Multiwavelength fiber laser with an intracavity polarizer

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.

mailto:[email protected]

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.


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 /


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.


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.


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.


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


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

References

[1] S. Yamashita, K. Hotate, Electron. Lett. 32 (14) (1996)

1298.

[2] H.L. An, X.Z. Lin, E.Y.B. Pun, H.D. Liu, Opt. Commun.

169 (1999) 159.

[3] D. Wei, T. Li, Y. Zhao, S. Jian, Opt. Lett. 25 (16) (2000)

1150.

[4] X.P. Dong, S. Li, K.S. Chiang, M.N. Ng, B.C.B. Chu,

Electron. Lett. 36 (19) (2000) 1609.

[5] J.-N. Maran, S. LaRochelle, P. Besnard, Opt. Commun.

218 (2003) 81.

[6] N. Park, P.F. Wysocki, IEEE Photon. Technol. Lett. 8 (11)

(1996) 1459.

[7] P.D. Humphrey, J.E. Bowers, IEEE Photon. Technol.

Lett. 5 (1) (1993) 32.

[8] N. Azami, A. Saıssy, M. De Micheli, G. Monnon, D.B.

Ostrowsky, Opt. Commun. 158 (1998) 84.

[9] S. Yamashita, T. Baba, Electron. Lett. 37 (16) (2001)

1015.

[10] U. Ghera, N. Konforti, M. Tur, IEEE Photon. Technol.

Lett. 4 (1) (1992) 4.


[11] N. Friedman, A. Eyal, M. Tur, IEEE J. Quantum Electron.

33 (5) (1997) 642.

[12] J.S. Krasinski, Y.B. Band, T. Chin, D.F. Heller, R.C.

Morris, P.A. Papanestor, Opt. Lett. 14 (8) (1989) 393.

[13] C.D. Poole, R.E. Wagner, Electron. Lett. 22 (19) (1986)

1029.

[14] B.L.Heffner, IEEEPhoton. Technol. Lett. 4 (9) (1992) 1066.

[15] S. Bielawski, D. Derozier, P. Glorieux, Phys. Rev. A 46 (5)

(1992) 2811.

[16] T. Chartier, A. Hideur, C. Ozkul, F. Sanchez, G. Stephan,

Appl. Opt. 40 (30) (2001) 5343.

[17] A. Hideur, T. Chartier, C. Ozkul, F. Sanchez, Opt. Lett.

26 (14) (2001) 1054.

[18] H.Y. Kim, E.H. Lee, B.Y. Kim, Appl. Opt. 36 (27) (1997)

6764.

[19] T. Chartier, F. Sanchez, G. Stephan, P. Le Boudec, E.

Delevaque, R. Leners, P.L. Francois, Opt. Lett. 18 (5)

(1993) 355.

Multiwavelength fiber laser with an intracavity polarizer

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