1) The document describes a methodology for measuring the Mueller matrices of monomode optical fibers under uniform strains using an optical polarimeter and Stokes-Mueller formalism. 2) A theoretical model based on coupled-mode equations is used to describe polarization evolution in uniformly perturbed fibers and relate it to physical fiber parameters like coupling coefficients. 3) Experiments are conducted to measure fiber Mueller matrices under different strains. The measured matrices are then analyzed to extract the physical coupling coefficients and assess the validity of the theoretical model.

1. J. Phys. D: Appl. Phys. 32 (1999) 1618–1625. Printed in the UK PII: S0022-3727(99)01288-7
Measurement of optical fibre
parameters using an optical
polarimeter and Stokes–Mueller
formalism
P Olivard, P Y Gerligand, B Le Jeune, J Cariou and J Lotrian
Laboratoire de Spectrom´ trie et Optique Laser, Universit´ de Bretagne Occidentale, 6,
e e
Avenue Le Gorgeu BP 809, 29285 Brest Cedex, France
E-mail: Pascal.Olivard@univ-brest.fr
Received 26 January 1999
Abstract. Although the method based on the Mueller matrix for the experimental
determination of optical-device polarization behaviour is a powerful tool, it has rarely been
applied to optical fibre. This paper introduces an experimental methodology for measuring
the Mueller matrices of monomode fibre under uniform strains. Using a theoretical model
derived from coupled-mode equations, we were able to first estimate the physical parameters
of the fibre, then to use them to both test the model validity and assess their influence on the
polarimetric behaviour of the fibre.
1. Introduction 2. Theoretical foundations
The polarization characteristics of optical fibre are of In the ideal case of an isotropic monomode optical fibre two
particular interest in systems like optical sensors or degenerate modes can propagate. This degeneracy indicates
interferometers using fibre. Although the Mueller matrix- that their propagation constants are identical, and so all the
based technique has rarely been used to describe polarization states of polarization launched at the input face of the guide
phenomena in monomode optical fibres, Mueller matrix will not be transformed during the propagation process. In
polarimetry is powerful for the experimental determination the case of external perturbations or of imperfections in the
of the polarization properties of optical devices like fibre core the degeneracy is lifted, the value of the propagation
linear or circular birefringence, dichroism, losses, rotary constant is modified and then the two modes are coupled. If
power or also depolarization. Thus, an experimental ax and ay are the amplitudes of these modes which are now
Mueller matrix enables one to predict the transformation z-dependent, the total electric field in the weakly guiding
of every state of polarization by propagation through approximation can be described by the following expression
the medium characterized by the matrix. Furthermore, where the time dependence exp(−jωt) is assumed and not
the coupled-mode theory well known in the Jones expressed:
space (amplitude terms), transformed into Stokes space
(energetic terms), has rarely been investigated experimentally E(x, y, z) = ax (z)Ex (x, y) + ay (z)Ey (x, y). (1)
[1, 2].
The present paper is organized as follows: the theoretical The coupling process can be described [3] by a set of two
development permitting the description of polarization first-order differential equations on the hypothesis that the
evolution in a uniformly perturbed fibre is recalled in variations be small versus z, the distance travelled along the
section 2 where the theoretical Mueller matrix used is guide. This set is written as follows
presented. Section 3 reports on some simulation results in the
case of elliptical birefringence, whereas section 4 describes d ax (z) k k12 ax (z)
= −j 11 (2)
the experimental method applied to measure the Mueller dz ay (z) k21 k22 ay (z)
matrix. Some experimental results and the methodology
to extract physical parameters, i.e. coupling coefficients, are where kij are the coupling coefficients and d/dz indicates the
presented in sections 5 and 6. Conclusions are presented in derivative with regard to z. This system can be directly solved
section 7. [4, 5], and its solutions are then expressed as a function of
0022-3727/99/141618+08$30.00 © 1999 IOP Publishing Ltd

2. Measurement of optical fibre parameters
both the initial conditions, i.e. the input state of polarization,
and the coupling coefficients: y y'
ax
= f (ax (0), ay (0), z, kij ). (3) τ x'
ay
Unfortunately, this formulation has some drawbacks for z x
carrying out experimental investigations: first, the quantities
are complex and thus difficult to measure and second,
depolarization is not considered. These considerations led us Figure 1. A representation of the laboratory (x, y) and rotary
to integrate this system in Stokes space where all quantities (x , y ) frame.
have an energy dimension. In consequence, the four Stokes
C
parameters can be expressed as a function of amplitude terms S
D
3
by the following relations: P1
∗ ∗
S0 (z) = ax (z)ax (z) + ay (z)ay (z) -45
∗ ∗
S1 (z) = ax (z)ax (z) − ay (z)ay (z)
H O
V
∗ ∗ S
2θ
S2 (z) = ax (z)ay (z) + ay (z)ax (z) 1
∗ ∗ P0
S3 (z) = j[ax (z)ay (z) − ax (z)ay (z)]. (4)
+45
Introducing the system (4) into (2) and assuming insignificant S
2
losses, which is consistent with the conservation of energy
along the fibre, i.e. dS0 /dz = 0 ⇒ S0 (z) = S0 (0) = C
G
constant, the differential system becomes
 dS1  Figure 2. Evolution of the eigenpolarization in the rotary frame
dz 0 2k2 2k1 S1 (z) versus the external twist rate.
 dS2  = −2k2 0 S2 (z) (5)
dz
dS3 −2k1 − 0 S3 (z) along the fibre or at its output versus perturbation variation.
dz
∗ At this level of development, thanks to an experimental
where K = k12 = k21 = k1 + jk2 and = k22 − k11 in the
measurement of the Stokes vectors, one can determine the
lossless case.
physical parameters of the fibre, i.e. the coupling coefficients,
Without depolarization, i.e. S0 = S1 + S2 + S3 the
2 2 2 2
procedure developed by Franceschetti and Smith [6] enables and then find whether the model used is valid or not. It should
one to solve this system and obtain the evolution of the be underlined that this measurement will be valid only in the
Stokes parameters versus the distance z. When the coupling case of a given incident state of polarization. For example,
coefficients are independent of z, and only in that case, the predicting from one measurement how another polarization
solutions take the following form state will behave will be quite impossible. The Mueller
matrix has allowed us to get round this disadvantage.
S0 (z) = S0 (0) = 1
Let us note that the output Stokes vector is linearly bound
ˆ ˆ
S1 (z) = S1 N + (S1 (0) − S1 N) cos(δβz) to the input one, leading thus directly to the Mueller matrix
±(Sˆ3 S2 (0) − S2 S3 (0)) sin(δβz)
ˆ in the following form:
ˆ ˆ 
S2 (z) = S2 N + (S2 (0) − S2 N) cos(δβz) 1 0 0
ˆ1 S3 (0) − S3 S1 (0)) sin(δβz)
ˆ 0 ˆ2 ˆ2
S1 + (1 − S1 )C ˆ ˆ ˆ
S1 S2 (1 − C) ± S3 S
±(S [M(z)] =  ˆ ˆ
 0 S1 S2 (1 − C) S3 S ˆ ˆ ˆ
S2 + (1 − S2 )C
2 2
ˆ ˆ
S3 (z) = S3 N + (S3 (0) − S3 N) cos(δβz) ˆ ˆ ˆ ˆ ˆ
0 S1 S3 (1 − C) ± S2 S S2 S3 (1 − C) S1 S ˆ
ˆ ˆ
±(S2 S1 (0) − S1 S2 (0)) sin(δβz) (6) 
0
ˆ ˆ ˆ
where N = S1 S1 (0) + S2 S2 (0) + S3 S3 (0) and S = ˆ ˆ ˆ ˆ
S1 S3 (1 − C) S2 S 
ˆ ˆ ˆ
[1, S1 , S2 , S3 ] is the Stokes vector of the eigenpolarization ˆ2 S3 (1 − C) ± S1 S  (9)
S ˆ ˆ
mode, i.e. the states of polarization that propagate without ˆ2 ˆ2
S3 + (1 − S3 )C
ˆ
transformation between the fibre ends. The Si components
are expressed as follows: with C = cos(δβz) and S = sin(δβz).
This matrix describes the most general case of elliptical
ˆ ˆ 2k1 ˆ 2k2 birefringence without losses and depolarization. As the
S1 = ± S2 = S3 = ± (7)
δβ δβ δβ measurement of the Mueller matrix allows one to predict
the behaviour of each incident state of polarization, the
δβ = 2 + 4|K|2 . (8) Poincar´ sphere representation can also be used to evaluate,
e
In equation (8), δβ is the propagation constant difference for example, the influence of various parameters such as the
of the two eigenpolarization modes. The set of equations (6) external strain on polarization evolution.
permits us to describe the evolution of the state of polarization Two particular cases will be discussed in the following:
1619

3. P Olivard et al
1
m11
0.8
m12
0.6
0.4 m13
0.2 m21
0 m22
-0.2 0
m23
-0.4 m31
-0.6
m32
-0.8
m33
-1
0.25 0.5 0.75 1 1.25 1.5
Twist rate (turns per meter)
Figure 3. Evolution of the Mueller matrix elements in the rotary frame versus the external twist rate.
2.1. Linearly-birefringent fibre C
D
S
3
P1
Let us suppose a monomode optical fibre in which there
is a linear birefringence β expressed in radians per length -45
unit, so that its fast axis makes an angle θ with respect to
the horizontal x-axis, the coupling coefficients are then [7]
O
2k11 = −2k22 = β cos 2θ and 2k12 = −2k21 = β sin 2θ .
H
V
S 2θ
1
One can easily compute the parameters contained in the
P0
theoretical Mueller matrix defined in (9) so that
+45
S
2k1 = β sin 2θ k2 = 0 2
C
= −β cos 2θ δβ = β. G
The eigenpolarization Stokes vectors are then [1, cos 2θ, Figure 4. Evolution of the eigenpolarization in the laboratory
frame versus the external twist rate.
sin 2θ, 0] and the Mueller matrix takes the following well
known form with C = cos 2θ and S = sin 2θ :
takes the following form:

1 0 0  
 0 C 2 + S 2 cos βz CS(1 − cos βz) 1 0 0 0
[M(z)] =   0 cos(gτ z) sin(gτ z) 0 
0 CS(1 − cos βz) S 2 + C 2 cos βz [M(z)] =  . (11)
0 −S sin βz C sin βz 0 − sin(gτ z) cos(gτ z) 0
 0 0 0 1
0
S sin βz  This is an ideal case. However, if the fibre exhibits a small
. (10)
−C sin βz intrinsic birefringence because of core imperfection, it will
cos βz have significant effects on the polarization properties of the
guide. Elliptical birefringence will occur which must be
This matrix corresponds to a linear birefringent network.
carefully considered.
This form is valid only in an invariant frame and with a
uniform birefringence, i.e. with independence of z. This
matrix can be used to study the effect of bending with regard 3. Elliptical birefringence
to the internal intrinsic birefringence of the fibre. However,
Elliptical birefringence results from the superposition of
in the calculation of the birefringence, one should be cautious
linear and circular birefringence. This becomes the case
of the beat length (transition at 2π ) and take into account the
when a linearly-birefringent monomode fibre is uniformly
possible rotation of the fibre [8] local axis (see section 3).
twisted. It then requires one to take into account the rotation
of the fibre axes which, in the laboratory frame x, y, z, makes
2.2. Circular-birefringent fibre the coupling coefficients dependent on z. As previously
mentioned, the integration method giving the Mueller matrix
An isotropic monomode fibre under uniform twist τ (in is not valid in this case. To circumvent this problem, one
radians per length unit) holds a circular birefringence gτ , must refer to the local frame of the fibre (x , y , z) depicted
where g is the elasto-optic coefficient of the material. In in figure 1. The coupling coefficients are then independent
such a case, the coupling coefficients can be expressed [9] of z and the Mueller matrix can be computed by the method
∗
as k11 = k22 = 0 and k12 = k21 = jgτ/2, so that k1 = 0, described above.
2k2 = gτ , = 0 and δβ = gτ . The eigenpolarization Let us consider a monomode fibre with a linear uniform
Stokes vectors are then [1, 0, 0, ±1] and the Mueller matrix retardance β expressed in radians per length unit so that its
1620

4. Measurement of optical fibre parameters
1
m11
0.8
m12
0.6
0.4 m13
0.2 m21
0 m22
-0.2 0
m23
-0.4 m31
-0.6
m32
-0.8
m33
-1
1 2 3 4 5 6 7 8
Twist rate (turns per meter)
Figure 5. Evolution of the Mueller matrix elements in the laboratory frame versus the external twist rate.
(a) (b)
Figure 6. Evolution of the state of polarization for two different incident states (in the laboratory frame). (a) Horizontal incident state:
[1, 1, 0, 0]. (b) Elliptical incident state: [1, 0.578, −0.21, 0.788] (θ = −10◦ , ε = 26◦ ).
vertical Quarter-wave Quarter-wave Horizontal
polarizer plate plate polarizer
ν ν'
Laser Sample Detector
P1 L1 L2 P2
Figure 7. The schematic layout of the polarimeter.
fast axis makes an angle θ with respect to the horizontal A similar procedure was also followed by Sakai and Kimura
x -axis. If this fibre is submitted to an external twist τ (in [4] in Jones space.
radians per length unit), the coupling coefficients can be Let [M] be the Mueller matrix in the laboratory frame
expressed in the rotary frame by and [M ] that in the rotary frame. The following relation can
then be written:
2k11 = −2k22 = β cos 2θ
∗ [M(z)] = [R(−2τ z)][M (z)]
2k12 = k21 = β sin 2θ − jτ (2 − g) (12)
and the theoretical Mueller matrix takes the general form [M (z)] = [R(2τ z)][M(z)]. (13)
of equation (9). In the most general case, experimental
In the same way, let [S] be the Stokes vector in the laboratory
Mueller matrices are measured in the laboratory frame, so
frame and [S ] that in the rotary frame. One can thus write
they are not directly comparable to the theoretical model
and this does not allow an easy estimation of the physical [S(z)] = [R(−2τ z)] (14a)
parameters. There are two possible ways of solving this
problem: either the theoretical matrices are converted from and consequently
the rotary to the laboratory frame or, conversely, experimental
results are converted from the laboratory to the rotary frame. [S(z)] = [R(−2τ z)][M (z)][S (0)]. (14b)
1621

5. P Olivard et al
P L P1 L1 O1 F O2 L2 P 2
M3
CAMERA
M1
M2
LASER
P, P1, P2 : Linear polarizers M 1 , M 2 , M 3 : Dielectric mirrors
L1 , L2 : Quarter - wave plates O1 , O2 : Injection optics
L : Half - wave plate F : Optical fiber
Figure 8. Experimental set-up.
1.20
0.80 m11
m12
0.40 m13
m21
0.00 m22
m23
-0.40 m31
m32
-0.80 m33
-1.20
0.00 3.14 6.28 α z (rad) 9.42
Figure 9. Evolution of experimental Mueller matrice coefficients (in the laboratory frame).
When considering that the rotary frame coincides with the monomode optical fibre of 1 m in length was considered and
laboratory frame at τ = 0 where there is no twist, the previous the following physical parameters were used: g = 0.16,
relation becomes β = 0.52 rad m−1 , θ = 0.52 rad. The Mueller matrix
coefficients were computed by introducing equation (12) into
[S(z)] = [R(−2τ z)][M (z)][S(0)]. (15)
(9) and applying relation (13). Equations (14) and (15)
Introducing equations (12) into (9) with the help of equations were used for the Stokes vectors. The simulation results
(13)–(15) enables one to simulate the evolution of every are presented by a couple of figures. Figures 2 and 3 were
coefficient of the Mueller matrix or of the Stokes vectors obtained in the rotary frame and figures 4 and 5 in the
in the local or laboratory frame. These simulations can be laboratory frame.
made versus z, the distance propagated in the guide, versus The arrows in figures 3 and 5 indicate some characteristic
the physical parameters such as β or versus the external twist points where the fibre can be considered as a pure polarization
strain τ . Thus, the Poincar´ sphere can be used to draw
e rotator. One should note that all the coefficients of the last row
the trajectory of the polarization state using the normalized and last column of the matrix (m33 being excepted) become
Stokes vector coefficients. rapidly negligible, and the rotation characteristics are then
preponderant. This fact was corroborated by the evolution
4. Simulations of the eigenstate of polarization which converged toward the
north of the sphere. The two trajectories simulated on the
To illustrate the developments described in the previous Poincar´ sphere (figure 6) indicate a reduction of oscillation
e
section we will now present some simulation results. A with twist and the rotation also appears to be preponderant.
1622

6. Measurement of optical fibre parameters
500 Therefore, the 64 measured intensities were expressed by the
(δβ)²
Experimental points
following matrix
400 Quadratic interpolation
k = [0.15]
[Ik ] = [B][m1 ] with (18)
l = [4i + j ]
300
where the coefficients of matrix B depend on the orientation
angles of the quarter-wave plates L1 and L2 . The mij (i, j =
200
0.3) were represented by
[m1 ] = ([B]T [B])−1 [B]T [Ik ] (19)
100
where []T indicates the transposed matrix.
0
This equation can then be computed for each
0.00 3.14 6.28 αz (rad) 9.42 measurement, and the expression ([B]T [B])−1 [B]T remains
valid from one experience to each other.
Figure 10. Evolution of (δβ)2 . Figure 8 displays the experimental set-up. The light
source is an ionized argon laser emitting at 514 nm. The
These simulation results have shown that the intrinsic detection is made by a charged coupled device (CCD) camera.
linear birefringence of the fibre can exert a strong influence The polarimeter was calibrated without the fibre and the
on the guided state of polarization even when this fibre is injection devices (O1 , O2 ). The axes of the various optical
submitted to a small twist strain. elements were aligned with an accuracy of 0.01◦ using a
dichotomous method at null intensity. A ‘χ 2 test’ [10] was
carried out to quantify the influence of noise on the Mueller
5. Experimental methodology and measurement matrix coefficients, which was thus minimized with respect
set-up to these coefficients. The estimated standard deviation on
each mij was then less than 0.5%.
The Mueller matrix was measured with an optical polarimeter
as schematized in figure 7. The input polarization encoding
6. Results and analysis
system is composed of a vertical (y-axis) linear polarizer fol-
lowed by a quarter-wave plate whose fast axis makes an angle The experimental results presented were obtained with a
ν with the y-axis. The output polarization decoding system 0.865 m long monomode optical fibre submitted to a uniform
is composed of a quarter-wave plate whose fast axis makes small twist, i.e. less than 1.5 turn. Figure 9 illustrates the
an angle ν with respect to the reference y-axis, followed by evolution of the experimental Mueller matrix coefficients
a linear horizontal polarizer. The two polarizers P1 and P2 (birefringent elements only). Some characteristic points like
were initially crossed in order to obtain a null intensity. those defined in section 3 are viewed, so the fibre is then
The Stokes vector S emerging from the last polarizer a pure rotator of polarization. The depolarization index of
can be expressed by the following matrix product each matrix was computed. As the depolarization index was
always greater than 0.95, the depolarization was considered
S = [P2 ][L2 ][M][L1 ][P1 ]S(0) (16)
as negligible. Moreover, it indicated that the theoretical
model defined above could be used both to describe evolution
where [P2 ], [L2 ], [L1 ] and [P1 ] are the Mueller matrices of
and to estimate the physical parameters.
each of the devices constituting the polarimeter, [M] is the
The experimental Mueller matrices represented by
unknown Mueller matrix and S(0) is the Stokes vector of the
equation (9) were transferred from the laboratory frame (see
incident light. The measurable intensity is enclosed in the
figure 9) to the rotary frame using equation (13). The physical
first term of the output Stokes vector and, in the case of perfect
parameters g and β are enclosed in δβ, which was easily
optical devices, i.e. quarter-wave plate linear retardance equal
computed to give the following expression:
to 90◦ , can be expressed as a function of the orientations of
3 3
both the two quarter-wave plates (ν, ν ) and the 16 unknown
δβ = tan−1 (mij − mj i )1/2 mii
Mueller matrix coefficients:
i=j =1 i=1
S0 (ν, ν ) = I (ν, ν ) = m00 + m01 C 2 + m02 CS + m03 S = (β 2 + τ 2 (2 − g)2 )1/2 . (20)
2 2
+(m10 + m11 C + m12 CS + m13 S)(−C ) These two parameters, g and β, were estimated with a
+(m20 + m21 C 2 + m22 CS + m23 S)(−C S ) quadratic method from the curve presented in figure 10. In
+(m30 + m31 C 2 + m32 CS + m33 S)(S ) (17) this particular case we obtained g = 0.147 and β = 0.46 rad.
The estimated elasto-optic coefficient was in good agreement
with C = cos 2ν, S = sin 2ν, C = cos 2ν and S = sin 2ν . with published results [5, 7, 9].
One needs only 16 equations to obtain the Mueller matrix The direction of the linear birefringent axis, θ , can be
coefficients. To minimize errors we used an overdetermined computed in the rotary frame and takes the following form:
system of 64 equations corresponding to 64 combinations of
the angles (ν, ν ). These angles were multiples of 22.5◦ so m31 − m13 m23 + m32
2θ = tan−1 = tan−1 . (21)
that they were capable of describing specific test positions. m23 − m32 m13 + m31
1623

7. P Olivard et al
1.00 1.00 1.00
0.80 0.80 0.80
0.60 0.60 0.60
0.40 0.40 0.40
0.20 0.20 0.20
m11
m12
m13
0.00 0.00 0.00
-0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42
-0 .4 0 -0 .4 0 -0 .4 0
-0 .6 0 -0 .6 0 -0 .6 0
-0 .8 0 -0 .8 0 -0 .8 0
-1 .0 0 -1 .0 0 -1 .0 0
1.00 1.00 1.00
0.80 0.80 0.80
0.60 0.60 0.60
0.40 0.40 0.40
0.20 0.20 0.20
m21
m22
m23
0.00 0.00 0.00
-0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42
-0 .4 0 -0 .4 0 -0 .4 0
-0 .6 0 -0 .6 0 -0 .6 0
-0 .8 0 -0 .8 0 -0 .8 0
-1 .0 0 -1 .0 0 -1 .0 0
1.00 1.00 1.00
0.80 0.80 0.80
0.60 0.60 0.60
0.40 0.40 0.40
0.20 0.20 0.20
m31
m32
m33
0.00 0.00 0.00
-0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42
-0 .4 0 -0 .4 0 -0 .4 0
-0 .6 0 -0 .6 0 -0 .6 0
-0 .8 0 -0 .8 0 -0 .8 0
-1 .0 0 -1 .0 0 -1 .0 0
Figure 11. Measured Mueller matrices versus the external twist αz (in rad). The curves represent the theoretical constructions obtained
with equation (13).
Experimental
theoretical
(a) (b)
Figure 12. Evolution of the state of polarization for two different incident states. Theoretical and experimental trajectories. (a) Horizontal
incident state: [1, 1, 0, 0]. (b) Elliptical incident state: [1, 0.578, −0.21, 0.788] (θ = −10◦ , ε = 26◦ ).
However, the value obtained from this expression is not 7. Conclusion
constant, and this implies that the assumption of uniform
linear birefringence is not valid. Consequently, to compare An experimental method to determine the physical
our measurements with the theoretical model, we computed parameters of a monomode optical fibre has been presented.
the mean value of θ and obtained θ = −0.52 rad. The theoretical model derived from the coupled-mode
With the estimated physical parameters we adjusted equations applied to the Stokes–Mueller formalism gave
the theoretical model and compared it to the experimental good results. It showed that the non-uniformity of the linear
results (figure 11). The agreement between theory and birefringence could influence polarization behaviour.
measurement was quite good. Nevertheless, there were The Stokes–Mueller formalism is a powerful tool for
discrepancies in the coefficients m13 , m23 , m31 and m32 ; they carrying out experimental studies of polarization phenomena
were likely to be due to the non-uniformity of the intrinsic in optical devices. The Poincar´ sphere, directly connected
e
linear birefringence of the fibre. The comparison between to the Stokes–Mueller formalism, also constitutes a useful
theoretical and measured trajectories on the Poincar´ sphere
e representation to describe the evolution of the polarization
(figure 12) showed identical rotary behaviour and a decrease state. The results reported here have shown that these
of the ellipticity with twist. This result corroborates the non- formalisms are very relevant in the characterization of optical
uniformity of the linear intrinsic birefringence. fibres.
1624

8. Measurement of optical fibre parameters
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e
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1625