are assumed to be narrow spectra and regarded as single wavelength, ignoring the non linear effects and influences of amplified spontaneous emission on the inverse population. Using the rate-equation theory under the steady state conditions, we can be governed the propagations of the light wave power in fiber cavity as follows [7]:

(1)

(2)

(3)

Equations (1)-(3), N_yt is the concentration of Yb³⁺, N_b(x,t) is the upper laser level population density, and represent the power of forward and backward propagation pump light, respectively. and represent the power of forward and backward propagation laser power, respectively. Γ_p and Γ_s are power filling factors. 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. τ is the lifetime of Yb³⁺ in upper level, α_p and α_s represent scattering loss coefficients of laser light and pump light respectively. The above Equations (1)-(3) at fiber front (x = 0) and fiber rear end (x = L) satisfy the boundary conditions for the laser as follows:

(4)

Where R_s1 = 0.98 and R_s2 = 0.04 the power laser reflectively of the reflector at x = 0 and x = L, respectively. And the boundary conditions for the pump propagation equation can be expressed as follows [8]:

(5)

(6)

(7)

Where P_p01 the launched pump power into the front end (x = 0) and P_p02 the launched pump power into the back end (x = L). R_p2 and R_p1 represent the reflectivity at pump wave length in forward and backward pumped fiber laser, respectively.

In the YDCF, heat is generated in the fiber core, and this heat is transferred from the fiber core to the inner clad by thermal conduction, and by thermal conduction and radiation to air-clad, in air clad no convection because the width air-clad is very small, which justified by Grashof number G_rδ in air clad given by [10] and by thermal conduction and radiation to other clad, and dissipated from the fiber surface to ambient air by convective and radiative heat transfer. Therefore the transverse temperature distributions in the YDCF at room temperature are governed by following thermal conductive equations in symmetric cylindrical coordinates [11].

(8)

Then we find:

(9)

(10)

(11)

(12)

Where a, b₁, b₂ and b represent the radii of the core, inner clad, air-clad and outer clad respectively. T₁(r), T₂(r), T₃(r) and T₄(r) represent the temperature in fiber core, inner clad, air-clad and outer clad, respectively. Q(x) represents the heat power density along the axial direction of the fiber core. Which mainly depends on the scattering and loss of pump light and given by:

(13)

Where, is absorption coefficient, and S is the quantum efficiency whose theoretical value is λ_p/λ_s. However, it cannot reach the theoretical value in practical applications. In Other regions of the fiber, the value of Q(x) is zero. Supposing perfect thermal connection among the cladding and inner core, the temperature and their derivatives are continuous at the boundary (r = a), therefore, the boundary conditions can be expressed as follows [12]:

For r = 0 we have:

(14)

For we have:

(15)

For we have:

(16)

For we have:

(17)

In Figure 2, we have demonstrates the temperature difference between the ambient air and the fiber surface is determined by the convective and radiation heat flow, and thus the boundary condition cross the fiber other cladding and the ambient air is given by the Newton’s law and the Stefan-Boltzmann law [13].

(18)

Where k₂ is the thermal conductivity of air, k₁ is the thermal conductivity of material, T_c is the temperature of ambient air, ε is the emissivity, σ is the Stefan-Boltzman constant and h_c is the convective coefficient which can be calculated by [14].

(19)

(20)

(21)

Where N_u, P_r and Gr are Nusselt, Prandtl numbers and Grashof, respectively. g and ν are the acceleration and Kinematic viscosity, the temperature in regions core, inner clad, air-clad and other clad can be derived by using Equations (9)-(12) subject to the boundary conditions (14)-(17) as follows:

(22)

(23)

(24)

(25)

Where T₀ represents the temperature of fiber axis (r = 0). Using Equations (22)-(25) subject to boundary condition (18), the relationship between the temperature of fiber surface T₄(b) and the ambient air T_c is given by Equations (26):

(26)

Substituting Equation (26) into Equations (22)-(25), we can obtain the temperature distributions in fiber core and other regions.

3. Simulation and Discussion

In our simulation, we have used the Common parameters given in the Table 1" target="_self"> Table 1. Figure 3 illustrates the output power as a function of fiber length (YDCL) for different pump schemes. It is shown that the increasing output power reaches the maximum value 174.7 W, for length l = 5 m in the forward pump scheme with reflection R_p2 = 0.98. In the backward pump scheme with reflection R_p1 = 0.98 the value reached is 178.7 W for the length l = 6.4 m, the maximum value 171.7 W, for length l = 7.4 m in the forward pump scheme with reflection R_p2 = 0, in the backward pump scheme with reflection R_p1 = 0, the value

reached is 177.7 W for the length l = 10.1 m and it is reaches the value 174.7 W for l = 10 m in the bidirectional pump scheme. Therefore, the forward pump with reflection is in good scheme to minimize the length of YDCF and the backward pump with reflection is the most suitable scheme to minimize the pumping energy.

The heat dissipation along axial, calculated by using Equation (13) and the axial distribution of temperature (r = 0) for different pump schemes have reported, respectively in Figures 4 and 5.

It is shown that the maximum temperature for the bidirectional pump scheme T_max = 90.89˚C is lower than that for the forward pump with (reflection R_p2 = 0.98 and R_p2 = 0) as T_max = 148.4˚C and than that the backward with (reflection R_p1 = 0.98 and R_p1 = 0) as T_max = 154.4˚C. The maximum difference is 57.51˚C.

The difference between the maximum and the minimum is: 40.14˚C in the case of bi-directional pumping, 109.19˚C in the case of pumping forward with R_p2 = 0.98, 115.38˚C in the case forward with R_p2 = 0, 115.19˚C in the case backward with R_p1 = 0.98 and 121.12˚C in the case back with R_p1 = 0. Therefore, the reflectivity of the pump has a role crucial is to minimize the temperature difference.

The radial distribution of the temperature is reported in Figures 6-10 as function of core radius for different pump schemes. Consequently, the effect of the radius core is negligible and the maximum temperature difference between a = 5 and a = 10 µm does not exceed 1˚C in (r = 0) for different pump schemes Figures (a). But in Figures (b), we have shown that the temperature distributions at the fiber in (r = 0) increases when the air-clad increases for different pump schemes. In Figures (c), we can see another scenario as the temperature of the fiber increases when the outer cladding radius increases.

4. Conclusion

In this paper, we have compared in detail the heat dissi-

pation and the temperature distribution for different pumps schemes. The results show that the temperature distribution for bidirectional pump mode is more even than that the forward pump with reflection R_p2 and backward pump with reflection R_p1. These two latter schemes are in agreement for saving pump powers and YDCF length. Moreover, the forward pump with R_p1 = 0.98 and the backward pump with R_p2 = 0.98 in less length resulting in high output power are thermally acceptable. Further experimental and theoretical studies are needed to explain the temperature distribution of fiber laser with different pump schemes.

REFERENCES

NOTES

^*Corresponding authors.