## Abstract

In this paper, the thermal-lens induced mode coupling in step-index large mode area fiber laser is systematically investigated. The pertinent mode coupling coefficient is studied firstly, to the best of our knowledge. It is demonstrated that the mode coupling can be induced by the thermal-lens induced waveguide changing along the active fiber. It is found that the mode coupling can be enhanced mainly by the thermally-induced mode distortion and refractive index variation, both of which will become severe with the large thermal load. The impacts of fiber configuration parameters on the mode coupling are discussed. It is found that in the straight fiber, the mode coupling in a larger-core fiber can be weakened when the thermal load is low, but it will become stronger when thermal-lens effect is severe enough. However, in the bent fiber, enlarging core size, reducing core numerical aperture (NA), or decreasing bend radius will all aggravate the mode coupling. Especially when NA is excessively reduced, the mode coupling will be dramatically raised even with a small thermal load. The pertinent study is significant for understanding the mode coupling phenomenon in high-power fiber lasers.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fiber lasers and amplifiers have been significantly progressed because of their excellent properties of flexibility, robustness, high efficiency, and have been widely used in industrial manufacturing and medical treatment [1,2]. Although nowadays the output power has been significantly promoted up to 10 kW with near-diffraction limitation [3], there are still several physical factors appeared limiting the further power scaling. Among these limitations, the thermal-lens effect is considered as an important issue [4], which is induced by the quantum defect in the active fiber. It will bring a temperature increment which is parabolic shape in the core and logarithmic shape in the cladding [5]. Correspondingly, a refractive index (RI) increment will be introduced through thermal-optic effect and change the waveguide structure [6,7].

The “thermal-lens” effect has been frequently investigated in the solid-state laser, which will result in the thermal beam focusing effect and is considered as the main factor lowering the beam quality [8–10]. The pertinent theoretical studies were also carried out with the ABCD matrix [9,10]. However, because of the small scale of fiber core, the thermal-lens effect was not paid sufficient attention until 2008 when Dawson revealed that the thermal-lens should be a determinant limitation on the power scalability of fiber laser in [4]. In the reference, a Gaussian beam focusing induced by the thermal-lens effect (similar to the case of solid-state laser) was taken into account, and a formula predicting thermal-lens limitation was also presented with the ABCD matrix method. The formula was utilized widely in later studies on the power scalability of fiber lasers [11–13].

After that, the thermal-lens effect was also investigated in some ultra-large mode area active fibers, such as the photonic crystal fiber (PCF), gain-guided index-antiguided (GG-IAG) fiber, and thermally guiding index-antiguiding-core (TG-IAG) fiber [14–17], etc. In these studies, the shrinking of transverse mode induced by the thermal-lens effect was taken into account, and its effects on the gain abstract (induced by the variation of mode overlap with the doped core because of the mode shrinking) and single-mode amplification were investigated. Besides, the adiabatic criterion of fundamental mode propagation in the TG-IAG fiber was studied in [18]. In [19], a normalized thermal lensing parameter was also proposed to characterize the effect of thermal-lens and revealed that HOMs can be supported by the single-mode fiber when the thermal load is large enough.

Parallel to the study of thermal-lens effect, another thermal effect named as thermally-induced transversal mode instabilities (TMI) also becomes heated recently [20–26]. TMI is a transverse-mode coupling phenomena present when the pump power is large enough, which will induce the severe power transferring from the fundamental mode (FM) to HOM(s) and degenerate the beam quality of fiber lasers [20–22]. This phenomena was firstly reported in 2010 [21], and later, was considered as one determinant limiting the power scalability of single-mode fiber lasers [13,20]. Although the physical origin of TMI is still under investigation, the current widely-accepted view is that the mode coupling of TMI is caused by the thermally-induced refractive index grating formed by the mode interference pattern between the FM and HOMs [24–26]. However, as a distinguished thermal effect, whether the thermal-lens effect can induce the mode coupling is still not so clear.

In this paper, the thermal-lens induced mode coupling in the step-index large mode area (LMA) active fiber is revealed for the first time, to the best of our knowledge. Different from the TMI, the thermal-lens induced mode coupling results from the variation of thermal load (and pertinent thermal-lens effect) along the absorption of pump light, which will form an irregular thermal-lens induced waveguide. In such an irregular waveguide, the mode coupling will take place. In this paper, the mode coupling effect is studied by investigating the coupling coefficient between the FM and HOMs. The content is arranged as follows. In Section 2, the mode field variation induced by thermal-lens effect is studied in both straight and bent fiber. In Section 3, the mode coupling is investigated, and its dependence on bend radius is given. In Section 4, the impacts of fiber configuration parameters are discussed. In Section 5, the conclusions will be drawn.

## 2. Thermal-lens induced mode field variation in step-index LMA fiber

The mode field distribution is one of the major aspects that will decide the mode coupling in the active fiber, thus before the analysis on mode coupling, we would like to firstly discuss the mode field variation. Moreover, considering that the local FM is most important in the active fiber, the variation of FM is primarily discussed in this section.

#### 2.1 Straight fiber case

With the fiber parameters shown in Table 1 and a given thermal load *Q*, the two-dimensional thermally-induced RI increment *Δn(x, y)* can be calculated based on the heat conduction equation [see Eq. (6) and Eq. (7) shown in [17]]. Then, the optical field of local FM (i.e. LP_{01} mode) *Ψ _{01}* can be obtained with the finite-element method (FEM).

To characterize the mode field variation with the thermal load, the effective mode area *A _{eff}* and the filling factor

*Γ*of the LP

_{01}mode are calculated by

*I(x,y)*represents the intensity of the transverse mode field. The results of fiber with parameters shown in Table 1 and various V values are shown in Fig. 1.

It can be found from Fig. 1 that with the increment of thermal load, the *A _{eff}* of the LP

_{01}mode drops while the filling factor

*Γ*of three fibers increases monotonously. With the thermal load increasing to around 100 W/m, the reduction of

*A*becomes much slower and the filling factor

_{eff}*Γ*almost reaches to 1. This can be easily understood by the shrinking of the mode field induced by the increment of thermal load. It also can be found that

*A*is larger with smaller V value when the thermal load is smaller than 100 W/m, but such a difference vanishes when the thermal load is larger than 150 W/m. This is because that when the thermal load is large enough, the thermal-lens induced waveguide will be more predominant and thus the difference caused by the original NA will be weakened. Besides, it also can be seen that the variation of

_{eff}*A*and

_{eff}*Γ*is more severe with smaller V (or NA) value, which suggests that the mode field is more sensitive to the thermal-lens effect.

#### 2.2 Bent fiber case

In a practical fiber laser system, bending of the active fiber is unavoidable because its length is generally tens of meters. When the active fiber is bent, the axisymmetric of waveguide will no longer be kept and the RI distribution can be transformed equivalently as [27]

where*R*is the bend radius,

*x*is the bend axis coordinate,

*n*and

_{0}(x, y)*Δn(x, y)*represent the two-dimensional RI distribution of the original fiber and the thermally-induced increment, respectively. Then, the mode field can be calculated with a given thermal load and bend radius. Figure 2 shows the LP

_{01}mode field of a 40/250 μm core/cladding diameter fiber with 0.04 NA (V value is 4.72) and bend radius of 50 cm, 30 cm, and 10 cm respectively. It can be found that the symmetric of mode field is broken and the mode distortion is more severe with smaller bend radius.

Figure 3(a) is given to quantitate the impacts of bend radius and thermal load on the *A _{eff}*. It shows that with the same thermal load,

*A*decreases monotonously with the reduction of bend radius and this reduction becomes more and more rapid when bend radius becomes smaller. It also can be found from Fig. 3(a) that the variation of

_{eff}*A*with the thermal load becomes gradually slighter when the bend radius is smaller. This is because the impact of bending becomes gradually stronger than the thermal-lens effect, and plays more important role on the mode distortion.

_{eff}Figure 3(b) shows the variation of bend radius *R _{m}* versus the thermal load corresponding to different degree of mode shrinking. It can be found in Fig. 3(b) that

*R*decreases monotonously with the increment of thermal load. To investigate the reason, the mode field variation of the LP

_{m}_{01}mode with

*R*= 10 cm is given in Fig. 4. It shows that with the increment of the thermal load, the mode distortion and the nonaxisymmetry caused by bending becomes weaker and the whole mode field gradually concentrates to the core. This means that the thermally-induced RI increment

*Δn*strengthens the confinement of optical field and improves the bending resistance of active fiber. Thus,

*R*can be smaller with larger thermal load.

_{m}Then, the impacts of fiber core diameter and V values on the *A _{eff}* and

*R*(corresponding to 30% mode shrinking) are shown in Fig. 5. It can be found that when V value is given, the thermal-lens induced mode shrinking is more severe and

_{m}*R*increases monotonously with the enlargement of core size. It means again that the fiber is more sensitive to the thermal load with larger core. Besides, it can be noted that

_{m}*R*also increases with the decrease of V value due to the weakening of bending resistance, but the difference is slight and can be negligible especially when

_{m}*Q*is larger than 100 W/m.

## 3. Mode coupling induced by thermal-lens effect

#### 3.1 Straight fiber case

In the above section the thermal-lens induced variation of LP_{01} mode is discussed, which suggests that the mode fields will vary with the thermal load and the variation should be different in different fibers. Since the thermal load is changing along the fiber (due to the pump absorption), the thermal-lens induced waveguide and the pertinent mode fields should also vary with the propagation distance *z*, which may induce the mode coupling. Thus, in this section, pertinent mode coupling will be investigated based on the mode field analysis in the former section.

Considering that the RI is changing with the propagation distance *z*, its distribution *n* and the thermally-induced RI increment *Δn* in Eq. (3) should be represented in three-dimension *n*(*x, y, z*) and *Δn*(*x, y, z*), respectively. Because *Δn* is axisymmetric in the straight fiber, the local LP_{01} mode should only couple to the higher-order modes with the same azimuthal symmetry, i.e. LP_{0m} modes [28,29]. Thus, in this section, the mode coupling between the LP_{01} and LP_{02} mode will be discussed. Considering a co-pumping or counter-pumping scheme and assumed that pump power is exponentially varying with the pump absorption coefficient *α _{p}*, the mode coupling coefficient can be given by [18]

*k*= 2π/

*λ*is the free-space wave-number and

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

_{s}*Ψ*and

_{j,m}*β*are the optical fields and propagating constants of

_{j,m}*j*

_{th}and

*m*

_{th}mode respectively. In the straight case, the

*j*

_{th}and

*m*

_{th}mode are referred to the LP

_{01}and LP

_{02}mode respectively and the mode coupling coefficient can be represented by

*C*. Here, the local mode can be adopted in the calculation because the index change is slowly varying along the fiber. Although the local modes are orthogonal to each other at each propagation position

_{12}*z*, the mode coupling coefficient can be non-zero because of the variation of thermally-induced refractive index (i.e.

*Δn*) caused by the thermal load variation induced by the pump absorption [18,28]. Generally,

*α*is a constant for a given active fiber. Then, the variation of coupling coefficient

_{p}*C*can be determined by the coefficient

_{12}*C’*. Therefore, we focus our discussions on the coefficient

_{12}*C’*in following parts.

_{12}Figure 6 gives the variation of *C’ _{12}* and (

*β*

_{01}

*-β*

_{02}) with the thermal load in a fiber with 0.04 NA and 40/250 μm core/cladding diameter. It can be seen from Fig. 6(a) that

*C’*increases rapidly with the thermal load increasing and then the increment becomes slower when

_{12}*Q*is larger than

*Q*(about 50 W/m in Fig. 6a). It means that the mode coupling is enhanced with the increment of

_{th}*Q*, and such an enhancement becomes weaker when

*Q*is large enough.

This variation can be understood by the help of Fig. 7. It should be noted from Fig. 7(a) that when *Q* is 0 W/m, the RI increment *Δn* is 0 (see yellow dashed line), which corresponds to an undisturbed waveguide and the mode coupling will not happen. Then, when *Q* increases initially, the mode shrinking will be induced, which makes the mode field more concentrated to the core center (see Fig. 7a from 0 W/m to 50 W/m). Then, together with the increment of *Δn*, the mode shrinking will enlarge the integral in Eq. (4) and make the coefficient *C’ _{12}* increased. However, when

*Q*is large enough, the mode shrinking and its effect will be weakened (see Fig. 7b from 50 W/m to 100 W/m and Fig. 4), and then the coefficient

*C’*will be increased mainly by the increment of

_{12}*Δn*, which makes the increment of

*C’*slower. Moreover, it also can be noted from Fig. 6(b) that the value of (

_{12}*β*

_{01}

*-β*

_{02}) increases monotonously with the increment of

*Q*, which is obvious not beneficial to the mode coupling. Therefore, it can be concluded that the increment of

*C’*shown in Fig. 6(a) is mainly induced by the mode field and RI variation [see the integral in Eq. (4)].

_{12}#### 3.2 Bent fiber case

As mentioned above, bending is generally unavoidable in a practical fiber laser. Thus, in this section, we will take the bend-induced mode field variation into account. Then, different from the straight case, the thermal-lens induced coupling between LP_{01} mode and the LP_{11e} mode should be primarily considered (the coupling between LP_{01} and LP_{11o} mode is not taken into account because it is much weaker [27,28]). The mode coupling coefficient *C _{01}* also can be calculated by Eq. (4) with the assumption that the bend radius

*R*is independent on

*z*and the bending effect on RI is much smaller than

*n*[18]. In this case,

_{0}*Δn*in Eq. (4) can just be given approximately by the thermally-induced RI variation, which is proportional to the thermal load

*Q*[18], and the bending effect only affects the coupling coefficient by changing the

*Ψ*and

_{01,11}*β*. Thus, the coefficient

_{01,11}*C'*

_{01}should be the function of both thermal load

*Q*and bend radius

*R*. Here, we also take fiber with 40/250 μm core/cladding diameter and 0.04 NA as a paradigm to investigate the variations of

*C'*

_{01}and (

*β*

_{01}

*-β*

_{11}), the results are given respectively in Fig. 8(a) and 8(b). Here, the bend radii are chosen larger than

*R*(8.5 cm) according to Fig. 3(b) to obtain a large enough

_{m}*A*of the LP

_{eff}_{01}mode.

Figure 8(a) shows that with various bend radii, the variations of *C' _{01}* are similar, which initially increase rapidly with the thermal load and then the increment become slower when the thermal load is beyond

*Q*(generally around 50 W/m). The reason can be found from Fig. 9(a), which shows the mode field variation with the thermal load when

_{th}*R*is 10 cm. It can be noted that when no thermal load presents (i.e.

*Q*= 0 W/m), the value of

*C'*is 0 and no mode coupling appears, although the distortion has been caused by bending (as shown with blue lines in Fig. 9a). The mode coupling can only happen when the thermal load is given. It means that the thermal load is indispensable for the occurrence of mode coupling. In spite of that, Fig. 8(a) also illuminates that the mode coupling can be enhanced by the reduction of bend radius, which means that the increment of

_{01}*C'*should also be related to the bend-induced mode distortion. As shown in Fig. 9(a), when

_{01}*Q*is smaller than 50 W/m (see the blue line), both the mode field distortion and the increment of

*Δn*will enhance the integral in Eq. (4). However, when

*Q*increases to larger than 50 W/m (see the yellow and red lines in Fig. 9a), the effect of thermal-lens will be so severe that the mode distortion caused by bending is gradually compensated and becomes weaker. Then, the pertinent contribution of the mode field distortion is lowered, and the increment of

*C'*mainly owns to the increment of

_{01}*Δn*, thus the enhancement of

*C'*with the thermal load is weakened. For larger bend radius, e. g.

_{01}*R*= 50 cm, the bend-induced mode field distortion is weaker, as shown in Fig. 9(b). Thus, its contribution to the mode coupling will be weakened, which reduces the value of

*C’*. Moreover, the value of (

_{01}*β*

_{01}

*-β*

_{11}) shown in Fig. 8(b) also increases monotonously with the increment of

*Q*and the reduction of

*R*, which proves again the decisive role of the mode field and RI variation.

## 4. Impacts of fiber parameters

In this section, the impacts of fiber core NA, core size and inner-cladding size on the thermal-lens induced mode coupling in both straight and bent cases will be discussed.

#### 4.1 Straight fiber case

The impact of fiber core NA on the coefficient *C’ _{12}* in a 40/250 μm core/cladding diameter fiber is firstly discussed in Fig. 10. It can be found that the value of

*C’*is smaller with larger core NA. This can be understood that the value of

_{12}*C’*is enhanced by both thermally-induced mode shrinking and increment of

_{12}*Δn*, as we illustrated in Section 3.1. Since the fiber with larger NA is less sensitive to the thermal load (see section 2.1), the mode shrinking should be slighter, which also can be witnessed by Fig. 11. Thus, the increment is slower and leads to a smaller value of

*C’*. Also because of its less sensitivity to the thermal load, the existence of

_{12}*Q*becomes weaker with larger NA, i.e. the increment rate gradually becomes more uniform and the monotonicity of the variation becomes stronger with 0.06 NA and 0.1 NA, as Fig. 10 shown. Therefore, the impact of NA can be summarized by the mode-guided ability. Increasing the core NA strengthens the mode guide in the fiber core and thus reducing the thermal-lens induced mode coupling.

_{th}The effects of the core and cladding size are given in Fig. 12, which shows that the effect of cladding size can be negligible (see Fig. 12b) while the core size is much more essential on the mode coupling (see Fig. 12a). It can be seen from Fig. 12(a) that although the value of *C’ _{12}* increases monotonously with the thermal load for 30-μm, 40-μm, 60-μm core fiber, the increment processes are different for these three fibers.

Figure 12(a) illuminates that the mode coupling in the fiber with larger core is stronger when the thermal load is large enough (e.g. the coefficient *C’ _{12}* in the 60-μm core fiber is larger than the cases of 30-μm and 40-μm core when the thermal load is larger than 75 W/m and 125 W/m, respectively). This is reasonable because that the thermal-lens effect is much more severe in the core than in the cladding, and the mode coupling induced by this effect should be more severe in fiber with larger core. In spite of that, it also can be noted that when the thermal load is smaller than 75 W/m, the value of

*C’*is larger with 30-μm core than 60-μm core. This is because the mode shrinking effect is more severe in smaller-core fiber due to the less confinement of mode, which plays most important role when

_{12}*Q*is smaller than 75 W/m (see the cases of 0 and 50 W/m in Fig. 13). These results reveal that in the straight case, the mode coupling in a larger core can be weaker when the thermal load is low, but it will become stronger when thermal-lens effect is severe enough.

#### 4.2 Bent fiber case

In this section, the impacts of fiber parameters on the thermal-lens induced coupling between LP_{01} and LP_{11e} mode will be investigated. Figure 14(a) shows the variations of *C’ _{01}* in a 40/250 μm core/diameter fiber with various core NA. It can be found from Fig. 14(a) that for 0.06-NA and 0.1-NA core, the variation monotonicity of

*C’*resembles to that in Fig. 8 and

_{01}*C’*is smaller with larger NA because of the weaker distortion of mode field. The distinguishing behavior is in the variation of

_{01}*C’*with 0.03 NA, where the monotonicity is changed at

_{01}*Q*(about 15 W/m) and a local maximum appears. When

_{th}*Q*is smaller than

*Q*,

_{th}*C’*with 0.03 NA increases dramatically, then it gradually reduces when

_{01}*Q*is beyond

*Q*and finally keeps almost unvaried when

_{th}*Q*is larger than 150 W/m.

Such a difference is also induced by the variation of mode fields. By comparing Fig. 15(a) with Fig. 9(a) (0.04 NA), it can be seen that when *Q* is small, the thermally-induced variation of mode field is more dramatic with 0.03 NA (see the blue and yellow line in Fig. 15a), which results in the rapid increment of *C’ _{01}*. However, when

*Q*is larger than 15 W/m, the thermally-induced variation of mode field becomes so weakened (see the orange line in Fig. 15a) and its effect on lowering the mode coupling or

*C’*is so serious that cannot be compensated by the thermally-induced RI increment. As a result, the value of

_{01}*C’*reduces. When the thermal load is large enough (see Fig. 15b) that the thermally-induced RI increment can compensate the effect of mode field variation, the value of

_{01}*C’*will keep almost unvaried.

_{01}In Addition, the variations of *C’ _{01}* in 0.03-NA fiber with various bend radii are also given in Fig. 14(b). It shows that the local maximum of

*C’*always exists even when the bend radius increases to 100 cm, although it is lowered with the increment of bend radius. Therefore, it is suggested that when the core NA is small, the strongest mode coupling can be induced at a certain thermal load.

_{01}Then, the impacts of fiber core and cladding diameter are discussed in Fig. 16. It shows that in the bent case, the cladding size also has negligible effect on the mode coupling (see Fig. 16b) and the influence of core size is more obvious (see Fig. 16a). It can be found from Fig. 16(a) that the mode coupling is always stronger with larger core. This can be understood by the weaker bend resistance of larger-core fiber. For example as Fig. 17 shows, the bend-induced mode distortion is much stronger with 60-μm core than with 30-μm core fiber. Then, together with the increment of *Δn*, the value of *C’ _{01}* with 60-μm core will be enhanced more rapidly.

It should be noted from Fig. 16(a) that the most distinguished variation is the case of 30-μm core, the coefficient *C’ _{01}* of which decreases (rather than increases) monotonously when the thermal load is beyond

*Q*(about 30 W/m). The occurrence of the local maximum also can be understood by the similar explanation to the result of the straight fiber with 0.03-NA and 40-μm core given in Fig. 14a, i.e. the dramatic thermally-induced variation of mode field when

_{th}*Q*is small, which can be verified from Fig. 17(b). It shows that the LP

_{11}mode field concentrates rapidly to the core center when

*Q*is smaller than 30 W/m (see blue and yellow lines in Fig. 17b). However, different from the situation with 0.03-NA and 40-μm core (see Fig. 14a), the value of

*C'*

_{01}keeps reducing rather than remaining invariable with further increment of the thermal load. This can be explained that although the V values of two fibers are equal (3.54), the fiber with smaller core is better bend resistant and its bend-induced mode distortion can almost been compensated at

*Q*(see red line in Fig. 17b). This will enhance the axisymmetry of two modes which weakens the coupling between LP

_{th}_{01}mode and LP

_{11}mode because of their different axisymmetry. Such weakening is so severe that cannot be well compensated by the RI increment, and results in the monotonous decreasing of the coefficient

*C’*.

_{01}From these results shown in bent case, it can be found that both lowering the fiber NA and enlarging the core size will enhance the thermal-lens induced mode coupling between LP_{01} and LP_{11} mode. Especially when NA is extremely low or fiber is too much bent, the mode coupling might be dramatically aggravated even when the thermal load is small.

## 5. Conclusions

The thermal-lens induced mode coupling in step-index LMA fiber laser is systematically investigated and the coupling coefficient is studied for the first time, to the best of our knowledge. It is predicted that the mode coupling can be induced by the irregular thermal-lens induced waveguide in the active fiber. The variations of mode coupling coefficient with the thermal load are discussed in straight and bent fiber, respectively. Following conclusions can be obtained.

- 1. Such a mode coupling cannot happen without the thermal-lens effect (i.e. the pertinent coefficient is zero with zero thermal load, see Fig. 6 and Fig. 8), which means that the mode coupling should be induced by the thermal-lens effect (this is also the reason why we name it as the “thermal-lens induced mode coupling”). Besides, the enhancement of thermal-lens effect (or equivalently, the increment of thermal load) will generally aggravate the mode coupling, although the most serious mode coupling can also be induced with a relatively low thermal load in bent fiber with small NA or core (see Fig. 14 and Fig. 16a).
- 2. No matter in the straight or bent fiber, the mode coupling can be enhanced by reducing core NA, which will lower the core confinement to the optical field and make the fiber modes more sensitive to the bend radius and thermal load. Especially in tightly bent fiber (i.e. the bend radius is small), the value of
*C'*might be dramatically raised even when the thermal load is small (see Fig. 14a). It is suggested that the fiber with larger NA should be more resistant to such a mode coupling._{01} - 3. It is revealed that the mode coupling can also be strengthened by enlarging the core size, which means that the fiber with larger core will be more sensitive to the thermal-lens induced mode coupling. However, in the straight fiber, such a strengthening of core size can only be realized when the thermal load is large enough (see Fig. 12a). Furthermore, in the bent fiber, it is also unexpectedly found that the coupling coefficient can be reduced by increasing the thermal load when the core size is small (see the case of 30-um core in Fig. 16a). Then, it is suggested that such a mode coupling can be suppressed by reducing the core size.

Here, we would like to address that all the discussions are carried out with the help of the coefficient of *C’ _{jm}* rather than the mode coupling coefficient

*C*. However, it should be noted that the mode coupling coefficient

_{jm}*C*is the one able to evaluate the coupling effect, and is also proportional to the pump absorption

_{jm}*α*[see Eq. (4)]. It means that the thermal-lens induced mode coupling will be more severe in the active fiber with larger pump absorption (e.g. in the fiber laser with a short cavity [30]). It is also implied that the value of

_{p}*C*can be several times of the

_{jm}*C’*value given in above sections if the pump absorption is large enough.

_{jm}Moreover, it should be noted that in a fiber laser, because of the variation of thermal load along the pump absorption (i.e. the thermal load is larger near the pump input end and become weaker and weaker along the propagation of pump light), the thermally-induced RI variation *Δn* should vary along the active fiber. It indicates that the value of *C _{jm}* and the corresponding mode coupling will also vary with the propagation position

*z*. Such a variation of

*C*should be taken into account when studying the thermal-lens induced mode coupling in a fiber laser. Besides, because the gain should be present with the pump absorption, the gain effect (such as the transverse spatial-hole burning [31]) should also be considered in the pertinent study in a fiber laser.

_{jm}Another point should be noted is that different from TMI studied in [20–26] which is induced originated from the thermally-induced gratings caused by the interference pattern between transverse modes, the thermal-lens induced mode coupling revealed in this paper has no relationship to the mode interference, and thus will be independent on the coherence of optical field. Thus, the thermal-lens induced mode coupling can be of great help for deeply understanding the mode coupling phenomenon in the high-power fiber laser or other sources (e.g. the superfluorescent fiber source) with the relative broad bandwidth (around or more-than nanometer level) and/or low coherence (actually, the mode coupling phenomenon have been observed and treated as one main limitation to the power scaling of these non-narrow band fiber sources [32,33]). Besides, the relationship of the thermal-lens induced mode coupling with TMI given in [7,19] will be another point needed to be further investigated. These pertinent further studies will be carried out in our successive work.

## Funding

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

## Acknowledgments

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

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