3.1 Numerical model

The comprehensive numerical model is based on the solution of laser rate equations together with wavelength-resolved propagation equations for power spectral density^[ Reference Grábner, Švejkarová, Aubrecht, Pokorný, Honzátko and Peterka^26]. It takes into account a great deal of physically relevant effects that are important for realistic modeling of high-power fiber lasers, such as amplified spontaneous emission, fiber temperature changes due to self-heating^[ Reference Grábner, Švejkarová, Aubrecht, Pokorný, Honzátko and Peterka^26] based on the analytic model for a temperature radial profile in the fiber^[ Reference Grábner, Peterka and Honzátko^27], temperature-dependent absorption and emission cross-section spectra^[ Reference Jiříčková, Grábner, Jauregui, Aubrecht, Schreiber and Peterka^28, Reference Jiříčková, Švejkar, Grábner, Jauregui, Aubrecht, Schreiber and Peterka^29], intrinsic fiber attenuation and cross-relaxation dependence on concentration.

3.2 Model parameters

High-power TDFL parameters applied in the numerical model are essentially the same as those reported in Ref. [Reference Grábner, Švejkarová, Aubrecht, Pokorný, Honzátko and Peterka26]. While a majority of the parameters, including, for example, spectroscopic parameters of Tm ${}^{3+}$ ions in silica, are known with sufficient accuracy, the parameters determining the exact relationship between fiber temperature and heat load depend on fiber cooling efficiency provided by a cooling system and thus they are problem-specific. In order to demonstrate the concept, the analytical model^[ Reference Grábner, Peterka and Honzátko^27] for temperature radial distribution in a layered medium is used with the following set of parameters: doped core with a radius ${a}_1=10$ μm, silica cladding ${a}_2=200$ μm, polymer coating ${a}_3=300$ μm, thermally conducting paste ${a}_4=500$ μm, aluminum heat sink ${a}_5= 501$ μm. Thermal conductivity values in different layers are ${k}_1={k}_2=1.38$ W m ${}^{-1}$ K ${}^{-1}$ , ${k}_3=0.18$ W m ${}^{-1}$ K ${}^{-1}$ , ${k}_4=0.15$ W m ${}^{-1}$ K ${}^{-1}$ , ${k}_5=238$ W m ${}^{-1}$ K ${}^{-1}$ . The background temperature is set to 23°C. Note that the analytical model uses cylindrical layers; in Section 4.2 we show, for comparison, the temperature distribution for real geometry calculated by a numerical model; see the inset in Figure 9.

3.3 Concentration profiles

Several concentration longitudinal profiles were selected to be analyzed and compared with a constant concentration case. The linear profile is defined as follows:

(5) $$\begin{align}{N}_\mathrm{t}\left(\zeta \right)={\overline{N}}_\mathrm{t}\left( a\zeta +b\right),\kern1em a=2\left(1-b\right), \end{align}$$

where $\zeta =z/L$ is the relative longitudinal coordinate, $a$ , $b$ are mutually dependent parameters and ${\overline{N}}_\mathrm{t}$ is the average concentration along the fiber, ${\overline{N}}_\mathrm{t}=\left(1/L\right){\int}_0^L{N}_\mathrm{t}(z)\mathrm{d}z$ . The step profile is defined as follows:

(6) $$\begin{align}{N}_\mathrm{t}\left(\zeta \right)={\overline{N}}_\mathrm{t}{a}_i,\kern1em {\zeta}_i\le \zeta <{\zeta}_{i+1},\kern1em i=1,2,\dots,\end{align}$$

where ${a}_i$ is the relative concentration in the $i$ th section and ${\zeta}_i$ is the relative coordinate of the $i$ th section beginning. The Gaussian profile is defined as follows:

(7) $$\begin{align}{N}_\mathrm{t}\left(\zeta \right)={\overline{N}}_\mathrm{t}C\exp \left(-\frac{{\left(\zeta -\mu \right)}^2}{2{\sigma}^2}\right),\end{align}$$

where $\mu$ and $\sigma$ are the mean value and standard deviation of the Gaussian profile, respectively, and $C$ is a normalization constant. The tanh profile is defined as follows:

(8) $$\begin{align}{N}_\mathrm{t}\left(\zeta \right)={\overline{N}}_\mathrm{t}\left(a+\left(b-a\right)\frac{1}{2}\left(1+\tanh \left(\sigma \left(\zeta -\mu \right)\right)\right)\right),\end{align}$$

where $a$ and $b$ are the minimum and maximum relative concentration and $\mu$ , $\sigma$ are shifting and scaling parameters, respectively. The inverse distance profile is defined as a modification of (Equation (4)) as follows:

(9) $$\begin{align}{N}_\mathrm{t}\left(\zeta \right)=\frac{1}{\sigma_\mathrm{pa}{\varGamma}_\mathrm{p}}\frac{1}{L^{\prime }-{L}^{\prime }{\zeta}^{\prime }},\end{align}$$

(10) $$\begin{align}{\zeta}^{\prime}=\frac{\zeta }{{\left(1+{\left|\zeta \right|}^p\right)}^{1/p}},\end{align}$$

(11) $$\begin{align}p=\frac{-\ln 2}{\ln \left(1-1/\left({N}_\mathrm{tm}{\sigma}_\mathrm{pa}{\Gamma}_\mathrm{p}{L}^{\prime}\right)\right)},\end{align}$$

where ${L}^{\prime}= cL$ , such that $c\le 1$ is a shorter-than-fiber-length distance, and ${N}_\mathrm{tm}$ is a maximal concentration parameter.

3.4 Temperature versus concentration profiles

In order to demonstrate the influence of the longitudinally inhomogeneous concentration on power and temperature distribution along the active fiber, the cladding-pumped TDFL example is analyzed.

In the analyzed example, the following parameters are assumed: fiber length $L=7$ m, core diameter ${d}_\mathrm{c}=20$ μm, cladding diameter ${d}_\mathrm{cl}=400$ μm, coating diameter ${d}_\mathrm{ct}= 600$ μm (note that ${d}_\mathrm{c}=2{a}_1$ , ${d}_\mathrm{cl}=2{a}_2$ , ${d}_\mathrm{ct}=2{a}_3$ ), numerical aperture (NA) = 0.3, average concentration ${\overline{N}}_\mathrm{t}=2.4\times {10}^{26}$ m ${}^{-3}$ , pump wavelength ${\lambda}_\mathrm{p}=790$ nm, signal wavelength ${\lambda}_\mathrm{s}=2000$ nm, reflectivities ${R}_1=0.99$ , ${R}_2=0.036$ and signal and pump overlap factors ${\varGamma}_\mathrm{s}=0.90$ , ${\varGamma}_\mathrm{p}=0.0025$ . The temperature-dependent cross-section spectra (the Cryo variant^[ Reference Jiříčková, Švejkar, Grábner, Jauregui, Aubrecht, Schreiber and Peterka^29]) are applied.

Figure 2 compares six longitudinal profiles of concentration ${N}_\mathrm{t}(z)$ with the same average concentration ${\overline{N}}_\mathrm{t}=2.4\times {10}^{26}$ m ${}^{-3}$ . The constant concentration profile (Figure 2(a)) leads to the maximum core temperature of 127.7°C, which is located near the beginning of a fiber. The maximum of the forward propagating signal power ${P}_\mathrm{f}\left({\lambda}_\mathrm{s}\right)$ is 592.1 W. Assuming the linear profile of concentration according to Equation (5) with $b=0.7$ (see Figure 2(b)), the maximum core temperature decreased to 105.8°C and its position shifted inside the fiber. The maximal signal power also decreased to 554.9 W. Assuming a step-wise concentration profile according to Equation (6) with parameters ${\zeta}_1=0$ , ${\zeta}_2=0.33$ , ${\zeta}_3=0.66$ , ${a}_1=0.7$ , ${a}_2=1$ , ${a}_3=1.3$ (see Figure 2(c)), the maximal temperature drops to 114.6°C but the discontinuities of temperature appear at the positions of abrupt changes in the concentration profile. Assuming the Gaussian profile according to Equation (7) with parameters $\mu =0.7$ , $\sigma =0.4$ (see Figure 2(d)), the maximal temperature is 114°C with a maximal signal power of 561 W. Assuming the tanh profile according to Equation (8) with parameters $a=0.25$ , $b=1.3$ , $\mu =0.3$ , $\sigma =6$ (Figure 2(e)), the maximal core temperature is 121.1°C with a maximal signal power of 562.9 W. The inverse distance profile according to Equation (11) with parameters $c=0.55$ , ${N}_\mathrm{tm}=5\times {10}^{26}$ (see Figure 2(f)), exhibits the greatest drop of maximum temperature to 98.9°C but clearly at the expense of the output power since maximal signal power is ${P}_\mathrm{f}\left({\lambda}_\mathrm{s}\right)=516.4$ W. Note that laser output power is ${P}_\mathrm{sout}={P}_\mathrm{f}\left({\lambda}_\mathrm{s},z=L\right)\left(1-{R}_2\right)$ .

Figure 2 Power and temperature distribution along the TDFL (pump power ${P}_\mathrm{p}=1000$ W): (a) constant concentration, (b) linear profile, (c) step profile, (d) Gaussian profile, (e) tanh profile and (f) inverse distance profile. All profiles are with the same average concentration ${\overline{N}}_\mathrm{t}=2.4\times {10}^{26}$ m ${}^{-3}$ ( $\sim$ 10,900 mol ppm). Notes: numerical values of heat load $Q$ [W/m] are on the temperature axis; ${P}_\mathrm{f}\left({\lambda}_\mathrm{p}\right)$ is the forward propagating pump power, ${P}_\mathrm{f}\left({\lambda}_\mathrm{s}\right)$ is the forward propagating signal power and ${P}_{\mathrm{b}}\left({\lambda}_\mathrm{s}\right)$ is the backward propagating signal power.

The relation between the laser output signal power and maximum core temperature along the fiber is depicted in Figure 3 where the profiles with the same average concentration are compared. The output signal power values were achieved by applying pump power from 200 to 2000 W in 100 W steps (see the dots). Clearly, the maximal fiber temperature limits the achievable output power, or in other words, the particular output power leads to different fiber temperatures with different concentration profiles. Note that the temperature differences are less pronounced in the case of coating temperature.

Figure 3 Maximal core temperature versus laser output signal power (for pump power ${P}_\mathrm{p}$ = (200:100:2000) W for different concentration profiles with the same average concentration ${\overline{N}}_\mathrm{t}=2.4\times {10}^{26}$ m ${}^{-3}$ ( $\sim$ 10,900 mol ppm) (circles) and for inverse distance and constant profiles with ${\overline{N}}_\mathrm{t}=2.9\times {10}^{26}$ m ${}^{-3}$ ( $\sim$ 13,100 mol ppm) (squares).

It is evident from the presented examples and Figure 3 that reducing maximal temperature is achieved only at the expense of efficiency (i.e., the ratio of output power to pump power). However, one can achieve the same output power with the same pumping power and still significantly reduce a temperature by using a higher average concentration. The inverse distance profile with parameters $c=0.4$ , ${N}_\mathrm{tm}=5\times {10}^{26}$ and average concentration ${\overline{N}}_\mathrm{t}=2.9\times {10}^{26}$ m ${}^{-3}$ is analyzed in Figure 4. The same maximal signal power is achieved as in the case of constant concentration ${\overline{N}}_\mathrm{t}=2.4\times {10}^{26}$ m ${}^{-3}$ , but here the maximal core temperature is reduced to 106.4°C, which is more than 20°C less than in the case of constant concentration (with the same output signal power). This inverse distance profile with higher average concentration is also shown in Figure 3 (see the square dots), to achieve a significant reduction of temperature with the same or slightly better efficiency in comparison with the case of a constant profile.

Figure 4 Power and temperature distribution along the TDFL (pump power ${P}_\mathrm{p}=1000$ W); inverse distance profile with the average concentration ${\overline{N}}_\mathrm{t}=2.9\times {10}^{26}$ m ${}^{-3}$ ( $\sim$ 13,100 mol ppm).

Note that a constant concentration profile with an increased average concentration would provide even higher output power but also significantly higher maximal heat load and temperature. Therefore, using a suitable inhomogeneous profile, one can set a better trade-off between power and temperature, or in other words, a higher output power can be achieved under the requirement that the prescribed maximum temperature limit is not to be exceeded.

For example, let us assume that the coating temperature ${t}_\mathrm{ct}$ has to be kept lower than 81°C. Taking this limit into account, the maximum signal power 574.4 W can be achieved with maximum core temperature ${t}_\mathrm{c}=106.9$ °C, maximum coating temperature ${t}_\mathrm{ct}=80.9$ °C for lower constant concentration ${N}_\mathrm{t}=2.05\times {10}^{26}$ m ${}^{-3}$ and increased fiber length $L=12$ m. Note that the output signal power is lower than in Figure 4 where the same temperature limits are fulfilled by using an inverse distance profile on a 7 m long fiber.

The numerical model of the fiber laser respects nonuniform pump absorption due to nonuniform population inversion along the fiber but assumes a homogeneous overlap factor of the pump field with the doped area. However, simulations of electromagnetic field propagation^[ Reference Grábner, Nithyanandan, Peterka, Koška, Jasim and Honzátko^30] in double-clad active fibers show that the overlap and resulting pump absorption profile are also dependent on the spatial distribution of the excitation field and on the cladding geometry.

4.1 Methodology

An in-house prepared double-clad fiber (SG1647) with inhomogeneous thulium concentration was used for the experiments. That allowed the selection of various sections with different concentrations with a minimal variation in the refractive index profile, thus reducing the splice losses between different parts of the fiber. However, it is essential to clarify that this inhomogeneity was not localized in the short fiber segments. The fiber was processed to a quasi-octagonal shape with core, pedestal and cladding (flat to flat) dimensions of 12, 31 and 132 μm, respectively. During the experiment, the fiber was placed on the aluminum plate and fixed using Kapton tape at several spots. This arrangement allowed for optimal conditions to observe heat distribution.

In the initial stage, the individual segments with Tm ${}^{3+}$ concentrations of 6900 (A), 11,800 (B), 13,700 (C) and 16,000 (D) molar ppm were measured separately. The concentration was calculated based on the measured cladding absorption and the cladding/core radius ratio. In a 100%–4% setup, shown in Figure 5, the resonator was formed by a high-reflectivity (HR) fiber Bragg grating (FBG), centered around 1940 nm (AFR/TeraXion), along with a perpendicularly cleaved end. The pump was provided by a laser diode (LD) operating at 792 nm (Aerodiode/BWT) protected from 2 μm radiation by fiber with high OH concentration (Hi OH, Thorlabs FG105UCA). Individual segments were measured up to a pump power of 30 W. A passive fiber (Nufern SM-GDF 10/130 μm) was utilized to minimize mode-field diameter mismatch between the active fiber and the various FBGs. The unabsorbed pump power was separated by a silicon filter or dichroic mirror. Using cut-back measurement, the maximal slope efficiency with respect to pump power was determined.

Figure 5 Experimental setup of the TDFL.

In the next step, two two-segmented lasers were prepared to compare segmented and uniformly doped active fibers in the resonator. Both had the same fiber length and similar average concentration and they contained a splice in between to minimize their influence. These splices were created using an optimized program on the splicer (Fitel S185) providing loss estimation based on image analysis as well. In addition, we tested three- and four-segmented lasers to demonstrate the character of the step-profile laser. All TDFLs were tested using the same setup as for the individual segments. During the characterization, the spectra were measured with an optical spectrum analyzer (Yokogawa AQ6375), and the surface temperature was monitored using a thermal camera (Micro-epsilon TIM40).

4.2 Results

Table 1 summarizes the slope efficiencies for individual segments. According to these results, segmented TDFLs were formed by splicing together individual segments. The particular compositions of the segmented TDFLs are listed in Table 2.

Table 1 Slope efficiency for individual segments pumped up to 30 W.

Table 2 Composition of formed segmented TDFLs; all units are in meters.

Figure 6 shows the performance of two-segmented TDFLs. The segmented laser’s slope efficiency was 65.3%, consistent with the uniformly doped one. However the two-segmented uniform TDFL did not last out the pump power above 40 W. The thermal camera image in Figure 7 illustrates well the fundamental principles of multi-segment laser operation. The temperature label indicates the highest value in the marked area. At the same pump power, the heat distribution along the segmented fiber was more even, allowing the fiber to handle higher pump powers.

Figure 6 Laser performance comparison of two-segmented and uniformly doped fiber. Both TDFLs have similar average Tm concentration and fiber length.

Figure 7 Thermal image of the (a) two-segmented TDFL and (b) two-segmented uniform TDFL under the same pump powers. The temperature of 110°C was reached just before the splice failure.

Additional segments were added to the laser configuration for further experiments. The three-segmented laser demonstrated performance reaching a slope efficiency of 62%, as shown in Figure 8(a). The thermal camera image in Figure 8(b) indicates a notably improved heat distribution. The image also shows that the splice between the first and second segments (S1-S2) was warmer than the splice where the pump power enters the first segment of the active fiber (FBG-S1). This observation suggests that the lengths and concentrations of the individual active fibers can be designed more effectively.

Figure 8 (a) Performance of the three-segmented TDFL. (b) Thermal camera image at a pump power of 28 W.

The temperature at the outer coating along the fiber determined from the thermal camera image is compared with the simulated fiber temperature in Figure 9. The temperature was calculated from the heat load $Q$ using the analytical model^[ Reference Grábner, Peterka and Honzátko^27] calibrated by Comsol heat transfer simulations of an active fiber placed on the cooled plate; see the inset of Figure 9. The temperature peaks were modeled by introducing ‘splice losses’ ${L}_\mathrm{s}=$ 0.02, 0.025 and 0.012 dB/cm at locations $z=$ 0, 55 and 162 cm along the fiber, respectively, based on the experimental observations. It is evident that in the first 50 cm of the fiber where $Q(z)\sim 4$ W/m, the temperature was significantly lower compared with the case of uniform fiber with the same average concentration (see dashed lines).

Figure 9 Measured (crosses) and simulated (lines) fiber temperature of the three-segmented TDFL at a pump power of 28 W versus uniform fiber with the same average concentration (dashed lines). Inset: simulated temperature distribution of active fiber laying on a cooling desk under core heat load $Q=20$ W/m.

Based on these findings, an attempt was made to further improve laser performance by integrating a fourth segment. In particular, segment B was placed at the second position to achieve a more gradual increase of Tm ion concentration while maintaining approximately the same resonator length and same total averaged concentration. It reduced the temperature of the splice S1-S2 (compare Figures 8 and 10) and segment C could be shortened, as the last splice was not at higher risk of heat damage. The laser exhibited a stable output power of 54 W and even greater improvements in slope efficiency exceeding 64% with the pump power up to 30 W and 62% for the high power test, shown in Figure 10. Furthermore, the threshold was lowered compared to individual segments with higher concentrations. See Figure 11 for a summary of the slope efficiency and threshold power.

Figure 10 (a) Performance of the four-segmented fiber laser. (b) Thermal camera image at a pump power of 33 W.

Figure 11 Summary of the measurements at pump power of up to 30 W.