Theoretical and Numerical Study of Stress and Thermo-Optic in Photonic Crystal Fiber Laser in Different Pump Schemes

In this paper, we present a study of thermal, average power scaling, change in index of refraction and stress in photonic crystal fiber lasers with different pump schemes: forward pump scheme, backward pump scheme, forward pump scheme with reflection of 98%, backward pump scheme with reflection of 98% and bi-directional pump scheme. We show that management of thermal effects in fiber lasers will determine the efficiency and success of scaling-up efforts. In addition, we show that the most suitable scheme is the bi-directional.

Recently a new type of the pump schemes with the high brightness semiconductor diode pump laser was studied [5].And the new type of photonic crystal fibers (PCFs) is attracting increasing interests because of its unique properties such as endlessly single-mode guiding, freedom of dispersion characteristics, and large mode area [6] [7].We also concentrate on PCFs in which a core doped with Yb 3+ surrounded by a lower index cladding, which is, surrounded by an air-clad region, in turn, surrounded by a second lower index cladding index.
By analytical and numerical calculation and using the finite-difference method (FDM), we have determined the expressions of temperature distribution in different regions of the photonic crystal fiber laser (PCF) along the axial and radial directions from the integration of the steady-state heat equation for an isotropic medium.Then we used the expressions previously derived for the temperatures in Regions I, II, III, and IV [8] in the results of the expressions for the stress components ( ( ) ) and the change in the index of refraction in Regions I, II, III, and IV.The results are compared in different pump schemes for giving the design guidelines to ensure maximum heat dissipation and saving pump powers.The calculated stress values are very small in different pump schemes, and will consequently have a negligible effect upon the index of refraction.
The resulting gradients are still small as the corresponding stress values.As a consequence, we expect that changes in the index of refraction due to the stress-optic effect will be negligible in different pump schemes, and thermally induced birefringence will be absent in fiber lasers (in all cases of pumping), the stress component ( ) z r σ increases with the increase of the length the fiber in the cases (forward pump schemes) and decreases along the fiber laser in the cases (backward pump schemes).These values are between −0.4 × 10 −6 kg/m 2 and −3.4 × 10 −6 kg/m 2 in the four primer cases.Moreover, in the case bi-directional pump scheme, its value is between −0.8 × 10 −6 kg/m 2 and −1.8 × 10 −6 kg/m 2 .And the values of the change in index of refraction increases in the cases of forward pump schemes and decreases in the cases of the backward pump schemes, along the fiber laser.For the bi-directional, its value is even smaller.

Average Temperature in PCF
As shown in Figure 1, a typical high power Yb 3+ -doped double-clad PCF laser consists of an Yb 3+ -doped double PCF with reflectors on both of the ends.The temperature distribution in a fiber reported in [8] is necessary for determining the radially varying index of refraction due to d d n T , the calculated stresses, and the change in index of refraction through the stress-optic effect.
We calculate the average temperature av T as T r r T r r T r r T r r T r where where the temperature expressions ( ) T r are given by [8].And

Stress Distributions
The length of the optical fibers is much greater than a typical fiber outside radius (b), we can invoke the planestrain approximation [10] in which the z strain everywhere.The radial, tangential, and z stresses ( ) z r σ can be found from [11].
in the case where the fiber end faces are free of traction where α , E and v are thermal expansion coefficient, Young's modulus, and Poisson's ratio, respectively.So- lution of ( 10)-( 12), using the expressions previously derived for the temperatures in Regions I, II, III, and VI [8], results in the following final expressions for the stresses in Regions I, II, III, and IV: where. ( It can be shown that ( 13)-( 24) satisfy the boundary conditions  13)-( 24) will be used in Section IV under to calculate the radially varying index of refraction in different pump schemes, due to the stress-optic effect.Note that the equations as derived above for the fiber stresses could also have been obtained by use of the deviation of the temperature distribution from the average and the Airy stress potential [11].

Index of Refractions
Using the expressions for the radial temperature distributions, reported [8] and for the stress distributions ( 13)-( 24), we tray now to make calculation of the radially varying radial and tangential index of refraction distributions.For comparison in different pump schemes, we prefer to cast the calculation of the induced change in the index of refraction in terms of material stresses, rather than strains.We begin by writing the changes in the indices of refraction: And ( ) ( ) ( ) And ( ) ( ) ( ) The calculation of the stress-induced index changes ( ) , , , , , 2 where B  and B ⊥ are the parallel and perpendicular stress-optic coefficients.Their values are

Summary
In this paper, we have investigated a comparison of stress and thermo-optic of photonic crystal fiber (PCFs) in different pump schemes.Using in these calculations a simple model of (PCFs) and finite differential method (FDM), we have revealed the temperature in the core of the fiber and by laws of heat transfer, we determined its value at the surface of the fiber and the stress value in the different regions of the fiber.In summary, regarding stress, thermo-optic and the change in index of refraction, their value does not have a great effect on the quality of the laser beam in different pump schemes, especially in the case of the bi-directional pumping.Hence, after this investigation, we proved the architecture of laser cavity that was the most convenient in specific condition.The bi-directional pump scheme is the most suitable because it has less thermal effects than the other cases.However, the backward pump with reflection of 98% and the forward pump with reflection of 98% save more energy than the backward pump with reflection of 0%, the forward pump with reflection of 0% and bi-directional pump scheme.

Figure 2
Figure 2 shows the heat flow mechanisms in fiber laser and the radial coordinate r and the tangential angle ϕ .The quantities

Figure 1 .
Figure 1.Schematic illustration of end pumped fiber lasers.

For
index changes for the r and ϕ components of the electric field.As before, the roman numerals refer to Regions I, II, III and IV.Here, we will calculate Regions I, II, III and IV, becomes

Figure 3 .
Figure 3. Distribution of longitudinal stress in PCF at 200 W pump of 940 nm load calculated by FDM analysis.(a) Forward pump schemes; (b) Forward pump with a reflectivity of 98%; (c) Backward pump schemes; (d) Backward pump with a reflectivity of 98%; (e) Tow-end pump schemes.

Figure 4 .Figure 5 .
Figure 4. Distribution of radial stress in PCF at 200 W of 940 nm pump load calculated by FDM analysis.(a) Forward pump schemes; (b) Forward pump with a reflectivity of 98%; (c) Backward pump schemes; (d) Backward pump with a reflectivity of 98%; (e) Tow-end pump schemes.crease along the fiber lasers in Figure 5(a) and Figure 5(b) from the order 1.2 × 10 −3 to 0.1 × 10 −3 .In Figure 5(c) and Figure 5(d) the values of change in index of refraction increase in the opposite order 0.1 × 10 −3 to 1.2 × 10 −3 .Nevertheless, in Figure 5(e) the value is in order of 3 × 10 −3 to 7 × 10 −3 .As result, we notice in side of