Optics Communications 253 (2005) 352–361
www.elsevier.com/locate/optcom
Multiwavelength fiber laser with an intracavity polarizer
Thierry Chartier a,*, Adrian Mihaescu a, Gilles Martel b, Ammar Hideur b,Francois Sanchez c
a Laboratoire d’Optronique, UMR CNRS 6082 FOTON, ENSSAT, 6 rue de Kerampont, 22305 Lannion Cedex, Franceb Groupe d’Optique et d’Optronique, CORIA UMR CNRS 6614, Universite de Rouen, Avenue de l�universite,
76801 Saint Etienne du Rouvray Cedex, Francec Laboratoire POMA UMR CNRS 6136, Universite d’Angers, 2 Bd. Lavoisier, 49045 Angers Cedex 01, France
Received 22 September 2004; received in revised form 21 April 2005; accepted 27 April 2005
Abstract
We present the experiment that allows multiwavelength operation and tunability of a neodymium-doped fiber laser
with an intracavity polarizer. Then, we present the experimental method to extract the Jones matrix of the fiber as a
function of the wavelength. Hence, using the round-trip operator method, we compare both experimental and theoret-
ical wavelength selection and find a excellent agreement.
� 2005 Elsevier B.V. All rights reserved.
PACS: 42.55.Wd; 42.81.Gs
Keywords: Fiber lasers; Birefringence; Polarization
1. Introduction
Multiwavelength fiber lasers have extensively
been studied in recent years [1–10]. They are of
great interest for various applications such as
wavelength-division-multiplexed communicationsystems, fiber sensors or instrument testing.
Many approaches have been proposed to obtain
0030-4018/$ - see front matter � 2005 Elsevier B.V. All rights reserv
doi:10.1016/j.optcom.2005.04.077
* Corresponding author. Tel.: +33 296469144; fax: +33
296370199.
E-mail address: [email protected] (T. Chartier).
multiwavelength operation from rare-earth-doped
fiber lasers. Most of these methods include intra-
cavity components such as Fabry–Perot etalon
[1], Mach–Zhender fiber interferometer [2], fiber
Bragg gratings [3], fiber loop mirrors [4],
acousto-optic frequency-shifter [5] or intracavitypolarizer [6–10]. The latter method is very attrac-
tive because of its simplicity. It consists to insert
a polarizer in the laser cavity containing the bire-
fringent gain fiber. Because of the wavelength
dependence of the phase shift induced by fiber
birefringence, wavelength selection occurs, leading
ed.
T. Chartier et al. / Optics Communications 253 (2005) 352–361 353
to multiwavelength operation. For example,
24-line multiwavelength operation over 17-nm
spectral range has been demonstrated with an er-
bium-doped fiber laser using this technique [6].
Assuming that the fiber of length L can be mod-eled by a linear retardation plate of birefringence
B, the spacing Dk between each line can simply
be calculated from [7–9,12]
Dk ¼ k202BL
; ð1Þ
where k0 is the central wavelength of the spectrum.
The advantage of this method is also to provide
tunability of the spectral lines by rotation of the
polarizer or some intracavity polarization control-
lers [7–11]. However, tunability of the laser lines ismore difficult to model. For example, the previous
theoretical approach which consists to assume that
the laser cavity contains only both a linear retarda-
tion plate and a polarizer fails to describe the tun-
ability when the polarizer rotates [11,12]. The
complicated nature of fiber birefringence imposes
a more realistic model. In [11], Friedman et al.
use the principal states of polarization (PSP) todescribe the tunability of a single-wavelength er-
bium-doped fiber laser with an intracavity pola-
rizer. The PSP, initially introduced to
characterize polarization mode dipersion in fiber-
based transmission systems [13], are extracted
using Jones matrix eigenanalysis [14].
The aim of our paper is to present a novel ap-
proach to describe the tunability of a multiwave-length fiber laser with an intracavity polarizer.
The principle is to use an experimental method
to extract the Jones matrix of the birefringent gain
fiber as a function of the wavelength and to apply
the round-trip operator of the cold cavity to ex-
plain wavelength selection. In contrast with previ-
Nd-doped fibe
M1
Laser diode
O1
Fig. 1. Schematic represe
ous work [11], this method does not need the use of
the PSP and is applied to a multiwavelength fiber
laser. In Section 2, we present the experiment on
the tunability of a multiwavelength neodymium-
doped fiber laser with an intracavity polarizer. InSection 3, the experimental method to extract the
Jones matrix of the fiber is presented. Section 4
is devoted to the theoretical prediction of the tun-
ability and comparison with experiment.
2. Experiment on the laser
2.1. Description of the laser
The set-up of the Nd-doped fiber laser with an
intracavity polarizer is shown in Fig. 1. We use a
500-ppm-Nd-doped fiber as the gain medium.
The Ge-doped-silica core has a diameter of 2.7
lm, the numerical aperture is 0.24 leading to a cut-
off wavelength of 850 nm. The fiber is 20-m-longand is wound on a drum. The intrinsic birefrin-
gence B of the fiber has been evaluated using the
magneto-optic method and is of the order of
10�5 [16]. The laser diode operates at 810 nm
and is focused in the fiber through the microscope
objective O1. Its maximum output power is 150
mW. The microscope objective O2 collimates the
output beam. The cavity is composed of two mir-rors M1 and M2. Mirror M1 has high reflectivity
around 1080 nm and high transmission at 810
nm. Mirror M2 is the output coupler with a reflec-
tivity of 80% around 1080 nm. The bulk rotatable
polarizer has an extinction ratio greater than 30
dB around 1080 nm. The orientation of its trans-
mission axis with respect to the laboratory frame
of reference is h. The output signal is detected withan optical spectrum analyzer (resolution of 0.01
r
M2
x
y
θ
polarizer (θ)
O2
z
ntation of the laser.
354 T. Chartier et al. / Optics Communications 253 (2005) 352–361
nm). The threshold of the laser is about 10 mW of
pump power. In the following, measurements will
be made at 40 mW of pump power for which the
laser delivers 0.5 mW.
2.2. Results
In the absence of polarizer in the cavity, the laser
operates along two linear and orthogonal polariza-
tions [15]. The output spectrum of the laser, is given
in Fig. 2(a). The spectral width is about 12 nm
(�3000 GHz) around 1086 nm. Since the free spec-
tral range of the cavity is about 5MHz, the spectrumcontains several hundred of thousands longitudinal
modes of the cavity. Nowavelength selection occurs.
When the polarizer is present in the cavity and
oriented along h, the output signal is polarized
1078 1080 1082 1084 1086 1088 1090 1092 1094
Wavelength (nm)
θθθθ = 0˚
θθθθ = 35˚
θθθθ = 50˚
θθθθ = 70˚
No polarizer(a)
(b)
(c)
(d)
(e)
Fig. 2. Spectra versus h.
along h and its spectrum reduces only into peaks.
Each peak of this multiwavelength-spectrum can
be tuned by rotating the polarizer. Figs. 2(b)–(e)
give examples of spectra for different values of
the polarizer angle h.In order to map the tunability of the spectrum,
we recorded spectra for h varying from 0� to 180�by step of 5�. For each spectrum, we measured the
central wavelength of each peak. Fig. 3 reports the
evolution of peak-wavelengths as a function of h.We note the periodicity of 90�. We also note that
for h around the particular values of 0� and 90�,the periodicity in the spectrum is 3 nm, in goodagreement with Eq. (1), while for h around 45�,the periodicity is 1.5 nm. Moreover, Fig. 3 shows
this unexpected phenomenon of splitting of the
peak wavelengths around 20� and 70�.Note that near threshold, the spectrum is nar-
rower and does not exhibit a sufficient number of
peaks to relate accurately this periodical phenom-
enon. Far above threshold the number of peaksdoes not increase significantly to justify a pumping
at maximum pump power.
2.3. Discussion
As previously mentioned in the introduction,
wavelength selection in a cavity containing both
a birefringent media and a polarizer is well under-stood [7–9,12]. Let us recall the basic principle. In
a laser cavity with no polarization-dependent loss
(no polarizer for example), the output polarization
state is fixed by the resonance condition stipulating
that a polarization state must remain unchanged
after one round-trip in the cavity. If the cavity
contains only one linear birefringent element, this
condition is satisfied for two polarization statesaligned with the eigenaxes of the birefringent ele-
ment, whatever the wavelength. If a polarizer is
now inserted in this cavity, one single polarization
state is fixed by the polarizer. Laser operation oc-
curs for polarizations that remain aligned with the
polarizer axis after one round-trip. A polarization
state that enters the birefringent media with a cer-
tain angle with respect to its eigenaxes experiencesa wavelength-dependent round-trip phase shift
equal to D/ = 4pBL/k0. To satisfy the resonance
condition, the lasing polarizations are the ones
0 30 60 90 120 150 180
1080
1082
1084
1086
1088
1090
1092
1094
Pea
kw
avel
engt
h(n
m)
Angle θ (deg.)
Fig. 3. Evolution of the peak-wavelengths versus h.
T. Chartier et al. / Optics Communications 253 (2005) 352–361 355
for which the phase shift is a multiple of 2p. Thisexplains wavelength selection in the spectrum of
a fiber laser and spacing Dk given by Eq. (1).
However, this simple interpretation fails to ex-
plain the three following experimental observa-
tions. First, if the polarizer is aligned with the
birefringence axis, no wavelength-selection should
occur. This point is not verified by the experiment.
Second, the spacing Dk should be independent ofthe angle h. In the experiment Dk varies from 1.5
to 3 nm. Third, no tuning of the lasing wave-
lengths should be observed when the polarizer is
rotated but rather a change in the loss of the lasing
polarizations [11,12]. However, tuning (and split-
ting) has been observed in the experiment.
The reason of this mismatching between theory
and experiment is due to the complicated nature offiber birefringence that cannot be described as a
simple linear retardation plate. We propose in
the following a more complete description of the
fiber birefringence that allows to predict the exper-
imentally observed laser spectrum.
3. Experiment on the fiber
3.1. Model for the fiber
In [16], we showed that a fiber of length L canbe described by the Jones matrix M of an elliptical
birefringent plate, i.e. a birefringent mediumwhose eigenpolarizations are elliptical. If M is
written in an appropriate basis of linear polariza-
tions (X,Y), M depends only on two parameters
a and b
M ¼a b
�b a�
� �ð2Þ
with
a ¼ cos�L2þ i
b�sin
�L2; ð3Þ
b ¼ � a�sin
�L2; ð4Þ
and
� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2
q. ð5Þ
Parameters a and b are related, respectively, to the
circular and linear birefringence of the fiber, � cor-responds to the total (elliptical) birefringence. All
of these parameters are expressed in radians/meter.
The basis (X,Y) has a particular signification:
polarizations X and Y coincide with the azimuth
of the elliptical eigenpolarizations of the fiber.
Note that if (X,Y) makes an angle /0 with respectto the laboratory basis (x,y), matrix of the fiber
writes, in the laboratory basis
ML ¼ Rð/0ÞMRð�/0Þ; ð6Þwhere R(/) is the rotation matrix of an angle /
356 T. Chartier et al. / Optics Communications 253 (2005) 352–361
Rð/Þ ¼cos/ � sin/
sin/ cos/
� �. ð7Þ
In summary, in the laboratory basis, three param-
eters define the Jones matrix ML of the fiber of
length L: a, b and /0.
3.2. Extracting parameters
In [16], we demonstrated the propriety that, for
any elliptical birefringent, there always exists an
input linear polarization (oriented along /in) that
exits the fiber linearly polarized (along /out). The
angle /0 is the bisector of the angle between /in
and /out and is given by
/0 ¼/in þ /out
2. ð8Þ
A method to extract a and b knowing /0 has
been proposed in [16]. In this paper a new meth-
od is presented. It simply consists to launch in
the fiber a linear polarization oriented at 45�with respect to /0 and to analyse the Stokesparameters of the corresponding output state of
polarization.
In the fiber basis (X,Y), this input state, ori-
ented at 45� of the basis axes, writes
Ein ¼1ffiffiffi2
p1
1
� �. ð9Þ
The corresponding output state is calculated as
follows:
Eout ¼ MEin; ð10Þwhere M is given by Eqs. (2)–(5). Using Appendix
A to calculate the Stokes parameters of Eout, we
find:
Fig. 4. Experime
S0 ¼ 1; ð11Þ
S1 ¼ � a�sin �L; ð12Þ
S2 ¼ cos �L; ð13Þ
S3 ¼ � b�sin �L. ð14Þ
We see that simple relations link Stokes parame-
ters to fiber parameters. From Eqs. (12)–(14), we
easily find:
� ¼ � 1
L½arccos S2 � 2kp�; ð15Þ
a ¼ ��S1
S21 þ S2
3
; ð16Þ
b ¼ ��S3
S21 þ S2
3
. ð17Þ
3.3. Experimental set-up
We present in Fig. 4 the experimental set-up
to measure the Jones matrix of the fiber as a
function of the wavelength in the range 1080–
1094 nm. The tunable laser source is a linearlypolarized ytterbium-doped double-clad fiber laser
similar to that described in [17] but delivering
here only few mW. The quarter-wave plate at
1080 nm transforms the linear polarization into
a circular one. The polarizer allows to inject
the appropriate linear polarization in the fiber.
This association of a quarter-wave plate and a
polarizer is convenient to control the input stateof polarization since it is wavelength indepen-
dent. The output state of polarization is
analyzed with a polarization analyzer (Thorlabs
polarimeter). We performed measurements in
ntal set-up.
-100
-50
0
50
100
φ in(d
eg.)
20
30
40
50
60
70
φ out
(deg
.)
-40
-20
0
20
40
60
φ 0(d
eg.)
1080 1082 1084 1086 1088 1090 1092 1094
-1.0
-0.5
0.0
0.5
1.0
S3
Wavelength (nm)
-1.0
-0.5
0.0
0.5
S2
-0.5
0.0
0.5
S1
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 5. Evolution of measured parameters versus wavelength.
T. Chartier et al. / Optics Communications 253 (2005) 352–361 357
the range 1080–1094 nm, with a step of 0.25 nm.
The procedure is the following. First, by rotating
the polarizer, we find the input linear polariza-
tion that exits the fiber linearly polarized. This
gives both angles /in and /out and we calculate/0 according to (8). Then, we inject in the fiber
the linear polarization oriented at 45� with re-
spect to /0 and measure the Stokes parameters
s1, s2 and s3 of the output polarization with
the polarimeter in the laboratory basis (x,y). A
simple transformation of s1, s2 and s3 will give
S1, S2 and S3 in the fiber basis. In order to
use these results to explain experimental tunabil-ity of the fiber laser, the very important point
here was not to move the fiber between both
experiments on the laser and on the fiber.
3.4. Results
Fig. 5 represent evolution of the six parameters
/in, /out, /0, S1, S2 and S3 as a function of thewavelength. First of all, let us point out the peri-
odicity of 3 nm for each parameter. Secondly, it
seems that each parameter obeys a simple law.
Dots represent experimental data while lines are
fitting functions. Fig. 5(a) represents the mea-
sured values of /in together with a fit by linear
functions of slope equal to �60�/nm. Fig. 5(b)
gives the measured values of /out and the corre-sponding fit by the sine curve of equation:
/out = 45 + 22.5sin (2pk/3 + p/3). Wavelength kis expressed in nm. Fig. 5(c) is deduced from data
of both Figs. 5(a) and (b) according to Eq. (8).
Figs. 5(d)–(f) represent the normalized Stokes
parameters S1, S2 and S3 of the output polariza-
tion state in the basis (X,Y) of the fiber. We used
Appendix B with the experimental values of /0 tochange the basis of the Stokes parameters. In Fig.
5(d), S1 is fitted by S1 = �0.3 + 0.6sin (2pk/3 + 5p/6) while in Fig. 5(e) S2 is fitted by
S2 = �0.4 + 0.6sin (2pk/3 + p/6). Fig. 5(f) repre-
sents S3 with its fitting func-tion deduced from
S1 and S2 according to: S20 ¼ S2
1 þ S22 þ S2
3 with
S0 = 1. Using experimental data of Fig. 5, we
can extract parameters a, b and � according toEqs. (15)–(17). Corresponding fitting function
for each parameter can also be calculated from
Eqs. (15)–(17) using the fitting functions given
above. Fig. 6 represents the extracted values of
aL, bL and �L. In Eqs. (15)–(17), the signs + or
� have been chosen in order to avoid discontinu-
ities in the evolution of parameters. We set the
value of k = 0, this leads to oscillations of �Laround p. In reality �L, which corresponds to
the total phase shift, is greater. For example, with
a birefringence of 10�5 and a fiber length of L =
20, �L should be of the order of 185 · 2p rad.
However, the value of k has no real importance
for our study since � appears in Eqs. (2)–(5) as
A
MLT(λ) PT(θ)
ML (λ) P(θ)
M1 M2
x
y
θ z
Fig. 7. Schematic representation of the cavity.1080 1082 1084 1086 1088 1090 1092 1094
−3π/2
−π
−π/2
0
π/2
π
3π/2
αLβLεL
α L,β
L,ε L
(rad
.)
Wavelength (nm)
Fig. 6. Extracted values of aL, bL and �L and their corre-
sponding fitting function.
358 T. Chartier et al. / Optics Communications 253 (2005) 352–361
the argument of trigonometric functions or
through ratios a/� or b/� which are independent
of k (see Eqs. (16) and (17)).
Figs. 6 and 5(c) show that both ellipticity and
azimuth of fiber birefringence evolves versus wave-
length. The fiber birefringence is generally ellipti-
cal but for some particular wavelengths (1080,
1083, 1086 nm, etc.) the fiber birefringence is linear(a = 0) and the value of the phase shift �L is such
that the fiber acts like a quarter-wave plate
(�L = p/2 or 3p/2). At this stage no explanation is
given to describe wavelength evolution of the fiber
birefringence. This point should be the subject of
further work.
4. Theory
4.1. The round-trip operator
The round-trip operator method describes, in
term of Jones matrices, the resonance condition
stipulating that a state of polarization must re-
mains unchanged after one round-trip in the lasercavity. This method has already been successfully
used to describe polarization properties of low-
power CW fiber lasers [18,19].
The laser cavity of Fig. 1 can be resumed to the
schematic representation of Fig. 7 if we are only
interested in polarization effects. We have assumed
that only two anisotropic elements are present in
the cavity: the fiber and the polarizer. Let ML
and P(h) be, respectively, the Jones matrices of
the fiber and the polarizer in the laboratory basis
(x,y) for light propagating from left to right
(forward direction). Matrix ML is given by Eq.
(6) and P(h) by
PðhÞ ¼ cos2h � sin h cos h
� sin h cos h cos2h
� �. ð18Þ
The Jones matrices of these components when
light propagates in the backward direction are
the transpose matrices MTL and PTðhÞ [18]. We
immediately note that PT(h) = P(h). The round-
trip operator C at point A is the Jones matrix expe-
rienced by a polarization starting from point A
and propagating through the cavity
C ¼ PðhÞMLMTLPðhÞ. ð19Þ
Using Eq. (6) for ML, we find
C ¼ PðhÞRð/0ÞMMTRð�/0ÞPðhÞ; ð20Þ
where M is the Jones matrix of the fiber in its own
basis in the forward direction and is given by Eqs.
(2)–(5).
Eigenvectors and eigenvalues of C represent,
respectively, the eigenpolarizations of the laser
at point A and the transmission of the eigenvec-tors through the cavity. Using experimental
results of Section 3, we can, at this stage, calcu-
late C for each angle h and each wavelength in
the range 1080–1094 nm. Moreover, using the
fitting functions found for a, b, � and /0 we
can simulate a continuous evolution of C versus
h and k.
0 30 60 90 120 150 180
1080
1082
1084
1086
1088
1090
1092
1094
Pea
kw
avel
engt
h(n
m)
Angle θ (deg.)
Fig. 9. Theoretical (lines) and experimental (crosses) evolution
of the laser spectrum peak-wavelengths versus h.
T. Chartier et al. / Optics Communications 253 (2005) 352–361 359
Note that the round-trip operator method is not
valid for high-power lasers or ultrashort pulsed
lasers because of the intensity dependance of
birefringence.
4.2. Simulations
It is obvious that one eigenvalue of C is always
null, corresponding to the eigenvector oriented at
90� of the transmission axis of the polarizer. Let cbe the non-zero eigenvalue corresponding to the
polarization oriented along the polarizer axis.
Modulus |c| corresponds to the transmission ofthis polarization after one round-trip through
the cavity. Fig. 8 represents the wavelength-
dependence of |c| for three particular values of
h. We note a strong modulation of |c|. Longitudi-nal modes for which the transmission coefficient
|c| is below a certain value cannot satisfy the oscil-
lation condition: gain = loss. Consequently, the
lasing wavelengths observed in the experimentalspectra correspond to the maximums of |c|. Note
that the round-trip operator analysis without the
intra-cavity polarizer does not give any wave-
length selection but a constant value equal to
unity for both eigenvalues of C. We have verified
that wavelength-selection always occurs for any
values of h. We have also verified that, for a sim-
ply linear birefringence model for the fiber, notuning occurs.
Tunability of the spectrum is also visible in
Fig. 8. In order to compare both experimental
1080 1082 1084 1086 1088 1090 1092 10940.0
0.2
0.4
0.6
0.8
1.0
20˚40˚60˚
| γ|
Wavelength (nm)
Fig. 8. Wavelength-dependence of |c| for three particular valuesof h.
and theoretical tunability, we performed simula-
tion of |c| from h=0� to 180� by step of 1�. Thenwe reported wavelengths corresponding to the
maximums of |c| as a function of h. Results are
shown in Fig. 9 together with the experimental
data of Fig. 3. We first note the good adequacy be-tween theoretical and experimental data. In partic-
ular, splitting of the peak wavelengths is clearly
visible. However, some discrepancies occur. In
particular, a red-shift of theoretical prediction
and the presence of additional wavelength around
h = 0�, h = 90� and h = 180�. The first one may be
due to a drift of fiber parameters between both
experiment on the laser and on the fiber, due forexample to temperature changes. The second one
may be attributed to the mismatch between exper-
imental data of Fig. 5 and corresponding fitting
functions.
However, to the authors knowledge, Fig. 9
represents the first direct comparison between
experimental and theoretical tunability of a multi-
wavelength fiber laser with an intracavitypolarizer.
5. Conclusion
We have presented multiwavelength operation
of a Nd-doped fiber laser with an intracavity pola-
rizer. We have shown tunability and splitting ofthe lasing peaks. In order to explain the experi-
mental observations, we have presented a novel
360 T. Chartier et al. / Optics Communications 253 (2005) 352–361
approach based on the Jones matrix analysis of
fiber birefringence and on the derivation of the
round-trip operator of the cavity. Excellent agree-
ment between theory and experiment validates the
principle of the method. Let us point out that thisapproach is not limited by the spectral range of
the laser and can be applied to other types of
multiwavelength fiber lasers based on polarization
selection. Possible extension of this work concerns
the comprehension of the wavelength evolution of
fiber birefringence.
Acknowledgements
The authors thank Pr. Pierre Pellat-Finet
(LAUBS, Lorient, France) for the idea to inject
a polarization oriented at 45�, Dr. Sebastien Coet-
mellec (CORIA, Saint-Etienne du Rouvray,
France) for helpfull discussions and Pr. Pascal
Besnard (UMR FOTON, Lannion, France) forhis comments on the manuscript.
Appendix A. Relations between Jones vectors,
Stokes parameters and ellipticity and azimuth
In the most general case, the Jones vector V rep-
resenting an elliptical state of polarization S, hastwo complex coordinates and can be written as
V ¼Ax expðiuxÞAy expðiuyÞ
!¼
ax þ ibxay þ iby
� �. ðA:1Þ
The Stokes parameters of S write:
s0 ¼ A2x þ A2
y ¼ a2x þ b2x þ a2y þ b2y ; ðA:2Þs1 ¼ A2
x � A2y ¼ a2x þ b2x � a2y � b2y ; ðA:3Þ
s2 ¼ 2AxAy cosðuy � uxÞ ¼ 2ðaxay þ bxbyÞ; ðA:4Þs3 ¼ 2AxAy sinðuy � uxÞ ¼ 2ðaxby � aybxÞ. ðA:5Þ
The ellipticity v and the azimuth / of S can be ex-
pressed as following:
sin 2v ¼ s3s0; ðA:6Þ
tan 2/ ¼ s2s1. ðA:7Þ
Appendix B. Change of basis for the Stokes
parameters
Let s0, s1, s2 and s3 be the Stokes parameters of
a polarization state S in a basis (x,y) of linearpolarizations. As a function of ellipticity v and azi-
mut / of S, they write:
s0 ¼ 1; ðB:1Þs1 ¼ s0 cos 2v cos 2/; ðB:2Þs2 ¼ s0 cos 2v sin 2/; ðB:3Þs3 ¼ s0 sin 2v. ðB:4Þ
Let us now consider a new basis (X,Y) of linear
polarizations that make an angle /0 with respect
to the basis (x,y). In this new basis the Stokesparameters of S write:
S0 ¼ 1; ðB:5ÞS1 ¼ S0 cos 2v cos 2ð/� /0Þ
¼ s1 cos 2/0 þ s2 sin 2/0; ðB:6ÞS2 ¼ S0 cos 2v sin 2ð/� /0Þ
¼ s2 cos 2/0 � s1 sin 2/0; ðB:7ÞS3 ¼ S0 sin 2v ¼ s3. ðB:8Þ
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