1. Introduction

The incredible power increase of single-mode fiber lasers well above the kW-level over the recent three years has revealed that the rare-earth-doped fiber constitutes a power scalable solid-state laser concept [1,2,3]. The main performance advantages of fiber lasers arise from their outstanding thermo-optical properties. In a fiber, the thermal load is spread over a considerable length resulting in a large ratio of surface to active volume leading to excellent heat dissipation. On the other hand, the beam quality of a fiber laser is mainly given by the refractive index profile of the doped core and is thus independent of the pump or laser power. Therefore, fiber lasers have the reputation that they are almost immune against any thermo-optical problems.

For optically pumped lasers the energy difference between the pump and laser photons, the so-called quantum defect, leads to a thermal load in the pumped region. Even the excellent heat dissipation capabilities of a fiber can not avoid an overall temperature increase and consequently a temperature gradient over the fiber cross section.

The overall temperature increase in a fiber laser potentially causes a fiber coating damage or a reduction of laser efficiency by thermal population of the lower laser level. However, the handling of this overall temperature increase is more or less a technological issue and can be solved by innovative cooling concepts. More critical is the internal temperature gradient, which cannot be avoided even by fiber cooling. This temperature gradient can cause a significant change of the refractive index profile and thus a significant change of guiding properties.

In general, the main performance limitation of fiber laser systems is nonlinearity in the fiber core itself. Over the recent years power and energy scaling of fiber lasers and amplifiers is made possible by so called low-numerical aperture large-mode-area fibers. Such fibers allow for a considerable increase of single-transverse-mode core area by a reduction of the refractive index step constituting the core. The increased mode area, and therefore reduced intensity in the core, lessens the nonlinearity of such a fiber. Besides the nonlinearity, a larger core has a higher damage threshold and better energy storage capabilities.

Rare-earth-doped intrinsically single-mode cores with diameters above 50 µm have been demonstrated [4]. Such fiber cores typically possess numerical apertures in the range of 0.03 to 0.06. As an example, a numerical aperture of 0.03 equals to a refractive index step of ~3·10^-4, the temperature induced index change of fused silica is +11.6·10^-6 per Kelvin [5]. Therefore, a gradient of few ten Kelvin inside the fiber, which is easily achieved in high power operation [6], leads to a temperature induced index change in the range of some percent of the index step itself.

In this contribution, we analyze the consequences of this thermally induced deformation of the refractive index profile for both step-index and photonic crystal fibers. A significant change in guiding properties with increased thermal load is revealed.

2. Basic considerations

The temperature distribution inside a fiber can be described by the well known heat transfer equation. For the cylindrical boundary conditions of a fiber the following analytical expressions for the temperature distribution T(r) in the core (thermal load region) and the cladding can be formulated [7]:

where T_cool is the cooling temperature that is kept constant in the following considerations, Q the induced heat per volume in the fiber, K is the thermal conductivity, h the convection coefficient, r_core and r_clad are the core and cladding radius, respectively. According to the discussion above this temperature distribution can be translated into a “thermal index”. The situation is illustrated in Fig. 1. The temperature profile has a parabolic shape in the core region and a logarithmic decay in the cladding region. Consequently, this leads to an index increase, with a bulge over the core region and a logarithmic distribution over the cladding.

 

Fig. 1. Illustration of the “cold” index profile (a), typical temperature profile according to Eq. (1) and Eq. (2) (b), temperature induced index deformation (c).

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In order to describe the influence of the temperature induced index change on modal properties numerical simulations have to be performed. There are several ways to simulate propagation or calculate eigenvalues of modes in a fiber or waveguide. Because of its simple implementation we have chosen a finite-difference algorithm, to solve the scalar Helmholtz equation as an eigenvalue problem [8]

where Ψ is the field distribution, β² the eigenvalue and ε(x,y) a two-dimensional arbitrary index-profile, so that its possible to describe the thermo-optical problems. Usually, the use of the Helmholtz equation is allowed for the analysis of index profiles with small refractive index changes. This is certainly not given for air-silica photonic crystal fibers, however it has been shown that this simplified approach delivers the very useful results even for photonic crystal fibers [8]. In order to analyze the problem systematically the computation of the index is split. In a first step we are concerned about the fiber geometry and it is implementation, while in a second step the temperature dependent index change is added.

In the following the change of modal properties will be discussed as a function of thermal load in units W/m. Therefore, the results are universally valid for all fiber lasers. As an example, in the case of ytterbium-doped fibers the thermal load is in the range of 6 to 16 % of the absorbed pump power depending on the specific pump and emission wavelength. For other dopants, such as Neodymium, Erbium or Thulium this value is significantly larger due to the larger quantum defect.

3. Definition of V-parameter, NA and cut-off condition in step-index (SIF) and photonic crystal (PCF) fibers

Our calculations make it possible to compare step-index fibers with photonic-crystal fibers with respect to their thermo-optical characteristics. In order to parameterize a definition of a V-parameter and numerical aperture is provided

with ${\mathit{NA}}_{\mathit{SIF}}=\sqrt{n_{\mathit{core}}^2-n_{\mathit{cladding}}^2}$ and ${\mathit{NA}}_{\mathit{PCF}}=\sqrt{n_{\mathit{core}}^2-n_{\mathit{FSM}}^2}$ as numerical aperture.

For conventional step-index fiber theory one can find the definition given above while a definition of a V-parameter for holey fibers is discussed in different ways [9,10]. The parameters for a step-index fiber are the core radius (r_core), the wavelength λ, the core and the cladding index (n_core, n_clad). Differing from this, in a PCF a fundamental space filling mode (FSM) is defined as the cladding mode of a infinitely expanded periodic air-hole structure with no hole missing. The effective core radius aeff is definied as Λ/√3, where Λ is the hole pitch [9]. Of peculiar interest is the cut-off condition for the first higher order mode which is defined as V=2.405 for SIF and PCF as well. However, this V-parameter criterion is not practical anymore considering such deformed refractive index profiles. Calculating the overlap of the intensity distribution of the first higher order mode with the core region (I_core/I_overall) is more useful, indeed this parameter is constant for SIF (=0.33) and PCF (=0.52) at the cut-off condition independent of the actual core design (radius, NA). Therefore, this criterion is used as single-mode to multi-mode transition.

4. Thermo-optical effects in step-index and photonic crystal fibers

A general result of the performed simulations is shown in Fig. 2. The movie shows the characteristic properties of the fundamental mode (LP₀₁-like) and the first higher order mode (LP₁₁-like) with respect to the thermal load in a photonic crystal fiber. On the one hand we are interested in the behaviour of the mode field diameter for the LP₀₁ mode and on the other hand in the change of the overlap of the LP₁₁ mode with the core region. As mentioned this overlap defines whether the higher order mode is a cladding or a guided mode. The initial situation shows an LP₁₁ cladding mode with an overlap of less than 0.15 and a LP₀₁ mode with a mode field diameter of approximately 30µm. For an increasing thermal load the guiding properties of a single-mode fiber are significantly changed. The higher order mode is more and more attracted by the core and finally becomes a guided mode (overlap >0.52). Thus, an originally single-mode fiber has turned into a multi-mode fiber. Meanwhile the mode-field diameter of the LP₀₁ mode is considerable reduced.

 

Fig. 2. Behaviour of the mode-field-diameter of the LP01 mode and the overlap of the LP₁₁ mode in dependence on the thermal load for a photonic crystal fiber.(NA=0.03, V=2.0). [Media 1]

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Figure 3 shows the results of a calculation for a SIF with NA=0.03 and NA=0.04. Illustrated is the overlap of the first higher order mode with the core region (in black) and the change of the mode field diameter (in red) subject to the thermal load. As shown, with increasing thermal load the overlap of the LP₁₁ mode increases (the confinement increases, see also Fig. 2) and reaches the single-mode limit at a certain value. The MFD changes to smaller values with a nearly linear slope. Lowering the V-parameter makes the fiber more insensitive to thermally induced refractive index profile deformations. On the other hand, lowering the V-parameter at a constant NA means also a lower mode-field diameter. As shown in Fig 3(b), for a SIF with NA=0.04 the single-mode limit is reached at significantly higher thermal loads as in the case of NA=0.03. However, at the same V-parameter such a design possesses a significantly smaller MFD and therefore higher nonlinearity.

 

Fig. 3. Simulated overlap of the LP₁₁ mode with the core region (black) and the MFD of the LP01 (red) for a SIF with a NA of 0.03 (a) and NA of 0.04 (b). The dashed line is the single-mode condition described in the text.

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Figure 4 shows the identical calculations for a PCF. For simplicity the calculations are done for a core consisting of one missing air-hole. Basically, similar results are obtained. Thus cores with stronger confinement are less sensitive to a temperature induced change of modal properties. At the same numerical aperture and V-parameter PCFs have a slightly larger mode field diameter and are even slightly less sensitive to thermally induced index deformation.

 

Fig. 4. Simulated overlap of the LP₁₁ mode with the core region (black) and the MFD of the LP₀₁ (red) for a PCF with a NA of 0.03 (a) and NA of 0.04 (b). The dashed line is the single-mode condition described in the text.

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To summarize the results Fig. 5 and 6 show the thermal load at the cut-off condition, i.e., the single-mode to multimode transition, and the average slope of the MFD change as a function of numerical aperture of the fiber core and for various V-parameters. As shown, a fiber with higher NA and lower V-parameter is more resistant to thermal load, the possible guidance of a higher order mode occurs at higher thermal loads. It has to be pointed out that both PCF and SIF have the same NA-dependent behaviour of the thermal load at cut-off, but photonic crystal fibers stay single mode for higher thermal load, the difference is approximately a factor 2. The slope of the mode-field diameter change reduces with higher NA but is nearly independent of the V-parameter of the fiber.

 

Fig. 5. Thermal load at cut-off condition for a SIF (a) and PCF (b) as a function of numerical aperture for different V parameters.

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Fig. 6. Simulated average slope of the MFD change for a SIF (a) and PCF (b) versus NA for different V-parameters.

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5. Discussion and conclusion

To emphasis the relevance of the presented calculations we consider an example of thermal load in a high power ytterbium-doped fiber laser. The extraction of 5 kW laser power from a 10 m long doped-fiber causes a thermal load in the range of 30 W/m to 100W/m depending on the quantum defect, which can be in the range from 6 to 16%. Due to the pump light distribution over the fiber length this value is considerably higher at the pumped fiber end. Therefore, the influence of the internal temperature gradient on the guiding properties of low-numerical aperture large-mode-area fibers has to be considered in the next generation of high power fiber lasers approaching the 10 kW level with diffraction-limited beam quality.

In summary, we have analyzed how thermal load affects guiding properties of step index and photonic crystal fibers. It is revealed that fibers with stronger confinement (higher NA, smaller V-parameter) are less sensitive to thermo-optical problems but have smaller mode-field diameters and therefore higher nonlinearity, lower energy storage capabilities and lower damage thresholds. However, single-mode fibers with extended dimensions are in general characterized by a weak confinement. Thus, if the extraction of high power levels from a fiber laser or amplifier requires a low-NA large-mode-area then a compensation of the herein discussed effects will be necessary.