OPTICS LETTERS 4039
to remark that, as previously pointed out in [7], the po-
larization attraction becomes less efficient as J
3
ðLÞ ap-
proaches 1. More precisely, it can be shown that the
system relaxes toward values of K belonging to an inter-
val whose length decreases to 0 as J
3
ðLÞ goes to 1.In
the limit J
3
ðLÞ¼1 (i.e., J
1
ðLÞ¼J
2
ðLÞ¼0), Eq. (2)no
longer depends on S
1;2
and H is a function of only K and
ϕ. This expression of H is not compatible with Eq. (
3).
This simply means that the corresponding stationary
state cannot belong to the line of singular tori in the
diagram ðH;KÞ.
In the particular case ϕ ¼ 0, the line of bitori in the
diagram ðH; KÞ reduces to a single point of coordinates
ðH ¼ −1;K ¼ 0Þ, which belongs to the boundary of the
diagram [see Fig.
1(b)]. According to our previous discus-
sion, this qualitative change should lead to a polarization
attraction toward a unique SOP on the Poincaré sphere.
This is confirmed by the theory and the numerical simu-
lations of Eq. (
1). They show that an arbitrary polariza-
tion state of the signal is attracted (at z ¼ L) toward a
specific SOP, which is determined by the injected pump
SOP, i.e., S
3
ðLÞ¼−J
3
ðLÞ, S
1
ðLÞ¼J
1
ðLÞ and S
2
ðLÞ¼
−J
2
ðLÞ. From a geometrical point of view, note that
the bitorus is transformed into a sphere when ϕ ¼ 0
(see Fig.
1).
Polarization attraction in RBFs. It is interesting to
note that RBFs exhibit a diagram ðH;KÞ very similar
to that of HBSFs with ϕ ¼ 0. We thus study RBFs by con-
sidering the model recently derived in [
9], in which the
counterpropagating interaction is ruled by Eq. (
1) with
I ¼ Diagð0; 0; 0Þ and J ¼ Diagð−1; 1; −1Þ. The stationary
system is Hamiltonian with H ¼ S
1
J
1
− S
2
J
2
þ S
3
J
3
.In
this expression of H, the three axes play a symmetric
role, a property that leads to three constants of motion,
K
1
¼ S
1
þ J
1
, K
2
¼ S
2
− J
2
, and K
3
¼ S
3
þ J
3
. As for
HBFs with ϕ ¼ 0, the diagram ðH;KÞ for RBFs exhibits
a unique singular point associated to a sphere belonging
to the boundary of the diagram [see Fig.
3(a)]. Any point
of this set that satisfies K
1
¼ K
2
¼ K
3
¼ 0 is a fixed point
with respect to the stationary Eq. (
1). Numerical compu-
tations show that this set plays the role of an attractor for
the spatiotemporal dynamics. Accordingly, the signal
beam is attracted (up to a sign change) by the SOP
of the pump, i.e., S
1
ðLÞ¼−J
1
ðLÞ, S
2
ðLÞ¼J
2
ðLÞ, and
S
3
ðLÞ¼−J
3
ðLÞ. This remarkable property has been con-
firmed by the space–time numerical simulations [see
Fig.
3(b)].
To conclude, note that, under rather general condi-
tions, for both HBSFs and RBFs the efficiency of the at-
traction process increases as the fiber length and the
powers of the beams increase. We underline that all
the above analysis can be extended to unequal signal-
pump powers: For HBSFs polarization attraction still oc-
curs along a line of polarization states on the Poincaré
sphere, while for RBFs the singular torus is shown to split
into two distinct singular tori for S
0
≠ J
0
, whose SOP
coordinates read S
1
¼ −ρJ
1
, S
2
¼ ρJ
2
, S
3
¼ −ρJ
3
, with
ρ ¼ S
0
=J
0
. For ρ ≃ 1 (within 10%), the simulations reveal
an attraction toward the two SOPs states, while for high-
er values of ρ the system exhibits a complex dynamics,
including periodic behaviors that will be the subject of
future investigations.
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Fig. 3. (Color online) Same as Figs. 1 and 2 but for the RBF.
The polarization attraction is due to the presence of a sphere of
singular points (a). All signal SOPs are attracted (up to a sign
change) toward the pump SOP [S
1
ðLÞ¼−J
1
ðLÞ, S
2
ðLÞ¼J
2
ðLÞ,
S
3
ðLÞ¼−J
3
ðLÞ], as confirmed by the numerical simulations (b)
with [L ¼ 20,
~
JðLÞ¼ð0:7; 0:7; 0Þ].
4040 OPTICS LETTERS / Vol. 36, No. 20 / October 15, 2011
![Fig. 2. (Color online) Polarization attraction toward a continuous line of polarization states for the HSBF: Numerical simulations of the spatiotemporal Eq. (1) on the Poincaré sphere [ϕ ¼ π=4 for (a) and ϕ ¼ π=5 for (b)]. The green and red dots denote respectively the initial (~Sð0Þ) and final (~SðLÞ) SOPs of the signal. The yellow dot denotes the fixed pump SOP: (a) ~JðLÞ ¼ ð0; 1; 0Þ, (b) ~JðLÞ ¼ ð0:7; 0:7; 0Þ for L ¼ 5. The blue line is calculated analytically from our method.](/figures/fig-2-color-online-polarization-attraction-toward-a-7c3mingd.webp)
![Fig. 3. (Color online) Same as Figs. 1 and 2 but for the RBF. The polarization attraction is due to the presence of a sphere of singular points (a). All signal SOPs are attracted (up to a sign change) toward the pump SOP [S1ðLÞ ¼ −J1ðLÞ, S2ðLÞ ¼ J2ðLÞ, S3ðLÞ ¼ −J3ðLÞ], as confirmed by the numerical simulations (b) with [L ¼ 20, ~JðLÞ ¼ ð0:7; 0:7; 0Þ].](/figures/fig-3-color-online-same-as-figs-1-and-2-but-for-the-rbf-the-1lqz4pen.webp)