## Abstract

The adiabaticity criterion of the thermally-guided very-large-mode-area (TG VLMA) fiber is presented based on the mode-coupling theory firstly, to the best of our knowledge. The requirement for the adiabatic propagation of fundamental mode is discussed systematically. It is revealed that the pump absorption plays the most important role and the adiabaticity criterion can be met as long as it is small enough. Then, the effects of the configuration parameters of TG VLMA fiber on the up-limitation of pump absorption for the adiabaticity criterion are investigated. It is found that for the straight TG VLMA fiber, reducing the initial refraction index and inner-cladding diameter and utilizing the bi-directional pumping scheme are beneficial to the adiabatic propagation of fundamental mode. The bent TG VLMA fiber is also studied. It is found that the bent fiber is much more difficult to meet the adiabaticity criterion than the straight one. The results show that even with the 100-cm bend radius, the pump absorption should be smaller than 1 dB/m to meet the adiabaticity criterion. It is suggested that enlarging the core-to-cladding ratio can be helpful for loosening the adiabaticity criterion of bent TG VLMA fiber. These pertinent results can provide significant guidance for understanding and designing the TG VLMA fiber and pertinent lasers and amplifiers.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Very-large mode area (VLMA) fiber is considered as a key device in power-scaling of high-power fiber lasers because of its important role in suppressing the non-linear effects [1–5]. In designing the VLMA fiber, the core problem is how to realize simultaneously the mode area increment and single-mode operation. However, solving this problem is not so easy, because the core diameter of VLMA fiber should be generally increased in order to increase the mode area, which makes it more and more difficult to keep the single-mode operation (the fiber normalized frequency V smaller than 2.404) [6, 7]. One straightforward way to realize the single-mode operation is to reduce the numerical aperture (NA) of fiber core. However, limited by the fabrication technology, the core NA cannot be unlimitedly reduced. Nowadays, the reported lowest core NA is 0.028 [7], and correspondingly, the core diameter should be no larger than 30 um to realize the strict single-mode operation.

In order to further increase the core diameter, one additional way is to increase the loss of high-order modes or the gain of fundamental mode to realize the single-mode operation. According to this way, various schemes were proposed, e.g., coiling [8, 9], gain-guided index antiguided (GG + IAG) fibers [10, 11], chirally coupled core fibers [12, 13], leakage channel fibers [14, 15], large-pitch photonic-crystal fibers [16, 17], etc. With these schemes, the mode-field diameter has been further scaled up to 50–100 μm with the single-mode operation. In spite of that, the micro-structure designs induced in some of these schemes bring in the difficulty of fabrication. Besides, the thermal effects (such as the thermal-lens effect and thermal instability) will also make a limitation to the increment of the mode-field diameter or mode area [18–22].

The thermally-guided (TG) VLMA fiber is a novel VLMA fiber [23]. Different from these schemes mentioned above, the waveguide of TG VLMA fiber is not formed by the initial index distribution, but formed by the thermal-lens effect [18–20]. It means that even when the initial index distribution is anti-guided (i.e., the core index is smaller than the inner-cladding index), the waveguide can also be thermally induced as long as the thermal load is large enough (assuming the thermal optical coefficient of fiber material is plus). The thermal-induced waveguide of TG VLMA fiber owns several advantages. Firstly, the fiber uses the thermal effect to form the waveguide, and thus, it may be more resistant to the thermal-limitation of mode area increment. Secondly, the ultra-low NA can be realized by the thermal-induced waveguide, which may induce the considerate proportion of fundamental mode spread into the inner-cladding. Then, the limitation of the core diameter to the mode area scaling can be weakened. Thirdly, the cylindrical symmetry of index distribution without any micro-structure also lowers the difficulty of the TG VLMA fiber fabrication.

In 2014, a proof-of-concept experiment of TG VLMA fiber was carried out [23]. By using a TG VLMA fiber with 42-μm core, the single-mode operation was realized with the mode diameter about 68 um. It should be noted that the mode diameter is more-than 60% larger than the core diameter, which demonstrates that the mode diameter can be greatly larger than the core diameter in the TG VLMA fiber. Later, the evolution of transverse modes with the thermal load in the TG VLMA fiber was numerically studied [24], and the numerical model of the TG VLMA fiber amplifier was also presented [25]. By numerically studying the TG VLMA fiber amplifier, it was revealed that the thermal load and the pertinent thermally-induced waveguide should vary along the TG VLMA fiber because of the decrement of the pump power. Then, a question comes out, i.e., under what condition can the single mode propagation be maintained or can the coupling between the fundamental mode and high-order modes be negligible in the thermally-induced waveguide of TG VLMA fiber?

In order to answer the above question, the adiabaticity criterion of TG VLMA fiber is investigated in this paper based on the mode-coupling theory. The paper is arranged as follows. In Section 2, the adiabaticity criterion will be deduced. In Section 3, the adiabaticity criterions of various TG VLMA fibers will be numerically investigated. In Section 4, the conclusions will be drawn.

## 2. Deduction of the adiabaticity criterion

Based on the mode-coupling theory, the power transferring between two modes is considered to be negligible if the following condition can be met [26], i.e.,

where*C*is the coupling coefficient of the

_{jm}*jth*and

*mth*modes;

*z*is the beat length of these two modes and can be calculated as (

_{b}*2π/|β*) where

_{j}-β_{m}|*β*are the propagation constants of the

_{j,m}*jth*and

*mth*modes, respectively. The coupling coefficient

*C*can be given as

_{jm}*Ψ*are fields of the

_{j,m}*j*th and

*m*th modes;

*k*= 2π/

*λ*is the free-space wave-number, and

_{s}*λ*is the free-space wavelength of the signal light;

_{s}*n*(

*x*,

*y*,

*z*) gives the three-dimensional index refraction; A

_{∞}is the infinite cross-section.

In the TG VLMA fiber, the index refraction *n*(*x*, *y*, *z*) consists of the initial index refraction *n _{0}* and the thermally-induced variation Δ

*n*(can be calculated with Eq. (4) in Ref [24].). Considering that the thermally-induced variation of refraction index Δ

*n*is much smaller than the initial refraction index, we can get that

*n*is proportional to the thermal load

*Q*which can be approximately calculated as

*α*is the absorption coefficient of the pump light;

_{p}*P*

^{±}

_{p}_{0}is the input co-pumping ( + ) and counter-pumping (-) power;

*λ*is the wavelength of pump light;

_{p}*L*is the length of TG VLMA fiber. Substituting Eqs. (3) and (4) into Eq. (2), it can be obtained as

*P*

^{+}

_{p}_{0}= 0 or

*P*

^{-}

_{p}_{0}= 0) and “<” corresponds to the bi-directional pump. Then, we can have thatIt should be noted that the coefficient

*C'*is a function of

_{jm}*Q*because the thermally-induced variation of index refraction Δ

*n*and the pertinent local mode fields

*Ψ*and propagating constants

_{j,m}*β*are all determined by

_{j,m}*Q*. In spite of that, the absorption coefficient

*α*and

_{p}*Q*can be varied independently because

*Q*is not only related to the absorption coefficient

*α*but also related to

_{p}*P*

^{±}

_{p}_{0}and (

*λ*/

_{p}*λ*) (see Eq. (4)). Then, by substituting (9) into (1), we can get that

_{s}Therefore, as long as the condition (10) can be satisfied at each position of TG VLMA fiber, the adiabatic propagation of fundamental mode can be realized regardless of the pumping scheme. It is also revealed that the adiabaticity criterion (10) should always be satisfied as long as the absorption coefficient *α _{p}* is small enough.

## 3. A paradigm of TG VLMA fiber

#### 3.1 The axisymmetric case

A paradigm of TG VLMA fiber will be discussed in this section. Here, we consider the TG VLMA fiber whose parameters are given in Table 1. Because the thermal-induced waveguide of TG VLMA fiber is axisymmetric [23], the local fundamental LP_{01} mode should couple only to the high-order modes with the same azimuthal symmetry, i.e., LP_{0m} modes [24]. Thus, the coupling between the fundamental LP_{01} mode and the LP_{02} mode should be first considered to study the adiabaticity criterion of TG VLMA fiber [26]. With the parameter values in Table 1 and a given thermal load, the thermally-induced variation of index refraction can be obtained, and then, the local LP_{01} and LP_{02} mode fields *Ψ _{01,02}* and their propagation constants

*β*can be calculated with the finite-element method [24]. With these results and (7), the values of

_{01,02}*C'*and

_{12}*z*can be obtained, and the adiabaticity criterion can be given with (10). Here, the wavelength of signal light is assumed as 1080 nm. Four values of the difference of initial index refraction Δ

_{b}*n*

_{0}(defined as [

*n*-

_{0cl}*n*] where

_{0co}*n*and

_{0cl}*n*represent the initial index refractions of the inner-cladding and active core, respectively) are considered, i.e., 0, 10

_{0co}^{−5}, 5 × 10

^{−5}and 10

^{−4}.

The variations of the coefficient *C' _{12}* and beat length

*z*with the thermal load

_{b}*Q*are given in Figs. 1(a) and 1(b), respectively. Figure 1(a) illuminates that the coefficient

*C'*should increase monotonously with the thermal load when the thermal load is large enough (larger than about 40 W/m). It is reasonable because the LP

_{12}_{01}and LP

_{02}mode fields are confined well in the active core with such large thermal load, and the fields will be more and more concentrated with the increment of

*Q*because of the thermal-lens effect, which makes the coefficient increased. This explanation can be witnessed by Fig. 1(c) which shows that when the thermal load is larger than 40 W/m, more-than 90% LP

_{01}mode field and more-than 60% LP

_{02}mode field are located in the active core (also see Fig. 1(e)), and both of these portions in the active core increase monotonously with the increment of thermal load. Figure 1(b) shows that when the thermal load is larger than 40 W/m, the beat length

*z*changes slowly with the thermal load, which means that the difference of

_{b}*β*

_{01}and

*β*

_{02}does not change too much.

In spite of that, the variation of the coefficient *C' _{12}* does not so monotonous with the thermal load when the thermal load is smaller than 40 W/m. Figure 1(a) shows that there is a fluctuation in the variation of

*C'*. Such a variation is also induced by the variations of LP

_{12}_{01}and LP

_{02}mode fields. From Fig. 1(c), it can be seen that although the filling factor of LP

_{01}mode increases monotonously with the thermal load, the filling factor of LP

_{02}mode does not vary monotonously, which is induced by the power transferring between the central spot and outer ring of LP

_{02}mode field (see [24] and Fig. 1(e)). Then, the variation of LP

_{02}mode field will result in the fluctuation of

*C'*. Figure 1(a) also revealed that more thermal load is needed to get the local maximum or minimum with a larger

_{12}*Δn*

_{0}, and correspondingly, the larger local maximum and minimum can be obtained. Figure 1(b) shows that the beat length

*z*increases firstly with the increment of the thermal load, and then, reaches to its maximum, after that, it decreases monotonously with the thermal load.

_{b}Figure 1(d) gives the variation of (*C' _{12} z_{b}*)

^{−1}with the thermal load. It can be found that the variation can be divided into four regions. Take the case of zero-

*Δn*

_{0}for example, with the increment of thermal load, the value of (

*C'*)

_{12}z_{b}^{−1}decreases monotonously when the thermal load is smaller than 3.0 W/m, and then, goes up till the thermal load reaches to 18 W/m, after that, it decreases again, and the pertinent slope becomes smaller when the thermal load is larger than 30 W/m. As a result, a minimum and maximum value is induced along the variation. It can also be found that these variations of (

*C'*)

_{12}z_{b}^{−1}values corresponding to various

*Δn*

_{0}are similar to each other, although these minimum and maximum values and pertinent positions are varied with

*Δn*

_{0}. Besides, together with Eq. (10), it can be known that the adiabaticity criterion should be determined by the local minimum value of (

*C'*)

_{12}z_{b}^{−1}which gives the case where the coupling between LP

_{01}and LP

_{02}modes is most serious. In order to understand these results given in Fig. 1(d), we investigate the evolution stage of each mode in the four regions by analyzing the variation of effective refractive index of each mode [24], which is given in Tab. 2 (The boundary value of 0 W/m means the mode has already become quasi-core confined). It can be seen that the local minimum value is mainly present within the quasi-core-confined mode region and the corresponding thermal load

*Q*is close to the core-confined boundary of LP

_{min}_{01}mode.

In Fig. 1(d), it should be noted that *Q _{min}* is small (generally smaller than 10 W/m). This result can be understood with Fig. 1(c). By comparing Fig. 1(d) with Fig. 1(c), it can be concluded that these local minimum values of (

*C'*)

_{12}z_{b}^{−1}are present when the filling factors of LP

_{01}and LP

_{02}mode are identical. Then, the fields of LP

_{01}and LP

_{02}mode are most similar to each other at the small thermal load (also see Fig. 1(e)), which makes the coupling between two modes is most severe. It is also interesting to found that the filling factors of two modes corresponding to the local minimum value almost keeps unvaried with

*Δn*

_{0}, and its value is about 0.3.

If we assume that the adiabaticity criterion will be satisfied as long as the pump absorption *α _{p}* is smaller than the value of (

*C'*)

_{12}z_{b}^{−1}, the adiabatic region can be given in Fig. 1(d). Figure 1(d) also illuminates that the local minimum value is lowered, and thus the adiabaticity criterion becomes more difficult to meet, with the larger value of Δ

*n*

_{0}. It can be found that the local minimum value is about 50 m

^{−1}(i.e., around 11.5 dB/m) corresponding to the zero-Δ

*n*

_{0}, and decreases to about 12 m

^{−1}(i.e., around 2.7 dB/m) corresponding to the 10

^{−4}- Δ

*n*

_{0}. It means that the adiabaticity criterion should be satisfied if the pump absorption

*α*is smaller than 12 m

_{p}^{−1}corresponding to the TG VLMA fiber with the value of Δ

*n*

_{0}smaller than 10

^{−4}. The variation of local minimum value of (

*C'*)

_{12}z_{b}^{−1}via Δ

*n*

_{0}is also given in Fig. 2(a). It can be found that the decrease of the local minimum value become slower with the increment of Δ

*n*

_{0}. In spite of that, Fig. 2(b) shows that the increment of

*Q*is approximately linear with the increment of Δ

_{min}*n*

_{0}.

Another thing should be noted that although the local minimum value of (*C' _{12} z_{b}*)

^{−1}is generally obtained with the thermal load smaller than 10 W/m, if the minimum thermal load is larger than 10 W/m along a TG VLMA fiber, the local minimum value determining the adiabaticity criterion will be enlarged, which will release the adiabaticity criterion requirement of the pump absorption. For example, as given in Fig. 1(d), to satisfy the adiabaticity criterion corresponding to the 5 × 10

^{−5}-Δ

*n*

_{0}, the pump absorption

*α*is smaller than 18 m

_{p}^{−1}which is the local minimum value of (

*C'*)

_{12}z_{b}^{−1}at the 3.6-W/m thermal load. However, if the minimum thermal load is 10 W/m in a TG VLMA fiber amplifier (can be easily realized with the bi-directional pump scheme or shorter fiber length), the local minimum value of (

*C'*)

_{12}z_{b}^{−1}is not 18 m

^{−1}anymore, but increased to 40 m

^{−1}(see Fig. 1(d)). Then, the pump absorption smaller than 40 m

^{−1}should be enough to meet the adiabaticity criterion, which means the adiabaticity criterion should be satisfied more easily.

#### 3.2 The non-axisymmetric case

Although the thermally-induced waveguide of TG VLMA fiber is axisymmetric, the symmetry cannot be kept if the bending or coiling is applied. Therefore, the adiabaticity criterion is discussed with the bending taking into account in this section. It should be noted that because of the presence of bending, the fundamental local mode cannot only couple to the LP_{0m} modes but can also couple to other high-order modes. Therefore, the coupling between the fundamental LP_{01} mode and the LP_{11} mode should be considered to study the adiabaticity criterion. It should be noted that in the bending case, Eq. (3) should be changed to

*R*represents the bend radius. Here, it is assumed that the bend radius should be a constant and not vary with z. However, because the inner-cladding radius is generally much smaller than the bend radius (about tens of centimeters), the second term in the bracket should be negligible. Therefore, the formula (7) can still be used to calculate the coefficient for the coupling of LP

_{01}mode and LP

_{11}mode, which we named as

*C'*here. In spite of that, it should be noted the bending will induce the variation of the optical fields of LP

_{01}_{01}and LP

_{11}modes, which will affect the coefficient

*C'*and eventually affect the adiabaticity criterion.

_{01}### 3.2.1 The evolution of transverse mode in the bent TG VLMA fiber

In order to understand the coupling of LP_{01} and LP_{11} modes in the bent TG VLMA fiber, we firstly make a brief discussion on the evolution of transverse modes with the thermal load. As shown in Fig. 3, the evolution of transverse mode with the thermal load in the bent TG VLMA fiber can be divided into three stages. In the first stage (see Fig. 4(a)), the thermal load is weak and the effective refractive index of transverse mode is located between the largest refractive index of inner-cladding and active core. Then, the transverse mode should propagate in the inner cladding and be named as the “inner-cladding confined” mode (in essence, such a transverse mode should be a whispering gallery mode in the inner-cladding [27]).

In the second stage (see Fig. 3(b)), with the increment of the thermal load and refractive index of active core, the effective refractive index of transverse mode is smaller than both the largest values of core and inner-cladding refractive index. Then, the optical field of the transverse mode will be affected by both the inner-cladding and active core. Thus, the transverse mode is named as the “transition” mode in this stage.

In the third stage (see Fig. 3(c)), the thermal load is large enough to make the effective refractive index of transverse mode larger than the refractive index of inner-cladding. Then, the transverse mode is confined in the active core, and thus, named as the “core confined” mode. Figure 4 gives the variation of the thermal load boundaries between three stages via Δ*n*_{0}, in which S1, S2, and S3 represents the first, second, and third stage, respectively. It is not surprise to see that more thermal load is needed to realize the core-confined mode with the larger Δ*n*_{0} and to realize the core-confined LP_{11} mode.

### 3.2.2 The adiabaticity criterion of bent TG VLMA fiber

With the optical fields and propagation constants of LP_{01} and LP_{11} modes calculated with the finite-element method, the value of (*C' _{01} z_{b}*)

^{−1}can be obtained and the adiabaticity criterion can be revealed with Eq. (10). Because of the initial anti-guided waveguide, TG VLMA fiber is sensitive to the bending, and then, the bend radius larger than 60 cm is considered here. Besides, only the coupling between LP

_{01}and LP

_{11e}mode is taken into account because the coupling between LP

_{01}and LP

_{11o}mode is so weak as to be negligible. This can also be witnessed by Fig. 5 which shows that the local minimum value of (

*C'*)

_{01}z_{b}^{−1}corresponding to LP

_{11o}(see Figs. 5(a)-5(c)) is much larger than that corresponding to LP

_{11e}. It is implied that the coupling between LP

_{01}and LP

_{11e}is more determinant. Thus, in this part, the LP

_{01}mode represents the LP

_{11e}mode and the discussion is carried out according to Figs. 5(d)-5(f).

From Fig. 5, it can be seen that similar to the case of straight TG VLMA fiber (see Fig. 1), the variations of (*C' _{01} z_{b}*)

^{−1}with the thermal load can also be divided into four regions. Besides, it can be revealed that when the thermal load is large enough (e.g., larger than 20 W/m and 25 W/m for zero and 10

^{−4}Δ

*n*

_{0}, respectively), the value of (

*C'*)

_{01}z_{b}^{−1}does not vary with the bend radius

*R*. It means that in this case, the thermally-induced waveguide should be sufficient to resist the bend effect on the coupling between two modes.

Figure 5 also illuminates that the local minimum values present with the thermal load lower than 20 W/m. It can be found from Fig. 5 that the local minimum value is much smaller than the case of straight fiber, which means that it is much more difficult to realize the adiabatic propagation of LP_{01} mode in the bent TG VLMA fiber. It is also revealed that the local minimum value decreases with the decrement of the bend radius, which means that the smaller bend radius make the coupling between two modes more dramatic. In spite of that, these results show that even with the 100-cm bent radius and zero-Δ*n*_{0}, the local minimum value is only 4.25 m^{−1}, which implies that the pump absorption *α _{p}* should be smaller than 0.98 dB/m to meet the adiabaticity criterion given by Eq. (10). Such a value of pump absorption of TG VLMA fiber is a little small as an active fiber. This value can be elevated by enlarging the bend radius or ensure the thermal load along the fiber large enough to avoid the region around the local minimum value (e.g., larger than 6.6 W/m for the case of 100-cm

*R*and zero-Δ

*n*

_{0}).

## 4. The adiabaticity criterion corresponding to various core and inner-cladding diameters

In the former sections, we discussed the adiabaticity criterion with a given core and inner-cladding diameter. In this section, the effects of core and inner-cladding diameter on the adiabaticity criterion will be discussed. Here, the initial index refraction difference Δ*n*_{0} is assumed as 0, 5 × 10^{−5} and 10^{−4}. Firstly, we would like to analyze the adiabaticity criterion of axisymmetric TG VLMA fiber, where the coupling between the LP_{01} and LP_{02} modes is taken into account.

Figure 6 gives the effects of core diameters. It can be found that the second region (comparing with Fig. 1) is weakened with the increment of core diameter. Figure 6(a) even shows that the second region disappear and the value of (*C' _{12} z_{b}*)

^{−1}decreases monotonously with the 80-um core diameter. These results can be understood by considering the mode evolution of LP

_{02}mode given in Figs. 6(d)-6(f). It can be found that the effect of core diameter on the LP

_{02}mode is more serious than that on the LP

_{01}mode, and the filling factor of LP

_{02}mode is enlarged with the increment of core diameter. Such a variation of LP

_{02}mode makes the maximum difference between the two modes (which does harm to the coupling between two modes and corresponds to the local maximum between the second and third region) weakened, and then, the second region is weakened. The monotonous decrement of (

*C'*)

_{12}z_{b}^{−1}with the 80-um core diameter (see Fig. 6(a)) can also be understood because there is no crossing (i.e., the equal case) between the filling factors of LP

_{01}and LP

_{02}mode. Furthermore, it is also revealed that the thermal load and pertinent filling factor of LP

_{01}and LP

_{02}mode corresponding to the local minimum value be enlarged with the increment of core diameter.

From Figs. 6(b) and 6(c), it can be seen that with the 5 × 10^{−5} and 10^{−4}-Δ*n*_{0}, the adiabaticity criterion is still determined by the local minimum value of (*C' _{12} z_{b}*)

^{−1}at the low thermal load. It can be known that the pump absorption should be smaller than 11 m

^{−1}and 9 m

^{−1}corresponding to the 60-um and 80-um core diameter, respectively, with Δ

*n*

_{0}lower than 10

^{−4}. Furthermore, it can also be seen that when the thermal load is larger than around 10 W/m, the value of (

*C'*)

_{12}z_{b}^{−1}decreases with the increment of core diameter, which means that the increment of core diameter makes the coupling between LP

_{01}and LP

_{02}mode more serious.

Figure 7 gives the effects of inner-cladding diameters. It can be found that the local minimum value of (*C' _{12} z_{b}*)

^{−1}and

*Q*decrease monotonously with the increment of inner-cladding diameter. These results can be understood with the variation of mode field induced by the change of inner-cladding diameter (see Figs. 7(d)-7(f)). Figure 7 shows that the thermal load corresponding to the local minimum value is still low and the LP

_{min}_{02}modes should be quasi-inner-cladding confined mode (see Table 2). Then, the LP

_{02}mode area should be enlarged with the increment of inner-cladding diameter. Note that, as mentioned in the section 3.1, the local minimum value is induced by the similarity of two modes enhancing their coupling. Therefore, a larger LP

_{01}mode area is needed to realize the local minimum value. Because the LP01 mode is quasi-core confined (see Table 2), a thermal load should be lowered to realize a larger mode area with a given core diameter. Then, a lower thermal load is needed to realize the local minimum value of (

*C'*)

_{12}z_{b}^{−1}with the increment of inner-cladding diameter. This explanation can also be witnessed by Figs. 7(d)-7(f), which shows that the filling factor corresponding to the local minimum value decreases monotonously with the enlargement of inner-cladding.

Besides, the decrement of the local minimum value also implies that these two modes are more similar and their coupling is more severe at the minimum point with the increment of inner-cladding diameter, and thus, it should be more difficult to meet the adiabaticity criterion. It can be found that even with the zero-Δ*n*_{0}, the pump absorption should have to be smaller than the small value of 11 m^{−1} to meet the adiabaticity criterion corresponding to the 250-um inner-cladding, and the small value will further decrease to 3.75m^{−1} with the 10^{−4}-Δ*n*_{0}. In spite of that, it can also be found that when the thermal load is beyond *Q _{min}*, the value of (

*C'*)

_{12}z_{b}^{−1}with larger inner-cladding diameter becomes larger again, and then, the negative effect of large inner-cladding will not exist.

Secondly, the adiabaticity criterion of bent TG VLMA fiber with various core and inner-cladding diameters will be discussed, and the results are given in Figs. 8 and 9. Here, the initial index refraction difference Δ*n*_{0} is assumed as 5 × 10^{−5}, and the coupling between LP_{01} and LP_{11e} mode coupling is considered.

It can be found that the local minimum value of (*C' _{01} z_{b}*)

^{−1}at the low thermal load increases with the increment of core-diameter or decrement of inner-cladding ratio. It means that the adiabaticity criterion can be more easily met with the larger core-to-cladding ratio of the TG VLMA fiber. It is also implied that the fiber with larger core-to-cladding ratio has more resistant ability to the bending. In spite of that, Fig. 8 illuminates again that with the 60-um or 80-um core diameter, the value of may be further decreased and smaller than the local minimum value when the thermal load large enough (e.g., larger than 100 W/m and 23 W/m with the 60-um and 80-um core diameter with 80-cm bend radius, respectively). It implies that the adiabaticity criterion may be determined by the high thermal load region when the core diameter is large enough.

## 5. Conclusion

The adiabaticity criterion of the TG VLMA fiber is presented based on the mode-coupling theory firstly, to the best of our knowledge. It is revealed that the adiabaticity criterion can be always satisfied as long as the pump absorption is small enough. Then, the adiabaticity criterions of various TG VLMA fibers are systematically discussed, and following conclusions can be obtained.

- 1) Reducing the value of Δ
*n*_{0}is beneficial to the adiabatic propagation of the fundamental mode. - 2) The adiabaticity criterion is generally determined by the coupling of two modes at the low thermal load (lower than 20 W/m) when the two modes are most similar to each other when the core diameter is smaller than 60 um. If the core diameter is too large, the adiabaticity criterion may be determined by the two modes coupling at the high thermal load (e.g., larger than 23 W/m with 80-um core diameter, see Fig. 8(b)).
- 3) For the straight TG VLMA fiber, reducing the inner-cladding diameter is beneficial to the adiabatic propagation of LP
_{01}mode, while the core diameter does not have so obvious effect. In spite of that, the coupling between LP_{01}and LP_{02}modes will be enhanced with the increment of core diameter when the thermal load is larger than 15 W/m, which is not beneficial to the adiabatic propagation of LP_{01}mode. - 4) For the bent case, the adiabaticity criterion becomes much stricter than the straight case because of the weak bending resistance of the TG VLMA fiber. It is found that the pump absorption α
_{p}should be smaller than 0.98 dB/m to meet the adiabaticity criterion even with the 100-cm bent radius and zero-*Δn*for the 40/170-um TG VLMA fiber._{0} - 5) For the bent case, the adiabaticity criterion can be released to some extent by increasing the core-to-cladding ratio which can improve the bending resistance of TG VLMA fiber. In spite of that, it is still not recommended to bend the TG VLMA fiber in the practical applications.
- 6) One effective way to loosen the adiabaticity criterion is making the thermal load along the fiber large enough to avoid the region around the local minimum value. It is implied that the bi-directional pumping scheme is beneficial to the adiabatic propagation of LP
_{01}mode - 7) For the straight case, it is recommended that
*Δn*should be smaller than 3 × 10_{0}^{−5}, and the inner-cladding diameter should be smaller than 170 um to ensure that more-than-5dB/m pump absorption (shorter than 4 m for totally 20-dB pump absorption) can meet the adiabaticity criterion.

In this work, the most severe case of pump absorption is considered. However, in actual fiber, the pump absorption would be softer and thus the theoretical results can sufficiently satisfy the adiabaticity criterion. Our discussions are limited to the 100-W/m thermal load because such a thermal load is large enough for the kilo-Watt pump power. The discussions on the bent radius limited to 100 cm because such a bent radius may be larger than the general scale of practical optical platform. Besides, these values are still sufficient for revealing the pertinent regulations. We believe that these results and conclusions can provide significant guidance for deeply understanding and designing the TG VLMA fiber and pertinent lasers and amplifiers. Furthermore, although the adiabaticity criterion Eq. (10) is used to analysis the TG VLMA fiber in this paper, it should be noted that its deduction is not dependent on the configuration of active fiber. Thus, it is also applicable in the studies on other sorts of active fibers and pertinent lasers and amplifiers.

## Funding

National Natural Science Foundation of China (NSFC) (61405249).

## Acknowledgments

W. Liu thanks Dr. S. Guo and Dr. Z. Li for helpful discussions.

## References and links

**1. **J. Nilsson and D. N. Payne, “High-power fiber lasers,” Science **332**(6032), 921–922 (2011). [CrossRef] [PubMed]

**2. **C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics **7**(11), 861–867 (2013). [CrossRef]

**3. **J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and A. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. **13**(3), 537–545 (2007). [CrossRef]

**4. **J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express **16**(17), 13240–13266 (2008). [CrossRef] [PubMed]

**5. **J. Cao, S. Guo, X. Xu, J. Chen, and Q. Lu, “Investigation on power scalability of diffraction-limited Yb-doped fiber lasers,” IEEE J. Sel. Top. Quantum Electron. **20**(5), 373–383 (2014). [CrossRef]

**6. **D. Jain, Y. Jung, P. Barua, S. Alam, and J. K. Sahu, “Demonstration of ultra-low NA rare-earth doped step index fiber for applications in high power fiber lasers,” Opt. Express **23**(6), 7407–7415 (2015). [CrossRef] [PubMed]

**7. **F. Kong, T. Hawkins, M. Jones, J. Parsons, M. Kalichevsky-Dong, C. Dunn, and L. Dong, “Ytterbium-doped 30/400 LMA Fibers with a recorded-low ~NA of 0.028,” in *CLEO: Science and Innovations*, OSA Technical Digest Series (CD) (Optical Society of America, 2016), paper SM2Q–2.

**8. **J. P. Koplow, D. A. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. **25**(7), 442–444 (2000). [CrossRef] [PubMed]

**9. **J. Li, J. Wang, and F. Jing, “Improvement of coiling mode to suppress higher-order-modes by considering mode coupling for large-mode-area fiber laser,” J. Electromagn. Waves Appl. **24**(8–9), 1113–1124 (2010). [CrossRef]

**10. **A. E. Siegman, Y. Chen, V. Sudesh, M. C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, “Confined propagation and near single-mode laser oscillation in a gain-guided, index antiguided optical fiber,” Appl. Phys. Lett. **89**(25), 251101 (2006). [CrossRef]

**11. **Y. Chen, T. McComb, V. Sudesh, M. Richardson, and M. Bass, “Very large-core, single-mode, gain-guided, index-antiguided fiber lasers,” Opt. Lett. **32**(17), 2505–2507 (2007). [CrossRef] [PubMed]

**12. **C. H. Liu, G. Chang, N. Litchinitser, D. Guertin, N. Jacobsen, and K. Tankala, “Chirally Coupled Core Fibers at 1550-nm and 1064-nm for Effectively Single-Mode Core Size Scaling,” in *CLEO: Lasers and Electro-Optics*, OSA Technical Digest Series (CD) (Optical Society of America, 2007) paper 1–2.

**13. **H. W. Chen, T. Sosnowski, C. H. Liu, L. J. Chen, J. R. Birge, A. Galvanauskas, F. X. Kärtner, and G. Chang, “Chirally-coupled-core Yb-fiber laser delivering 80-fs pulses with diffraction-limited beam quality warranted by a high-dispersion mirror based compressor,” Opt. Express **18**(24), 24699–24705 (2010). [CrossRef] [PubMed]

**14. **L. Dong, J. Li, and X. Peng, “Bend-resistant fundamental mode operation in ytterbium-doped leakage channel fibers with effective areas up to 3160 µm2,” Opt. Express **14**(24), 11512–11519 (2006). [CrossRef] [PubMed]

**15. **W. S. Wong, X. Peng, J. M. McLaughlin, and L. Dong, “Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers,” Opt. Lett. **30**(21), 2855–2857 (2005). [CrossRef] [PubMed]

**16. **F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. **36**(5), 689–691 (2011). [CrossRef] [PubMed]

**17. **J. Limpert, F. Stutzki, F. Jansen, H.-J. Otto, T. Eidam, C. Jauregui, and A. Tünnermann, “Yb-doped large-pitch fibres: effective single-mode operation based on higher-order mode delocalisation,” Light Sci. Appl. **1**(4), e8 (2012). [CrossRef]

**18. **D. Brown and H. J. Hoffman, “Thermal, stress, and Thermo-Optic effects in high average power double-clad Silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. **37**(2), 207–217 (2001). [CrossRef]

**19. **F. Jansen, F. Stutzki, H. J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express **20**(4), 3997–4008 (2012). [CrossRef] [PubMed]

**20. **L. Dong, “Thermal lensing in optical fibers,” Opt. Express **24**(17), 19841–19852 (2016). [CrossRef] [PubMed]

**21. **A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express **19**(11), 10180–10192 (2011). [CrossRef] [PubMed]

**22. **B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express **20**(10), 11407–11422 (2012). [CrossRef] [PubMed]

**23. **F. Jansen, F. Stutzki, H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “High-power thermally guiding index-antiguiding-core fibers,” Opt. Lett. **38**(4), 510–512 (2013). [CrossRef] [PubMed]

**24. **L. Kong, J. Cao, S. Guo, Z. Jiang, and Q. Lu, “Thermal-induced transverse-mode evolution in thermally guiding index-antiguided-core fiber,” Appl. Opt. **55**(5), 1183–1189 (2016). [CrossRef] [PubMed]

**25. **J. Q. Cao, W. B. Liu, J. B. Chen, and Q. S. Lu, “Modeling the single-mode thermally guiding very-large-mode-area Yb-doped fiber amplifier,” Wuli Xuebao **66**(6), 64201 (2017).

**26. **J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices Part 1: Adiabaticity criteria,” IEEE Proceedings J. **138**(5), 343–354 (1991). [CrossRef]

**27. **R. T. Schermer, “Mode scalability in bent optical fibers,” Opt. Express **15**(24), 15674–15701 (2007). [CrossRef] [PubMed]