1. Introduction

High-power Yb-doped fiber lasers with diffraction-limited beam quality have attracted much attention because of their wide applications such as cutting, melting, material processing, beam combining, etc. In last two decades, the near-diffraction-limited Yb-doped fiber laser has experienced a rapid development and its output power has been up-scaled from several Watts to 20 kW [1–5].

In spite of that, its power scalability is also attractive, because majority of its applications are determined by its output power. In the high-power fiber lasers, the master-oscillator power amplifier (MOPA) configuration is generally utilized, the power scalability of which is mainly determined by the fiber amplifier. In 2008, J. Dawson et al. analyzed the power scalability of diffraction-limited fiber lasers and amplifiers. It was predicted that with the pump wavelength around 976 nm, the power limit of diffraction-limited fiber lasers should be about 36 kW which is determined by four physical factors, i.e., pump brightness (PB), surface damage, thermal lens (ThL) and stimulated Raman scattering (SRS) [6]. In the literature, a numerical model was also presented which set a theoretical frame for studying the power scalability of fiber lasers and amplifiers. With the help of this model, the power scalability of 1018-nm tandem-pumped fiber lasers and amplifiers was also studied in [7].

Later, another thermal effect, known as the transverse mode instability (TMI), is considered as an additional physical limitation, because it will cause the failure of diffraction-limited operation when the output power is large enough [8–20]. In 2016, an analytic formula was reported for predicting the TMI-limited output power [16], which was derived based on the stability analysis of the fundamental mode amplification in fiber amplifier [17]. By using this formula, it was predicted that the power limit was reduced to 28 kW for 976-nm laser diode (LD) pumping and 52 kW for 1018-nm tandem pumping because of the TMI limitation [18–20].

In aforementioned literatures, the great majority of studies were focused on the 976-nm LD pumping and 1018-nm tandem-pumping conditions. With the development of fiber lasers, the wavelength of tandem-pumping source can cover the range from 976 nm to 1050 nm or even longer with the brightness sufficiently improved compared with the LD source [21–23]. Then, the dependence of power limit on the pump wavelength becomes attractive. [24] developed the numerical model given in [6] and presented a wavelength-dependent model. Different from the model in [6], the active fiber length was not an independent variable, but the function of pumping and signal wavelengths in the wavelength-dependent model [24]. With the wavelength-dependent model, it was revealed that the power limit of diffraction-limited fiber amplifier can be elevated by lengthening the pumping wavelength, but larger core was required to achieve the power limit. However, it was pity that the TMI limitation was not taken into account in [24]. Moreover, although some qualitative discussions have been given in [24], the specific requirements on the active fiber (i.e., DCYF) for achieving the power limit or certain target power, which are of great importance for designing high-power fiber lasers and amplifiers, are still not so clear.

Therefore, in this paper, we focus our study on the requirements needed for up-scaling the output power of diffraction-limited Yb-doped fiber amplifiers by considering both the ThL and TMI effects. With the wavelength-dependent model, the specific requirements on DCYF for achieving the power scaling limit and the 20-kW target output power are numerically studied in detail. It can be found that these requirements of DCYF are strongly dependent on the pump wavelength and brightness. The effect of signal wavelength on these requirements will also be investigated. This paper is arranged as follows: The numerical model will be briefly introduced in Section 2. In Section 3, the requirements on DCYF to achieve the power limit and 20-kW target output power will be discussed based on the numerical model. Both the ThL- and TMI-limited cases will be considered in this section. The conclusions will be summarized in Section 4.

2. Introduction of numerical model

The physical factors limiting the power up-scaling of fiber lasers can be summarized as the PB, nonlinear effects, optical damage and thermal effects [6,20,24]. The nonlinear effects include the SRS and SBS effects which give the power limitations to non-narrow-band and narrow-band fiber lasers, respectively. In this paper, we focus our study on the SRS effect because of the better power scalability of non-narrow-band fiber laser [6]. The thermal effects consist of the thermal fracture, melting, ThL and TMI. In former studies, it has been revealed that the power scalability of Yb-doped fiber amplifier should be determined by two latter effects [6,20,24]. Therefore, we only take the ThL and TMI effects into account. Moreover, the fiber optical damage is mainly determined by the surface damage of fiber end with the intensity of laser beam beyond the damage threshold [6]. Nevertheless, by using the high-power end-cap, the output beam can be rapidly expanded and dramatically lower the intensity on the output surface of end-cap. Therefore, it is assumed that the end-cap can make sufficient beam expansion to ensure the intensity lower than the damage threshold. Then, the fiber damage can also be neglected. Based on above considerations, the power limitations studied in this paper are the PB, SRS, ThL and TMI. The PB- and SRS-limited powers can be given as [6]

Here, the analytic formula presented in [19] is adopted to predict the TMI-limited power. Because of the similarity of formulas, the ThL- and TMI-limited powers can both be given as [6,19,20]

By calculating the C₁ value, it is noteworthy that the C₁ cannot be always smaller than 1, which implied that the TMI should not be always more preponderant than the ThL effect in limiting the power of fiber amplifiers. It is found that the TMI can only be dominant with the normalized frequency (V value) in the range from 2.405 to 4.85 (noting that the TMI cannot happen in the single-mode fiber, see Appendix A), and otherwise, the ThL effect will become the dominant limitation instead. Therefore, in the following discussions, the ThL- and TMI-limited cases will be considered separately.

Moreover, by studying the variation of C₁ with the core radius (see Appendix A), it is also found that the variation of C₁ is negligible when the core diameter is larger than 10 µm. It should be noted that in the practical high-power fiber amplifiers (multi-kW or even more), the core diameter of DCYF is generally larger than 10 µm. Then, a good approximation can be made to take C₁ as a constant in the model. In spite of that, it should be noted that the constant value of C₁ should be also vary with the value of normalized frequency V. For example, the C₁ value is about 0.6 with the V value equal to 3 [18,20], and about 84% with the V value equal to 4 (see Appendix A). In the following discussions, the value of C₁ will be taken as 0.6 for the TMI-limited case, as was done in [18,20], which can bring convenience to make comparison with the results in [18,20]. Moreover, the coefficient C₁ will also be kept in the following derivations to reveal the effect of TMI on the power limits and pertinent requirements.

Till now, the formulas for predicting four power limits have been given. In spite of that, the dependences of these power limits on the pump and signal wavelengths are still not clear. In order to give the wavelength-dependent model, the function of active fiber length should also be introduced [24], i.e.,

Table 1. Parameters used in the calculation

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3. Results and discussions

3.1 Power limits and pertinent requirements

Firstly, we would like to analyze the variation of power limit with the pump and signal wavelengths. Here, the considered range of signal wavelength is from 1060 nm to 1090 nm, because out of the range, the amplified spontaneous emission will become another factor limiting the power of signal light, which is not included in the numerical model [21]. The range of pump wavelength considered here is from 900 nm to 1050 nm. As discussed in [6,20], the power limit is determined by the SRS and thermal effects (either ThL or TMI), and can be given as

Here, it should be noted that the power limit is related to pump and signal wavelengths because of the existence of η_heat determined by the quantum defect. It can also be seen that the power limit is proportional to the value of √C₁, which shows the TMI effect on the power limit. Figure 1 gives the variation of power limit with the pump and signal wavelengths in the ThL-limited case. The pertinent value in the TMI-limited case can be easily obtained by multiplying the factor of √C₁. In the calculation, the value of G is also assumed as 10 with the consideration of serious gain saturation that may be induced by such high signal power in the active fiber [6], although the assumption maybe a little conservative in fiber amplifiers of relatively lower power level (e.g., multi-kW level). In spite of that, the effect of G can also be easily predicted with Eq. (6). It can be found that with G increased from 10 to 100, the power limit will increase to √2 times.

 

Fig. 1. Variation of power limit with the pump wavelength corresponding to various signal wavelengths in the ThL-limited case.

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From Fig. 1, it can be seen that the power limit increases monotonously with lengthening the pump wavelength or shortening the signal wavelength, because of the reduction of quantum defect and η_heat. It is surprising to find that with the smallest η_heat (corresponding to the 1050-nm pump and 1060-nm signal wavelengths), the power limit can be as large as 108.5 kW in the ThL-limited case (C₁= 1) and 84 kW in the TMI-limited case (C₁= 0.6). From Fig. 1, it can also be found that the effect of signal wavelength becomes weaker with the shortening of pump wavelength. With the pump wavelength shortened to 976 nm, the power limit only increases from ∼33/26 kW to ∼37/29 kW in the ThL/TMI-limited case, when the signal wavelength is shortened from 1090 nm to 1060 nm.

3.1.1 Requirement on the core diameter

Then, we would like to discuss the requirement of core diameter (i.e., 2a) to achieve the power limit, because the core diameter plays an important role in power up-scaling of diffraction-limited fiber amplifiers [6,20,24]. On the one hand, the core diameter can determine the power limit by affecting the SRS, ThL and TMI effects (see Eqs. (2) and (3)) [6,20]. On the other hand, the core diameter is also related to the transverse mode controlling for the diffraction-limited operation of fiber amplifiers, because it is well known that the smaller the core diameter is, the more easily the transverse modes can be controlled for diffraction-limited operation. In this view, it is hoped that the power limit can be achieve with the core diameter as small as possible. Here, instead of the core radius, the core diameter is used in the following discussions, because it is more generally used in the designing of DCYFs.

From [6,20], it can be known that to achieve the power limit, the large-enough core diameter should be required. Then, with Eq. (1), the smallest core diameter needed for achieving the power limit (defined as the required core diameter) can be given as

Here, it should be noted that the form of Eq. (7) is different from the pertinent formula given in [6] (see Eq. (18) in [6]) by introducing the cladding-to-core area ratio R_Cl__Co of DCYF. Equation (7) indicates that the required core diameter is inversely proportional to the square root of the term of (I_pumpR_Cl__CoNA²).

Then, Eq. (7) seems to reveal some requirements for achieving the power limit. For example, the 30-µm core diameter seems applicable for achieving the 108.5-kW largest power limit corresponding to the 1050-nm pump and 1060-nm signal wavelength with the 600-µm-diameter and 0.445-NA inner-cladding, if the pump brightness can be about 0.77 W/µm²/sr which is not too large for the tandem-pumping source. However, it should be addressed that these requirements obtained with Eq. (7) are not sufficient to guarantee the achievement of power limit, because besides the core diameter, the requirement on the active fiber length is still needed [6]. Corresponding to the required core diameter given by Eq. (7), the required active fiber length should be given as [6]

Here, it should be noted that the active fiber length should also satisfy Eq. (5). Thus, together with Eq. (5), we can get the required cladding-to-core area ratio (see Appendix B), i.e.,

From Eq. (9), it can be found that the required cladding-to-core ratio $R_{Cl\_Co}^{opt}$ is related to the PB, inner-cladding NA and dopant concentration, but not related to the coefficient C₁. Thus, $R_{Cl\_Co}^{opt}$ should have the same value for both the ThL- and TMI-limited cases. Substituting Eq. (9) into Eq. (7), the required core diameter can be calculated. Here, it should be reminded that the cladding-to-core ratio should not be smaller than 1. Thus, if the $R_{Cl\_Co}^{opt}$ value calculated with Eq. (9) is smaller than 1, it should be given as 1 instead in the calculation.

Figure 2 gives the variations of required core diameter (2a_opt). Here, the case of 1060-nm signal wavelength is firstly considered in which the largest power limit can be achieved. It can be found that the minimum value of required core diameter is present at the pump wavelength of 976 nm corresponding to the largest absorption cross section of Yb-ion [27]. Then, the variation of required core diameter can be divided into two regions. The first one corresponds to the pump wavelength no longer than 976 nm. In this region, the largest power limit (see Fig. 1) can be achieved with the smallest core diameter at the pump wavelength of 976 nm. As mentioned above, the smaller core can be more helpful for transverse mode controlling for diffraction-limited beam quality. Thus, the 976 nm should be the optimum pump wavelength in this region.

 

Fig. 2. Contour plots of core diameter with the coordinates of pump wavelength and brightness under different dopant concentrations. The signal wavelength is 1060 nm. The values of dopant concentration are 2.5 × 10²⁵m^-3 (a), 5 × 10²⁵m^-3 (b), 7.5 × 10²⁵m^-3 (c) and 10²⁶m^-3 (d), respectively. The red star in (d) marks the required core diameter (about 36.5 µm) corresponding to 1018-nm pump wavelength and 20-W/µm²/sr PB.

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The other region corresponds to the pump wavelength longer than 976 nm. In this region, it can be found that the core diameter increases monotonously with the lengthening of pump wavelength, although the power limit also increases, correspondingly (see Fig. 1). It is indicated that in this region, the increment of power limit with lengthening the pump wavelength can only be achieved with the cost of enlarging the core diameter which makes it more challenging to obtain the diffraction-limited beam quality.

It can also be seen that the required core diameter can be reduced by enlarging the PB and dopant concentration. In spite of that, such reduction of core diameter is limited. Figure 2(d) shows that even with the PB of 20 W/µm²/sr and dopant concentration of 10²⁶m^-3, the required core diameter is also as large as 103.8 µm to achieve the largest 108.5-kW power limit with the 1050-nm pump wavelength.

From Fig. 2, it is astonishing to see that with the 976 nm pump wavelength, the 37-kW power limit can be achieved with the 20-µm core diameter, as long as the pump brightness can be large enough. It is required that the PB should be larger than 6.44 W/µm²/sr with the 5 × 10²⁵-m^-3 dopant concentration, and larger than 2.92 W/µm²/sr with the 10²⁶-m^-3 dopant concentration. Such brightness is too large for the LD pump source, but can be achieved with the fiber laser as the tandem-pumping source. Nowadays, the kW-level fiber laser operating near 980 nm has been demonstrated, and 1.1 kW output power was achieved with the Yb-doped fiber with the 105-µm-diameter 0.12-NA core [28]. The brightness is about 2.81 W/µm²/sr which is very close to the requirement. The higher brightness can also be expected by optimizing the fiber lasers near 980 nm [29]. Moreover, if the bi-directional pumping scheme can be utilized, the brightness of the fiber laser near 980 nm is sufficient enough to meet the requirement with the 10²⁶-m^-3 dopant concentration (the requirement of pump brightness is lowered to 1.46 W/µm²/sr with bi-directional pumping scheme). It is implied that the fiber lasers near 980 nm should be an important option as tandem-pumping source for up-scaling the output power of diffraction-limited fiber amplifiers, especially considering the smaller core diameter needed for achieving the diffraction-limited beam quality.

Another attractive pump wavelength is the 1018 nm which is used as the tandem-pumping wavelength in the IPG’s 10-kW near-diffraction-limited fiber laser [30]. From Fig. 2, it can be found that to achieve the pertinent power limit (about 53 kW corresponding to the 1018-nm pump wavelength, see Fig. 1), the core diameter should be larger than 36.5 µm even with the 10²⁶-m^-3 dopant concentration and 20-W/µm²/sr PB (see the red star mark in Fig. 2(d)). It means that the active fiber with 37-µm core diameter should be promising to achieve the 53-kW output power by using 1018-nm pump wavelength as long as the pump brightness and dopant concentration are large enough. In spite of that, it should be noted that the required core diameter will be enlarged to 40, 45, and 55 µm when the dopant concentration decreases to 7.5 × 10²⁵, 5 × 10²⁵ and 2.5 × 10²⁵m^-3, respectively (see Figs. 2(a)–2(c)). Thus, it can be concluded that besides the high PB, the high dopant concentration is still indispensable for reducing the required core diameter.

Figure 3 gives the variations of required core radius corresponding to various signal wavelengths. Here, the 10²⁶-m^-3 dopant concentration is considered in the case of which the smallest core diameter can be required. From Fig. 3, it can be found that the effect of signal wavelength is much weaker than the pump wavelength, especially when the pump wavelength is shorter than 980 nm. Taking the 976 nm for example, to achieve the power limit with 20-µm core diameter, the required PB reduces from 2.92 W/µm²/sr to 2.27 W/µm²/sr with the signal wavelength lengthened from 1060 nm to 1090 nm.

 

Fig. 3. Contour plots of required core diameter with the coordinates of pump wavelength and brightness under different signal wavelengths. The signal wavelengths are 1070 nm (a), 1080 nm (b) and 1090 nm (c), respectively.

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With further lengthening the pump wavelength, the effect of signal wavelength can become stronger. It can be found that the smaller core radius is required with the increment of signal wavelength. For example, with the PB of 20 W/µm²/sr and 1050-nm pump wavelength, about 103.8-µm core diameter is required to achieve the power limit (about 108.5 kW) with 1060-nm signal wavelength, while about 58.4-µm core diameter is required to achieve the power limit (about 56.6 kW) with the 1090-nm signal wavelength. Such difference of core diameter requirements is partly induced by the decrement of power limit with lengthening the signal wavelength (see Eq. (7)), and partly induced by the difference of gain characteristics with these signal wavelengths (see Eqs. (9) and (10)). Thus, from Fig. 3, it can be known that the smaller core diameter is required with the longer signal wavelength, although the pertinent power limit also decreases, correspondingly.

Above results on the core diameter requirements are obtained by considering the ThL-limited case (C₁= 1). In spite of that, from Eqs. (6) and (7), it can be known that the TMI affects these requirements of core diameter by simply changing the power limit. Therefore, the required core diameters obtained in the ThL-limited case can be simply transferred to the ones in the TMI-limited case by multiplying a factor of (C₁)^1/4 (about 0.88 for C₁= 0.6). Thus, in the TMI-limited case, the required core diameter should become smaller, because the pertinent power limit is lower than the ThL-limited case. Here, it should be reminded that C₁ should be smaller than 1 in the TMI-limited case.

3.1.2 Requirement on the inner cladding diameter

In this subsection, the requirement on inner-cladding diameter of DCYF will be investigated. In the numerical model given in [6], the inner-cladding diameter is not included explicitly, where the Yb-doped fiber length is taken as an independent variable. However, with the wavelength-dependent model used here, the requirement of the inner-cladding diameter can be revealed, because the cladding-to-core ratio R_Cl__Co is taken as an independent variable by introducing the function of active fiber length given by Eq. (5). Then, by studying the requirement of the cladding-to-core ratio, the required inner-cladding diameter can be given. It should be noted that the inner-cladding diameter and cladding-to-core ratio should not be too small in order to couple sufficient pump light. Therefore, it is hoped that the required inner-cladding diameter or cladding-to-core ratio can be large enough in the practical design of DCYF.

Besides the inner-cladding diameter, there is still another parameter related to the pump light coupling, i.e., the NA of inner-cladding (see Eq. (1)). Nevertheless, the NA of inner-cladding is mainly determined by the materials of inner- and outer-cladding, which makes it difficult to change the NA of inner-cladding arbitrarily. Thus, we mainly discuss the requirement of inner-cladding diameter in this subsection with the NA kept as a constant of 0.445 (see Table 1) that was also used in former studies [6,24]. The effect of NA will be discussed in brief at the end of Section 3. From Eq. (7), it can be known that the required inner-cladding diameter can be given as

It is interesting to find that the required inner-cladding diameter is not related to the dopant concentration. However, it should be noted that it is still related to the pump and signal wavelengths because of involving the wavelength-dependent power limit (see Eq. (6)). Then, the required inner-cladding diameter should be enlarged by lengthening the pump wavelength or shortening the signal wavelength which can elevate the power limit, because more pump light needs to be coupled for higher power limits. Equation (12) also shows that the larger inner-cladding diameter is needed with the lower PB I_pump in order to couple enough pump power.

Figure 4 gives the variations of required inner-cladding diameter. From Fig. 4(a), it can be found that to achieve the largest 108.5-kW power limit, the required inner-cladding diameter should be also about 117.8 µm which is close to the value of required core diameter (about 103.8 µm, see Fig. 2(a)). It means that the required cladding-to-core ratio should be only about 1.29 which is very close to 1. Here, it should be noted that the dopant concentration of 10²⁶m^-3 is used. If the dopant concentration is reduced to 5 × 10²⁵m^-3, the required cladding-to-core ratio should be further reduced to 1 (see Eq. (9)), which means that the core-pumping scheme should be used. However, it should be noted that with the core-pumping scheme, the core NA should be as large as the inner-cladding NA (i.e., 0.445) to couple enough pump power. Then, the core-pumping requirement dose not only bring the difficulty in the fabrication of DCYF with so large doped core, but also bring the difficulty in the pump light coupling in a practical fiber amplifier. Moreover, it is still very difficult to achieve the diffraction-limited beam quality with such core-pumping scheme. Therefore, the core-pumping requirement is not so applicable for the high-power diffraction-limited fiber amplifiers. It is also suggested that besides the required core diameter and active fiber length given by Eqs. (7) and (8), the inner-cladding and cladding-to-core ratio should also be carefully considered in the designation of a practical fiber amplifier.

 

Fig. 4. Contour plots of required inner-cladding diameter with the coordinates of pump wavelength and brightness corresponding to the signal wavelengths of 1060 (a), 1070 (b), 1080 (c) and 1090 nm (d), respectively.

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In spite of that, even with the dopant concentration of 10²⁶m^-3, the required cladding-to-core ratio (i.e., 1.29) is also very small which still makes the pump light coupling very challenging. One way to enlarge the cladding-to-core ratio is further increasing the dopant concentration (see Eq. (9)). However, this way is not so effective because of the limitation of dopant concentration in the fabrication of Yb-doped fiber. Another way is to lower the PB (see Eq. (9)) with the cost of enlarging the core diameter. From Figs. 4(a) and 2(a), it can be known that the required inner-cladding diameter increases to 200 µm with the core diameter increased to about 127.6 µm, if the required pump brightness is lowered from 20 to 6.95 W/µm²/sr. These results remind us that, although the increment of PB is helpful for reducing the required core diameter, the cladding-to-core ratio and inner-cladding diameter will also be lowered, corresponding, which may bring the difficulty in coupling the pump light. Thus, the core diameter, inner-cladding diameter and PB should be balanced very carefully with the consideration of state-of-art pump coupling technology.

Figure 4(d) gives the variation of required inner-cladding diameter with the 1090-nm signal wavelength. In this case, the largest power limit is reduced to 56.6 kW with the 1050-nm pump wavelength and the 58.4-µm core diameter is required (see Fig. 3(c)). Then, in order to achieve this power limit, the required inner-cladding diameter is about 85 µm. Although the corresponding value of cladding-to-core ratio is larger than 2 and the core-pumping scheme can be avoided, it is still very challenging to couple enough pump light into such a small inner-cladding. If the required inner-cladding diameter increases to 200 µm, the core diameter should be increased to about 84.3 µm with the PB no smaller than 3.61 W/µm²/sr.

By comparing Fig. 4 with Fig. 2, it can be found that with shortening the pump wavelength, the decrement of required core diameter is faster than that of required inner-cladding diameter in the region of pump wavelength larger than 976 nm. It is indicated that the required cladding-to-core ratio is also enlarged with the shorter pump wavelength in this region. Taking the 1018-nm pump wavelength for example, with the 200-µm inner-cladding diameter, the required core diameter is reduced to about 55.2 µm and 50.3 µm with the PB values of 3.39 W/µm²/sr and 2.7 W/µm²/sr corresponding to the 1060-nm and 1090-nm signal wavelength, respectively. Then, the cladding-to-core ratio is enlarged to 13.1 and 15.8, respectively. Here, it should be noted that with the 1018-nm pump wavelength, the power limit is also reduced to 53 kW and 42 kW corresponding to the 1060-nm and 1090-nm signal wavelength, respectively.

By comparing Figs. 2 and 4, it can also be found that the largest cladding-to-core ratio is required at the pump wavelength of 976 nm. Take the 1060-nm and 1090-nm signal wavelengths for example. It can be found that for the 1060-nm signal wavelengths, 20-µm core diameter and 181.2-µm inner-cladding diameter with the PB of 2.92 W/µm²/sr are required to achieve the 37-kW power limit, and for the 1090-nm signal wavelength, 20-µm core diameter and 194.4-µm inner-cladding diameter with the PB of 2.27 W/µm²/sr are required to achieve the 33-kW power limit. It means that the required cladding-to-core ratio is increased to 82.1 and 94.5, respectively.

Here, it should be reminded that although the required inner-cladding diameter is independent to the dopant concentration, the required cladding-to-core ratio is related to the dopant concentration. From Eq. (9), it can be known that the required cladding-to-core ratio becomes smaller with the reduction of dopant concentration. This result is reasonable because with the reduction of dopant concentration, the smaller cladding-to-core ratio is required to obtain the strong-enough pump absorption. Correspondingly, the larger PB is also required. For example, with the 976-nm pump and 1060-nm signal wavelengths, the required cladding-to-core ratio is reduced to 37.1 (corresponding to 20-µm core diameter and 121.9-µm inner-cladding diameter) with the PB of 6.45 W/µm²/sr, if the dopant concentration is reduced to 5 × 10²⁵m^-3.

At last, we would like to make some discussion on the TMI-limited case. Similar to the case of core diameter, the TMI effect on the required inner-cladding diameter is also induced by changing the power limit (see Eq. (6)). Therefore, the requirements on the inner-cladding diameter obtained with the ThL-limited case can be easily transferred to the ones in the TMI-limited case by multiplying the factor of (C₁)^1/4. It should also be reminded that the cladding-to-core ratio shares the same requirement in both of the ThL- and TMI-limited cases (see Eq. (9)).

3.2 Requirements for achieving a target power of 20 kW

In Section 3.1, we focused our discussion on the requirements to achieve the power limit. However, it should be noted that the power limit varies with the pump and signal wavelengths. In the designation of a practical fiber laser, it is a target output power that is generally given at first, and then, the requirements to achieve the target power are considered. Therefore, in this section, we will take the 20-kW target power for example and discuss the pertinent requirements. The requirements for achieving other target powers can also be investigated with the similar method. Here, it should be noted that the target power should be no larger than the power limit.

To achieve the 20-kW output power, the required core diameter should be determined by the PB and SRS limitations [6,20], and can be given as

Figure 5 gives the variations of required core diameter. Here, the 1070-nm signal wavelength is utilized which is more widely used in the studies of power scalability of Yb-doped fiber lasers and amplifiers [6,20,24]. In fact, it will be found later that the effect of signal wavelength is very weak.

 

Fig. 5. Contour plots of required core diameter with the coordinates of pump wavelength and brightness under different dopant concentrations. The values of dopant concentrations are 2.5 × 10²⁵ (a), 5 × 10²⁵ (b), 7.5 × 10²⁵ (c) and 10²⁶m^-3 (d), respectively.

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By comparing Fig. 5 with Fig. 2, it can be found that because the 20-kW target power is smaller than the power limit, the smaller core diameter is needed. It can also be found that the required core diameter is reduced with the increment of PB. In spite of that, it is still astonishing to see that only 10-µm core diameter is needed to achieve the 20-kW output power, if the 976-nm PB can be increased to 14.6 W/µm²/sr with the 10²⁶-m^-3 dopant concentration.

In spite of that, with the help of Eq. (14), it can be known that the required inner-cladding is very small, and its diameter is only about 59.2 µm, which will make the pump light coupling very difficult. In order to increase the inner-cladding diameter to 200 µm, the required core diameter should be enlarged to about 17.8 µm, with the required pump brightness lowered to 1.28 W/µm²/sr. If the inner-cladding diameter can be further enlarged to 400 µm with the PB lowered to 0.3 W/µm²/sr, the required core diameter can be enlarged to about 25.2 µm. These results indicate that the 26/400-µm DCYF should be well enough for achieving the 20-kW output power, as long as the 976-nm pump brightness is no smaller than 0.32 W/µm²/sr. Here, it should be noted that these results are obtained with the dopant concentration of 10²⁶m^-3. If the dopant concentration is lowered, the larger core diameter should be required. For example, with the dopant concentration lowered to 5 × 10²⁵m^-3, the 21.3-µm and 29.8-µm core diameters are required corresponding to 200-µm and 400-µm core diameters, respectively.

From Fig. 5, it can also be found that with further lengthening the pump wavelength beyond the 976 nm, the required core diameter will increase monotonously. Here, we will make some discussion on the 1018-nm pump wavelength which is generally used in the tandem-pumping scheme [7,20]. Figure 5 shows that the required core diameter with the 1018-nm pump wavelength is obviously larger than that with the 976-nm pump wavelength, although it can be reduced to some extent by increasing the dopant concentration. Besides, it can also be reduced by increasing the PB, but with the cost of reduction of inner-cladding diameter (see Eq. (14)). It should be noted that the inner-cladding should not be too small for the pump light coupling. Then, if the 200-µm-diameter inner-cladding is required, about 42.2-µm core diameter is required with the 10²⁶-m^-3 dopant concentration, and the value will be enlarged to about 50.4 µm and 60.4 µm with the dopant concentration of 5 × 10²⁵m^-3 and 2.5 × 10²⁵m^-3, respectively. If the required inner-cladding diameter is enlarged to 400 µm, about 59.3-µm and 70.6-µm core diameters are required with the 10²⁶m^-3 and 5 × 10²⁵m^-3 dopant concentration, respectively. Compared with the case of 976-nm pump wavelength, not only the larger core diameter, but also the smaller cladding-to-core ratio is required. Certainly, the current development of the 1018-nm fiber lasers makes it not difficult to meet the pertinent requirement on the 1018-nm pump brightness (about 1.28 W/µm²/sr and 0.32 W/µm²/sr for 200-µm and 400-µm inner-cladding diameters, respectively) by using the tandem-pumping scheme. However, transverse mode controlling and pump light coupling will become more challenging with the larger core diameter and smaller cladding-to-core ratio, respectively.

The effect of signal wavelength on the required core diameter can be revealed with Fig. 6. By comparing with Fig. 5(d), it can be found that the effect of signal wavelength is much weaker than that of pump wavelength, although the effect can be enhanced to some extent with the long-enough pump wavelength. Taking the 1018-nm pump wavelength for example, corresponding to the 200-µm inner-cladding diameter, the required core diameter is only changed from 42.9 µm to 41.5 µm with the signal wavelength varying from 1060 nm to 1090 nm. Then, the variation of required cladding-to-core ratio (varying from 21.8 to 23.2) is also very weak. It is implied that to achieve a target output power, the option of pump wavelength is much more important than the signal wavelength.

 

Fig. 6. Contour plots of required core diameter with the coordinates of pump wavelength and brightness under different signal wavelengths. Here, the dopant concentration is 10²⁶m^-3. The signal wavelengths are 1060 nm (a), 1080 nm (b) and 1090 nm (c), respectively.

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Here, we would like to remind that these requirements discussed above are only determined by the PB and SRS, but not by the ThL and TMI effects, because the target power is smaller than the power limit and not determined by the ThL or TMI effect [6,20]. Thus, these requirements given above should be applicable in both of ThL- and TMI-limited cases. It is also means that the ThL- or TMI-limited power should be larger than the target power limited by the SRS and PB [6,20]. However, this result seems not coincident to some experiment observations demonstrating that the TMI threshold can be lower than the SRS threshold in high-power fiber lasers and amplifiers [8,10,31–34]. One possible reason for this disagreement should be that some assumptions used in the derivation of TMI threshold (see [17]) cannot be satisfied in experiment. For example, it is assumed that the high-order mode (generally the LP₁₁ mode) involved in TMI effect should be the small perturbation which may be too small compared with the cases of those experiments. As a result, the TMI threshold observed in these experiments was obviously lower than the TMI threshold predicted by Eqs. (3) and (4).

In spite of that, the importance of Eqs. (3) and (4) is revealing that the TMI threshold can be sufficiently improved, as long as the high-order mode can be weak enough, to achieve the 20-kW output power [17,20]. In fact, recent studies have demonstrated that the TMI threshold can be elevated by suppressing the high-order mode (e.g., increasing the loss or decreasing the gain of high-order mode [9,10,33–36]), although the achieved TMI threshold was still obviously less than 20 kW. It is also well-known that the reduction of core diameter can also be of great help for elevating the TMI threshold. However, further studies are still needed to suppress the TMI effect for achieving such high output power.

At last, we would like to discuss the effect of inner-cladding NA on these requirements. From Eqs. (13) and (14), it can be known that the required core and inner-cladding diameters should be increased by reducing the NA, and the required cladding-to-core ratio should also be increased (see Eq. (9)), correspondingly. The reason is that the reduction of NA will lower the ability of pump light coupling of DCYF, and then, the larger inner-cladding and core will be needed for coupling enough pump light and providing strong-enough pump absorption, respectively. The increment of cladding-to-core ratio is mainly induced by the weaker SRS effect induced by the larger core. The weaker SRS effect allows the longer active fiber, which makes the required pump absorption lowered to some extent. Moreover, the Eq. (14) also shows that the required inner-cladding diameter is inversely proportional to the NA. It is meant that if the NA is reduced to 0.2225, the required inner-cladding diameter should be doubled with the PB unvaried. Here, it should be noted that the required core and inner-cladding diameters will not be changed as long as the value of the term (I_pumpNA²) is unvaried (see Eqs. (13) and (14)). Therefore, the requirements on the core and inner-cladding diameters given above will not change with the variation of inner-cladding NA, as long as the proper PB I_pump can be adopted to make the value of (I_pumpNA²) unvaried.

4. Discussions

By studying the requirements on the DCYF to achieve the power limits (see Subsection 3.1), it is found that the smallest core diameter and largest cladding-to-core ratio are required with the 976-nm pump wavelength, which can bring the convenience in the transverse mode controlling (for the diffraction-limited beam quality) and pump light coupling, respectively. In spite of that, the power limit achieved by the 976-nm pump wavelength is about 37 kW and 30 kW in the ThL-limited and TMI-limited cases, respectively, with the 1060-nm signal wavelength. Higher power limit can be achieved by lengthening the pump wavelength. It is found that in the ThL-limited case, about 108.5-kW power limit can be achieved by using the 1050-nm pump and 1060-nm signal wavelengths.

However, together with the elevation of power limit, the larger core diameter and smaller cladding-to-core ratio are required, which make the transverse mode controlling and pump light coupling more challenging. For example, to achieve the 108.5-kW power limit, it is required that the core diameter should be larger than 103.8 µm and the cladding-to-core ratio should be only about 1.29, even with the 10²⁶-m^-3 dopant concentration. Such small cladding-to-core ratio will make the pump light coupling very difficult. If the larger cladding-to-core ratio is needed for pump light coupling, the required core diameter should be further enlarged. Practically, it is very hard to obtain the diffraction-limited beam quality by using a conventional DCYF with such a large core. The Yb-doped micro-structured fiber may provide a more promising solution [37–40]. Reference [38] showed that the single-mode operation can be achieved by the passive large-pitch fiber with more-than 200-µm mode field diameter. In Ref. [40], it had been demonstrated that the diffraction-limited beam quality (M² factor of about 1.2) can be achieved with the 80-µm core diameter by using the rod type Yb-doped PCF. In spite of that, the single-mode operation in the Yb-doped very-large-mode-area fiber is still very challenging, and the long-term endeavor on this topic is still needed. Another way to reduce the core diameter is to further increase the dopant concentration of Yb-ion. However, this way is not so effective because of the limitation of the dopant technology in the fabrication of DCYF.

The requirement on the inner-cladding diameter of DCYF is also studied. It is found that the larger inner-cladding diameter is required with the increment of power limit which needs more pump light. It is also revealed that the required inner-cladding should decrease monotonously with the increment of PB. However, it should be noted that the inner-cladding diameter cannot be too small for the practical pump light coupling. Thus, it is suggested that the inner-cladding diameter and PB should be carefully designed with the consideration of pump light coupling technology.

In the TMI-limited case, the power limit will be lowered compared with the ThL-limited case. Correspondingly, the required core and inner-cladding diameters of DCYF will also become smaller. In spite of that, the requirement on the cladding-to-core ratio will keep the same as that in the ThL-limited case, as long as the PB and dopant concentration are unvaried.

The requirements to achieve a target output power (smaller than the power limit) are also investigated. Here, the target output power is given as 20 kW. In this case, it is found that the 976 nm is the optimum pump wavelength with which the smallest core diameter and largest cladding-to-core ratio are required. It is astonishing to find that the DCYF with 18-µm-diameter core and 200-µm-diameter inner-cladding is potential to generate the 20-kW output power, as long as the 976-nm PB can be larger than 1.28 W/µm²/sr. It is also revealed that the DCYF with 26-µm-diameter core and 400-µm-diameter inner-cladding is also applicable as long as the 976-nm PB can be larger than 0.32 W/µm²/sr. However, these required PB values are too large to be met by the current LD source, because the pump brightness is not only determined by the single diode emitter, but also by the brightness degeneration caused by the pump light combining and coupling technology. In spite of that, the fiber lasers around 976 nm can provide a promising solution as the tandem-pumping source for elevating the pump brightness. Moreover, in order to output such high power from so small core diameter, the end-cap should be used and well designed to protect the fiber end from optical damage (see also Section 2).

Another pump option is the 1018-µm tandem-pumping source, which can also provide high-enough pump brightness. However, the required core diameters are enlarged to 43 µm and 60 µm for the 200-µm and 400-µm inner-cladding diameters, respectively, which are much larger than the case of 976-nm pump wavelength. Then, the mode controlling will become more challenging to achieve the diffraction-limited beam quality with the 1018-nm tandem-pumping source.

It should be noted that to achieve the target power smaller than the power limit, these requirements are not determined by the TMI effect, because the TMI threshold predicted by Eq. (3) is larger than the target power. As discussed above, this conclusion is only valid with the assumption of small perturbation of high-order mode such as the LP₁₁ mode, which gives an addition requirement on the high-order mode suppressed in the designation of DCYF. It is well-known that the smaller the core diameter is, the more easily the high-order mode can be suppressed. Thus, the 976-nm pump wavelength with the smallest required core diameter should be the most convenient to meet the requirement on the high-order mode suppression, as long as the PB is large enough. The effects of signal wavelengths (ranging from 1060 nm to 1090 nm) on these requirements are also studied, and it is found that the effect of signal wavelength is much weaker than that of pump wavelength.

Here, it should be noted that these requirements are related to the absorption and emission cross sections of Yb-ion which can be varied to some extent with different co-dopant elements such as Al, P or Ce [41–43]. For example, the increment of absorption cross section can reduce the required core diameter or enlarge the require cladding-to-core ratio. In spite of that, the regulars of these requirements will be still valid, although the specific values may be varied to some extent, if these cross sections of Yb-ion are not the same as the values used in this paper.

In addition, although only the requirements of 20-kW target power are discussed in this paper, the pertinent results can be used to predict the requirements of other target powers. From Eqs. (9), (13) and (14), it can be known that the target output power can only affect the requirements on the core and inner-cladding diameters, but not affect the cladding-to-core ratio. Then, taking the 10-kW target power for example, the required core and inner-cladding diameters will change to 1/√2 of these values obtained for the 20-kW target power with the other requirements on the PB and dopant concentration unvaried (see Eqs. (13) and (14)). Then, it means that to achieve 10-kW output power with 1018-nm pump wavelength, the required core diameter can be reduced to 30 µm with the 10²⁶-m^-3 dopant concentration (the required core diameter is 42.2 µm for 20-kW target power, see Fig. 5(d)) and the required length of active fiber (see Eq. (8)) should be about 16.7 m. These results agree well with the IPG’s 10-kW near-diffraction-limited fiber laser reported in Ref. [30] where the 30-µm core diameter and 15-m active fiber length were used with the 1018-nm pump wavelength. In spite of that, the experimental study on the power scalability of fiber lasers and amplifiers is currently very rare, which should be paid more attention to and carried out in the future.

At last, we would like to make a brief discussion on the effect of gain G. Equations (6), (9) and (12) indicate that the power limit and its required values of cladding-to-core ratio and inner-cladding diameter will be increased with the increment of G. However, for achieving a target output power, the required cladding-to-core ratio will be increased with the increment of G (see Eq. (9)), but the required inner-cladding diameter keeps unvaried (see Eq. (14)), which makes the required core diameter decreased (see Eq. (13)).

5. Conclusions

In this paper, the requirements on DCYF for power scaling of diffraction-limited fiber amplifiers are studied with the wavelength-dependent numerical model. By calculating the power scaling limit, it is revealed that the power limit increases monotonously with the reduction of difference between the pump and signal wavelengths because of the weaker thermal effects (i.e., ThL or TMI) induced by the quantum defect. It is found that the power limit can be enlarged to more-than 100/80 kW in the ThL/TMI-limited case with the 1050-nm pump and 1060-nm signal wavelengths. Then, the requirements on the DCYF to achieve these power limits are studied. It is found that the requirements are strongly dependent on the pump wavelength and brightness. The results show that the smallest core diameter and largest cladding-to-core ratio are required with the 976-nm pump wavelength, but the corresponding power limit is only about 37/30 kW in the ThL/TMI-limited case with the 1060-nm signal wavelength. Higher power limit can be achieved by lengthening the pump wavelength with the cost of larger core diameter and smaller cladding-to-core ratio. The DCYF requirements to achieve the 20-kW target output power are also investigated, and it is found that the required core diameter can be reduced to smaller-than 20 µm if the 976-nm pump brightness can be large enough. We believe that the results and conclusions obtained in this paper can provide significant guidance on the designation of DCYF to up-scale the output power of diffraction-limited Yb-doped fiber lasers and amplifiers. These analyses can also be extended to other sorts of fiber lasers and solid-state lasers.

Appendix A: Estimation of the coefficient C₁

From Eq. (4), it can be known that the value of C₁ is related to the configuration of DCYF. For a step-index fiber, U_mn can be approximately given as [17,25]

(15)$${U_{mn}}(\textrm{V} )= U_{mn}^0{e^{ - ({1/\textrm{V}} )}}$$

where V is the normalized frequency of fiber; the value of $U_{mn}^0$ is 2.405 for LP₀₁ mode and 3.832 for LP₁₁ mode. Then, the value of n_eff can be given as [17,25]

(16)$${n_{eff}}\textrm{ = }\sqrt {n_{co}^2 - {{({{{{\lambda_s}U_{01}^{}} / {2\pi a}}} )}^2}}$$

where n_co is the core index of Yb-doped fiber. It should be noted that the TMI can be preponderant in limiting the power of fiber amplifiers, only when the value of C₁ is smaller than 1. In this case, the normalized frequency V should satisfy that

(17)$$2.405 < \textrm{V} < \frac{4}{{\ln \{{{{({U_{11}^0} )}^2}[{{{({U_{11}^0} )}^2} - {{({U_{01}^0} )}^2}} ]} \}- \ln ({4{\pi^2}{n_{co}}} )}}$$

The lower limitation is resulted from the physical reality that TMI cannot be present in the single-mode fiber. Then, it can be calculated that the V value should be smaller than 4.87 (with the n_co of 1.45) in the TMI-limited case.

The variation of C₁ with the core diameter is also given in Fig. 7. It can be seen that although C₁ varies with the core diameter, the variation is negligible when the core diameter is larger than 10 µm. It should be noted that the practical multi-kW fiber lasers are generally realized with the core radius larger than 10 µm. Then, a good approximation can be made to take C₁ as a constant. It is also coincident with the conclusion obtained in [18–20] that the ratio of TMI limited power to the thermal lens limited one is close to a constant 0.6 with V = 3. Besides, it can be seen that the value should be larger than 1 when V reaches to 5 which agrees with the theoretical prediction given by Eq. (17) (i.e., V should be smaller than 4.87).

 

Fig. 7. The variation of C1 with the core diameter. The pertinent constants are 0.940, 0.841, 0.729, 0.603, 0.462 for the V values of 4.5, 4, 3.5, 3, 2.5, respectively.

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Appendix B: Derivation of required cladding-to-core area ratio

By letting Eq. (5) equal to Eq. (8), it can be got that

(18)$$A{B_2}{({R_{Cl\_Co}^{opt}} )^2} + \frac{{{G_s}}}{{\Gamma _s^2}}{B_1}R_{Cl\_Co}^{opt} - \frac{{16{B_3}{N_0}}}{{{I_{pump}}N{A^2}}} = 0$$

By solving Eq. (18), the required cladding-to-core ratio can be given as

(19)$$R_{Cl\_Co}^{opt}\textrm{ = }\frac{\textrm{1}}{{2A{B_\textrm{2}}}}\left[ {\sqrt {\frac{{{G_s}^2}}{{\Gamma _s^4}}B_1^2 + \textrm{64}\frac{{A{B_3}{B_\textrm{2}}{N_0}}}{{{I_{pump}}N{A^2}}}} - \frac{{{G_s}}}{{\Gamma _s^2}}{B_\textrm{1}}} \right]$$

where B₁, B₂ and B₃ are given as Eqs. (10) and (11), respectively. Since the cladding-to-core ratio should be no smaller than 1, we can have that (assuming that B₂ is positive)

(20)$${N_0} \ge \left( {\frac{{A{G_s}}}{{16\Gamma _s^2}}\frac{{{B_\textrm{1}}}}{{{B_3}}} + \frac{{{A^2}B_2^{}}}{{16{B_3}}}} \right){I_{pump}}N{A^2}$$

which gives the requirement of dopant concentration to make the cladding-to-core ratio no smaller than 1. It should be noted that the requirement is related to the PB and inner-cladding NA of DCYF. This result can be understood with the help of Eq. (8). From Eq. (8), it can be known that the shorter active fiber length is required with the increment of PB and NA. Then, the higher dopant concentration is needed to provide high-enough pump absorption.

Funding

National Natural Science Foundation of China (U20B2058).

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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