1. Introduction

High power fiber lasers have earned a wide variety of application in industry, science and defense due to high conversion efficiency, excellent beam quality, and robustness [1, 2]. The Large-Mode-Area (LMA) double-cladding gain fibers lead to an exponential evolution of the output power of fiber lasers over the past two decades [3]. However, the evolution is suffering from a sudden halt [4] due to transverse mode instability (TMI). The TMI occurs suddenly once the output power of fiber laser exceeds a certain value (several hundred to several thousand watts), which exhibits obvious threshold-like behaviors [5]. When the TMI emerges, the beam quality of fiberlaser is getting worse suddenly, which has become a major limitation for further enhancing the output power of fiber lasers [1, 4]. Therefore, it is very important to understand the underlying physics of TMI and find out efficient strategies to suppress it.

Since TMI was firstly observed in 2011 [5], a series of important characteristics of TMI have been reported, such as obvious threshold-like behavior [5], temporal fluctuations in kHz frequency towards chaos with increased power [6–8], the TMI threshold and fluctuation frequency sensitive to fiber parameters such as core-cladding ratio [6], and so on. An initial physical interpretation for TMI was that a self-induced long period refractive index grating (RIG) could cause energy transfer between two transverse modes due to quantum-defect heating [9]. Shortly after, Smith et al. pointed out that it must be a phase shift between RIG and the inter-mode interference pattern for efficient energy coupling within transverse modes [10]. The RIG was considered as moving grating rather than stationary grating and a frequency detuning between two transverse modes is necessary for the moving gratings [11]. A Semi-static numerical model in fiber amplifiers was proposed and explains the effects of different cooling configurations on the threshold based on full numerical simulation [12]. Afterwards, a static TMI model in double-pass fiber amplifiers was also established [13]. Very recently, M. N. Zervas proposed the first TMI threshold formula by linear stability analysis in spatial domain in fiber amplifiers and pointed out that the relative phase between the fundamental mode and the transverse perturbation significantly affects the local TMI gain [14, 15].

Despite the proposed theories can explain the main characteristics of TMI, however, the suddenness and reversibility of TMI are still not clearly explained. Furthermore, the analysis results based on these theories only provides qualitative agreement with experimental results. Therefore, there remains the need for further work to provide clarity on the validity of the theoretical model. Moreover, the characteristics of TMI, such as spontaneity and criticality, indicate that TMI is a non-equilibrium phase transition phenomenon and has a striking resemblance to those found in other research fields such as hydrodynamics [16], plasma physics [17], nonlinear chemical reactions [18]. Therefore, we can try to explain this problem from the perspective of non-equilibrium phase transition.

In this paper, we established a general dynamics model (containing time) of TMI by modifying the heat equation and dimensionless transformation. A special phase transition model of a TMF oscillator was established by applying this general dynamic model to TMF oscillator case based on eigenmodes expansion analysis (spectral method). Theoretical analysis shows that there is a reversible phase transition point in this TMF oscillator model, which can well explain the abrupt and reversible change of TMI. Finally, an analytical threshold formula of TMI was given to calculate the TMI threshold in the TMF oscillator with help of linear stability analysis near the phase transition point. The calculated results are consistent with the reported experimental results. The relationship between the TMI threshold and several parameters was also discussed in detail such as laser wavelength, pump wavelength, core radius, cladding radius, etc. This theoretical model will provide a clear explanation for TMI, which will be useful to understand and suppress TMI.

 

Fig. 1 Schematic of the double-cladding Yb-doped fiber. a and b are the core radius and cladding radius of the active fiber, respectively.

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2. General model

Here, a resonant cavity based on double-cladding Yb-doped fibers was considered, as shown in Fig. 1. The total length of the cavity is L, while a is the core radius and b is the cladding radius of the active fiber. The laser is assumed to be operated at single-frequency state and the pump light is transmitted in the cladding and evenly distributed along the fiber. Based on scalar wave equation of electric field [19]:

Eq. (2) can be rewritten as:

The temperature change T can be given by heat equation:

It is worth noting that this quantum-defect heat expressed by the above polynomial expression (Q) is more accurate than that by pump power ($\propto P_0$) [21] or signal intensity change ($\propto \frac{\partial {\left|F\right|}^2}{\partial z}$) [14]. The reason is that the heat source is obtained in general cases, rather than in special cases (such as steady-state case $\frac{\partial }{\partial t}=0$). Furthermore, unlike the expression in previous investigation, this modified polynomial expression is also easy for the following eigenmodes expansion analysis according to phase transition theory.

By setting $D=N_2-{\sigma }_{as}{\left({\sigma }_{es}+{\sigma }_{as}\right)}^{-1}N$ and using the results from Appendix A, taking the results from Appendix A, Eqs. (5), (7) and (8) can be transformed to the following dimensionless equations:

3. TMI model in TMF oscillator

In this section, by controlling the V-parameter of the fiber, one case on TMI emerges as a two-mode fiber (TMF) oscillator. Based on eigenmodes expansion analysis (spectral method, see also in Appendix B) and the results of Appendix C, a phase transition model of TMI in the TMF oscillator can be expressed as:

In the following section, we look for a phase transition point based on Eqs. (11a)-(11d) with help of linear stability analysis near the single-mode (${\text{LP}}_{01}$) solution.

When pump power is low, the fiber oscillator generally operates in the ${\text{LP}}_{01}$ mode. Correspondingly, Eqs. (11a)-(11d) must have a single-mode stationary solution. Therefore, by setting $\frac{\partial }{\partial t}=0$, $f_0\ne 0$ and $f_1=0$, the single-mode stationary solution can be obtained from Eqs. (11a)-(11d):

The linear stability of this single-mode solution is obtained by substituting $f_0=f_0^c+e^{\lambda t}\delta f_0$, $f_1=e^{\lambda t}\delta f_1$, $d_0=d_0^c+e^{\lambda t}\delta d_0$ and ${\widehat{\tau }}_1=e^{\lambda t}\delta {\widehat{\tau }}_1$ into Eqs. (11a)-(11d) and linearizing Eqs. (11a)-(11d) for small δ to obtain the eigenvalue problem:

Four eigenvalues are obtained in total, which can be ordered as:

Eigenvalue λ_n with the lowest n or the largest $\text{Re}\left({\lambda }_n\right)$ is the so-called critical one. In other words, the stability of this single-mode solution is determined by λ₁ under small perturbation. λ₁ can be expressed as:

According to linear stability analysis, when $\text{Re}\left({\lambda }_1\right)>0$, the single-mode solution becomes unstable. Therefore, the unstable condition of the single-mode stationary solution is as following:

Therefore, there is a phase transition point in the vicinity of the single-mode stationary solution. This phase transition point can well explain the sudden change characteristic of TMI. Furthermore, this phase transition process is reversible according to the above analysis, which also can explain why TMI is reversible effect. It is worth to mention that the above-mentioned two points are basic characteristics of TMI, but previous theories cannot clearly explain them.

4. Threshold discussion

Based on the above analysis, the RIG is initially triggered at $D_0=D_0^{\text{th}}$, suggesting the pump power is equal to threshold power. Using the definition of D₀ in appendix A and substituting it to $D_0=D_0^{\text{th}}$, an analytical threshold formula of TMI can be obtained:

Table 1. Parameters used in calculation

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Here, the TMI threshold value is estimated for a typical fiber oscillator comprising a commercial fiber (Nufern, LMA-YDF-20/400-9M) and a pair commercial fiber Bragg gratings (FBGs, HR-99.6%, OC-8.7%), which is also used in the experimental study of fiber oscillators [22]. By using the parameters values in Table 1, we can obtain $P_0^{\text{th}}=2.2 \text{kW}$ according to Eq. (18). This TMI threshold is quite close to the experimental pump power value $2.6\text{kW}$ (slope efficiency∼73%) in counter-pump scheme [22]. It is worth noted that the overlap factors $c_i\left(i=0,2,\cdots ,15\right)$ in Table 1 are obtained when $a=10\mu \mathrm{m}$ and $b=200\mu \mathrm{m}$. The lifetimes ${\tau }_{{\text{LP}}_{01}}$ and ${\tau }_{{\text{LP}}_{11}}$ in Table 1 is calculated according to the reflectivity of two fiber gratings and the bending losses of two modes. The bending losses of the two modes is mainly determined by the selected window of bending loss [23].

In the following section, the relationship between the TMI threshold and parameters is discussed. As seen in Eq. (18), the TMI threshold power is proportional to the thermal conductivity (K) and inversely proportional to the thermo-optic coefficient (κ), suggesting higher the thermal conductivity, the lower is temperature caused by quantum-defect and the higher power threshold. Similarly, larger is thermo-optic coefficient, smaller is the refractive change caused by the temperature and the higher power threshold. This relationship is consistent with the TMI threshold relationship in fiber amplifiers [14, 15].

Eq. (18) also presents the TMI threshold power inversely proportional to quantum-defect $\hslash \left({\omega }_p-{\omega }_0\right)$. Although there is a simple inverse relationship between the threshold and quantum-defect, the threshold is still complicatedly dependent on the pump and the signal frequency. Because the ${\sigma }_{es}$, ${\sigma }_{as}$, ${\sigma }_{ep}$, ${\sigma }_{ap}$ of Eq. (18) are also dependent on the pump and signal frequency (or wavelength). In order to gain abetter understanding on the relationship with the signal and pump wavelength, we can calculate the TMI threshold value according to Eq. (18). Parameters used in calculation are also seen in Table 1.

 

Fig. 2 (a) Threshold power vs the signal wavelength (λ_p = 976 nm); (b) Threshold power vs the pump wavelength in the long-wavelength pumping technology scheme (λ₀ = 1.08 μm). Other parameters: a = 12 μm, b = 250 μm, ${\tau }_{{\text{LP}}_{10}}=84 \text{ns}$, ${\tau }_{{\text{LP}}_{11}}=18 \text{ns}$.

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Fig. 2(a) shows that the TMI threshold power decreases with the increasing of signal wavelength. This result has also been experimentally confirmed [24]. The main reason for this change is the increase of quantum-defect with the increasing of signal wavelength. Indeed, when ${\lambda }_p=const=976 \text{nm}$, it can be obtained by Eq. (18):

Note that we have assumed ${\sigma }_{es}\gg {\sigma }_{as}$, ${\sigma }_{es}{\sigma }_{ap}\gg {\sigma }_{as}{\sigma }_{ep}$ and ignored the effect of signal wavelength change on the integral constants $c_i\left(i=0,2,\cdots ,15\right)$ in the derivation of Eq. (19). Therefore, the TMI threshold can be improved by reducing the signal wavelength.

Another effective way to improve the TMI threshold is to adopt long-wavelength pump, such as the tandem-pumping technology [25]. Fig. 2(b) shows that the TMI threshold dramatically increases with the increasing of pump wavelength. Similarly, when ${\lambda }_0=const=1.08 \mu \mathrm{m}$, the relationship between the TMI threshold and the pump wavelength $P_0^{\text{th}}\propto {\left({\lambda }_0-{\lambda }_p\right)}^{-1}{\sigma }_{ap}^{-1}\left({\lambda }_p\right)$ can be obtained by Eq. (19). Therefore, there are two reasons for the surprising increase of the TMI threshold with pump wavelength, namley ${\lambda }_0-{\lambda }_p$ and ${\sigma }_{ap}\left({\lambda }_p\right)$. But the effect of ${\sigma }_{ap}\le \left({\lambda }_p\right)$ on the TMI threshold is much greater than that of ${\lambda }_0-{\lambda }_p$, For example, when pump wavelength ${\lambda }_p=1020 \text{nm}$ is adopted, the TMI threshold is increased by 33 times comparing with ${\lambda }_p=976 \text{nm}$. Among them, ${\lambda }_0-{\lambda }_p$ makes 1.7% contribution to increase of the TMI threshold, whereas ${\sigma }_{ap}\left({\lambda }_p\right)$ makes 98.3% contribution. Therefore, the main reason for surprising increase of the TMI threshold by long-wavelength pumping is reduction of absorption cross section of pumped light. It is worth to mention that the important effect of ${\sigma }_{ap}\left({\lambda }_p\right)$ on the TMI threshold have not been clarified in previous studies. Furthermore, this result is supported by the tandem-pumping scheme, by which IPG Photonics generates a 10-20 kW single-mode beam by utilizing ${\lambda }_p=1018 \text{nm}$ [26].

 

Fig. 3 (a) Threshold power vs core radius (b = 250 μm). (b) Threshold power vs cladding radius (a = 12 μm). Other parameters: λ_p = 976 nm, λ₀ = 1.08 μm, ${\tau }_{rmL{\mathrm{P}}_{10}}=84 \text{ns}$, ${\tau }_{{\text{LP}}_{11}}=18 \text{ns}$.

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Figs. 3(a) and 3(b) shows the TMI threshold as function of core radius and cladding radius. It can be observed that the TMI threshold decreases with the increasing of core radius, whereas the TMI threshold increases with the increasing of cladding radius. The relationship between threshold and core radius is qualitatively consistent with the results in [15], which is expressed as $P_0^{\text{th}}\propto a^{-2}$. The above two relationships can be explained by the overlap ratio of temperature mode and laser mode. The TMI threshold is dependent on overlap integral $c_i\left(i=0,2,\cdots ,15\right)$ due to $P_0^{\text{th}}\propto c_3{c}_5{c}_{11}{\left(c_0{c}_{10}{c}_{15}\right)}^{-1}$. This bears a resemblance to the stimulated Brillouin scattering threshold, which depends on the overlap ratio between the laser mode and the excited acoustic mode [27]. When the core radius is reduced or the cladding radius is increased, the overlap ratio between the temperature mode and the laser mode is reduced. As a consequence, the temperature effect on the laser becomes smaller and the TMI threshold increases. Furthermore, the relationship between threshold and cladding radius can be approximately expressed as $P_0^{\text{th}}\propto b^4$ (see also Appendix C) in the TMF oscillator. Therefore, we can increase the threshold mainly by increasing the cladding radius. This is due to reducing the core radius may result in strong nonlinear effects.

 

Fig. 4 (a) Threshold power vs lifetime of LP₀₁ (${\tau }_{{\text{LP}}_{11}}=18 \text{ns}$). (b) Threshold power vs lifetime of LP₁₁ (${\tau }_{{\text{LP}}_{01}}=84 \text{ns}$). Other parameters: λ_p = 976 nm, λ₀ = 1.08 μm, a = 12 μm, b = 250 μm.

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Figs. 4(a) and 4(b) shows the TMI threshold as function of the lifetime of ${\text{LP}}_{01}$ and ${\text{LP}}_{11}$ in the cavity. It can be seen that the TMI threshold increases with the increasing of the lifetime of ${\text{LP}}_{01}$, whereas the TMI threshold decreases with the increasing of the lifetime of ${\text{LP}}_{11}$. This can be explained by mode competition between ${\text{LP}}_{01}$ and ${\text{LP}}_{11}$. The longer the lifetime of ${\text{LP}}_{01}$ or the shorter the lifetime of ${\text{LP}}_{11}$, the stronger the competition ability of ${\text{LP}}_{01}$ has. As a result, the higher TMI threshold can be obtained, vice versa. As seen in Figs. 4(a) and 4(b), the TMI threshold is increased by 35 W when the lifetime of ${\text{LP}}_{01}$is changed by 10 ns (from 80 ns to 90 ns), whereas the TMI threshold is increased by 1274 W when the lifetime of ${\text{LP}}_{01}$ is changed by 10 ns (from 25 ns to 15 ns). Therefore, the TMI threshold can be mainly improved by reducing the lifetime of${\text{LP}}_{11}$ mode such as reducing the fiber bending radius, adopting fiber Bragg gratings with low reflectivity for ${\text{LP}}_{11}$ mode, which can be also useful to suppress the TMI in fiber oscillators.

5. Conclusion

In this paper, a phase transition model for understanding TMI in a TMF oscillator was established. Theoretical analysis shows that there is a reversible phase transition point in this TMF oscillator model, which can well explain the characteristic of sudden and reversible change of TMI. It is worth to mention that phase transition was used to understand TMI in fiber oscillators at the first time, which can provide a more clear explanation for TMI than previous studies. Furthermore, an analytical threshold formula of TMI was given to calculate the TMI threshold in TMF oscillator with help of linear stability analysis near the phase transition point. The calculated results are consistent with the reported experimental results. The relationship between the TMI threshold and several parameters was also discussed in detail such as laser wavelength, pump wavelength, core radius, cladding radius, lifetime of modes, etc. As a consequence, the TMI threshold can be dramatically increased by the increasing of pump wavelength. The main reason for surprising increase of the TMI threshold by long-wavelength pumping is reduction of absorption cross section of pumped light. The TMI threshold in the TMF oscillator is also found to be proportional to the fourth power of the cladding radius. Thus, the TMI threshold can be increased mainly by the increasing of cladding radius. Moreover, the TMI threshold also can be improved by reducing the lifetime of ${\text{LP}}_{11}$ mode. This theoretical model and findings will be useful to understand and suppress the TMI in fiber oscillators.