Heat transport in solid and air-clad fibers for high-power fiber lasers

B. Zintzen; T. Langer; J. Geiger; D. Hoffmann; P. Loosen

doi:10.1364/OE.15.016787

1. Introduction

Several studies have been conducted on the heat transport in the active fiber of fiber lasers [1–3]. However, the different approaches do not emphasize on explaining the influence of the fiber parameters on the heat transfer. In this work we present an analytic approach on the heat transfer problems in the fiber which provides a deeper insight into the governing parameters and the possibility to optimize the system.

2. Method

For one-dimensional heat conduction under steady state conditions with no internal source the appropriate form of the heat equation in radial coordinates is Eq. (1) with λ as the thermal conductivity.

\frac{1}{r} \frac{d}{dr} (λ r \frac{dT}{dr}) = 0

Integrating two times leads to Eq. (2) [4]:

T (r) = C_{1} \ln r + C_{2}

Introducing the boundary conditions T(r=r₁)=T_s1 and T(r=r₂)=T_s2 with r₁ as the inner and r₂ as the outer radius of the layer leads to Eq. (3).

T (r) = \frac{T_{s 1} - T_{s 2}}{\ln (r_{1} ⁄ r_{2})} \ln \frac{r}{r_{2}} + T_{s 2}

Differentiation and application of Fouriers law leads to Eq. (4) for the heat flow per unit length q’ through an arbitrary layer of the fiber.

q' = \frac{2 π λ (T_{s 1} - T_{s 2})}{\ln (r_{2} ⁄ r_{1})}

This equation suggests an interesting concept for the treatment of the heat flow through a fiber layer. In analogy to diffusion of electrical charge the heat flow can be described using equation (5) [4]:

T_{s 1} - T_{s 2} = {R'}_{t} \cdot q'

The temperature difference is analog to the electrical voltage which drives the heat flow through a thermal resistance. The conductive thermal resistance per unit length R′_tcond of a hollow cylinder can be expressed as:

R_{tcond}^{'} = \frac{\ln (r_{2} ⁄ r_{1})}{2 π λ}

This expression differs from the similar expression used for the same purpose in [5], and delivers different results particular for layers close to the fiber core. With Eq. (6) the thermal resistance of a pump cladding with 400 µm outer and 20 µm inner diameter is 0.345 mK/W, Eq. (4) from [5] delivers 0.21 mK/W. The thermal resistance is thus 65 % higher for this layer close to the core than given in [5]. The deviation gets smaller for layers further away from the core.

A similar approach can be applied for the convective cooling of the fiber surface. The thermal resistance of the surface R′_tconv is given in Eq. (7) with α as the convective heat transfer coefficient:

R_{tconv}^{'} = \frac{1}{α 2 π r}

Table 1 exemplarily gives the thermal resistances of the fiber layers of two types of fibers.

Table 1. Thermal resistances for two optical fibers

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The table shows clearly, which layers have a major impact on the heat transport from the fiber. Using Eq. (6) the parameters of the critical layers can be varied to optimize the performance of the system. The thermal resistances of the surface cooling show the potential of direct water cooling compared to air cooling

The maximum temperature in the coating is the critical temperature which limits the acceptable heat load. It is calculated for a given heat load by adding up the thermal resistances of the layers between coolant and inner coating surface and applying Eq. (5).

3. Application to Air-Clad Fibers

The method can also be applied to air-clad fibers. The thermal resistance of an air-clad is found by calculating the thermal resistances of the bridges R′_tbridge and the air chambers R′_tair separately and combining these with:

R_{tac}^{'} = \frac{1}{\frac{1}{R_{tair}^{'}} + \frac{1}{R_{tbridge}^{'}}}

The thermal resistance of the bridges is found as in [5] by:

R_{tbridge}^{'} = \frac{1}{nb λ}

with l as the length, b as the width and n as the number of the bridges. The thermal resistance of the air chambers is determined by calculating the resistance of a cylindrical layer of air with a thermal conductivity of 0.023 W/(mK). It is not as in [5] calculated assuming convection because in the air chambers the product of Grashof- und Prandt- number is smaller than 1000. According to [6] the effect of convection is negligibly in this case. The results of the calculation are shown in Table 2.

Table 2. thermal resistances of air-clad fibers

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Compared to the thermal resistance of 2.08 mK/W for fiber 1 and 0.46 mK/W for fiber 2 given in [5], these values show a much better thermal conductivity of the air-clad even though radiative heat transport was neglected. The deviation is slightly compensated by the overrating of the thermal conductivity in solid layers in [6], however, it appears, that the critical heat loads for air-clad fibers can be increased from the numbers given there.

4. Optimization of the active fiber

It is worth noting, that the thermal resistance of the 40/700 fiber with air cooling is smaller than that of the thinner fiber with the thinner coating. This observation leads to the question, if a thicker coating might be advantageous in the case of convective cooling. Increasing the outer radius expands the cooling surface but the conductive thermal resistance of the coating increases as well. The overall thermal resistance of the cladding can be expressed as sum of the conductive of the layer and the convective resistance at the surface [4]:

R_{tot}^{'} = R_{cond}^{'} + R_{conv}^{'} = \frac{\ln (r_{coat} ⁄ r_{p})}{2 π λ} + \frac{1}{2 π α r_{coat}}

For the optimum cladding radius the following condition applies:

\frac{d R_{tot}^{'}}{dr} = 0

which leads to:

\frac{1}{2 π λ r_{coat}} - \frac{1}{{2 π α r_{coat}}^{2}} = 0

and finally to:

r_{coat} = \frac{λ}{α}

Second derivation shows, that a minimum was found.

Figure 1 shows the conductive and convective resistance and their sum of the outer coating of the 40/700-fiber for a convection heat transfer coefficient of 200 W/(m²K).

Fig. 1. Thermal resistances of coating and coating surface und the sum of both

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Eq. (11) shows that the optimization is only sensible for air cooling. With a thermal conductivity of polymers in the range of 0.3 W/(mK) and a convection heat transfer coefficient 2*10⁴ W/(mK) for a fluid velocity of 1 m/s Eq. (13) leads to an inner coating radius smaller than the pump cladding radius.

For conductive cooling of the fiber it is of interest to analyze the influence of the outer glass diameter on the heat transport while maintaining a constant coating thickness. While increasing the thickness of the pump cladding (or adding an additional glass cladding), the thermal resistance of the glass cladding increases while the thermal resistance of the coating decreases due to an enlarged circumference.

Following the same path as described above the result for the optimum outer glass diameter r_p can be identified with Eq (14):

r_{p} = d_{coat} (\frac{λ_{p}}{λ_{coat}} - 1)

5. Packaging of Splices

Measurements show that the local heat loads in splices are much higher than in the active fiber. There are a number of defects which cause losses in splices, for double clad fibers the recoat is an additional source of loss and heat. Heat generation may be different for different splices in the same fiber and it is not possible to accurately simulate the heat transport in FEM-models as the lateral and radial distribution of heat generation is not known. However, one-dimensional modeling gives insight into the question whether it is advantageous to embed the splice in a transparent medium with a comparatively low thermal conductivity or to use an absorbing medium with higher thermal conductivity. Obviously, if the complete loss is scattered out of the fiber and the surrounding medium does not absorb the radiation, the best packaging is a transparent medium with an arbitrarily thermal conductivity. However, if a part of the loss is absorbed in the fiber or the recoat it is not obvious, if the thermal conductivity of the surrounding medium or the thermal resistance of the boundary layer towards the cooling has the stronger impact on the overall performance. The model we use to examine this question is that of a fiber with a 20 µm core and a 400 µm pump cladding. Instead of adding a additional layer of a transparent medium we assume that the recoat of the splice is transparent for heat-pump-and signal-radiation. The recoat is surrounded by a 50 µm-layer of heat conducting paste to ensure good thermal contact to the water-cooled fiber holder.

The model and the equivalent thermal circuit diagram are depicted in Fig. (1). q′₁ is the heat generated in the fiber or at the surface of the fiber, q′₂ is the heat generated by absorption of the scattered light at the boundary layer of the recoat.

Fig. 2. Model and equivalent thermal circuit diagram for the simulation of the heat transport from recoated splices

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The equations to calculate the respective temperatures are:

T_{src} - T_{c} = (q_{1}^{'} + q_{2}^{'}) \cdot (R_{thcp}^{'} + R_{tcu}^{'} + R_{tconv}^{'})

T_{sp} - T_{src} = q_{1}^{'} \cdot R_{trc}^{'}

If q′₁ is zero the maximum temperature occurs at the boundary layer between recoat and surrounding medium, if q′₁ is not zero, the maximum temperature is at the surface of pump cladding. We varied the thickness of the recoat with a varying fraction of q′₁ of the total heat-flow. The applied thermal conductivities of the layers are 1.38 W/(mK) for the glass part, 0.3 W/(mK) for the recoat, 2 W/(mK) for the heat conducting paste and 380 W/(mK) for the splice holder with an outer radius of 10 mm. The thermal resistance of the convective water cooling was calculated for a plane of 10 mm width and a convective heat transfer coefficient of 4000 W/(m²K) which corresponds to a fluid velocity of 1 m/s. The results are given in Fig. 3. The applied heat flow per unit length q′_tot=q′₁+q′₂ was 4255 W/m.

Thickness of recoat in µm

Fig. 3. Maximum temperature in recoat depending on the recoat thickness and the share of q′₁ of the total heat flow

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Figure 3 shows that up to a share of about 2.5 % of q′₁ an optimum recoat thickness can be found. If the heat flow from pump cladding exceeds 3 % of the total heat flow, the optimum coating thickness is 0 µm. This result corresponds to experimental results found for fused fiber bundles [7] and applies to arbitrary total heat loads and thermal conductivities of the layers other than the recoat. Only the absolute maximum temperature changes when these parameters change. However, if a recoat with a thermal conductivity of glass can be implemented, the percentage of q′₁ for which an optimum recoat thickness greater than 0 can be found, increases to 15 %. 3-dimensional FEM-modeling shows, that these percentages are still valid, when the 3-dimensiomal heat flow is taken into account, only the absolute temperatures decrease.

6. Accuracy of the model

The accuracy of the model is strongly dependent on the availability of data on the thermal conductivity of the polymer. This information is not supplied by fiber manufacturers and the thermal conductive coefficient for e.g. different types of polyamide varies in the range of 0.24–0.33 W/(mK) [8]. The cross linking achieved during fiber production or recoating in the case of splices may additionally influence the thermal conductivity. Further more the thermal conductivity of polymers may be dependent on temperature [8], a fact which was not taken into account in this paper as even accurate mean values for room temperature were not available. However, the model consists of linear equations, errors of the input parameters are linearly transferred to deviations of the thermal resistance. 20 % error margin for the conductivity of the recoat leads to same percentage of uncertainty concerning the thermal resistance of this layer. The error in a given thermal resistance influences the maximum temperature proportionally to the weight of this resistance in the system.

It is also possible, that the thermal contact resistance between the layers of the fiber is of significance. In this case, when the thermal contact resistance between layers is known, it can simply be added to the model. This can also be done for the radiative heat transport from surfaces. For the definition of a radiative thermal resistance we refer to [4].

Radiation as heat transport mechanism was neglected in this model for reasons of simplicity. Furthermore its share of the heat flow from surfaces and inside the solid material is two magnitudes smaller than that of conduction and convection within the limited temperature range defined by thermal stability of the polymer materials. Transparency of the fiber materials for infrared radiation may also influence the results of thermographic measurements of the fiber. For the aspects to be considered for such measurements we refer to [9].

7. Conclusion

We demonstrate an analytic approach for the thermal analysis and optimization of the design of solid and air-clad fibers for high-power fiber lasers. Even if the material parameters are not exactly known, the approach gives useful insight to the governing parameters of the heat transport. We show that under certain conditions the fiber design can be optimized for cooling by increasing the cladding or coating radius. The analysis of the heat transport in splices shows that if only small amount of heat is generated inside the fiber or at the fiber surface, the recoat thickness should be as low as technically feasible.

Acknowledgements

This work is supported by the German Federal Ministry of Education and Research (BMBF).

References and links

1. D. C. Brown and H. J. Hoffman, “Thermal Stress and Thermo-Optic Effects in High Average Power Double-Clad Silica Fiber Lasers,” IEEE J. Quantum Electron. 37, 2, (2001). [CrossRef]

2. N. A. Brilliant and K. Lagonik, “Thermal effects in a dual-clad ytterbium fiber laser,” Opt. Lett. 26, 1669–1671, (2001). [CrossRef]

3. Y. Wang, C.-Q. Xu, and H. Po “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt.Commun. 242, 487–502 (2004). [CrossRef]

4. F. Incropera and D. DeWitt, Fundamentals of Heat and Mass Transfer, (John Wiley & Sons, Hoboken, 2002).

5. J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express 11, 2982 (2003). [CrossRef] [PubMed]

6. Verein Deutscher Ingenieure, VDI-Wärmeatlas, Berechnungsblätter für den Wärmeübergang, 6. Edition, (VDI-Verlag GmbH Düsseldorf, 1991).

7. F. Seguin, A. Wetter, L. Martineau, M. Faucher, C. Delisle, and S. Caplette “Tapered fused bundle coupler package for reliable high optical power dissipation,” Proc. SPIE 6102, 61021N, (2006). [CrossRef]

8. W. Kuester, “Die Waermeleitfähigkeit thermoplastischer Kunststoffe,” Heat and Mass Transfer 1, 121–128 (1968).

9. B. Zintzen, A. Emmerich, J. Geiger, D. Hoffmann, and P. Loosen, “Effective Cooling for High-Power Fiber Lasers,” Proc. Fourth International WLT-Conference on Lasers in Manufacturing, Munich, (2007).

Parameter	40/700 fiber	20/400 fiber
Core radius	20 µm	10 µm
Pump cladding radius	360 µm	200 µm
Radius coating 1	490 µm	250 µm
Radius coating 2	750 µm	-
Thermal conductivity Coating 1	0.35 W/(mK)	0.24 W/(mK)
Thermal conductivity Coating 2	0.24 W/(mK)	-
R′_tp	0.333 mK/W	0.345 mK/W
R′_tcoat1	0.140 mK/W	0.147 mK/W
R′_tcoat2	0.282 mK/W	-
R′_ttotal	0.755 mK/W	0.492 mK/W
R′_conv (air, 20 m/s)	0.490 mK/W	0.902 mK/W
R′_conv (water, 0.1 m/s)	0.037 mK/W	0.064 mK/W

Parameter	fiber 1	fiber 2
Radius pump cladding	150 µm	270 µm
Bridge length l	50 µm	20 µm
Bridge width b	400 nm	350 nm
Number of bridges n	42	90
R′_tair	3,12 mK/W	0,84 mK/W
R′_tbridge	2,17 mK/W	0,46 mK/W
R′_tac	1,28 mK/W	0,3 mK/W

Parameter	40/700 fiber	20/400 fiber
Core radius	20 µm	10 µm
Pump cladding radius	360 µm	200 µm
Radius coating 1	490 µm	250 µm
Radius coating 2	750 µm	-
Thermal conductivity Coating 1	0.35 W/(mK)	0.24 W/(mK)
Thermal conductivity Coating 2	0.24 W/(mK)	-
R′_tp	0.333 mK/W	0.345 mK/W
R′_tcoat1	0.140 mK/W	0.147 mK/W
R′_tcoat2	0.282 mK/W	-
R′_ttotal	0.755 mK/W	0.492 mK/W
R′_conv (air, 20 m/s)	0.490 mK/W	0.902 mK/W
R′_conv (water, 0.1 m/s)	0.037 mK/W	0.064 mK/W

Parameter	fiber 1	fiber 2
Radius pump cladding	150 µm	270 µm
Bridge length l	50 µm	20 µm
Bridge width b	400 nm	350 nm
Number of bridges n	42	90
R′_tair	3,12 mK/W	0,84 mK/W
R′_tbridge	2,17 mK/W	0,46 mK/W
R′_tac	1,28 mK/W	0,3 mK/W

Heat transport in solid and air-clad fibers for high-power fiber lasers

Abstract

1. Introduction

2. Method

3. Application to Air-Clad Fibers

4. Optimization of the active fiber

5. Packaging of Splices

6. Accuracy of the model

7. Conclusion

Acknowledgements

References and links

Cited By

Figures (3)

Tables (2)

Equations (16)

Optics Express