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http://dx.doi.org/10.4236/opj.2015.511030

How to cite this paper: Rouchdi, N., Abouricha, M., Boulezhar, A. and Kriraa, M. (2015) Comparative Study of the Birefrin-gence in Photonic Crystal Fiber Lasers. Optics and Photonics Journal, 5, 320-325.

http://dx.doi.org/10.4236/opj.2015.511030

Comparative Study of the Birefringence in

Photonic Crystal Fiber Lasers

N. Rouchdi

1

, M. Abouricha

1*

, A. Boulezhar

1

, M. Kriraa

2

1Laboratory of theoretical and Applied Physics, Faculty of Sciences-Ain Chok, Hassan II Casablanca University, Casablanca, Morocco

2Thermal Group LPMMAT, Department of Physics, Faculty of Sciences-Ain Chok, Hassan II Casablanca University, Casablanca, Morocco

Received 27 June 2015; accepted 17 November 2015; published 20 November 2015

Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

In this paper, we study the birefringence in photonic crystal fiber lasers PCFs and in conventional fiber lasers in the bi-directional pump scheme in the linear cavity laser. We show that the value of birefringence in photonic crystal fibers is smaller than that of conventional fiber lasers [1].

Keywords

Pump Schemes, PCF Lasers, Fiber Lasers, Yb-Doped Fiber Lasers, Thermal Effects in Fiber Lasers

1. Introduction

Photonic crystal fibers (PCFs) are attracting increasing interests because of their unique properties such as: en-dlessly single-mode guiding, freedom of dispersion characteristics, and large mode area [2] [3]. In this work, we focus on PCFs in which a core doped with Yb3+ is surrounded by a lower index cladding, which is, surrounded by an air-clad region, in turn, surrounded by a second lower index cladding index.

We use the rate equation for finding the expressions of the temperature dependence in Regions I, II, III, and IV [4] [5]. In order to get this dependence we utilize the results of the expressions for the stress components (

σ

r

( )

r ,

σ

ϕ

( )

r , and

σ

y

( )

r ), the change in the index of refraction and therefore the birefringence with the

re-sult of the birefringence in Regions I, II, III, and IV. Then, the obtained rere-sults are compared for different pump schemes for giving the design guidelines to ensure maximum heat dissipation and pump powers.

2. Theoretical Model

Figure 1displays a schematic illustration of Yb doped PCFs [6]. For the convenience of analysis, the pump

(2)

light and output laser can be expressed as follows:

( )

(

)

( )

( )

( )

p

p ap ep b ap yt p p p

P y

N x N P y P y

y σ σ σ α

±

± ±

 

± = Γ + − × −

∂ (1)

( )

(

) ( )

( )

( )

s

s as es b as yt s s s

P y

N y N P y P y

y σ σ σ α

±

± ±

 

± = −Γ + − × −

∂ (2)

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

, ,

p p ap p p ap

b

yt b p p yt b

co co

p p ap

yt b yt b p p

co

p p ap b

yt b p p

co

N y

N N y P y P y N N y

t hcA hcA

N N y N N y P x t P y

hcA

N y

N N y P x P y

hcA

λ σ λ σ

λ σ λ σ τ + − + − + − Γ Γ           = + Γ       −     +   Γ       −    + −   (3)

where Nb(y) is the upper laser level population density, Nyt is the concentration of Yb3+, Pp

( )

,y

and Pp

( )

,y

represent the power of forward and backward propagation pump light, respectively. Ps

( )

,y

and Ps

( )

,y

represent the power of forward and backward propagation laser power, respectively, σas and σes are the laser

ab-sorption and emission cross section, σap andσep are the pump light absorption and emission cross-section

[image:2.595.201.425.84.174.2]

respec-tively, Γpand Γs are the power-filling factors [7].

Figure 2shows the injection mode in fiber laser PCFs. and the radial coordinate r and the tangential ϕ. The quantities a b c, , and e are the radius of the core, the inner cladding, the air-clad and the outer cladding, respec-tively. The temperature distribution in a fiber reported in Ref. [4] is necessary for determining the radially vary-ing index of refraction dn dT, and the calculate stresses. By using Equations (1)-(3), Abouricha et al. [4]

demonstrated that the temperature expressions of the stress components can be written as: [1] [8]

( ) ( )

2

( )

2

( )

0 0 1 1 d d 1 e r r E

r T r r r T r r r

b r α σ ϑ   =

− 

 (4)

( ) ( )

2

( )

2

( )

( )

0 0 1 1 d d 1 e r E

r T r r r T r r r T r

b r ϕ α σ ϑ   = − −

− 

 (5)

In the case where the fiber end faces are free of traction

( )

( )

( )

y r r r ϕ r

σ

=

σ

+

σ

(6)

where

σ

r

( )

r ,

σ

ϕ

( )

r , and

σ

y

( )

r are of the radial, tangential, and azimuth, stress components, respectively, and𝛼𝛼, 𝐸𝐸and 𝑣𝑣 are the thermal expansion coefficient, Young’s modulus, and Poisson’s ratio, respectively.

The length of optical fibers is much greater than that of a typical fiber outside radius (b). Thus, we can invoke the plane-strain approximation [9] in which the y strain component everywhere. We also can define the bire-fringence ∆nI II III IVB, , ,

( )

r , given by.

( )

( )

( )

, , , , , , , , ,

I II III IV I II III IV I II III IV

B r

n r n r nϕ r

∆ = ∆ − ∆ (7)

[image:2.595.147.525.217.383.2]
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[image:3.595.90.536.79.374.2]

Figure 2. Transversal injection of the PCFs.

( )

30

(

( )

( )

)

2

I I I

B r

n

n r B σ r σϕ r

∆ = − (8)

( )

30

(

( )

( )

)

2

II II II

B r

n

n r B σ r σϕ r

∆ = − (9)

( )

03

(

( )

( )

)

2

III III III

B r

n

n r B σ r σϕ r

∆ = − (10)

( )

03

(

( )

( )

)

2

IV IV IV

B r

n

n r B σ r σϕ r

∆ = − (11)

B=BB (12)

where B and B⊥ are the parallel and perpendicular stress-optic coefficients. Their values are

8 2

4.5 10× − cm kg,

8 2

27.7 10× − cm kg respectively [1]. The numerical values of E and v are E=73 GPa=7.45 kg cm2 and 0.16

v= [5].

Equations (8)-(11) show that the fiber birefringence depends only on the thermally induced stresses.

Using the expressions of the temperature dependence in Regions I, II, III, and VI [4], and Equations (8)-(11) we calculate the birefringence in different regions of cavity laser

3. Simulation and Discussion

For the purposes of illustration, unless indicated otherwise, the parameters used in the simulation are λp = 975

nm, λs = 1080 nm, R1s = 0.98, R2s = 0.04, L = 5 m,

τ

=0.8 ms,

24 2

2 10 m

ap

σ

= ×

,

σ

ep 2 10 24m2

= × ,

27 2

3.1 10 m

as

σ

= ×

,

σ

es 4.2 1025m2

= × , Nyt=1.6 10× 26 , Γ =p 0.0012 , Γ =s 0.8 ,

3 1

5 10 m

s

α

= × − −

,

3 1

3.1 10 m

p

α

= × − −

fiber core diameter D = 10 µm and NA = 0.05.

The boundary conditions for the laser propagation equation can be written as:

( )

0 1

( )

0

s s s

P+ =R P− (13)

( )

2

( )

s s s

PL =R P+ L (14)

In addition, for the pump:

In the bidirectional pump scheme:

( )

0 100 W and

( )

100 W

P = P L = (15)

In the forward pump scheme:

( )

0 200 W and

( )

0

P = P L = (16)

(4)
[image:4.595.171.457.241.442.2]

diminishing their values in the forward pump scheme. Therefore, the second advantage of the bidirectional pump scheme is to have a high quality of laser beam then the forward pump scheme and then in conventional fiber lasers [1].

[image:4.595.169.459.474.692.2]

Figure 3. Distributions of pump and laser powers along the PCFs in the bidirec- tional pump scheme.

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[image:5.595.122.510.87.345.2]

Figure 5. Birefringence in PCF lasers doped by ytterbium in case of the bi-directional pump scheme.

Figure 6. Birefringence in PCF lasers doped ytterbium in case of the forward pump scheme.

4. Conclusion

In this paper, we have compared the birefringence in photonic crystal fibers (PCFs) with the birefringence in conventional fibers.

[image:5.595.117.512.370.631.2]
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[3] Knight, J.C., Briks, T.A., Cregan, R.F., et al. (1998) Large Mode Area Photonic Crystal Fiber. Electronics Letters, 34, 1347-1349. http://dx.doi.org/10.1049/el:19980965

[4] Abouricha, M., Boulezhar, A. and Habiballah, N. (2013) The Comparative Study of the Temperature Distribution of Fiber Laser with Different Pump Schemes. Open Journal of Metal, 3, 64-71.

http://dx.doi.org/10.4236/ojmetal.2013.34010

[5] Bass, M., Ed. (1995) Handbook of Optics (Sponsored by the Optical Society of America), Vol. II, Devices, Measure-ments, and Properties. McGraw-Hill, New York.

[6] Brown, D.C. (1998) Nonlinear Thermal Distortion in YAG Rod Amplifiers. IEEE Journal of Quantum Electronics, 34, 2383-2392. http://dx.doi.org/10.1109/3.736113

[7] Limpert, J., Schreiber, T., Liem, A., Nolte, S., Zellmer, H.,Peschel, T., Guyenot, V. and Tünnermann, A. (2003) Ther- mo-Optical Properties of Air-Clad Photonic Crystal Fiber Lasers in High Power Operation. Optic Express, 11, 2984. [8] Eggleston, J.M., Kane, T.J., Kuhn, K., Unternahrer, J. and Byer, R.L. (1984) The Slab Geometry Laser—Part I: Theory.

IEEE Journal of Quantum Electronics, 20, 289-301. http://dx.doi.org/10.1109/JQE.1984.1072386

[9] Boley, B.A. and Weiner, J.H. (1985) Theory of Thermal Stresses. Reprint Edition, Kreiger, Melbourne.

[10] Abouricha, M., Boulezhar, A., El Mouden, M. and Kriraa, M. (2014) Theoretical and Numerical Study of Stress and Thermo-Optic in Photonic Crystal Fiber Laser in Different Pump Schemes. Open Journal of Metal, 4, 120-130.

Figure

Figure 2 demonstrated that the temperature expressions of the stress components can be written as: ing index of refraction tively
Figure 2. Transversal injection of the PCFs.
Figure 3. Distributions of pump and laser powers along the PCFs in the bidirec- tional pump scheme
Figure 5. Birefringence in PCF lasers doped by ytterbium in case of the bi-directional pump scheme

References

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