Photonic crystal fibers (PCFs) are attracting increasing interests because of their unique properties such as: endlessly 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 (, , and), the change in the index of refraction and therefore the birefringence with the result of the birefringence in Regions I, II, III, and IV. Then, the obtained results are compared for different pump schemes for giving the design guidelines to ensure maximum heat dissipation and pump powers.

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

light and output laser can be expressed as follows:

where N_b(y) is the upper laser level population density, N_yt is the concentration of Yb³⁺, and represent the power of forward and backward propagation pump light, respectively. and represent the power of forward and backward propagation laser power, respectively, σ_as and σ_es are the laser absorption and emission cross section, σ_ap and σ_ep are the pump light absorption and emission cross-section respectively, Γ_p and Γ_s are the power-filling factors [7] .

Figure 2 shows the injection mode in fiber laser PCFs. and the radial coordinate r and the tangential. The quantities and e are the radius of the core, the inner cladding, the air-clad and the outer cladding, respectively. The temperature distribution in a fiber reported in Ref. [4] is necessary for determining the radially varying index of refraction, 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]

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

where, , and are of the radial, tangential, and azimuth, stress components, respectively, and , and 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 birefringence, given by.

Therefore, the expressions in different regions of the fiber can be given by [10] :

where and are the parallel and perpendicular stress-optic coefficients. Their values are, respectively [1] . The numerical values of E and v are and [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

For the purposes of illustration, unless indicated otherwise, the parameters used in the simulation are λ_p = 975 nm, λ_s = 1080 nm, R_1s = 0.98, R_2s = 0.04, L = 5 m, , , , , , , , , , fiber core diameter D = 10 µm and N_A = 0.05.

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

In addition, for the pump:

In the bidirectional pump scheme:

In the forward pump scheme:

Using analytical and numerical calculations, the finite-difference method (FDM) and a simple model of (PCFs), we have determined the distributions of pump and laser along the PCFs in the forward and bidirectional pumped YPCFs, and the birefringence in PCF laser dope ytterbium in cases the forward and bi-directional pump scheme respectively as function of the radius.

Figure 3 and Figure 4 show the distributions of pump and laser along the PCFs in the forward and bidirectional pumped YPCFs. All the launched pump powers are the same: 200 W. We notice an increase of the speed of forward signal power along the PCFs in both cases (the forward and bidirectional pump schemes), and it can be seen that this increase slows down along the YPCFs for the forward pumped lasers. As a result, the forward pump power and inverse population correspondingly decrease. The bidirectional pump scheme helps to the equivalently distribution of the pump power, avoiding the optical and thermal damages.

Both Figure 5 and Figure 6 show the birefringence as function of the radius in PCF lasers doped ytterbium in the bidirectional pump scheme and forward pump scheme respectively. Their values do not have a great effect on the quality of laser beam in these pump schemes. However, the birefringence in bidirectional pump scheme is 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] .

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

In summary, regarding the birefringence, their value in PCFs are less than that found in conventional fiber lasers [1] and do not have a great effect on the quality of the laser beam in different pump schemes, especially in the bi-directional pumping. Hence, after this comparison, we optimized the x position of the transversal pump in the laser cavity (PCFs) which was the most convenient in specific conditions.

N. Rouchdi,M. Abouricha,A. Boulezhar,M. Kriraa, (2015) Comparative Study of the Birefringence in Photonic Crystal Fiber Lasers. Optics and Photonics Journal,05,320-325. doi: 10.4236/opj.2015.511030