## Abstract

We report on the investigation of the thermo-optical behavior of air-clad ytterbium-doped large-mode-area photonic crystal fiber lasers. Analytical and numerical models are applied to calculate the heat distribution and induced stresses in a microstructured fiber. The results are compared to conventional double-clad fiber lasers and design guidelines are provided to ensure maximum heat dissipation and scalability to power levels of several kWs.

©2003 Optical Society of America

## 1. Introduction

Rare-earth-doped fibers are a promising gain medium for high power laser radiation with excellent beam quality. This has been demonstrated by several experiments in the continuous-wave [1,2,3] and pulsed regime [4,5]. Recently, a new class of optical fibers based on a wavelength-scale morphological microstructure running down their length has come up [6]. Microstructuring the fiber adds several attractive properties to conventional fibers, e.g. high nonlinearity due to the tight confinement and a shift of zero-dispersion wavelength towards the visible spectral range.

The gain medium of a fiber laser can be constituted by introducing a rare-earth ion doping into the core of the microstructured fiber. Even the large-mode-area core design (low nonlinearity) [7] and the double-clad concept [8] have been transferred to such fibers, with the promising feature of a high numerical aperture of the inner cladding. This is achieved by surrounding the inner cladding with a web of silica bridges which are substantially narrower than the wavelength of the guided radiation. Numerical apertures of up to 0.8 are reported [9]. The air-cladding leads to a second important feature, no radiation has contact with the coating material, which makes these fibers pre-destinated for high power operation.

Recently, the extraction of 80 W output power from an ytterbium-doped air-clad large-mode-area microstructured fiber has been demonstrated [10]. The 2.3 m long fiber laser emits single transverse mode radiation at around 1070 nm with a slope efficiency of 78%. Even at this power level (35 W/m) no thermo-optical distortions are observed. The specifications of this fiber (PCF 1) are summarized in Fig. 1.

This power level could even be scaled up using a fiber (PCF 2) with an increased diameter of the inner cladding (properties are summarized in Fig. 2). The fiber laser is pumped from both sides by two fiber coupled diode lasers emitting at 976 nm. A fiber length of 4 m was sufficient to ensure that the entire launched pump radiation is absorbed. At a launched pump power of 360 W we generated 260 W of output power in a single transverse mode with a slope efficiency of 73%. The output characteristic is shown in Fig. 3. The extracted power per fiber length corresponds to 65 W/m, and is, therefore, in the same order of magnitude as the highest values reported for conventional double-clad fiber lasers. This power level is achieved without any thermo-optical problems, reduction in slope efficiency, beam quality or degradation of the coating.

In this contribution, we investigate the thermal management of an air-clad microstructure fiber laser for the first time. The main drawback of these fibers seems to be that the air-cladding acts as a thermal insulation layer and interrupts the heat dissipation from the inner to the outer cladding. A simple analytical model and the finite-element method (FEM) are used to calculate the temperature distribution in such a fiber. Additionally, thermally induced stresses are investigated. Such an analysis is a prerequisite for the evaluation of the power scalability of these fibers. The analysis leads to the conclusion that the performance of air-clad microstructured fiber lasers is not limited by thermo-optical problems.

## 2. Theory of thermal heat dissipation analysis

Heat is generated in the rare-earth-doped fiber core mainly due to the quantum defect between the pump and laser photons. This heat is transported to the surface of the fiber by thermal conduction through fused silica and the coating material. In addition there is convective and radiative heat flow in the chambers of the air-clad region. Heat dissipation from the coating to the ambient air also involves convective and radiative mechanisms. These processes are illustrated in Fig. 4.

#### 2.1 Heat flow balance at the fiber surface

The generated heat per unit fiber length is balanced with the convective and radiative heat flow from the outer surface into the ambient air. The convective heat flow from a cylinder is given by

where ΔT is the temperature difference, dA is an unit area element and α_{k} is the heat transition coefficient. This coefficient is defined by [10]

where d is the diameter of the cylinder and C_{1} a temperature dependent coefficient. The relevant values for C_{1} are summarized for air and water in Table 1.

The radiative heat flow from the surface of the fiber is given by the well-known Stefan-Boltzmann law

where σ=5.6705·10^{-8} W/(m^{2}·K^{4}) is the Stefan-Boltzmann constant, ε the emission factor (assumed to be 0.95 for silica) and T_{1} - T_{2} is the temperature difference. Therefore, the balance between the thermal load inside the fiber and the convective and radiative heat flow determines the temperature difference between the fiber surface and the ambient air.

#### 2.2 Heat flow balance inside the fiber

The temperature distribution inside the fiber is determined by conductive heat flow in the fused silica parts and convective and radiative heat flow in the air chambers of the air-clad. The conductive heat flow in a hollow cylinder with inner radius R_{1} and outer radius R_{2} is given by

where k is the thermal conductivity,
$\overline{\mathrm{dA}}$
the average cross section of flow, L the distance of the heat flow, d*l* the unit fiber length and ΔT the temperature difference from the inside to the outside of the cylinder. The inner cladding, outer cladding and the coating layers are considered as hollow cylinders. Calculations are carried out using k=1.37 W/(m K) for fused silica and k=0.2 W/(m K) for the coating material (acrylate) [12].

The conductive heat transport in the silica bridges of the air-clad is given by

The convective and radiative heat flows in the air chambers are governed by Eqs. (1) and (3) respectively. Therefore, the temperature difference across the air-cladding is determined by the balance between the thermal load and the conductive, convective and radiative heat flow through the air-cladding.

#### 2.3. Finite-element method (FEM) applied to the thermal behavior of PC fibers

Additionally to the simple (analytical) treatment discussed above, a numerical simulation of the two-dimensional temperature distribution across the fiber cross-section was carried out using finite-element method [13] to solve the stationary heat conduction equation. Heat generation was considered constant across the inner core of the fiber. Convective and radiative heat transports are considered as a boundary condition at the fiber to ambient air interface. In the air-cladding, both mechanisms were accounted for by coupling all surface elements in each cell to an additional free-floating node, which represents the air temperature in the respective cell. Furthermore, the FEM analysis was applied to calculate the thermally induced mechanical stresses inside the photonic crystal fiber.

## 3. Simulated thermal behavior of air-clad microstructured fiber lasers

The thermal behavior of the two ytterbium-doped air-clad microstructured fibers described in section 1 has been investigated using both the simple model and full numerical simulations (FEM).

The first fiber (PCF 1) was operated with an extracted power of 35 W/m which results in a thermal load of approximately 5 W/m. Averaging the thermal load over the fiber length is a valid approximation because the fiber laser is pumped from both sides. Table 2 shows the calculated temperature differences across the specific sections of this fiber at 5 W/m thermal load using the simple model (the temperature of the ambient air is used as the boundary condition). For comparison the temperature distribution in a conventional double-clad fiber (DCF) with identical parameters (replacing the air-clad by fused silica) is also shown in Table 2.

While the temperature differences across the inner and outer claddings and the coating layer are marginal, significant temperature differences of 10.4 K and 93.5 K occur in the air cladding and at the fiber surface respectively. This shows that the main barrier for the heat dissipation is the fiber surface even with the large air-filling ratio of the air-cladding. Therefore, the overall temperature difference between the core and the ambient air does not significantly differ from that in a conventional double-clad fiber. At 5W/m thermal load the convective and radiative heat flows at the fiber surface are 3.76 W/m and 1.24 W/m respectively.

It is important to note that due to the small temperature difference across the air-clad the convective and the radiative heat flows in the chambers of the air-clad are just 0.13 W/m and 0.09 W/m respectively. The conductive heat flow is 4.78 W/m, which is more than 95% of the overall heat flow.

The temperature difference between the core and ambient air/fiber surface as a function of the thermal load for the air-clad fiber (PCF 1) in comparison with that in a conventional double-clad fiber is illustrated in Fig. 5.a and 5.b. These calculations reveal that thermal loads above 15 W/m (extracted power ~100 W/m) produce fiber temperatures approaching the critical value of the coating material of approximately 300°C. However, this is essentially the same for both fiber types. The main difference between both types of fiber is in the temperature drop inside the fiber, shown in Fig. 5.b. Nevertheless, the temperature difference in the inner cladding does not cause a refractive index change that could influence the guiding properties of the core, because the refractive index temperature dependence is approximately 1.2·10^{-6} 1/K.

The thermal behavior derived from the simple model is confirmed by using the FEM. Figure 6 shows the simulated temperature distribution in PCF 1 at 5 W/m thermal load. The results of the FEM are in good agreement with those from the simple model and the heating of the core is calculated to be 111 K and 105 K respectively.

Additionally, the FEM analysis provides the possibility to simulate the thermally induced stresses in the air-clad fiber. Again, we restricted the analysis to a two-dimensional cross section of the fiber. For the mechanical model we chose elements with a “generalized plain strain” formulation which corresponds to homogeneous strain and zero average axial stress over the fiber cross section. This condition corresponds to a freely suspended fiber. Temperature profiles obtained from the thermal analysis (see. Fig. 6) were transferred as loads to the mechanical model.

In the air-clad fiber, stress is caused primarily by the temperature difference between the inner and outer claddings. Because of its higher temperature, the inner cladding tries to expand more than the outer one. This effect causes compressive stress in the bridges through the air cladding as well as compressive and tensile axial stresses in the inner and outer cladding layers respectively. Figure 7 shows the thermally induced distribution of radial stress in the PCF 1 at 5 W/m thermal load. The bending of the silica bridges shown in the figure is strongly exaggerated. The maximum tangential displacement is just 12 nm over a bridge length of approximately 50 µm. In this fiber structure the maximum radial and axial stresses amount to 1.1 MPa and 0.6 MPa respectively. These values are at least one order of magnitude below the critical tensile stress in fused silica which is greater than 10 MPa [12].

The thermal behavior of the second air-clad microstructured fiber (PCF 2) is slightly different due to its different design, in particular the parameters of the air-clad. The extracted power is as high as 65 W/m (see section 1) which corresponds to an average thermal load of 9 W/m. Table 3 shows the calculated temperature differences across the specific sections of this fiber at 9 W/m thermal load in comparison with a conventional double-clad fiber with identical dimensions using the simple model.

Several important results are revealed. The temperature difference across the air-cladding is significantly reduced compared to PCF 1, even at the higher value of thermal load. This is due to the reduced length and increased number of silica bridges in the air-clad region. Therefore, the fraction of radiative and convective to conductive heat flow through the air-cladding is less than 2%. The temperature at the fiber surface is significantly larger (approaching 200°C) due to the higher thermal load and the smaller fiber diameter. As stated in section 1, this value of thermal load is experimentally obtained without any thermo-optical problems, reduction in slope efficiency or beam quality and degradation of the coating.

The behavior is very similar to that of a conventional fiber as illustrated in Fig. 8. The difference in overall fiber temperature is only in the range of 3% at 20 W/m thermal load.

The presented analysis leads to the following design guidelines for air-clad microstructured fibers to ensure maximum heat dissipation. According to Eq. (5) the length of the silica bridges should be as short as possible and the number as high as possible. A tradeoff has to be found regarding the thickness of the bridges. A reduction in thickness leads to a higher numerical aperture of the inner cladding where an increased thickness leads to an improved conductive heat flow.

The diameter of the outer cladding should be as large as possible. According to Eq. (3) the radiative heat flow scales with the diameter d and according to Eqs. (1) and (2) the convective heat flow scales with d^{0.75}. For example, for a 20 W/m thermal load the PCF 2 (outer cladding diameter=360 µm) has a fiber core to ambience temperature difference of 297 K. For an outer cladding diameter of 700 µm this temperature difference would be reduced to 200 K.

If the temperature of the fiber has to be significantly reduced, water cooling could be the solution. According to Eqs. (1) and (2) and Table 1 the heat transition coefficient of water is two orders of magnitude larger than that of air. For example, at 20 W/m thermal load and air cooling the PCF 2 has a fiber core to ambience temperature difference of 297 K, applying water cooling this temperature difference is reduced to 27 K.

## 4. Conclusion

In conclusion, we have investigated the thermal management of air-clad microstructured fibers in comparison with conventional double-clad fibers. The calculations using a simple model and FEM have revealed that the temperature in the fiber core is determined primarily by heat transport through the outer surface of the fiber. If the dimensions of the air-clad are properly designed the temperature profile is comparable to a conventional double-clad fiber. Additionally, the FEM analysis has revealed that thermally induced stresses are not critical for the thermal loads considered. Thus, the air-clad region does not establish a limitation of power scaling capabilities of microstructured fiber lasers if properly designed. The presented analysis leads to the conclusion that air-clad microstructure fibers are likely to be scalable to power levels of several kWs.

## Acknowledgments

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

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