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Abstract
An empirical TMI threshold formula is derived based on a recently developed model and used to analyze the power-scaling performance of ytterbium-doped silica glass and YAG (Y3Al5O12) and lutetia (Lu2O3) single-crystalline fiber amplifiers. Overall, the single-crystalline fiber lasers are found to scale potentially to higher average powers due to their higher thermal conductivities compared to silica glass. This work serves as a useful extension to earlier works and shines significant new light on optimal fiber and amplifier designs for maximum average output power with TMI considered.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High power diffraction-limited fiber lasers have found a wide range of industrial, scientific, and defense applications. It is now well understood that transverse mode instability (TMI) is the major limitation to further average power scaling [1–7]. The current record output power of 10-20 kW was reached with TMI mitigated by a tandem-pumping technique [8].
Slightly over a decade ago, a study of power scaling limit was performed for ytterbium-doped silica fiber lasers considering a number of physical (e.g., core melting), thermal (e.g., thermal fracture, thermal lensing), and optical (e.g., nonlinear, optical damage, pump brightness) limits [9]. This study was later extended to non-silica glass and single-crystal YAG fiber lasers [10–12]. TMI was not well understood at the time and was not included in these studies, thereby significantly over-estimating output powers. Recently, a simple and approximate TMI threshold formula was developed based on an instability analysis [13]. The formula is almost entirely based on fiber parameters, ignoring most amplifier parameters, which are well known to have significant impacts on TMI. The formula was used in a recent study of power scaling [14].
More recently, an accurate TMI model has been developed based on combining a stimulated thermal Rayleigh scattering (STRS) model and a quasi-three-dimensional (3D) fiber amplifier model [15]. The STRS approach has been well established in several theoretical studies [3,4,15,16]. It is also supported by direct experimental observations [17]. The addition of the quasi-3D fiber amplifier model enables the evaluation of gain saturation both across the core and along the fiber to accurately assess its impact on TMI. The model is bench-marked to the full 3D model in Ref. [5]. It is very efficient and, more importantly, agrees well with experimentally measured TMI thresholds, adding significant credence to its accuracy [15]. Gain saturation was found to be a dominating factor in TMI and can explain many of the experimentally observed behaviors.
In this work, a simple TMI threshold formula is derived based on this new TMI model so that the TMI threshold for broad ranges of parameters can be calculated quickly based on a single TMI threshold simulation and known dependence of TMI threshold on fiber and amplifier parameters. Power scaling analysis is performed for ytterbium-doped silica and single crystalline YAG (Y3Al5O12) and Lutetia (Lu2O3) fiber lasers with an accurate consideration of TMI. The single-crystalline fiber lasers have attracted much interest recently due to their high thermal conductivity [18–24]. Single-crystalline Lutetia fiber lasers are especially interesting due to their potential for high ytterbium doping levels without detrimental reductions in thermal conductivity.
The silica fiber laser is studied for several pumping schemes and their power scaling potential is found to be significantly limited by TMI. The TMI threshold is found to be significantly higher in the single-crystalline fiber lasers due to the much higher thermal conductivities. Overall, the single-crystalline fiber lasers are found to scale potentially to higher average powers. This work refines existing prediction of power scaling limit and provides much needed guidance on material, fiber, and amplifier designs for further power scaling with the consideration of TMI.
2. Theory
Our recent work has shown that gain saturation is a dominating factor in TMI and can easily explain many experimentally observed behaviors of TMI [15]. The study also reveals that TMI is determined by the total heat load of the amplifier only in the absence of gain saturation. TMI can be significantly suppressed in the presence of strong gain saturation as already noted in Refs. [5,7], where only a small fraction of the total heat load contributes towards TMI. Gain saturation is therefore a significant factor to consider for higher TMI threshold.
To evaluate power scaling limits with TMI considered, a simple formula for TMI threshold is necessary to consider its dependence on seed power, core diameter, and cladding diameter so that it can be evaluated quickly for a large range of fiber and amplifier configurations. The nonlinear coupled-mode equations governing the coupling between the fundamental mode and higher order modes due to STRS can be written as [4],
If one is only interested in estimating the TMI threshold, defined as P11(L) = 0.01P01(L), the first term on the right-hand side of Eq. (1a), representing the depletion of the fundamental mode due to TMI, can be ignored, resulting in
Combining Eqs. (2) and (3), considering α01 can be ignored, yields
In this regime, from Eq. (1a) and ignoring fundamental mode depletion due to TMI, leads to
The TMI threshold can be derived in this case as
For the rest of the analysis, mode dependent losses, α01 and α11, are ignored to simplify the problem. This leads to an underestimate TMI threshold, especially for smaller core diameters (< 20µm), where higher-order-mode loss can be significant. Mode dependent gain due to gain overlap g01 and g11 are considered in this model.
In this case, the model shows that the TMI threshold is independent of fiber length if the total small-signal pump absorption is kept constant [15]. The TMI threshold’s dependence on seed power was also studied in Ref. [15] for various values of x0 for a co-pumping configuration. The results have shown that TMI threshold’s dependence on x0 in this case is determined by
It is worth noting that this dependence on x0 is expected to be dependent on losses α01 and α11 and gains g01 and g11, see Eq. (6). Even in cases where the losses can be ignored, the gains are expected to be dependent on dopant profile and mode profiles. This is especially true near the LP11 mode cut-off. In this study, the same x0 is used throughout and this dependence is not used.
The TMI threshold versus core diameter, 2ρ, was also studied in Ref. [15] for a pump wavelength of 976 nm and a seed wavelength of 1064 nm. The Yb-doped double-clad fiber is Nufern’s LMA-YDF-25/400-M fiber. The fiber length is adjusted in tandem to keep the total small-signal cladding pump absorption at 20 dB. The seed power is 50W. Losses α01 and α11 are ignored. The central part of the core, up to 80% of the radius, is Yb-doped. The value of x0 employed is 3 × 10−12.
The simulated data is re-plotted in Fig. 1(a) along with a fit of $\propto \mathrm{\rho}$-2 for the counter-pumping case and a fit of $\propto \mathrm{\rho}$-1.4 for the co-pumping case. The fits work reasonably well. The high TMI regime is of greater interest in this study of power scaling limits. For the same reason, only counter-pumping is considered in this study as it provides a higher TMI threshold. The fitting error in the high TMI regime is ∼10% (see 15µm core diameter in Fig. 1(a)).
Fig. 1. (a) The simulated TMI threshold versus core diameter, 2ρ, for both co- and counter-pumping cases. The data is from Ref. [15]. The fits are $\propto \mathrm{\rho}$-2 and $\propto \mathrm{\rho}$-1.4, respectively, for counter- and co-pumping cases. (b) The simulated TMI threshold versus cladding diameter 2R for counter-pumping case. The fit is ∝R2. (c) TMI threshold versus seed power for counter pumping configurations. The fit is −0.00246*Pseed2 + 4.02*Pseed + 1710, where Pseed = P01(0). Total small-signal cladding pump absorption at 976 nm is 20 dB and the seed power is 50W at 1064 nm unless otherwise specified. Losses α01 and α11 are ignored. The Yb-doped double-clad fiber is a Nufern LMA-YDF-25/400-M. Central part of the core up to 80% of radius is doped. x0 is 3 × 10−12. For (a) and (b), length is adjusted to keep total pump absorption constant.
The TMI threshold versus cladding diameter, 2R, is simulated for the counter-pumping case and is plotted in Fig. 1 (b) along with a fit of ∝R2. All parameters are the same as in Fig. 1 (a) apart from core diameter being fixed at 25µm and cladding diameter varied while fiber length is adjusted in tandem to keep the total small-signal cladding pump absorption at 20 dB. The fit is reasonable, and the fitting error is just a few percent at the high TMI thresholds.
The TMI threshold versus seed power is also simulated for the counter-pumping case and is shown in Fig. 1 (c) along with a fit of −0.00246*Pseed2 + 4.02*Pseed + 1710, where Pseed = P01(0). The core and cladding diameters are the nominal 25µm and 400µm, respectively. The fit is good over the whole range.
Since losses α01 and α11 are ignored for all the simulations in Fig. 1, the observed TMI dependence is mostly determined by gain saturation. Mode-dependent gain due to the respective mode overlap with the gain media, g01 and g11, are considered in this model. It is expected to only be of significant impact for smaller core diameters where the LP11 mode approaches cut-off leading to large TMI suppression due to the diminishing LP11 mode overlap with the gain media [4].
Gain saturation is determined by the balance of pumping rate σpaIp (the product of pump absorption cross section and pump intensity) and signal emission rate σseIs (the product of signal emission cross section and signal intensity). In the simplest case, gain saturation is expected to be dependent on pump and signal intensity, inverse-quadratically scaling with cladding radius, R, and core radius, ρ, respectively. A large core diameter reduces the signal intensity while a large cladding diameter lowers pump intensity, and gain saturation is therefore expected to scale as (R/ρ)2. Since TMI is dominated by gain saturation with the TMI threshold increasing with gain saturation [15], it is also expected to scale as (R/ρ)2 to first order.
Interestingly, this 1/ρ2 dependence was also observed from the instability analysis in Refs. [13,14]. In the model employed here, gain saturation both across the core and along the fiber is considered, the overall effect is a 3D averaging of local effects, which can deviate from this simple analysis. Mode-dependent gain can also lead to some deviation near the LP11 mode cutoff. The observed TMI scaling as (R/ρ)2 in Fig. 1 for the counter-pumping case is therefore not surprising. The signal intensity is linearly dependent on seed power and TMI threshold is therefore expected to scale linearly with seed power to first order. The observed dependence of −0.00246*Pseed2 + 4.02*Pseed + 1710 is, unsurprisingly, largely linear.
Combining the TMI threshold’s dependence on x0, core size, cladding size, and seed power Pseed, one obtains an empirical formula for TMI threshold as
The constant K can be obtained by the simulation of TMI threshold for a reference fiber. Each of the reference fiber with known R and ρ is simulated for the given Pseed and x0 listed in Table 1 to find $P_{th}^{TMI}$. K, the only unknown in Eq. (8), can then be determined, and is given for each case in the caption of relevent figures.
Table 1. Nominal values used in this work unless otherwise specified.
If the amplifier gain G is known, $P_{th}^{TMI}$=G${P_{seed}}$, Eq. (8) can be re-written as
This quadratic equation can be solved to find the TMI threshold by ignoring the negative solution as non-physical once K is determined by the simulation of a reference fiber.
3. Power scaling of silica fiber lasers
In the first studies of power scalability for Yb-doped fiber lasers, limits associated with thermal fracture, melting, thermal lensing, stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), optical damage, and pump brightness were considered [9]. SRS, SBS, pump brightness, thermal lensing, and optical damages were found to be the most relevant. Here, SRS, pump brightness, thermal lensing, and optical damage are evaluated in addition to TMI. SBS can be mitigated efficiently by spectral broadening and is not well characterized for the single-crystalline fibers. It is ignored in this study.
The optical damage threshold used in Ref. [9] was only 10W/µm2. Considering 10-20 kW diffraction-limited power has been achieved in 30µm core fiber, this is equivalent to an upper bound of ∼44W/µm2. This value is clearly too low and a damage intensity of 100W/µm2 is employed in this work as an estimate. There is a general lack of optical damage threshold data for CW lasers. Bulk damage thresholds using a nanosecond pulsed YAG laser at 1064 nm are however available as follows: ∼4.75 kW/µm2 for silica using 8 ns pulses [25], up to ∼500W/µm2 for Nd:YAG using 8 ns pulses [26], and ∼250W/µm2 for undoped ceramic YAG using 4 ns pulses [27]. Surface damage thresholds are typically much lower but can be mitigated by beam-expanding end caps.
Fiber lasers at kW levels, in general, have laser gains well over the value of 10 used in Ref. [9]. Here, a gain of 25 is used as a more reasonable current estimate. This has the impact of reducing the effective nonlinear fiber length and increasing nonlinear thresholds. The pump brightness of 0.021W/µm2/Sr used in Ref. [9] is also much lower compared to diodes available today. For example, ∼200W in 100µm pump delivery fibers with a 0.22 NA (numerical aperture) are currently available. Considering bi-directional pumping, this yields a pump brightness of 0.34W/µm2/Sr. The core pump absorption is increased in the current work to 1000 dB/m as commercial fibers already achieve about ∼600 dB/m. Cladding diameter contours are also included in this study. Above a cladding diameter of 800µm, the fiber becomes very difficult to coil. Further, since the fibers considered herein have only partially doped cores, the doped radius instead of core radius is used to calculate pump brightness limit. Apart from the TMI threshold, all the other limits were evaluated using the formulae in Ref. [9].
In the first case, a Yb-doped fiber laser, counter-pumped at 976 nm and operated at 1064 nm is studied. The reference fiber was Nufern’s LMA-YDF-25/400-M. The same fiber was used in the previous analysis [15]. The TMI threshold in this fiber was carefully measured for a range of amplifier configurations in Ref. [28], which provides a great opportunity to verify the model employed here [15]. The Yb3+ absorption and emission cross section in silica fibers has been refined from that used in Ref. [15] (see Fig. 2(a)). The absorption cross-sections at the three pump wavelengths used in this study of 976 nm, 990 nm, and 1018 nm, respectively, are 2.6 × 10−24 m2, 3.22 × 10−25 m2, and 9.36 × 10−26 m2. To achieve the best fit to the experimental TMI data in Ref. [28], the Yb-doped area occupies up to 80% of radius in the core center and x0 = 3 × 10−12. This x0 is revised from the value used in Ref. [15]. The absorption and emission cross-section data of an ytterbium-doped phosphosilicate glass was used in Ref. [15]. We have decided to use the cross-section data of an ytterbium-doped aluminosilicate host in this study to more accurately reflect the fiber studied and consequently revised x0 value. These values are used for the entire study.
Fig. 2. Yb3+ absorption and emission cross-sections used in this study (a) silica fiber, (b) YAG, and (c) Lutetia.
For counter-pumping case, the simulated TMI threshold is 1945W for 50W seed at 1064 nm and 20 dB total small signal absorption at the pump wavelength of 976 nm for the 10 m reference fiber length. See Table 1 for details of parameters used herein.
The power scaling limit in this case is given in Fig. 3(a). For shorter fibers, TMI dominates. In this regime, the pump core absorption for a fixed core size and fiber length to achieve 20 dB total small-signal pump absorption limits the cladding size and, consequently, lowers TMI threshold. For a given fiber length, the ratio of cladding size and core size R/ρ is fixed to maintain the 20 dB total small-signal pump absorption, leading to TMI threshold being independent of core diameter from Eq. (8). SRS dominates at small core sizes and long fiber length. The maximum power in this case is just ∼5 kW for a 400µm-cladding fiber with a ∼17.5µm core and ∼16 m length.
Fig. 3. Power scaling limit of counter-pumped Yb-doped silica fiber laser, (a) seed wavelength of 1064 nm (K = 0.1138) and (b) seed wavelength of 1030 nm (K = 0.3753). Pump wavelength is at 976 nm. SRS: stimulated Raman scattering, TMI: transverse mode instability. The dark blue contour lines represent cladding diameters and light blue contour lines represent powers.
Moving the seed wavelength to 1030 nm increases the signal emission cross-section. For a fixed pump, this increases gain saturation, which scales with the ratio of signal emission cross-section to pump absorption cross-section, σse/σpa, and, therefore, provides TMI suppression. Our previous study has shown that gain saturation provides most of the TMI suppression in this case, with the reduction in quantum defect only playing a minor role [15].
In the second case, the seed wavelength is moved to 1030 nm, while leaving all else the same. For counter-pumping at 976 nm, the simulated TMI threshold is 6414W for a 50W seed at 1030 nm for the reference fiber. The power scaling limit in this case is given in Fig. 3(b). The maximum power is increased to ∼13.1 kW in a 400µm-cladding fiber with a ∼22µm core and ∼10 m length.
In the third case, the pump wavelength is 990 nm while keeping the seed at 1064 nm. Moving the pump off the absorption peak significantly lowers the pump absorption cross section and pumping rate, and, therefore, increases gain saturation, which scales with σse/σpa. The downside of this is lowering of pump core absorption from 1000 dB/m at 976 nm to about 124 dB/m at 990 nm. The simulated TMI threshold for the reference fiber (Nufern LMA-YDF-25/400-M) is 15289W for 50W seed at a wavelength of 1064 nm with the fiber length increased to 80.8 m. The power scaling limit in this case is given in Fig. 4(a). There is a small region at small core diameters (less than 21µm) and shorter fiber lengths (5-21 m) where the pump brightness limits power-scaling. A maximum power of 11.4 kW can be achieved in a 400µm-cladding fiber with a ∼36µm core diameter and ∼30 m length.
Fig. 4. Power scaling limit of counter-pumped Yb-doped silica fiber laser at seed wavelength of 1064 nm and pump wavelength of 990 nm (K = 0.8946), core pump absorption at 976 nm is (a) 1000 dB/m and (b) 4000 dB/m. PB: pump brightness. The dark blue contour lines represent cladding diameters and light blue contour lines represent powers.
In the fourth case, the third case is reconsidered but with a core absorption at 976 nm increased to 4000 dB/m. The increase of ytterbium doping level may be justified in this case due to the lower photodarkening associated with an off-resonance pumping scheme. In this case, the core pump absorption at 990 nm is 495.2 dB/m. A maximum power of 19.3 kW can be achieved in a 400µm-cladding fiber with a ∼29µm core diameter and ∼11.5 m length.
The smaller core diameter and shorter fiber compared to those in the third case makes this scheme more practical. Given the current available commercial core diameter of 30µm, ∼20 kW can be achieved with this scheme with ∼12 m length and potentially commercially available pump diodes. This is a much simpler architecture for 20 kW diffraction-limited fiber laser than the tandem-pumping scheme used today [8]. Due to the much lower pumping rate in this case, the laser operates at a lower inversion and the scheme is more appropriate for a seed at longer wavelengths.
Tandem pumping has been used to achieve record 10-20 kW diffraction-limited powers [8]. This case is modeled here by shifting the pump wavelength to 1018 nm and the seed wavelength to 1070 nm. The even smaller core pump absorption of 36 dB/m at 1018 nm, given the core pump absorption of 1000 dB/m at 976 nm, requires a reduction of cladding diameter to 200µm for the 25µm core reference fiber to allow for a reasonable fiber length of 69.5 m and to achieve the 20 dB total small-signal pump absorption. For counter-pumping at 1018 nm, the simulated TMI threshold is 6776W for a 50W seed at 1070 nm for the reference fiber. The model employed herein indicates that the higher TMI threshold, compared to the 976 nm pumping scheme, is mostly due to the higher gain saturation because of the higher σse/σpa, not due to a reduction in quantum defect as has been commonly thought [8].
For the tandem-pumped 10 kW fiber laser in Ref. [8], forty-seven (47) 270W fiber lasers at a wavelength of 1018 nm were combined using a 57/1 combiner. The maximum pump brightness with fifty-seven (57) 1018 nm fiber lasers can be estimated to be ∼26W/µm2/Sr considering bi-directional pumping into cladding diameter of 100µm. This 26W/µm2/Sr is employed as the available pump brightness at 1018 nm for this study. The power scaling limit in this case is given in Fig. 5(a). A maximum power of 11 kW can be achieved in a 400µm-cladding fiber with a ∼48µm core diameter and ∼57 m length.
Fig. 5. Power scaling limit of counter-pumped Yb-doped silica fiber laser at seed wavelength of 1070 nm and tandem pump wavelength of 1018 nm (K = 1.5859), core pump absorption at 976 nm is (a) 1000 dB/m and (b) 4000 dB/m. The dark blue contour lines represent cladding diameters and light blue contour lines represent powers.
If one similarly argues that the ytterbium doping level can be increased in this case to core pump absorption of 4000 dB/m at 976 nm, corresponding to core pump absorption of 144 dB/m at 1018 nm, due to the less photo-darkening because of the off-resonance pumping scheme, then the power scaling limit is shown in Fig. 5(b). Under these conditions, a maximum power of 18.7 kW can be achieved in a 400µm-cladding fiber with a ∼39µm core diameter and ∼22 m length while 11 kW can be achieved in a 200µm-cladding fiber with a ∼24µm core diameter and ∼14 m length. The 10 kW tandem silica fiber laser in Ref. [8] employed a 15 m length of 30µm-core fiber. The cladding diameter was not given, but a “reduced cladding size” was mentioned. The fiber may be a 30/200 (core/clad diameter, in micrometers) fiber in Ref. [8], which would be close to the 24/200 predicted in the model used herein. For 20 kW output power in a 400µm-cladding fiber, a slightly higher ytterbium doping level is necessary.
4. Power scaling of single-crystalline fiber lasers
As discussed in the previous Section, TMI is a major limit to power scaling of silica fiber lasers. Rare-earth doped crystals are also well-known laser gain media. Yttrium aluminum garnet, YAG (Y3Al5O12), has a much higher thermal conductivity of 8.6W/m/K (2% Yb) than the 1.38W/m/K of silica glass [29]. Lutetium sesquioxide, Lutetia (Lu2O3), has an even higher thermal conductivity of 12.3W/m/K [30,31] and can potentially incorporate as much as 50% rare earth ions comparing to the ∼4% in YAG, prior to reductions in thermal conductivity. YAG is known to have a Raman gain coefficient of 10−12 m/W [32], an order of magnitude higher than that of silica. This same value is assumed for Lutetia in this study although the actual number is not known.
Recently, there has been some success in growing single-crystalline fibers, notably the rare-earth-doped core and undoped cladding structure demonstrated in Ref. [21,22]. There has been a recent surge of interest for developing double-clad single-crystalline fiber lasers including a crystalline pump cladding [18–24], raising the hope that such fibers may become available soon.
The much higher thermal conductivity of such laser crystals, relative to silica glass, is expected to markedly increase the TMI threshold since the STRS coupling coefficient is inversely proportional to thermal conductivity [4,15]. YAG and Lutetia also have lower thermo-optic coefficients, dn/dT [29,30], which can also promote increased TMI thresholds as STRS coupling coefficient is proportional to dn/dT [4]. It is therefore very interesting to extend the previous study of power scaling limits to double-clad single-crystalline fiber in Refs. [10–12] to include TMI.
TMI thresholds of Yb:YAG and Yb:Lutetia double-clad fiber lasers are studied first for the reference fibers detailed in Table 1. The absorption and emission cross-section of Yb:YAG and Yb:lutetia are taken from Ref. [33] and are plotted in Fig. 2(b) and (c), respectively. The ratio of signal emission cross section to pump absorption cross section, σse/σpa, is much higher for YAG than silica. This increases the gain saturation and, consequently, TMI threshold. For the reference 25/400 fiber, the simulated TMI threshold is ∼258 kW, some two orders of magnitude higher than that of silica fiber. A higher seed power of 500W was used to account for the much higher TMI threshold. For Lutetia, σse/σpa is similar to that of a silica fiber laser. Accordingly, the TMI threshold is almost entirely due to the much higher thermal conductivity. For the reference 25/400 fiber, the simulated TMI threshold is ∼88 kW, over an order of magnitude higher than that of silica fiber laser, but only one-third that of the reference YAG fiber laser, despite the higher thermal conductivity and lower dn/dT values. A detailed analysis shows that the much higher σse/σpa ratio of a YAG fiber laser can alone increase the TMI threshold to ∼20 kW, assuming all else is equal to a silica fiber laser.
In this power scaling study, both single-crystalline fibers are assumed to have cores that are fully doped with ytterbium across the core diameter. The maximum core absorption of Yb:YAG is set at 2000dB/m at 968.8 nm, corresponding to a 4 atom percent Yb-doping concentration. Yb:Lutetia has a nearly four times larger pump absorption cross section at 975.6 nm (see Fig. 2) and can potentially be Yb-doped up to 50%. The maximum pump absorption for Yb:Lutetia is set to 7600 dB/m at 975.6 nm, corresponding to a Yb-doping level of 2 atom percent. This is much lower than what is possible but sufficient to keep the fiber laser long enough to avoid optical damage.
The power scaling limit of Yb:YAG fiber laser is shown in Fig. 6(a). Power scaling is not limited by TMI, but, instead, by SRS and thermal lensing. The results are similar to those in Refs. [10,11], where the TMI limit was not included. The maximum output power is ∼40 kW. The scaling limit of Yb:Lutetia is given in Fig. 6(b). In this case, TMI does limit the power scaling, with a maximum power of 47 kW. Fiber length in both cases is much shorter that those of silica fibers. This is especially true for Lutetia and makes compact multi-kW fiber lasers possible. Those short fibers can be kept straight, therefore removing the 800µm cladding diameter limit.
Fig. 6. Power scaling limit of counter-pumped Yb-doped (a) YAG (K = 9.2645) and (b) Lutetia (K = 4.7882). The dark blue contour lines represent cladding diameters and light blue contour lines represent powers.
5. Conclusions
A simple TMI formula is developed in this work to account for core size, cladding size, seed power and x0 = P11(0)/P01(0), enabling a rapid evaluation of TMI threshold once a constant is obtained using a single simulation run on a reference fiber. The formula was then used to study the power scaling limits of Yb-doped silica, YAG, and Lutetia fiber lasers. Not surprisingly, the Yb silica fiber laser is strongly limited by TMI with maximum powers much lowered than those predicted in previous studies [9,14]. For silica fibers, pumping schemes can be used to mitigate TMI, mainly by increasing gain saturation. Several pumping schemes were investigated for Yb silica fiber lasers.
At the conventional pumping wavelength of 976 nm and seed wavelength of 1064, the maximum diffraction-limited output power was found to be limited below ∼5 kW by a combination of SRS and TMI in 400µm-cladding fibers with core pump absorption of 1000 dB/m. When the seed wavelength is moved to 1030 nm to mitigate TMI, the maximum diffraction-limited output power is increased to ∼13 kW.
TMI can also be mitigated by shifting the pump wavelength off the absorption peak. With pump wavelength at 990 nm, the laser operates at a lower inversion, due to the much lower pumping rate, and the seed is therefore moved back to 1064 nm. The maximum diffraction-limited output power is ∼11 kW in 400µm-cladding fibers. The cost is a much longer fiber of larger core, comparing to the first two cases. By increasing the maximum core absorption from 1000 dB/m to 4000 dB/m at 976 nm, the fiber is shortened, and core size reduced with a maximum diffraction-limited output power of ∼19 kW in 400µm-cladding fibers. If the high doping level is feasible, this scheme can achieve diode-pumped 10-20 kW diffraction-limited power in 400µm-cladding fibers without using the cumbersome tandem pumping scheme commercially employed today.
Tandem pumping at 1018 nm and a seed at 1070 nm also was studied for two cases, the maximum core pump absorption of 1000 dB/m and 4000 dB/m at 976 nm. The maximum outpower for the first case is ∼11 kW and for the second case ∼19 kW in 400µm-cladding fibers. A 15 m, 30µm-core diameter fiber was used in the 10 kW tandem silica fiber laser in Ref. [8]. Cladding size was not disclosed but was noted as having a “reduced cladding size”. In the first case, 10 kW cannot be achieved with this fiber, but in the second case, a maximum power of 11 kW is predicted herein for a 200µm-cladding fiber with a ∼24µm core diameter and ∼14 m length, close to the fiber used in Ref. [8]. Our model therefore provides a reasonable prediction for the results in [8]. Our model also indicates that the higher TMI threshold in this case is mostly due to higher gain saturation not a reduction in quantum defect as has been commonly thought.
Found here, for the first time to the best of the authors’ knowledge, is that a single-crystalline YAG fiber has a TMI threshold over two orders of magnitude higher than equivalent silica fiber, due to a combination of high thermal conductivity and high σse/σpa. Power scaling is limited by a combination of SRS and thermal lensing, not by TMI, to ∼40 kW, consistent with previous studies [10,11].
Lutetia also has a much-improved TMI threshold, but at one-third that of YAG despite the higher thermal conductivity and lower dn/dT due to the much smaller σse/σpa. The maximum diffraction-limited power is ∼ 47 kW, limited by SRS, thermal lensing, and TMI. It is worth noting that Raman gain coefficient of Lutetia is not known currently and has been assumed it to be the same as that of YAG. Both YAG and Lutetia potentially enable very short (∼1 m or less) muti-kW fiber lasers, which can be kept straight and therefore removes the limit on upper cladding size.
Mode-dependent losses have been ignored in this current analysis, which will underestimate TMI threshold especially for smaller cores (<25µm). However, this effect is expected to be small for larger cores. Lower x0 = P11(0)/P01(0) is also possible using a seed laser with less noise. This can increase TMI threshold but only by a limited amount due to the logarithmic dependence. Using more optimized bidirectional pumping scheme can potentially increase TMI threshold by up to 30% [15,28]. Higher seed power and consequently lower amplifier gain can also be used to further increase TMI threshold but limited in practice. The power limit in this study is therefore meant to be an estimate and it is expected that somewhat higher power is likely if the additional parameters discussed above are further optimized.
Funding
Space and Missile Defense Command (W9113M16D004, W9113M18FD003); SMDC PAO#2134, approved for public release on 14 Dec 2022.
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A.V. Smith and J.J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011). [CrossRef]
2. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012). [CrossRef]
3. K.R. Hansen, T.T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37(12), 2382–2384 (2012). [CrossRef]
4. L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013). [CrossRef]
5. A.V. Smith and J.J. Smith, “Increasing mode instability threshold of fiber amplifiers by gain saturation,” Opt. Express 21(13), 15168–15182 (2013). [CrossRef]
6. S. Naderi, I. Dajani, T. Madden, and C. Robin, “Investigation of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21(13), 16111–16192 (2013). [CrossRef]
7. K.R. Hansen and J. Laegsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22(9), 11267–11278 (2014). [CrossRef]
8. H. Injeyan and G.D. Goodno, High-Power Laser Handbook (The McGraw-Hill Companies, 2011), chap. 18.
9. J.W. Dawson, M.J. Messerly, R.J. Beach, M.Y. Shverdin, E.A. Stappaerts, A.K. Sridharan, P.H. Pax, J.E. Heebner, C.W. Siders, and C.P.J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008). [CrossRef]
10. J. Dawson, M. Messerly, J.E. Heebner, P.H. Pax, A.K. Sridharan, A.L. Bullington, R.J. Beach, C.W. Siders, C.P.J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on non-silica materials,” in Proc. SPIE 7686, Laser Technology for Defense and Security VI (SPIE, 2010), paper 768611.
11. T.A. Parthasarathy, R.S. Hay, G. Fair, and F.K. Hopkins, “Predicted performance limits of yttrium aluminum garnet fiber lasers,” Opt. Eng. 49(9), 094302 (2010). [CrossRef]
12. D. Li, P. Hong, S. Meissner, and H. Meissner, “Power scaling analysis of crystalline fiber waveguides based on RE3+-doped YAG cores,” in Proc. SPIE 9744. Optical Components and materials XIII (SPIE, 2016), paper 974411.
13. M.N. Zervas, “Transverse mode instability analysis in fiber amplifiers,” in Proc. SPIE 10083, Fiber Lasers XIV: Technology and Systems (SPIE, 2017), paper 100830 M.
14. M.N. Zervas, “Transverse mode instability, thermal and power scaling in Yb3+-doped high-power fiber amplifiers,” Opt. Express 27(13), 19019–19041 (2019). [CrossRef]
15. L. Dong, “Accurate Modeling of Transverse Mode Instability in Fiber Amplifiers,” J. Lightwave Technol. 40(14), 4795–4803 (2022). [CrossRef]
16. C.R. Menyuk, J.T. Young, J. Hu, A.J. Goers, D.M. Brown, and M.L. Dennis, “Accurate and efficient modeling of the transverse mode instability in high energy laser amplifiers,” Opt. Express 29(12), 17746–17757 (2021). [CrossRef]
17. F. Kong, J. Xue, R.H. Stolen, and L. Dong, “Experimental Observation of Quantum-defect-assisted Polarization Mode Coupling in a Fiber Amplifier,” Optica 3(9), 975–978 (2016). [CrossRef]
18. I. Martial, S. Bigotta, M. Eichhorn, C. Kieleck, J. Didierjean, N. Aubry, R. Perfetti, F. Balembois, and P. Georges, “Er:YAG fiber-shaped laser crystals (single-crystal fibers) grown by micro-pulling down: Characterization and laser operation,” Opt. Mater. 32(9), 1251–1255 (2010). [CrossRef]
19. W. Kim, C. Florea, C. Baker, D. Gibson, L. B. Shaw, S. Bowman, S. O’Connor, G. Villalobos, S. Bayya, I. D. Aggarwal, and J. S. Sanghera, “Single crystal fibers for high power lasers,” in Proc. SPIE 8547, High-Power Lasers 2012: Technology and Systems (SPIE, 2012), paper 85470 K.
20. B. T. Laustsen and J. A. Harrington, “Fabrication and optical properties of single-crystal YAG fiber optics,” in Proc. SPIE 8235 (SPIE, 2012), paper 823505.
21. G. Maxwell, N. Soleimani, B. Ponting, and E. Gebremichael, “Coilable single crystal fibers of doped-YAG for high power laser applications,” in Proc. SPIE 8733 (SPIE, 2013), paper 87330.
22. G. Maxwell, B. Ponting, E. Gebremichael, and R. Magana, “Advances in single crystal fibers and thin rods grown by laser heated pedestal growth,” Crystals 7(1), 12 (2017). [CrossRef]
23. Y. Li, Z. Zhang, I. Buckley, J. K. Miller, E. G. Johnson, C. D. Nie, J. A. Harrington, and R. Shori, “Investigation of the amplification properties of Ho:YAG single crystal fiber,” in Proc. SPIE 9342 (SPIE, 2015), paper 934205.
24. C.D. Nie, S. Bera, and J.A. Harrington, “Growth of single-crystal fiber optics,” Opt. Express 24(14), 15522–15527 (2016). [CrossRef]
25. A.V. Smith and B.T. Do, “Bulk and surface laser damage of silica by picosecond and nanosecond pluses at 1064 nm,” Appl. Opt. 47(26), 4812–4832 (2008). [CrossRef]
26. D. E. Zelmon, K.L. Schepler, S. Guha, D. Rush, S.M. Hegde, L.P. Gonzalez, and J. Lee, “Optical properties of Nd-doped ceramic yttrium aluminum garnet,” in Proceedings of SPIE, 5647 (SPIE, 2005), pp. 255–264.
27. J.F. Bisson, Y. Feng, A. Shirakawa, H. Yoneda, J. Lu, H. Yagi, T. Yanagitani, and K.I. Ueda, “Laser Damage Threshold of Ceramic YAG,” Jpn. J. Appl. Phys. 42(Part 2, No. 8B), L1025–L1027 (2003). [CrossRef]
28. T. Li, C. Zha, Y. Sun, Y. Ma, W. Ke, and W. Peng, “3.5 kW bidirectionally pumped narrow-linewidth fiber amplifier seeded by white-noise-source phase-modulated laser,” Laser Phys. 28(10), 105101 (2018). [CrossRef]
29. R. L. Aggarwal, D. J. Ripin, J. R. Ochoa, and T. Y. Fan, “Measurement of thermo-optic properties of Y3Al5O12, Lu3Al5O12, YAlO3, LiYF4, LiLuF4, BaY2F8, KGd(WO4)2, and KY(WO4)2 laser crystals in the 80–300 K temperature range,” J. Appl. Phys. 98(10), 103514 (2005). [CrossRef]
30. P. A. Loiko, K. V. Yumashev, R. Schödel, M. Peltz, C. Liebald, X. Mateos, B. Deppe, and C. Kränkel, “Thermo-optic properties of Yb:Lu2O3 single crystals,” Appl. Phys. B 120(4), 601–607 (2015). [CrossRef]
31. J.H Mun, A. Jouini, A. Yoshikawa, J.H Kim, T. Fukuda, and J.S. Lee, “Thermal and optical properties of Yb-doped Lu2O3 single crystal grown by the micro-pulling-down method,” J. Ceram. Process. Res. 14(5), 636–640 (2013).
32. A.A. Kaminskii, H. Rhee, O. Lux, H.J. Eichler, S.N. Bagayev, H. Yagi, K. Ueda, A. Shirakawa, and J. Dong, “Stimulated Raman scattering in “garnet” Lu3Al5O12 ceramics – a novel host-materiel for Ln- and TM-lasant ions,” Laser Phys. Lett. 8(6), 458–464 (2011). [CrossRef]
33. R. Peters, C. Krankel, K. Petermann, and G. Huber, “Broadly tunable high-power Yb:Lu2O3 thin disk laser with 80% slope efficiency,” Opt. Express 15(11), 7075–7082 (2007). [CrossRef]
