2 Fiber characterization and experimental setup

The confined-doped fiber preform is fabricated through modified chemical vapor deposition (MCVD) in conjunction with the chelate gas deposition technique. The preform is subsequently drawn into optical fiber with the core and inner-cladding diameters of approximately 40 and 250 μm, respectively. The relative doping ratio is designed to be approximately 0.75 in order to balance the gain tailoring effect and nonlinearity suppression^[ Reference Wu, Li, Xiao, Huang, Yang, Pan, Leng and Zhou ^{30 ]}. The refractive index profile of the as-fabricated fiber is characterized and is shown in Figure 1(a), where the core diameter is approximately 39.82 μm and the diameter of the doped region is approximately 30.14 μm. Therefore, the relative doping ratio of the fabricated fiber is approximately 0.757, which is close to the design parameter. The numerical aperture (NA) of the fiber core is approximately 0.074. A cross-section photograph of the confined-doped fiber is shown in Figure 1(b). The inner cladding is an octagonal shape with a flat-to-flat diameter of approximately 250 μm. The cladding absorption coefficient of the fiber is measured to be approximately 0.4 dB/m at 1018 nm in the low-power regime.

Figure 1 (a) Refractive index profile of the fiber across the core region; (b) cross-section photograph of the confined-doped fiber.

The experimental setup of the tandem-pumped confined-doped fiber amplifier is based on a master oscillator power amplifier scheme, as depicted in Figure 2. The seed laser is a conventional fiber oscillator employing ytterbium-doped fiber (YDF) with a core/inner-cladding diameter of 20/400 μm and core NA of approximately 0.065. In this fiber, only the LP₀₁ and LP₁₁ modes can be supported. The reflectance values of the high-reflectivity fiber Bragg grating (HR FBG) and output coupling fiber Bragg grating (OC FBG) are more than 99.9% and approximately 10% at approximately 1080 nm, respectively. The YDF is approximately 14-m-long and is spirally coiled on a water-cooled heat sink at approximately 20°C, the coiling diameter of which gradually increases from 10 cm (close to the HR FBG) to 16 cm (close to the OC FBG). The seed fiber laser operates at approximately 1080 nm and can deliver up to approximately 300 W output power. The 1018 nm pump laser module is based on the 7×1 beam combining technique^[Reference Li, Xiao, Leng, Chen, Xu, Wu and Zhou^31], which consists of seven individual 1018 nm fiber oscillators. Each fiber oscillator exploits optical fiber with a core/inner-cladding diameter of 15/130 μm and is capable of delivering approximately 350 W output power. The seed laser and the pump laser are injected into the confined-doped fiber through a (6+1)×1 pump and signal combiner (PSC). The core/inner-cladding diameters of the PSC input and output pigtail fiber are 20/130 and 40/250 μm, respectively. The length of the confined-doped fiber is approximately 36 m. A cladding mode striper (CMS) is spliced after the confined-doped fiber to remove the cladding light. Finally, the laser is output through an end cap. The pigtail fibers of the CMS and the end cap are all based on germanium-doped fiber (GDF) with a core/inner-cladding diameter of 40/250 μm.

Figure 2 Experimental schematic of the confined-doped fiber amplifier (PSC, pump and signal combiner; YDF, ytterbium-doped fiber; CMS, cladding mode stripper).

3 Results and discussion

The seed laser power is set to approximately 70 W, and the output power from the confined-doped fiber amplifier as a function of the pump power is shown in Figure 3(a). The output power increases almost linearly with the pump power and reaches approximately 5.85 kW at the pump power of approximately 7.18 kW, corresponding to an optical-to-optical (O-O) efficiency of approximately 80.5%. The beam quality factor evolution of this confined-doped fiber amplifier as a function of the output power is shown in Figure 3(b). The beam quality factor first improves from 1.87 at approximately 500 W to 1.54 at approximately 3.21 kW. Then, the beam quality starts to degrade upon the onset of TMI at approximately 4.98 kW, reaching approximately 2.04 at the maximum output power. The output power could be further improved, but the beam quality shows dramatic degradation, making it less attractive for further power scaling.

Figure 3 (a) The output power and optical-to-optical efficiency as a function of the pump power; (b) the beam quality factor evolution as a function of the output power.

According to previous studies, increasing the power of the seed laser could lead to a higher TMI threshold^[ Reference Huang, Tao, Shi, Meng, Zhang, Ma, Wang and Zhou ^{32 ,} Reference Tao, Wang, Zhou and Liu ^{33 ]}. With the aim of maintaining good beam quality of the confined-doped fiber amplifier at a higher power level, the impacts of the seed laser power on the TMI threshold of this confined-doped fiber amplifier are experimentally investigated. The seed laser power is successively set to approximately 70, 110, 140, 180 and 220 W, respectively. The TMI threshold and the extracted power (defined as the power difference between the TMI threshold and the seed laser power) at different seed powers are presented in Figure 4. As the seed power increases from approximately 70 W to approximately 220 W, the TMI threshold of the fiber amplifier increases from approximately 4.98 to 7.38 kW, and the extracted powers are respectively approximately 4.91, 5.82, 6.60, 7.17 and 7.16 kW at the seed powers of approximately 70, 110, 140, 180 and 220 W, respectively. Therefore, as the seed laser power increases, the TMI threshold improves significantly. However, when the seed laser power reaches 180 and 220 W, the extracted power slightly decreases. In this case, further increasing the seed laser power would not contribute to a much higher TMI threshold.

Figure 4 The TMI threshold and extracted power as a function of seed laser power.

In previous studies, the seed laser power does not have such a remarkable effect on the TMI threshold^[ Reference Tao, Wang, Zhou and Liu ^{33 ]}. However, in the current study, the impact of the seed laser power seems very significant. Since the TMI threshold could be affected by a series of factors, such as (i) the seed laser power^[ Reference Huang, Tao, Shi, Meng, Zhang, Ma, Wang and Zhou ^{32 ,} Reference Tao, Wang, Zhou and Liu ^{33 ]}, (ii) the temporal stability/noise of the seed laser^[ Reference Hansen, Alkeskjold, Broeng and Lægsgaard ^{34 ]}, (iii) the linewidth of the seed laser^[ Reference Ren, Lai, Wang, Li, Song, Chen, Ma, Liu and Zhou ^{35 ]} and (iv) the mode purity of the seed laser^[ Reference Li, Huang, Xiang, Liang, Lin, Xu, Yang, Wang and Jing ^{10 ]}, in order to find the dominant influencing factor, comparative investigations are systematically carried out.

To make the comparative experiments rigorous and reliable, we need to change one of the output characteristics of the seed laser each time without affecting the rest. Consequently, the bending loss is introduced to the seed laser for power control while not affecting the temporal and spectral properties. The output pigtail fiber of the seed laser is coiled to introduce the bending loss, and a CMS is integrated after the seed laser to remove the cladding light caused by the fiber bending, as shown in Figure 5. Considering that for fiber lasers based on fiber with a core/inner-cladding diameter of 20/400 μm, bending the fiber not only affects the output power but also may affect the mode purity of the laser. Hence, it is necessary to analyze the loss of HOMs in different bending states.

Figure 5 The experimental schematic of the power controllable seed laser based on the bending loss mechanism.

To this end, the spatially and spectrally resolved imaging (S²) method^[ Reference Nicholson, Yablon, Ramachandran and Ghalmi ^{36 ]} is used to study the mode contents of the 20/400 μm fiber used in the seed laser at different bending diameters. In the measurement, a tunable fiber laser (TFL) and a charge-coupled device (CCD) are used, and the measurement system is shown in Figure 6. The fiber under test (FUT) is the same type as the fiber used for the fiber Bragg grating of the seed laser, and the length of its bending region is kept around approximately 1.1 m. The single-mode output pigtail of the tunable laser light source and the FUT is spliced with a slight core offset to excite the HOM components. Then the laser light output from the FUT is magnified by a 20× microscope objective (MO) lens and enters the CCD for spot intensity distribution measurement. In the experiment, the FUT is coiled with bending diameters of approximately 35, 20, 15 and 8 cm. Under each bending diameter of the FUT, the center wavelength of the tunable laser is gradually adjusted from 1075 to 1115 nm, with an interval of 0.1 nm, and the corresponding spot intensity distribution images are collected at each wavelength. The data processing method for the S² measurement results refers to Ref. [Reference Chen, Huang, Yang, Xi, An, Yan, Chen, Pan and Zhou37].

Figure 6 The experimental schematic of the S² measurement (TFL, tunable fiber laser; FUT, fiber under test; MO, microscope objective; CCD, charge-coupled device).

The measurement results at different bending diameters are shown in Figure 7. By reconstructing the peaks at different differential mode group delays (DMGDs), it is found the DMGD range from approximately 0.06 to 0.22 ps/m corresponds to the LP₁₁ mode, which is further used for calculating the multiple-path interference (MPI) of the LP₁₁ mode. When the bending diameter is 35 cm, the MPI intensity is –13.88 dB. Under this circumstance, the intensity of the MPI is mainly determined by the HOM generated by the offset splicing. As the bending diameter decreases to 15 cm, the corresponding MPI decreases to –19.08 dB, indicating the HOM power is decreasing. When the bending diameter is further reduced to approximately 8 cm, the MPI drops to –37.60 dB, at which point the light intensity of the LP₁₁ mode is too weak to reconstruct. The above measurement results show that as the fiber bending diameter decreases, the LP₁₁ mode content decreases significantly, and when the bending diameter is less than 8 cm, a proportion of the LP₁₁ mode is already negligible. Therefore, when changing the output power of the seed laser by bending the fiber, the HOM components in the seed laser can be simultaneously suppressed.

Figure 7 S² measurement results when the bending diameter of the FUT is (a) 35 cm, (b) 20 cm, (c) 15 cm and (d) 8 cm.

Firstly, the seed laser is set to approximately 180 W (the corresponding pump current is 3.5 A). By coiling the fiber to bending states A–D (as listed in Table 1, where the state N represents ‘no bending’), the output power is subsequently decreased to 140 W (bending state A), 110 W (bending state B), 90 W (bending state C) and 70 W (bending state D) by coiling the pigtail fiber of the seed laser. In this case, the operation state of the seed laser remains unchanged, and thus the influences of the seed laser temporal property and linewidth can be ruled out. As a result, the extracted power is measured to be 7.17, 5.42, 4.54, 4.55 and 4.53 kW, when the seed laser power is 180, 140, 110, 90 and 70 W, respectively, as shown in Figure 8. The corresponding pump power is approximately 8.88, 6.73, 5.63 and 5.63 kW , and the corresponding O-O efficiency of the fiber amplifier remains around 80.6%. The extracted power remains almost unchanged when the seed laser power is between 70 and 110 W, which means the TMI threshold increment almost equals the increased seed laser power.

Figure 8 Extracted power at different seed laser powers.

Table 1 Bending diameter/length and the seed laser power at different bending states.

Then, we keep the bending diameter (bending state B) fixed and change the output power by tuning the pump current of the seed laser. When the output power is respectively set to 110 W (pump current: 3.5 A), 180 W (pump current: 4.4 A) and 220 W (pump current: 5.0 A), the extracted power of the confined-doped fiber amplifier is corresponding to 4.54, 4.53 and 4.53 kW, respectively, as shown in Figure 8. In this case, the corresponding linewidths of the seed laser are 2.32, 2.57 and 2.78 nm. Therefore, there is almost no dependence of the TMI threshold on the seed laser linewidth in this broad linewidth fiber amplifier. According to the above results, the influence of the seed laser linewidth and output power on the dramatic TMI threshold change can be basically excluded in the current laser system.

Furthermore, the influences of the HOM components in the seed laser on the TMI threshold are theoretically analyzed. In previous studies, the higher the HOM contents in the seed laser, the lower the TMI threshold^[ Reference Ward, Robin and Dajani ^{38 –} Reference Tao, Wang and Zhou ^{40 ]}, where the HOM components were regraded to be coherent with the fundamental mode. The seed laser used in this work is a broadband oscillator, and it can be considered that the fundamental mode could only interfere with the HOM component (HOM-A) with a small frequency shift, thus affecting the TMI threshold. While the fundamental mode cannot interfere with the HOM component (HOM-B) with no frequency shift, the fundamental mode and HOM-B are superposed incoherent and amplified separately in the amplifier, as assumed by Chu et al. ^[ Reference Chu, Tao, Li, Lin, Wang, Guo, Wang, Jing and Tang ^{41 ]}.

Since only the LP₀₁ and LP₁₁ modes are supported in the pigtail fiber of the seed laser, only the above two modes are considered in the simulation. A theoretical mode for the confined-doped fiber amplifier is established based on the rate equations while considering the transverse spatial-hole burning^[ Reference Jiang and Marciante ^{42 ]}. In the simulation, it is assumed that the fiber amplifier operates below the TMI threshold. The nonlinear mode coupling caused by the thermally refractive index grating and mode interference between different transverse modes are neglected. In the proposed simulation, the population inversion in the steady state can be described as follows:

(1) \begin{align}&{n}_2\left(r,\phi, z\right)=\notag\\ &\quad n\left(r,\phi \right)\frac{P_\mathrm{p}(z){\varGamma}_\mathrm{p}\left(r,\phi \right){\sigma}_{\mathrm{ap}}{\lambda}_\mathrm{p}+\sum \limits_{k}{P}_{k}(z){i}_{k}\left(r,\phi \right){i}_{\mathrm{dope}}{\sigma}_{\mathrm{as}}{\lambda}_\mathrm{s}}{P_\mathrm{p}(z){\varGamma}_\mathrm{p}\left(r,\phi \right)\left({\sigma}_{\mathrm{ap}}+{\sigma}_{\mathrm{ep}}\right){\lambda}_\mathrm{p}+\frac{hc}{\tau }+\sum \limits_{k}{P}_{k}(z){i}_{k}\left(r,\phi \right){i}_{\mathrm{dope}}\left({\sigma}_{\mathrm{as}}+{\sigma}_{\mathrm{es}}\right){\lambda}_{\mathrm{s}}},\end{align}

where r and $\phi$ are the radial and azimuthal coordinates, z is the fiber position, n is the Yb³⁺ dopant concentration, P _p is the pump power, P_k is the power of the mode k (k = 1, 2, where 1, 2 represent the LP₀₁ and LP₁₁ modes, respectively), λ _s/p is the signal/pump laser wavelength, σ _as/ap and σ _es/ep are respectively the absorption and emission cross-sections of the signal/pump laser, h is the Planck constant, c is the velocity of light in vacuum, τ is the spontaneous emission lifetime of Yb³⁺, $\varGamma_{\mathrm{p}}$ is the overlap factor of the pump laser, i _dope is the doping distribution, which takes up 75% of the core diameter in the central part, and i_k is the normalized mode intensity distribution of mode k, which is acquired through the finite element method.

The gain of each mode is given by the following:

(2) \begin{align}g\left(r,\phi, z\right)=\left({\sigma}_{\mathrm{as}}+{\sigma}_{\mathrm{es}}\right){n}_2\left(r,\phi, z\right)-{\sigma}_{\mathrm{as}}n\left(r,\phi, z\right),\end{align}

(3) \begin{align}{g}_{k}(z)=\iint g\left(r,\phi, z\right){i}_{k}\left(r,\phi \right) r\mathrm{d}r\mathrm{d}\phi,\end{align}

where $g\left(r,\phi, z\right)$ is the transverse distribution of the fiber amplifier gain.

The propagation equation is given by the following:

(4) \begin{align}\frac{\partial {P}_{k}(z)}{\partial z}={g}_{k}(z){P}_{k}(z).\end{align}

Assuming the fiber amplifier operates below the TMI threshold, according to the theory proposed by Dong^[ Reference Dong ^{43 ]}, the total gain of HOM-A ( ${G}_\mathrm{A}\left(z,\varOmega \right)$ ) and the corresponding output power ( ${P}_\mathrm{A}(L)$ ) can be obtained as follows:

(5) \begin{align}{G}_\mathrm{A}\left(z,\varOmega \right)={g}_2(z)-2\operatorname{Im}\left[{C}_{21}\left(z,\varOmega \right)\right]{P}_1(z),\end{align}

(6) \begin{align}&\operatorname{Im}\left[{C}_{21}\left(z,\varOmega \right)\right]=\nonumber\\&{\iint}_{{r},\phi }{\psi}_1^{\ast }{\psi}_2\sum \limits_{v=0}^{\infty}\sum \limits_{l=1}^{\infty}\left\{\frac{k_0\left({\lambda}_\mathrm{s}/{\lambda}_\mathrm{p}-1\right)\varOmega \rho {C}_0{k}_\mathrm{T}{J}_{v}\left({k}_{vl}r\right)} {\left[{\left(\kappa {k}_{vl}^2\right)}^2+{\left(\varOmega \rho {C}_0\right)}^2\right]{\iint}_{r,\phi }{J}_{v}^2\left({k}_{vl}r\right){\cos}^2\left( v\phi \right)}\right.\nonumber\\&.\left[\cos (v\phi) {\iint}_{r,\phi}\frac{g}{g_0}g{\psi}_1{\psi}_2^{\ast }{J}_{v}\left({k}_{vl}r\right)\cos (v\phi)\right.\nonumber\\& + \left.\left.\sin (v\phi) {\iint}_{{r},\phi}\frac{g}{g_0}g{\psi}_1{\psi}_2^{\ast }{J}_{v}\left({k}_{vl}r\right)\sin (v\phi) \right]\right\},\end{align}

(7) \begin{align}{P}_\mathrm{A}(L)=\underset{-\infty }{\overset{\infty }{\int }}{P}_\mathrm{A}\left(0,\varOmega \right)\exp \left({\tilde{G}}_\mathrm{A}\left(\varOmega \right)\right) \mathrm{d}\varOmega,\end{align}

where C ₂₁ is the nonlinear coupling coefficient^[ Reference Li, Huang, Wu, Pan and Zhou ^{44 ]}, ${k}_\mathrm{T}$ , $\rho$ , ${C}_0$ and $\kappa$ are the thermal-optical coefficient, density, specific heat capacity and thermal conductivity of silica, respectively, ${g}_0$ is the small-signal gain, ${\psi}_{1(2)}\left(r,\phi \right)$ is the normalized electric field distribution of the LP₀₁₍₁₁₎ mode, $\varOmega$ is the Stokes frequency shift between HOM-A and the fundamental mode, which is negative and makes Im(C ₂₁) negative as well, and ${P}_\mathrm{A}\left(0,\varOmega \right)$ is the initial power of HOM-A, which originates from quantum noise and is approximately 10^–16 W^[ Reference Smith and Smith ^{45 ]}. When the power of HOM-A takes up 5% of the total output power^[ Reference Tao, Wang, Zhou and Liu ^{33 ]}, it is regarded that TMI occurs.

The seed power is set to 100 W, and the mode purity varies from 0.6 to 1. Other parameters are listed in Table 2.

Table 2 The main parameters of the fiber amplifier.

The simulation results are shown in Figure 9. As the mode purity of the seed laser decreases, the TMI threshold of the amplifier gradually increases, as shown in Figure 9(a): in the case of pure fundamental mode seed laser injection, the TMI threshold of the amplifier is approximately 3.60 kW, and the output laser is in the pure fundamental mode as well. When the mode purity of the seed laser drops to 0.6, the TMI threshold rises to approximately 6.16 kW (with ~4.77 kW fundamental mode power and ~1.39 kW LP₁₁ mode power). Compared with the pure fundamental mode seed laser case, the TMI threshold of the amplifier has increased by approximately 71.11%. At the same time, the output power of the fundamental mode laser is approximately 4.77 kW; compared with the pure fundamental mode injection case (~3.60 kW), the power of the fundamental mode has increased by approximately 32.50%. Moreover, as the mode purity of the seed laser decreases from 90.0% to 60.0%, the mode purity of the output laser decreases from 92.42% to 77.44%, but it is still higher than that of the seed laser owing to the gain-filtering effect, as shown in Figure 9(b).

Figure 9 (a) TMI threshold and the power of the fundamental mode upon the onset of TMI and (b) the mode purity of the output laser as a function of the seed laser mode purity (simulation results).

As validated by the simulation, the TMI threshold is significantly improved by introducing HOM content in the seed laser. On the one hand, the introduction of HOMs would inevitably compete with the fundamental mode. According to Equation (5), this will result in the power reduction of P ₁(z) and lead to a smaller total gain of HOM-A ( ${G}_\mathrm{A}\left(z,\varOmega \right)$ ). On the other hand, the introduced HOMs would directly compete with HOM-A, which will change the distribution of g(x,y,z) and reduce the extractable pump gain and nonlinear coupling gain of HOM-A. Figure 10 shows the simulation results of the pump gain distribution of HOM-A along the fiber when the seed laser mode purities are 0.6 and 1 under the pump power of 1 kW. For other parameters refer to Table 2. In this condition, the pump gain of HOM-A with the seed laser mode purity of 0.6 is always smaller than in the case of seed laser mode purity of 1, indicating a better suppression of the TMI effect. Therefore, both factors contribute to a higher TMI threshold.

Figure 10 The pump gain distribution of HOM-A along the fiber when the seed mode purities are 0.6 and 1.

It is to be noted that this is a preliminary theoretical analysis of the TMI threshold in a confined-doped fiber amplifier seeded by laser with a certain amount of HOM content. The actual mode dynamics could be much more complicated. For instance, more HOMs could be excited at the splicing point (resulting from core NA mismatch/core offset splicing between the passive fiber and the confined-doped fiber). Therefore, other HOMs could also compete with HOM-A, and when the TMI occurs, besides the mode coupling between the fundamental mode and LP₁₁ mode, there could be mode coupling between the fundamental mode and other HOMs, and even mode coupling between HOMs. Therefore, the current simulation could reveal the possible mechanism for the dramatic change in TMI threshold, but differences exist between the simulated and experimental results.

Moreover, we are aware of the reported experimental study by Chu et al. ^[ Reference Chu, Tao, Li, Lin, Wang, Guo, Wang, Jing and Tang ^{41 ]}. In that work, by introducing HOM content into the fiber amplifier system through offset splicing, the TMI threshold was increased from approximately 3.0 to 3.5 kW, where the static beam quality degradation first appeared before the onset of TMI. However, in the current work, the beam quality is not degraded before the TMI occurs. According to the simulation results in Figure 9(b), the mode purity of the output laser from the amplifier is 77.44%, 81.79%, 86.65% and 92.42% when the corresponding mode purity of the seed laser is 60.0%, 70.0%, 80.0% and 90.0%, respectively. Therefore, the mode purity of the output laser is improved compared with the seed laser owing to the gain-filtering effect of the confined-doped fiber. Consequently, the confined-doped fiber amplifier could take full advantage of the seed laser control technique, that is, introducing HOM contents into the amplifier for TMI mitigation without seriously degrading the beam quality.

According to the above experimental and simulation results, the seed power of 180 W (without bending the seed laser pigtail fiber) is used for further power amplification. Under this circumstance, the output power as a function of the pump power of the fiber amplifier is as shown in Figure 11. The maximum output power of 7.49 kW is obtained at the pump power of 9.07 kW, which is the highest output power ever reported in a forward pumped confined-doped fiber amplifier. The O-O efficiency of the fiber amplifier is generally stable, which is approximately 80.6% at the maximum output power. The normalized output spectra at different output powers are depicted in Figure 11(b), and the 3-dB linewidth of the output laser increases from 2.32 nm of the seed laser to 4.08 nm at the maximum output power. The wing-like broadening of the output spectra is due to the intermodal four-wave mixing. Asymmetric spectral broadening is also observed, and the broadening in the long-wavelength side is much more severe, which results from the temperature rise induced red-shifting within the fiber. The signal-to-Raman ratio of the output spectrum at the maximum output power is approximately 31 dB. Therefore, further power scaling is limited by the stimulated Raman scattering (SRS).

Figure 11 (a) Output power as a function of the pump power; (b) output spectra at different output powers.

The beam quality factors of the fiber amplifier at different output powers are shown in Figure 12. The beam quality factor is M ² = 2.22 at the output power of approximately 500 W, and is improved to M ² = 1.75 at the output power of 4.98 kW owing to the gain-filtering effect of the confined-doped fiber. Then, the beam quality factor remains around 1.8 during the power scaling process and reaches 1.83 at the output power of 7.49 kW.

Figure 12 The beam quality factor evolution as a function of the output power.