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Abstract
In this work, the impact of fiber bending and mode content on transverse mode instability (TMI) is investigated. Based on a modified stimulated thermal Rayleigh scattering (STRS) model considering the gain competition between transverse modes, we theoretically detailed the TMI threshold under various mode content and bending conditions in few-mode fibers. Our theoretical calculations demonstrate that larger bending diameters increase the high order mode (HOM) components in the amplifier, which in turn reduces the frequency-shifted Stokes LP11o mode due to the inter-mode gain competition mechanism, thus improving the TMI threshold of few-mode amplifiers. The experimental results agree with the simulation. Finally, by optimizing the bending, an 8.38 kW output tandem pumped fiber amplifier is obtained with a beam quality M2 of 1.8. Both TMI and stimulated Raman scattering (SRS) are well suppressed at the maximum power. This work provides a comprehensive analysis of the TMI in few-mode amplifiers and offers a practical method to realize high-power high-brightness fiber lasers.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-power Yb-doped fiber laser technology has earned a solid reputation as a power-scalable laser concept with excellent beam quality and enabled a wide variety of applications ranging from industrial processing, medical treatment to scientific research [1–4]. Among the efforts toward high-power high-brightness fiber laser, the tandem pumping scheme is one of the promising approaches, owing to its intrinsic advantages of abundant injectable power [5], good thermal management [6,7], low gain for unwanted HOMs [8]. However, some detrimental effects, such as the SRS and TMI, make it challenging to achieve high output power while maintaining good beam quality [9–11].
Due to the low absorption cross-section of Yb ions at 1018 nm, a tandem pumped fiber laser usually employs long gain fiber for sufficient pump absorption, resulting in a relatively low SRS threshold. Lots of studies have been reported on SRS mitigation strategies in tandem pumping, such as optimizing gain fiber to improve pump absorption so that short fiber is allowed [12–14], and filtering the initial Stokes light of the seed laser [15,16]. However, for a 30 µm core of Yb-doped fiber (YDF), the SRS threshold is still limited to ∼5 kW level in forward tandem pumped amplifier [15–18]. To achieve higher output power, a common and practical method is to enlarge the fiber core for better SRS suppression. In this way, 10 kW- level forward tandem pumped fiber amplifiers have been demonstrated out of ∼ 50 µm fiber core [19–21]. However, the beam quality was not good due to the lack of mode control means in such large cores. Backward pumping permits good SRS suppression in relatively thin fiber cores and is therefore an effective approach for high power and good beam quality. In our previous work, we demonstrated a 5 kW backward tandem pumped fiber amplifier from 25/250 µm double-clad YDF. No SRS is observed and near single-mode beam quality is achieved at the maximum power [22].
TMI is another obstacle to high-power high-brightness lasers. Exceeding the TMI threshold, energy would dynamically transfer between transverse modes, resulting in an abrupt degradation in beam quality [23,24]. Although the physical origins of TMI are still controversial, it is believed to link with interference or energy coupling between two or more transverse modes in the gain fiber [25–27], and various practical strategies to mitigate TMI have been developed [28–37]. Among them, bending the gain fiber is the most convenient and economical method. Bending the gain fiber would change the loss of the HOMs [38–40], and subsequently change the interference features between transverse modes, thereby affecting the TMI threshold. Previous studies showed that tight bending benefits TMI suppression [32]. Later on, it is found that the bending may not apply to few-mode fibers. Several experimental phenomena indicated that the larger the bending diameter, the higher the TMI threshold [35,36]. However, there has been no detailed theoretical analysis that could explain the positive correlation between TMI threshold and bending diameter so far. Moreover, most experiments studying the TMI were conducted in a forward laser diode pumping scheme, investigations on the TMI effect in multi-kilowatt backward tandem pumping are more beneficial to the power scalability, which has never been reported before.
In this work, a theoretical model is built to analyze the impact of fiber bending and the evolution of HOMs on the TMI threshold. Theoretical calculations suggest that increasing bending diameters in single-mode lasers result in higher HOMs content and lower TMI thresholds. Conversely, for few-mode laser operation, larger bending diameters would lead to a higher proportion of HOMs while a higher TMI threshold. In the experiment, by increasing the bending diameter, we overcome the TMI barrier and achieve an 8.38 kW output power and beam quality M2 of 1.8 based on the backward tandem pumping scheme. Further power scaling is limited by available pump power.
2. Theoretical discussion
Fiber bending introduces mode-dependent bending losses. Previous studies showed that small bending diameters contribute to high bending loss of HOMs and thereby favor high output beam quality [41]. However, recent research indicated that tight coiling is more prone to TMI in few-mode fibers, resulting in an abrupt degradation in beam quality [35,36]. Since there is a contradiction between the effect of gain fiber bending on the TMI threshold and beam quality, it is necessary to understand the physical mechanism of the intrinsic link between them. In this section, a theoretical model is built to describe the TMI evolution under various bending diameters in few-mode running amplifiers.
2.1 Theoretical model
In most previous TMI calculations, the input signal laser is usually considered as ideal LP01 mode, while the impact of HOMs content from the signal laser on the TMI is ignored. In our model, based on the conventional STRS model [42], we further consider the coexistence of multiple modes in the gain fiber and introduce the transverse modes competition effect into the model.
Some assumptions are made to treat the impact of HOMs on TMI. Firstly, we assume that the operating laser power is below or near the TMI threshold, and the nonlinear coupling process does not affect the mode evolution of fiber amplifier. Secondly, since the modes walk-off time propagating through the gain fiber length is usually larger than the coherence time [43] in broadband fiber laser studied in this work, it is reasonable to assume that different inter-modes originating from the seed are incoherent [28]. In this case, the amplified HOMs only affect the excited-state population and the modes gain distribution while could not directly couple with the LP01 mode. The initial HOMs with Stokes frequency shift are assumed to come from the quantum noise. Finally, the thermal lensing effect is ignored for simplicity.
The excited-state population and power distribution of transverse modes along the gain fiber are given by
From Eqs. (1)–(3), the excited-state population and gain distribution can be obtained, which provides a basis for solving the TMI threshold using the STRS model [44]. Due to the largest nonlinear coupling gain between LP01 and LP11 modes, dynamic mode coupling would first occur between them [45]. Furthermore, since the bending loss of LP11o mode is smaller than that of LP11e mode, it is reasonable to only consider the power evolution of Stokes LP11o mode for simplicity. The total gain ${\tilde{G}_{stokes}}(\Omega )$ and output power ${\tilde{P}_{stokes}}(L)$ of Stokes LP11o mode can be expressed as
Here, $\varOmega $ is the Stokes frequency shift between LP11o mode and LP01 mode. $\chi $ is the nonlinear coupling coefficient in the form of Ref. [46], L is the fiber length, ${\tilde{P}_{stokes}}(0,\Omega )$ donates the Stokes LP11o mode power with a frequency shift of $\varOmega $ at the fiber position of z = 0. The power of the initial Stokes LP11o quantum noise is set at $4.6 \times {10^{ - 16}}$ W [45]. In the simulation, when the Stokes LP11o mode ${\tilde{P}_{stokes}}(L)$ increased to 5% of the total output power of all transvers modes ($\sum\limits_{k = 1}^6 {{P_k}(L)}$), the output power at this point is defined as the TMI threshold power.
2.2 Numerical simulations
Bending fiber affects the TMI threshold by modulating the HOMs loss. We'll start by investigating the effect of the mode loss coefficient on the TMI threshold before proceeding with bending. To simplify the calculation, we assume that the seed only contains LP01 and LP11o modes, with initial seed powers of 90 W and 10 W, respectively. This assumption is reasonable for high beam quality seeds in the experiment. Additionally, we define the loss of the LP01 mode as 0.001 dB/m, while the loss range of the LP11o mode is from 0 to 5 dB/m. The fiber parameters used in the simulation are shown in Table 1. The absorption/emission cross-sections employed come from the measured data of our homemade YDF.
Table 1. The main parameters of the gain fiber.
Figure 1 shows the TMI threshold and HOM proportion at TMI threshold power versus the LP11o loss coefficient for forward/backward pumping schemes. In general, as the LP11o loss increases, the TMI threshold first decreases and then increases, while the HOM proportion continues to decrease. When the LP11o mode loss coefficient ${\alpha _2}$ is over 0.5 dB/m, the total loss exceeds 20 dB due to a long gain fiber (∼40 m) used in the tandem pump. Consequently, the output HOM content nearly vanishes, leading to single-mode laser output. In this case, the TMI threshold gradually increases as ${\alpha _2}$ increases, which is in line with the traditional theory of TMI in single-mode fiber lasers [47–49].
Fig. 1. (a) The TMI threshold and (b) HOM proportion at TMI threshold power as a function of LP11o mode loss coefficient in the forward (blue) and backward (red) tandem pumped amplifiers.
When ${\alpha _2}<0.5$ dB/m, there exists a relatively high LP11o mode content in the amplifier, thus the laser is operating in the few-mode state. It should be noted that the LP11o mode here comes from the seed laser, which is considered incoherent with the LP01 mode and cannot directly participate in the formation of the thermal gratings [28]. However, it does compete with the Stokes LP11o mode (comes from quantum noise), thereby reducing the gain of Stokes LP11o mode. In this scenario, as ${\alpha _2}$ decreases, Stokes LP11o mode would obtain less gain, more pump powers are required to satisfy the dynamic coupling condition of TMI and thereby lifting the TMI threshold. The simulation results within this range agree with the latest research findings on the TMI threshold for few-mode fiber lasers [28,35,36,50,51].
Bending fiber is one of the simplest and most convenient ways to control HOMs loss. We calculated the bending loss of the 30/250 µm gain fiber (NA = 0.06) by the finite element method [39,40], as shown in Table 2.
Table 2. The bending loss of transverse modes at different bending diameters
Based on the mode-dependent bending losses, we are able to explore the TMI threshold related to bending. Figure 2 shows the TMI threshold and HOMs proportion at TMI threshold power versus bending diameters under forward/backward pumping schemes. The seed laser is set at 100 W. The initial LP01 mode accounts for 90% of the seed laser. The rest five HOMs account for 2% power respectively. For tight bending of 10 cm diameter, the output HOMs are well suppressed (see Fig. 2(b)) due to high bending loss. As the bending diameter increased, bending loss decreases and HOMs would obtain more gain. When the bending diameter is over 30 cm, bending loss is negligible and gain competition between transverse modes becomes stable, thus the output HOMs proportion almost stays unchanged. In addition, the HOMs proportion in backward pumping is higher than that in forward pumping for a fixed bending diameter. It is due to that HOMs contents would obtain more total gain in backward pumping cases, which has been demonstrated in [41].
Fig. 2. (a) The TMI threshold and (b) HOMs proportion at TMI threshold power as a function of bending diameters in the forward (blue) and backward (red) tandem pumped amplifiers.
TMI threshold almost shows the same variation trends as output HOMs proportion (see Fig. 2(a)). As the bending diameter increases from 10 cm to 40 cm, the TMI threshold increases from 2556 W to 3499 W in the forward pump and from 3424 W to 6232 W in the backward pump. It is worth noting that even under extremely small bending diameters (d = 10 cm), there is no increase in the TMI threshold. This seems to contradict the previous analysis that suggested high HOMs loss increases the TMI threshold. In fact, this is due to the LP11o mode loss (0.15 dB/m) still being too low to effectively suppress TMI at bending diameter of 10 cm. If the bending diameter is further reduced to 8 cm, the LP11o mode loss increases to 2.24 dB/m, which could meet the conditions for TMI suppression. However, the LP01 loss would also increase to 0.12 dB/m, and the efficiency of the laser would decrease sharply, making it challenging to achieve high power laser output while maintaining high efficiency. Therefore, for 30/250 µm NA = 0.06 gain fibers, it is impractical to suppress TMI by means of HOMs loss mechanisms. Instead, a more viable approach would be to increase the proportion of HOMs to improve the TMI threshold.
To further reveal the relationship between the gain competition and pumping directions, we examine the evolution of the Stokes gain and HOMs proportion along the fiber under forward/backward pumping cases, as shown in Fig. 3. The initial LP01 mode accounts for 90% of the seed laser. The rest five HOMs account for 2% power respectively. The bending diameters are 10 cm and 40 cm, respectively. It is shown that the local Stokes gain shows negative correlations with HOMs proportion in both pumping cases. In addition, a different Stokes gain response develops when the bending diameter changes in the forward and backward pumping scheme. As shown in Fig. 3(a), when the bending diameter increases from 10 cm to 40 cm in forward pumping, the local Stokes gain decreases, especially near the seed input end where the heat load is maximum. As a result, the total Stokes gain decreases by 22.2%, from 111.3 dB to 86.6 dB. In the backward pumping case (see, Fig. 3(b)), as the bending diameter increases from 10 cm to 40 cm, the local Stokes gain decreases rapidly all along the fiber, especially at the fiber end where the heat load is maximum. In this case, the total gain decreases by 35.3%, from 87.8 dB to 56.8 dB, which is a more pronounced gain reduction compared to forward pumping. Therefore, the difference in TMI thresholds for forward and backward pumping becomes progressively larger as the bending diameter increases, as shown in Fig. 2(a).
Fig. 3. The Stokes gain and HOMs proportion distribution along the gain fiber in the (a) forward and (b) backward tandem pumped amplifiers when the seed/pump powers are 100/2000 W with bending diameters of 10 cm and 40 cm.
The mode contents of the seed also affect the TMI threshold. Figure 4 shows the TMI threshold and the output HOMs proportion as a function of the bending diameter in the backward tandem pumped amplifiers with the different seed mode contents. When the seed laser is ideal LP01 mode, the TMI threshold is around 3112 W, almost independent of bending diameter. As the initial seed HOMs ratio increases, the positive correlations between the TMI threshold and the bending diameters are valid, while the TMI threshold at a certain bending diameter is significantly increased. At the maximum bending diameter of 40 cm, the TMI threshold increases from 6232 W to 10607 W when the initial seed HOMs proportion increases from 0.1 to 0.3. Despite the high output HOMs components when large bending diameter and low beam quality seed are employed (see Fig. 4(b)), the simulation results indicate that 10 kW level TMI-free fiber laser is possible based on the backward tandem pumped scheme.
Fig. 4. (a) The TMI threshold and (b) corresponding output HOMs proportion as a function of bending diameters in the backward tandem pumped amplifiers with initial seed HOMs proportion of 0, 0.1, 0.2, and 0.3 (each HOMs has the same power).
3. Experimental setup and results
Based on the feasible scheme given by theoretical calculations, we proceeded to the high-power setup. Figure 5 shows the schematic diagram of the backward tandem pumped laser system. The seed laser is a 100 W-level monolithic laser oscillator operating at ∼1080 nm. The output fiber of the seed laser has a core/cladding diameter of 15/130 µm. Aiming at suppressing the SRS effect, a homemade chirped and tilted fiber Bragg grating (CTFBG) is integrated after the seed. A mode field adaptor (MFA) is employed before the amplifier to reduce the transmission loss of the seed laser.
The gain medium of the amplifier is a homemade double-clad YDF. The core diameter of the gain fiber is 30 µm (NA = 0.06). The inner cladding diameter is 250 µm (NA = 0.46). The nominal pump absorption factor of the fiber is ∼0.37 dB/m@1018 nm. For sufficient pump absorption, about 40 m YDF is employed. The pump sources are five groups of power-combining-based 1018 nm fiber laser modules. Each module consists of seven 300 W level 1018 nm fiber lasers [52]. In that case, more than 10000 W pump laser can be produced. The pump laser is injected into the gain fiber through a homemade (6 + 1) × 1 backward pump and signal combiner (BPSC). The input and output signal ports of the BPSC are 30/250 µm fibers with core/clad NA of 0.06/0.46. Its six pump fiber ports have core/clad diameters of 135/155 µm with core NA of 0.22, which are identical to the pump sources’ pigtail fibers. The pump light coupling efficiency of the BPSC is more than 98%, and the insertion loss of the signal power is less than 1%. A cladding light stripper (CLS 1) is spliced in the front end of YDF to remove the residual pump laser. Another cladding light stripper (CLS 2) is employed after the BPSC to get rid of the unwanted cladding signal laser. Finally, a quartz block holder (QBH) is utilized for beam delivery. All the fiber components are actively cooled by the water chiller.
The output beam is collimated by a collimator (CO). Then, it is divided into two beams via a highly reflective mirror (HRM) with > 99.9% reflectivity. The low power transmitted laser goes into the laser quality monitor (LQM) for beam quality measurement. The high power reflected laser is collected by the power meter (PM). An optical spectrum analyzer (OSA) and a photodetector (PD) are used to detect the spectra and time domain data.
Limited by available water-cooled plate, we coil the gain fiber on a spiral water-cooled plate instead of a cylindrical one for heat dissipation. Therefore, we cannot control the bending diameter to a fixed value, but a certain range. To evaluate the impact of bending diameter on the laser performances, comparative experiments are systematically carried out. To be noted, in all cases, the signal laser is input from the position where the bending diameter is small, and the pump laser is injected reversely from the large bending port.
3.1 Laser performances with small bending diameters
First, the bending diameter of the signal input YDF is 10 cm, and gradually increased to 24 cm at the output end. The seed laser is operating at ∼100 W. To confirm the occurrence of TMI, we monitored the time-domain signal through a PD during the experiment. Figure 6 shows the time trace and corresponding Fourier spectra near the TMI threshold. At the operation of 3400 W, the temporal signal of the output laser is stable and no typical frequency peaks appear in the frequency domain, indicating that the TMI does not occur at this power level. When the output power increases to 3540 W, obvious temporal intensity fluctuations emerge and there are feature frequency components in the range of 0–4 kHz. Therefore, the TMI threshold for tight bending is around 3540 W.
Fig. 6. The time trace (inset) and corresponding Fourier spectra near the TMI threshold when the bending diameters of the gain fiber are 10–24 cm. (a) At 3400 W. (b) At 3540 W
Fig. 7. Laser output properties when the bending diameters of the gain fiber are 10-24 cm. (a) Output power and efficiency as a function of the pump power. (b) Evolution of the beam quality factor at different powers.
Figure 7(a) shows the output power and optical-to-optical (O-O) efficiency as a function of the pump power. The output power increases linearly to 4810 W when 5929 W pump lasers are injected, with an O-O efficiency of ∼80%. Interestingly, beyond the TMI threshold, no significant power roll over can be observed. It is mainly attributed to relatively large bending diameters (>15 cm) at the output end of the gain fiber, in which the bending loss of LP01, LP11o, LP11e mode is quite low, as shown in Table 2. In this case, HOMs arising from the TMI could transmit in the fiber core. Figure 7(b) shows the beam quality factor (M2) at different output powers. Thanks to the tight bending at the input end of the gain fiber, we obtained a near single mode seed laser (M2 = 1.3) from 30 µm fiber core. Below the TMI threshold, the M2 increases slowly with the output power. After the emergence of TMI, M2 experiences a sharp increase, degrading from 1.59 at 3750 W to 1.87 at 4810 W. The beam quality is likely to get worse if the output power continues to increase. To achieve excellent beam quality at high power, TMI must be suppressed.
3.2 Laser performances with moderate bending diameters
Then, the bending diameters of the YDF are increased to 24-32 cm from the input end to the output end. The seed laser is also set at 100 W. Figure 8 shows the time trace and its corresponding Fourier spectra near the TMI threshold. At the output power of 5440 W, the time trace is stable and no characteristic peak is observed in the Fourier spectra, indicating no TMI occurred. When the output power reaches 5530 W, the temporal signal fluctuation can be clearly observed, and the characteristic frequency components distribute in the range of 0∼4 kHz, indicating the occurrence of TMI. Therefore, by increasing the bending diameter of the gain fiber from 10-24 cm to 24-32 cm, we increase the TMI threshold from 3540 W to 5530 W.
Fig. 8. The time trace (inset) and corresponding Fourier spectra near the TMI threshold when the bending diameters of the gain fiber are 24 cm – 32 cm. (a) At 5440 W. (b) At 5530 W
Figure 9(a) shows the output power and O-O efficiency versus the pump power. The laser is amplified to 6010 W, with an O-O efficiency of ∼80%. Similarly, the O-O efficiency does not show an obvious decline beyond the TMI threshold, because the HOMs arising from TMI can be supported in 30/250 µm gain fiber under this bending conditions. Figure 9(b) shows the M2 evolutions at different output powers. The M2 increases slowly from 1.55 of the seed to 1.68 at 2000 W. Then it maintains well with the increment of the output power, varying from the 1.67@2990 W to 1.7@5440 W. After the occurrence of TMI, M2 increases dramatically, degrading from 1.71@5530 W to 2.0@6010 W.
Fig. 9. Laser output properties when the bending diameters of the gain fiber are 24–32 cm. (a) Output power and efficiency versus the pump power. (b) Evolution of the beam quality factor at different powers.
3.3 Laser performances with large bending diameters
Last, the bending diameters of the YDF are increased to 32-37 cm from the input end to the output end. Figure 10(a) shows the output power and O-O efficiency versus the pump power. The laser is finally amplified to 8380 W, with an O-O efficiency of ∼80%. Figure 10(b) shows the time trace and corresponding Fourier spectrum at 8380 W. No characteristic periodic fluctuation or typical frequency peaks can be observed, indicating that the TMI does not occur at this power level. Figure 10(c) shows the laser spectra. The 3 dB linewidth increases from ∼1.04 nm of the seed laser to ∼4.68 nm at 8380 W. The signal-to-Raman ratio is ∼40 dB at the maximum power, demonstrating a good SRS suppression of the system. Judging from the power curve with linear growth and output spectrum with a high signal-to-Raman ratio, there is great room for further power scaling if more powerful pump lasers are provided.
Fig. 10. (a) Output power and efficiency versus the pump power. (b) The time trace (inset) and corresponding Fourier spectrum at the maximum power. (c) Output spectra at operation power of 70 W and 8380 W
The beam quality M2 evolutions at different powers are depicted in Fig. 11(a). The M2 of the seed laser output from QBH is 1.6. During the power scaling process, the beam quality M2 is relatively well-preserved before the operation of 4000 W, ranging from 1.55 to 1.64. Then the M2 gradually increases from 1.6@4980W to 1.8@8380W (see Fig. 11(b)). A possible reason for beam quality degradation might be the high thermal load of the BPSC at high power level, which may cause beam distortion [53]. Compared to beam quality variation in relatively tight coiling cases (Fig. 7(b) and Fig. 9(b)), one can see that no abrupt beam quality deterioration appeared when the bending diameter is over 32 cm. But the cost is the worse overall beam quality before the TMI occurrence. How to balance the beam quality and TMI threshold would be the next goal of our research.
Fig. 11. (a) Beam quality factor M2 at different output powers. (b) Screenshot of the beam quality measurement software at the operation of 8380 W.
4. Conclusion
In conclusion, we have shown that the fiber bending diameter and mode content are critical parameters for TMI suppression. The calculations indicate that the impact of fiber bending diameter and mode content on the TMI threshold differs between single-mode and few-mode fiber lasers. For few-mode fiber lasers, the TMI threshold is positively correlated with the HOMs proportion. Therefore, increasing the bending diameter or decreasing the seed beam quality would increase the TMI threshold, albeit a consequential deterioration in the output beam quality, which is in good agreement with claims in Ref. [28,35,36,50,51]. In the experiment, when the bending diameters of the gain fiber were increased from 10-24 cm to 32-37 cm, the TMI threshold was increased from 3.54 kW to more than 8.38 kW. At the maximum power of 8.38 kW, the signal power was ∼40 dB higher than the Stokes light power and the beam quality factor M2 was 1.8. Further power scaling was limited by the available pump power. We hope this work can help gain a deeper insight into the physical mechanism of TMI and provide valuable guidelines for achieving high-power and high-brightness fiber lasers.
Funding
National Key Research and Development Program of China (2022YFB3606000); National Natural Science Foundation of China (62035015); Hunan Provincial Innovation Construct Project (2019RS3018); Innovative Research Groups of Hunan Province (2019JJ10005).
Disclosures
The authors declare 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.
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