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Abstract
With both radiation effects and thermal effects taken into consideration, a multiphysics thermal model concerning high-power Yb-doped fiber lasers operated with post-irradiated active fibers is established. Radiation-related parameters, including propagation losses, refractive indexes and lifetime, are considered. And, with the temperature profile of the active fiber, temperature-dependent parameters, including absorption and emission cross-sections, refractive indexes and lifetime, are updated every loop to simulate the output parameters. Simulation results show that radiation induces great changes to the thermal profiles of the active fiber. And severe performance degradation of high-power Yb-doped fiber lasers are recorded, featuring a remarkable drop in output power and an even steeper decline in the transverse mode instability threshold, which is a predominant limitation at high radiation doses. With a deposited radiation of 100 Gy, an output decline of about 50% and a mode instability threshold drop over 85% are observed. And it’s shown that, with the exploited active fiber, it’s hardly possible for the investigated fiber laser to generate stable single-mode output at kilowatt levels with accumulated radiation doses beyond 50 Gy. At low radiation doses within 20 Gy, to maintain safe and stable single-mode operation of the laser system, longer active fibers with lower absorption coefficients are preferred despite a small rollover of the output power.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Boasting unmatchable merits including compact package, simple construction, excellent beam quality and high conversion efficiency, fiber lasers have enjoyed a rapid development for decades with high-power Yb-doped fiber lasers (YDFLs) as the most eye-catching representative [1–3]. Along with the expeditious advance in fiber laser technology, their applications are expanding from conventional fields to nuclear plants, accelerators and space, where they would face with adverse radiation environments.
Extensive studies have demonstrated that, in radioactive conditions, active fiber is the most sensitive section of the laser system [4–10]. Generally, there are three major macroscopic effects for radiation-silica interactions, namely radiation induced attenuation (RIA), radiation induced compaction (RIC) and radiation induced emission (RIE). And, it was found that different types of radiation sources, including gamma-ray, X-ray, electrons, protons and neutrons, cause comparable effects at the same radiation dose in silicate fibers [10–12]. In high-power fiber laser systems, RIA and RIC are the two main factors that cannot be neglected as they are closely related to the propagation losses and signal overlapping factor of the active fiber. RIA corresponds to an increase of extra propagation loss in the fiber. Experiments show that significant reduction in the optical transmission is induced by radiation in active fibers due to Al and P co-doping [8–10,13–16]. Consequently, this extra loss in transmission would pose negative effects on the performance of fiber systems, especially for high-power fiber systems. RIC corresponds to refractive index (RI) change of the irradiated fiber [9,13,17–19]. And, researches show that the core and the cladding experience different degrees of RI variation [19] which would alter the overlapping factor of the propagating laser signal. Besides RIA and RIC, the lifetime of doped rare-earth ions is also a radiation sensitive factor that needs to be considered. And, strong decrease is recorded for the lifetime of Yb ions under different radiation conditions [12,20].
Practically, in most applications, before the laser system is ready for service at the designated spot, it has already been exposed to a certain level of radiation. Thus, it’s of vital significance to investigate the performance of fiber systems with post-irradiated active fibers, which is usually denoted as the OFF configuration. Up to now, radiation effects on the performance of low and moderate power level (below 50 W) fiber systems have been investigated intensively, especially for Er/Yb codoped fiber amplifiers (EYDFAs) [21–28]. Experiments show that, exploiting standard Er/Yb codoped fibers (EYDFs) without radiation-hardening or radiation-resistant treatment, typical low power EYDFAs within 25 dBm show a significant gain decrease over 70% at ∼400 Gy [22–23], while, for watt level amplifiers, the decrease is only about 30% [24–25]. Even better performance was reported by A. Ladaci for 20 W amplifiers in 2018, where a gain drop smaller than 30% was recorded at 1100 Gy, representing the current state of the art for related researches [28]. Similar degradation was also reported in low power Tm-doped fiber lasers with active fibers exposed to gamma radiation. And, an output power decline higher than 50% was recorded at an accumulated dose of 500 Gy [29–30].
Recently, L. Mescia and C. Campanella developed a multiphysics model to investigate and analyze the performance of EYDFAs in radiation environments [31–32]. This novel model takes comprehensive considerations of the thermal effects in fiber systems, and, provides a useful tool in the design, analysis and optimization of EYDFAs. Modeling shows that extra thermal effects caused by space radiation would strongly degrade the performance of fiber amplifier systems.
However, for high-power Yb-doped fiber (YDF) systems, up to now, only a few reports have been published [33–34]. In 2018, the impact of radiation on the transverse mode instability (TMI) of Yb-doped fiber amplifiers was investigated by Y. S. Chen [33]. After exposing the YDF to 100 Gy gamma radiation, the maximal stable laser output dropped from 367.6 W to 170.5 W, while the laser efficiency shrank from 74.7% to 41.4%. And, at radiation doses over 255 Gy, the optical-optical efficiency keeps below 25%, demonstrating lethal performance deterioration. Similar phenomenon was also reported by Y. Wang in kW level YDFLs based on homemade Yb-doped aluminophosphosilicate fiber and commercial YDF [34].
In this report, we focus on the theoretical investigation on the performance of kilowatt level high-power YDFLs operating with post-irradiated active fibers. A multiphysics thermal model is developed with various radiation dependent and thermal dependent input parameters introduced. Simulations indicate significant deterioration in the output power, efficiency and TMI threshold of high-power YDFLs. And, comparisons with published experimental data show high consistence regarding the operation efficiency. At low radiation levels, laser systems with low absorption active fibers would be a better option to keep a relatively high TMI threshold.
2. Theoretical model
2.1 Conventional model
The conventional model is a versatile model for continuous wave (CW), strongly pumped fiber lasers developed by I. Kelson and A. A. Hardy [35–36], which has been widely validated and exploited for the investigation of CW fiber lasers [37–40]. With this model, the output characteristics of the fiber laser and the optical power distribution inside of the active fiber can be obtained by solving three groups of equations, namely the rate equation, the propagation equation and the boundary equation.
In the conventional model, the overlapping factors, the cross-sections and the lifetime are all set to be constant and uniform along the whole active fiber. Thus, the rate equation can be written as [35–36]
For CW operation, we have $\frac{{d{N_1}(z)}}{{dt}} = 0$. Then, ${N_1}(z)$ can be attained.
In the active fiber, $P_p^ \pm$ and $P_s^ \pm$ are confined by the following propagation equation [35–36]
And, the boundary conditions for the propagation equation are
Equations (1)–(3) can be easily resolved with finite difference method by dividing the active fiber equally into M slices.
2.2 Thermal and radiation effects
The conventional model provides a convenient way for the design and optimization of CW fiber lasers. However, for high-power fiber systems, thermal effects should be considered as the temperature varies along the active fiber due to an inhomogeneous optical power distribution. In fact, major physical parameters including the absorption and emission cross-sections, the signal overlapping factor and the lifetime of the Yb ions, are all temperature dependent.
More importantly, for systems with irradiated active fibers, radiation effects also play an important role. Radiation not only induces extra propagation losses to the pump and laser signal through RIA, which would alter the optical power profile inside the active fiber, but also affects the signal overlapping factor via RIC. Besides, the lifetime of the Yb ions is also sensitive to radiation. Thus, the conventional model should be modified for the investigation of high-power fiber laser systems with irradiated active fibers.
2.2.1 Heat load and heat conduction
In the active fiber, the heat load Q (W/m) contains two terms, namely the quantum defect term and the propagation loss term. The first term is originated from the energy gap between the pump and the laser signal photons. And, this part of energy is lost via fast non-radiative decay processes which result in a temperature increase of the active fiber. With a quantum defect ratio of $1 - \frac{{{\lambda _p}}}{{{\lambda _s}}}$, the quantum defect term ${Q_{QD}}$ can be expressed as
As for the second term, both the pump and laser signal make contributions to the propagation loss. However, as the RIA is mainly generated in the doped area, only propagation losses in the fiber core should be counted. Thus, the propagation loss term ${Q_{PL}}$ can be written as
Then, the overall heat load becomes
With the heat load distribution in Eq. (6), the temperature profile of the active fiber can be derived through the heat conduction equation. Concerning the heat conduction of the active fiber, following assumptions are made: 1) the fiber is actively cooled and the surface temperature of the fiber is constant at a coolant temperature ${T_c}$; 2) only the fiber core is active for heat generating. As the active fiber is divided into M equal slices in the simulation, for a single small fiber segment at position ${z_0}$, the heat load $Q({z_0})$ can be taken as uniform, representing the overall heat load of the whole segment. Thus, at steady-state, for a single fiber segment, the heat conduction equation for the double-clad fiber can be written as [41–44]
Radially, the temperatures and their derivatives must be continuous across the boundaries. And, at the very center of the fiber core, the derivative of the temperature should be finite. Thus, we have [41]
Solving Eqs. (7) and (8), the radial temperature distribution $T(r)$ of the fiber at position ${z_0}$ can be obtained [41–42]
Repeating this process throughout the whole active fiber with the heat load distribution $Q(z)$, the radial and longitudinal distribution of the temperature $T(r,z)$ can be attained. An, the whole process is like doing a tomography to the fiber to achieve the cylindrical temperature profile.
2.2.2 RI and overlapping factor
For a Gaussian beam propagating in an optical fiber, the overlapping factor of the laser signal ${\Gamma _s}$ is expressed as [45]
where w is the beam width parameter. For step-index fibers, w can be resolved with the standard waveguide equation.RI is related to both the temperature through thermo-optic effect and the radiation through RIC. When taking thermal and radiation effects into consideration, the uneven temperature distribution inside the active fiber and the radiation would certainly lead to a change to the index profile of the fiber.
The temperature variation induced RI change $\Delta n(T)$ can be expressed as [42–44]
The radiation induced RI change $\Delta n(D)$ can be derived from the first order defect generation kinetic model as [19,46]
where ${I_s}$ and ${D_s}$ are the maximal RI change and the related saturating dose. Within several kGy level, $\Delta n(D)$ can be simplified by a linear expression [17–19] where $\kappa ^{\prime}$ is the RIC coefficient.Thus, the index profile of the irradiated and heated step-index fiber becomes
Since the temperature rise holds a parabolic expression at the radial direction, as given in Eq. (9), the RI profile becomes parabolically modified. Comparison between a typical standard step-index profile and the parabolically modified profile is shown in Fig. 1. As can be found, in high-power fiber lasers, due to the thermo-optic effects, the step-index profile show graded distribution in the fiber core and the inner cladding separately. And, this variation makes the calculation of w a big challenge.
Fig. 1. Comparison between typical standard step-index profile and the parabolically modified RI profile in high-power fiber lasers.
Fortunately, a novel perturbation analysis technique proposed by L. Dong provides a convenient way to solve this issue [44,47]. The relatively small variation in RI distribution depicted in Fig. 1 can be taken as a small perturbation to the standard waveguide equation. Thus, we have
For Gaussian approximation, the relationship between the original beam width ${w_0}$ and the perturbed one w can be derived through waveguide simulation with Eq. (15). Thus, the modified overlapping factor of the laser signal along the fiber, ${\Gamma _s}(D,z)$, can be obtained.
2.2.3 Cross-sections and lifetime
Absorption and emission cross-sections of YDFs are also temperature dependent parameters [48–51]. For the two energy levels of Yb ions involved, the excited energy level, 2F5/2, contains three Stark sublevels, while the ground energy level, 2F7/2, holds four Stark sublevels. And, the absorption and emission processes involve all relevant subtransitions between 2F5/2 and 2F7/2. As the populations of each sublevels are governed by the Boltzmann distribution, at high temperatures, a redistribution of population among different energy levels occurs, which will surely change the absorption and emission characteristics of the fiber. Generally, as the temperature rises, both the absorption at 976 nm and the emission at 1070 nm will drop due to a reduction of populations at the lowest Stark sublevels of 2F5/2 and 2F7/2.
At the same time, the lifetimes of these energy levels also vary against the temperature. Taking the decay of Stark sublevels into consideration, the lifetime of the 2F5/2 energy level can be modeled as [48,52]
Besides, radiation also affects the lifetime of rare earth ions. Experiments in [12] and [20] show that the lifetime of the 2F5/2 energy level of Yb ions presents a power law relationship against the radiation dose within a dose range of 105 Gy. Thus, for a given radiation dose D, the lifetime $\tau (D)$ can be written as
where $\gamma$ is the radiation-lifetime coefficient of Yb ions. ${\tau _0}$ is the initial lifetime of Yb ions before irradiation. Combining Eqs. (16) and (17), we haveAt last, with the aforementioned thermal and radiation effects taken into consideration, a modified multiphysics thermal model is established.
2.3 Numerical algorithm
The algorithm exploited to carry out this modeling is depicted in Fig. 2. As can be found, the whole simulation process is linked by an iterative loop concerning the calculation of the optical powers. Firstly, with a given radiation dose, input parameters including RI profile, overlapping factor, propagation losses, absorption/emission cross-sections and lifetime are derived at room temperature. And, together with a given pump power, fiber laser simulation is carried out by solving the rate equation and the propagation equation under certain boundary conditions. Here, all input parameters are taken as constant along the whole active fiber. Then, primary power distributions along the fiber, $P_p^ \pm (D,z)$ and $P_s^ \pm (D,z)$, and the output power ${P_{out}}(D)$ of the fiber laser can be obtained. Secondly, temperature profile $T(D,r,z)$ of the active fiber is generated through thermal simulation by solving the heat conduction equation. As absorption/emission cross-sections, signal overlapping factor and lifetime are temperature dependent and vary along the fiber, these input parameters should be updated with the newly attained temperature profile. Thus, an iteration process is introduced to make sure that these temperature and position dependent parameters are immediately updated every time a new temperature profile is derived. And this iteration loop would not break to the end until the system reaches an “equilibrium state” where the calculated output powers of two adjacent iterations show limited relative difference no bigger than $\delta $, which is set to be 1×10−4 in the numerical modeling. Optical powers at different radiation levels are obtained by changing the input radiation dose.
Fig. 2. Diagram of the numerical algorithm. Data in the dashed box is the major input parameters of the model.
It should be pointed out that, in the determination of cross-sections and lifetime, the mean temperature across the fiber core at position z is taken as the effective temperature. And, in the derivation of the overlapping factor ${\Gamma _s}(D,T,z)$, the cylindrical temperature profile $T(D,r,z)$ is used to give a cylindrical RI profile $n(D,r,z)$.
2.4 TMI threshold
Another major factor directly related to thermal effects in high-power fiber lasers is the TMI threshold [53–54], which can also be simulated with this modified thermal model.
It has been proven that it is feasible to estimate the TMI threshold of widely different high-power fiber systems with a constant value of average heat load along the active fiber. Comparisons between simulations and experiments give a constant threshold average heat load of ${Q_{th}} = 34\textrm{ }W/m$ which is adequate for most fiber systems [54]. With Eq. (6), the overall heat load of a fiber laser system can be derived as
However, it should be noted that, here, in the calculation of the average heat load along the active fiber, only fiber sections contributing to the overall heat load could be considered. In practical simulation, only fiber sections with a pump power bigger than 1% of the total launched pump are taken as effective. Thus, the average heat load can be written as
where ${L_{eff}}$ is the effective active fiber length.For the simulation of the TMI threshold, another iterative loop is build. Inside the TMI threshold loop, the launched pump power increases every loop to test whether the generated average heat load at the “equilibrium state” meets the threshold requirement. And, the increase step of the pump is set to be 1 W in the simulation.
3. Experiments and simulation results
3.1 Parameters used in the simulation
To make the input parameters of this model more reliable, major parameters including the radiation dependent propagation losses and the temperature dependent cross-sections of the active fiber are obtained directly through experiments.
3.1.1 Propagation losses at different radiation doses
On-line measurement is conducted to investigate the propagation loss of the exploited YDF at 1070 nm as a function of radiation dose. The measurement was taken with a 15 m long YDF exposed to 60Co radiation at room temperature with a radiation dose rate of 0.5 Gy/s. As shown in Fig. 3, the propagation loss increases monotonously with the rise of the accumulated radiation dose due to RIA. And, the inset shows that, within 200 Gy, the RIA exhibits a linear relationship with the radiation dose in the log-log scale, validating the power law relationship between RIA and the deposited radiation [4–10]. However, beyond 200 Gy, the RIA saturates with a smaller slope.
Fig. 3. Measured propagation loss at 1070 nm as a function of radiation doses. Inset shows the propagation loss in the log-log scale.
In the simulation, the propagation loss for the pump is set as equal to that at 1070 nm considering the close location of the pump wavelength and the lasing wavelength. However, it should be noted that, this measured propagation loss at room temperature are only applicable for the simulation of the OFF configuration operation, as in the ON configuration, where the active fiber is pumped during the whole irradiation process, the combination of radiation and temperature leads to complicated and unpredictable responses of the active fiber [28,55].
3.1.2 Cross-sections at different temperatures
Absorption and emission cross-sections are key parameters affecting the performance of high-power fiber lasers. To obtain these parameters, experiments are carried out with a section of YDF heated in an incubator. And, the absorption data is acquired at different temperatures with a white light source, while the corresponding emission cross-sections are derived by the McCumber theory [56–58]. The cross-sections at 976 nm are illustrated in Fig. 4(a). As can be found, with the rise of the temperature, the absorption and emission cross-sections at 976 nm both drop monotonously. With a total temperature rise of ∼ 370 K, the absorption cross-section at 976 nm shrinks about 20%, while the emission cross-section experienced a slightly bigger decline of about 28%. These values are comparable with the reported results in [58] for single mode aluminosilicate YDFs where a temperature rise of 200 K leads to a cross-section fall of over 35%.
Cross-sections at 1070 nm are depicted in Fig. 4(b). Different from that at 976 nm, the absorption cross-section at 1070 nm shows a proportional relationship against the temperature with a small rise of about 10% from room temperature to about 670 K. What’s astonishing is that, the emission cross-section at 1070 nm witnessed a remarkable decline of over 90% when the fiber is heated over 600 K, which would definitely pose a severe impact on the performance of the laser system at high temperatures. Similar degradation was also recorded in phosphate glass, where a temperature rise from 293 K to 403 K leads to an emission cross-section fall of over 70% around 1050 nm [59]. Comparisons with the cross-sections at 976 nm show that the emission cross-section at 1070 nm is the most sensitive to temperature rises.
3.1.3 Other parameters
The RIs of the fiber core and the cladding are measured through experiments. And, the thermo-optic coefficient of pure silica is taken for the fiber cladding. As for the doped fiber core, the thermo-optic coefficient ${\kappa _{co}}$ can be derived with the additivity model described in [60] and [61]. Here, the exploited YDF is an aluminosilicate fiber. And, the fiber core mainly contains two compounds, namely SiO2 and Al2O3. As the thermo-optic coefficient of Al2O3 is very close to SiO2, the effective thermo-optic coefficient of the doped fiber core is set to be the same with that of silica [60–61].
Other parameters exploited in the simulation are taken from published literatures, as listed in Table 1.
Table 1. Physical parameters exploited in the simulation.
3.2 Results and discussion
In the simulation, 15 m long 20/400 double-clad YDF is exploited with a core NA of 0.075 and an inner cladding absorption coefficient of 1.34 dB at 976 nm. The active fiber is divided into 3000 equal segments to ensure a proper convergence. And, 2 kW 976 nm pump is launched into the cavity with a forward pump configuration. Water cooling with a heat convective coefficient H of 1000 W/(m2·K) is employed for heat management of the laser system unless otherwise stated.
3.2.1 Thermal profiles
First, thermal effects of this laser system are investigated. Figure 5 depicts the temperature map of the fiber core at $r = 0$ and the heat load distribution at different radiation doses. As can be found in Fig. 5(a), the maximal temperature rise of the fiber core at 500 Gy is over 100 K. The corresponded temperature rise of the outer cladding surface is about 48 K which is within the safe operating range of the coating [62], demonstrating that a heat convective coefficient H of 1000 W/(m2·K) is sufficient for this laser system.
Fig. 5. Fiber core temperature map and the heat load distribution at different radiation doses: (a) temperature profile; (b) heat load distribution.
Besides, it can be seen that, the temperature of the fiber core changes greatly along with the accumulation of the radiation dose. At low radiation doses within 50 Gy, the temperature drops monotonously along the active fiber, while, at higher radiation doses, the temperature peak shifts to about 5 m away from the pump facet. The physical origin for the shift of the temperature peak is the variation of the heat load distribution inside of the fiber core at high radiation doses.
Figure 5(b) gives the detailed distributions of the quantum effect induced heat load ${Q_{QD}}$, propagation loss induced heat load ${Q_{PL}}$ and the overall heat load Q at three different radiation doses. As can be found, at low radiation doses within 50 Gy, the overall heat load distribution is mainly determined by ${Q_{QD}}$. And, the pump facet of the fiber bears the highest heat load. With the further rise of the radiation dose, ${Q_{QD}}$ drops moderately with a relatively small variation in distribution, while, ${Q_{PL}}$ rises remarkably due to RIA, making ${Q_{PL}}$ the dominant factor determining the overall heat load distribution of the fiber. As a result, beyond 50 Gy, the peak of Q shifts away from the facet along with the peak shift of ${Q_{PL}}$, which is caused by the optical power distribution inside the fiber core, as given in Eq. (5). Thus, it can be concluded that the shift of the temperature peak at high radiation doses should be attributed to the variation of the overall optical power distribution inside of the fiber core.
With the temperature profile of the fiber core, the cross-section profiles can be derived through the experimentally recorded cross-section versus temperature results in Fig. 4. And, the obtained cross-section profiles are shown in Fig. 6. As can be found, the absorption cross-section at 976 nm and the emission cross-section at 1070 nm show similar distribution characteristics since they both present a reciprocal relationship with the rise of the temperature. The maximal variation of the absorption cross-section at 976 nm is about 12%, while that of the emission cross-section at 1070 nm reaches 65%.
Fig. 6. Cross-section maps under different radiation doses: (a) absorption cross-section profile at 976 nm; (b) emission cross-section profile at 1070 nm.
Figure 7 illustrates the profile of the RI change inside the active fiber under different radiation doses. It should be noted that the change is induced not only by the thermo-optic effect but also RIC. In the active fiber, the maximal RI change, which is recorded in the fiber core, increases from 0.54×10−3 at 0 Gy to 1.10×10−3 at 500 Gy. And, similar to the core temperature profile, for radiation doses higher than 50 Gy, the position of the maximal RI change moves from the pump facet to about 5 m away from the pump end.
Fig. 7. Profile of the RI change inside the active fiber under different radiation doses: (a) 0 Gy; (b) 200 Gy; (c) 500 Gy.
With the RI profile, the perturbed waveguide equation can be resolved. And, the effective mode field area (MFA) of the laser signal can be attained, as shown in Fig. 8. Comparisons with the temperature map in Fig. 5(a) show that the MFA shrinks at high temperatures, as the RI increases with the temperature. And, this phenomenon is known as the thermal lensing effect in fibers [44,47]. At room temperature, the calculated effective MFA at 1070 nm is about 197 µm2, while, in this simulation, the minimal MFA hits 140 µm2, giving a maximal focusing ratio of near 30%.
3.2.2 Output power and efficiency
Then, the output characteristics of the YDFL under different radiation doses are examined, as presented in Fig. 9. Figure 9(a) gives the output power of the YDFL as a function of the pump power at different radiation doses. As can be seen, for a certain radiation level, the output still shows a linear relationship with the launched pump power, which is consistent with the experimental results in high-power YDF amplifiers [33] and Tm-doped fiber lasers [29–30].
Fig. 9. Output power characteristics at different radiation doses: (a) output power versus pump power; (b) optical-optical efficiency.
However, at a fixed pump, the output power drops exponentially with the accumulation of radiation dose, and, so does the optical-optical efficiency of the laser system as shown in Fig. 9(b). For 2 kW pump, the output experienced a huge decline of 79% from 1739W at 0 Gy to 363 W at 500 Gy. In Fig. 9(b), experiment data from [33] is also plotted with blue dots for comparison. It can be found that, the simulated optical-optical efficiency and the experiment data both drop exponentially with the rise of radiation dose from about 80% at 0 Gy to well below 25% beyond 250 Gy. Besides, the simulation fits the experiment records quantitatively, further validating the veracity of this thermal model. Similar degradation of a kilowatt level YDFL was also reported in [34] where a deposited radiation dose of 117 Gy leads to a fatal slope efficiency drop from 84% to 29%. The decline of the output power and optical-optical efficiency should be mainly attributed to two factors. The first one is the dramatic shrink of the absorption cross-section of the pump and the emission cross-section of the laser signal at high temperatures. And, the second one is the extra losses of the pump and the laser signal induced by RIA.
Simulation results of this thermal model are also compared with the conventional model where thermal effects are not considered. In the thermal model, the decline of the absorption cross-section at the pump wavelength and the emission cross-section at the lasing wavelength would surely lead to a smaller output compared to the conventional model. The relative errors of the output power between these two models are depicted in Fig. 10(a) with different convective heat transfer coefficients. As shown in Fig. 10(a), the relative error is related to both the radiation dose and the cooling conditions. A monotonous rise of the relative error is recorded with the accumulation of radiation dose, as larger extra heat load is induced into the fiber core by RIA at higher radiation doses. As for the cooling conditions, the relative error reaches about 6% at poor cooling conditions with an H value of 200 W/(m2·K), while, for adequate heat management with an H value of 1000 W/(m2·K), the deviation between these two models becomes negligible with a relative error of only ∼2%.
Fig. 10. Relative errors and fiber core temperature distributions with different convective heat transfer coefficients: (a) relative errors compared to the conventional model under different radiation doses; (b) comparison of fiber core temperature distributions of the thermal model and the conventional model under 500 Gy radiation dose. TM: thermal model; CM: conventional model.
The origin of the relative error lies in the different fiber core temperature distributions of these two models. Figure 10(b) compares the fiber core temperature distributions of the thermal model and the conventional model under 500 Gy radiation dose at different cooling conditions. As can be seen, firstly, in the conventional model, the recorded fiber core temperature is generally higher than that generated with the thermal model. A maximal temperature difference of about 100 K is recorded with a convective heat transfer coefficient of 200 W/(m2·K). In the conventional model, as the major parameters are taken at room temperature, the calculated optical power is relatively larger than that at high temperatures as both the absorption cross-section of the pump and the emission cross-section of the signal decrease with the temperature. As shown in Fig. 2, the simulation results of the conventional model are, in fact, the results of the first iteration in the thermal model. And, these results are exploited to update the temperature-dependent input parameters for the calculation of the second iteration. After several iteration loops, the generated optical powers of the laser system become stable. And, consequently, a comparatively lower temperature can be expected compared with the conventional model.
Secondly, in both models, lower convective heat transfer coefficients lead to higher temperature rises in the fiber core, especially for the conventional model. Thus, the temperature differences between these two models expand at lower convective heat transfer coefficients, which would result in bigger variations of the input parameters including cross-sections, signal overlapping factor and lifetime, and, thus, larger relative errors.
However, it should be noted that, in practical operations, a convective heat transfer coefficient of 200 W/(m2·K) would not be sufficient to support the safe operation of the laser system at 2 kW pump, as it’s highly possible that the fiber jacket would be severely damaged. In fact, simulations show that the maximal temperature of the outer cladding surface reaches about 400 K even for pristine fibers, while, in the cross-section measurement experiments, the polymer coating was burnt into black when the fiber is heated to 400 K.
3.2.3 TMI threshold
TMI is an important factor limiting the output capability of high-power fiber systems. Here, the TMI threshold is investigated under different radiation doses as shown in Fig. 11. It can be found that, similar to the evolution of the output power and optical-optical efficiency in Fig. 9, the TMI threshold also decreases with the rise of the radiation dose, as higher radiation doses cause stronger heat load. Figure 11(a) compares the TMI threshold and the output power of laser systems with post-irradiated active fibers. And, two different pump powers, namely 2 kW and 3 kW, are exploited. As can be found, at low radiation doses within 200 Gy, the TMI threshold exhibits a much steeper drop than the output with a decline ratio of 93% from 4205 W at 0 Gy to 317 W at 200 Gy. And, beyond that, it drops gradually to 175 W at 500 Gy. The TMI threshold crosses with the output power of the 2 kW pump laser system at about 35 Gy, which means that, beyond a radiation dose accumulation of 35 Gy, the output of the fiber laser would become unstable with TMI. And, this part of the output is marked as dashed in Fig. 11(a). For the laser system with 3 kW pump, this transform happens at about 14 Gy. And, the stable single-mode output is limited to about 2320 W.
Fig. 11. TMI threshold variation along radiation doses: (a) output power versus TMI threshold; (b) output profile and the TMI area.
At certain radiation levels, to maintain stable single-mode operation of the fiber laser system, the output power should be controlled below the TMI threshold. Thus, the TMI region should be specified as shown in Fig. 11(b). This figure illustrates the output profile and the TMI region as a function of the pump power and the radiation dose. The color map defines the stable operation zone of this laser system with different output power levels labeled by contours. And, the boundary separating the color map and the TMI area defines the maximal pump power that can be launched for different radiation levels. As can be found, with the rise of the deposited radiation dose, the pump power should be tuned downward exponentially to avoid the appearance of TMI. And, kilowatt single-mode output is hardly accessible for a fiber laser system with its active fiber exposed to a radiation dose level beyond 50 Gy.
3.2.4 Active fiber length
The output characteristics and the TMI threshold of the laser system versus different fiber lengths are also investigated, as depicted in Fig. 12. Here, it should be noted that, in the simulation, as the fiber length varies, the doping concentration of the active fiber changes to ensure the active fiber provides the same total pump absorption, while keeping the size parameters of the fiber as constants. Figure 12(a) presents the evolution of the output power against the active fiber length at different radiation doses. As can be found, there exists an optimal active fiber length for the highest output power at different radiation doses. And, as radiation accumulates, the optimal length shortens gradually. This can be explained by the increasing overall propagation loss for longer active fibers.
Fig. 12. Output power and the TMI threshold of the laser system versus different fiber lengths: (a) output power; (b) TMI threshold.
Figure 12(b) depicts the variation of the TMI threshold along with the active fiber length. As can be seen, within the investigated length range, the TMI threshold rises with the increase of the active fiber length at low radiation doses within 20 Gy. At 0 Gy, the TMI threshold rise almost linearly with a slope of about 120 W/m. However, at high radiation levels, the TMI threshold shows a relatively steady evolution against the variation of the active fiber length.
As given in Eq. (6), at low radiation doses, the overall heat load is dominated by the quantum defect term ${Q_{QD}}$, which is closely related to the pump power distribution along the active fiber. With a constant total pump absorption, a longer active fiber corresponds to a lower doping concentration, thus, a smaller pump absorption coefficient, which, in turn, leads to a more homogeneous optical power distribution along the active fiber. And, with a constant convective heat transfer coefficient along the active fiber, consequently, this leads to a much lower overall heat load in the fiber. Thus, at low radiation doses, when the RIA is comparatively small, a longer active fiber could support higher TMI threshold, which is consistent with the results in [54] and [63]. And, it can be indicated that, for practical applications encountering low radiation dose situations, a slightly longer active fiber can be utilized to support the stable single-mode operation of the fiber laser system although the output power might experience a mild decline. At high radiation doses, thanks to a large RIA, the propagation loss term ${Q_{PL}}$ becomes the dominant factor of the heat load. The first consequence brought by this change is the remarkable drop of the TMI threshold due to the extra heat load induced by RIA. And, secondly, with a constant total pump absorption and a slowly shrinking laser signal, as shown in Fig. 12(a), the overall optical power distribution becomes relatively steady for different fiber lengths, leading to a relatively stable ${Q_{PL}}$ at different fiber lengths. And, consequently, the TMI threshold becomes less sensitive to the active fiber length.
4. Conclusions
A multiphysics thermal model concerning kilowatt level high-power YDFLs with post-irradiated active fibers is developed with both radiation effects and thermal effects taken into consideration. Major parameters including radiation dependent propagation losses and temperature dependent cross-sections of the active fiber were measured through experiments. Thermal effects, output power, TMI threshold and active fiber length are examined for different radiation doses. Simulations show that radiation is an important factor affecting the thermal profiles inside the active fiber. And, severe deterioration of the output power and optical-optical efficiency is observed at a radiation dose of several hundreds of Gy. TMI threshold is the main factor limiting the stable output capability of fiber laser systems at high radiation levels, as it exhibits a much faster decline with the accumulation of radiation dose. Thus, to maintain safe and stable single-mode operation of the laser system, the TMI area is specified defining the maximal pump power that can be launched for a given deposited radiation dose. For applications at low radiation levels, active fibers with relatively lower absorption coefficients are preferred at the cost of a small rollover of the output power.
This thermal model provides a useful tool for the investigation and prediction of the performance of high-power fiber systems working in adverse radiation environments. And, with this thermal model, the TMI region can be defined as a reference for the proper operation of high-power fiber systems. To tackle with the radiation induced performance degradation issues, optimization of high-power fiber systems should be considered. And, more importantly, active measures should be taken to protect keys components of high-power fiber systems from various radiations.
Funding
Chinese Academy of Sciences (Hundred-Talent Program); The State Key Laboratory of Laser Interaction with Matter (SKLLIM1709, SKLLIM1801Z).
Acknowledgments
The authors would like to express their gratitude to Prof. Chunlei Yu from Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Prof. Jianxiang Wen from Shanghai University, Prof. L. Dong from Clemson University, Prof. Haitao Guo from Xi’an Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, and Prof. Zujun Wang from the State Key Laboratory of Intense Pulsed Radiation Simulation and Effect for helpful discussions.
Disclosures
The authors declare no conflicts of interest.
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