Simplified expression for transverse mode instability threshold in high power fiber lasers

Haobo Li; Liangjin Huang; Liangjin Huang; Liangjin Huang; Hanshuo Wu; Hanshuo Wu; Hanshuo Wu; Hanshuo Wu; Xiaolin Wang; Xiaolin Wang; Xiaolin Wang; Xiaolin Wang; Pu Zhou

doi:10.1364/OE.511666

1. Introduction

Transverse mode instability (TMI) is a dynamic inter-mode coupling phenomenon which originates from the thermo-optical effect [1,2]. The mode interference pattern modulates the fiber refractive index through the thermo-optical effect, generating the refractive index grating [3]. The variational phase shift between the mode interference pattern and the refractive index grating results in a rapid energy transfer between the fundamental mode and the higher-order mode. When the average output power of the fiber laser exceeds the TMI threshold power, the output beam quality will degrade sharply [4]. Currently, a large amount of theoretical and experimental evidence shows that the TMI effect has become one of the main limitations in the average power improvement of high power and high brightness fiber lasers [5–12].

Over the past decade, a large number of numerical and semi-analytical theoretical models have been developed which play an important role in revealing the TMI physical mechanism and mitigating TMI [13–22]. In particular, the semi-analytical theoretical model not only greatly reduces the calculation time, but also has the opportunity to obtain some simple TMI threshold expressions [10,16,23,24]. These TMI threshold expressions can quickly calculate the TMI threshold power, and clearly show the influence of some fiber parameters and laser parameters on the TMI threshold, which is helpful for fiber design and TMI mitigation. Among these TMI threshold expressions, the most popular one is proposed by Zervas, which has no undetermined coefficients and only requires some key parameters for TMI threshold estimation [21]. However, it also shows some inconsistencies with some experimental phenomena. For example, the TMI threshold is negatively correlated with fiber normalization constant V experimentally [9,25], but the TMI threshold is positively correlated with V value in the expression. Although other threshold expressions that can correctly simulate the effect of the V value on TMI, they also include undetermined coefficients making the calculation difficult [10,23].

Therefore, it would be of great significance to develop a TMI threshold expression with calculation simplicity while consistent with general experimental results, which could help to accelerate the fiber design and mitigate the TMI effect. To this end, we will consider the advantages of the above threshold expression and propose an optimized TMI threshold expression including more fiber and laser parameters while excluding undetermined coefficients.

Here, we establish a semi-analytical TMI theoretical model based the on two-beam coupling theory in Section 2, and then explore the effects of phase matching gain and phase mismatch gain on the higher-order mode gain in conjunction with previous work. In section 3, we will propose and verify the developed simple TMI threshold expression based on the higher-order mode gain expression. The influence of some special parameters on the TMI threshold and the power scaling is simulated and analyzed in Section 4.

2. Theoretical model

In high power fiber laser systems, we consider the simplest case, where there are only LP₀₁ and LP₁₁ modes, and the LP₀₁ mode power is much larger than LP₁₁ mode power. The electric field ${E_s}(r,\varphi ,z,t)$ in the fiber can be approximated by the normalized linear polarization mode

(1)$${E_s}(r,\varphi ,z,t) = \sum\limits_x {{A_x}(z,t){\psi _x}(r,\varphi )\textrm{exp} (i{\beta _x}z - i{\omega _x}t)} + c.c.$$

where subscripts x = 1 and 2 respectively represent LP₀₁ and LP₁₁ modes, A is the electric field amplitude, ${A_1} \gg {A_2}$, $\psi $ is the normalized transverse mode field distribution, $\beta $ is the propagation constant, and $\omega $ is the circular frequency.

In the step-index fiber, the mode field distribution can be expressed as

(2)$${\psi _x}(r,\varphi ) = \frac{1}{{{N_x}}}\left\{ \begin{array}{l} {J_{x - 1}}({u_x}r)\cos (x\varphi ),\textrm{ }r \le {r_{core}}\\ \frac{{{J_{x - 1}}({U_x})}}{{{K_{x - 1}}({W_x})}}{K_{x - 1}}({w_x}r)\cos (x\varphi ),\textrm{ }r > {r_{core}} \end{array} \right.$$

(3)$$N_x^2 = (3 - x)\pi [\int\limits_0^{{r_{core}}} {J_{x - 1}^2({u_x}r)rdr} + \int\limits_{{r_{core}}}^{{r_{clad}}} {\frac{{J_{x - 1}^2({U_x})}}{{K_{x - 1}^2({W_x})}}K_{x - 1}^2({w_x}r)rdr} ]$$

where U² +W² = V², U = ur_core, W = wr_core. U can be approximated as U₁ = 2.405exp(-1/V), U₂ = 3.832exp(-1/V) [26].

The signal light intensity is expressed as

(4)$$I = {I_0} + {I_\varepsilon } = {A_1}A_1^\ast \psi _1^2 + {A_2}A_2^\ast \psi _2^2 + [{A_1}A_2^\ast {\psi _1}{\psi _2}\textrm{exp} (i\Delta \beta z - i\Omega t) + c.c.]$$

(5)$${I_\varepsilon } = {A_1}A_2^\ast {\psi _1}{\psi _2}\textrm{exp} (i\Delta \beta z - i\Omega t) + c.c.$$

where $\Delta \beta = {\beta _1} - {\beta _2}$, the circular frequency difference $\Omega = {\omega _1} - {\omega _2}$. When ${I_\varepsilon }$ begins to change in the fiber, a dynamic refractive index grating will occur. When $\Omega > 0$, the grating moves in the same direction as the signal transmission direction. In contrast, when $\Omega < 0$, the grating moves in the opposite direction to the signal transmission direction.

The dynamic refractive index grating $\Delta {n_\varepsilon }$ can be expressed through heat conduction equation and thermo-optic effect

(6)$$\Delta {n_\varepsilon } = {k_T}{T_\varepsilon }$$

(7)$$\rho {C_0}\frac{{\partial {T_\varepsilon }}}{{\partial t}} - \kappa {\nabla ^2}{T_\varepsilon } = {Q_\varepsilon }$$

(8)$${Q_\varepsilon } \approx {q_D}\frac{{{g_0}}}{{{{({1 + {I_0}/{I_{sat}}} )}^2}}}{I_\varepsilon }$$

(9)$${g_0} = \frac{{{P_p}({\sigma _{ap}}{\sigma _{es}} - {\sigma _{ep}}{\sigma _{as}})/(h{\nu _p}{A_p}) - {\sigma _{as}}/\tau }}{{{P_p}({\sigma _{ap}} + {\sigma _{ep}})/(h{\nu _p}{A_p}) + 1/\tau }}{N_{Yb}}$$

(10)$${I_{sat}} = [{{P_p}({\sigma_{ap}} + {\sigma_{ep}})/h{\nu_p}{A_p} + 1/\tau } ]\frac{{h{\nu _s}}}{{{\sigma _{as}} + {\sigma _{es}}}}$$

where ${k_T}$ is the thermal-optical coefficient, $\rho$, ${C_0}$, and $\kappa$ are the core density, specific heat capacity, and thermal conductivity, respectively. ${Q_\varepsilon }$ is the thermal density, ${T_\varepsilon }$ is the temperature change, ${q_D} = {\lambda _s}/{\lambda _p} - 1$ is the quantum defect coefficient, ${\lambda _{s(p)}}$ is the signal (pump) wavelength. ${g_0}$ is the small-signal gain, ${I_{sat}}$ is the saturation intensity, ${P_p}$ is the pump power, ${\sigma _{as(p)}}$ is the signal (pump) absorption cross-section, ${\sigma _{es(p)}}$ is the signal (pump) emission cross-section, ${\nu _{s(p)}}$ is the signal (pump) frequency, h is the Planck constant, ${A_p}$ is the cladding area, and ${N_{Yb}}$ is the total Yb³⁺ concentration.

The temperature change ${T_\varepsilon }$ is obtained by solving equations (7)∼(10) using Green’s function method:

(11)$${T_\varepsilon }(r,\varphi ,z,t) = \sum\limits_{v = 0}^\infty {\sum\limits_{l = 1}^\infty {\frac{{\int\limits_0^{2\pi } {\int\limits_0^{{r_{clad}}} {\frac{{{q_D}{g_0}{I_\varepsilon }(r^{\prime},\varphi ^{\prime},z,t){J_v}({k_{vl}}r^{\prime})\cos v(\phi - \phi ^{\prime})}}{{{{({1 + {I_0}(r^{\prime},\varphi^{\prime},z,t)/{I_{sat}}} )}^2}}}r^{\prime}dr^{\prime}} } d\phi ^{\prime}}}{{(\kappa k_{vl}^2 - i\varOmega \rho {C_0})\int\limits_0^{2\pi } {\int\limits_0^{{r_{clad}}} {J_v^2({k_{vl}}r){{\cos }^2}(v\phi )rdrd\phi } } }}{J_v}({k_{vl}}r)} }$$

where ${J_v}$ is the Bessel function of order v, ${k_{vl}}$ is the l-th positive root of the equation ${J_v}({k_{vl}}R) = 0$.

Equation (11) is composed of many Bessel functions, which is too complex and makes the calculation difficult. Therefore, some approximations are very necessary. The LP₀₁ mode and LP₁₁ mode are expressed by Bessel functions of order 0 and order 1, respectively. The Gaussian function is usually used to approximate the LP₀₁ mode, so ${J_0}(x)$ can be approximated as

(12)$${J_0}(x) \approx \textrm{exp} ({{{ - {x^2}} / 3}} )$$

Based on the equation ${J_1}(x) ={-} {J_0}^\prime (x)$, ${J_1}(x)$ can be expressed as $2x\textrm{exp} ({{{ - {x^2}} / 3}} )/3$. However, the approximation ${J_1}(x) \approx 2x\textrm{exp} ({{{ - {x^2}} / 3}} )/3$ is not very accurate because of the error between the Gaussian function and ${J_0}(x)$. To ensure the accuracy, the approximate function of ${J_1}(x)$ is rewritten as

(13)$${J_1}(x) \approx x\textrm{exp} ({{{ - {x^2}} / 6}} )/2$$

For the fiber amplifier with strong gain saturation effect, the temperature ${T_\varepsilon }$ can be expressed as

(14)$${T_\varepsilon }(r < {r_{core}}) = \frac{{{q_D}{g_0}{u_2}I_{sat}^2N_1^3}}{{b{N_2}P_1^2({\kappa {b^2} - i\Omega \rho {C_0}} )}}{A_1}A_2^\ast {J_1}(br)\cos \varphi {e^{i(\Delta \beta z - \varOmega t)}} + c.c.$$

where $b = \sqrt {u_2^2 - 6u_1^2} $, the denominator $\kappa {b^2} - i\Omega \rho {C_0}$ is an imaginary number, indicating that there is a phase shift between the refractive index grating and the signal light.

The variation of the signal light ${E_s}(r,\varphi ,z,t)$ satisfies the nonlinear scalar wave equation ${\nabla ^2}{E_s} - ({{1 / {{c^2}}}} )({{{{\partial^2}{n^2}{E_s}} / {{\partial^2}t}}} )= 0$. Substituting the Eq. (14) into the nonlinear wave equation, the LP₁₁ mode amplitude transmission equation based on the orthogonality of modes can be written as

(15)$$\frac{{d{A_2}}}{{dz}} = \frac{1}{2}({g_2} - {\alpha _2}){A_2} + CA_1^\ast {A_1}{A_2} + {C^\ast }{A_1}{A_1}A_2^\ast {e^{2i\Delta \beta z}}$$

(16)$$C = i\frac{{\pi {q_D}{g_0}{k_0}{k_T}{u_2}N_1^2I_{sat}^2\Gamma }}{{bN_2^2P_1^2({\kappa {b^2} + i\Omega \rho C} )}}$$

(17)$$\Gamma \approx \int\limits_0^{{r_{core}}} {{J_0}({u_1}r){J_1}} ({u_2}r){J_1}(br)dr$$

where the influence of the refractive index grating in the cladding region on the nonlinear coupling is weak and neglected. According to the above approximations of the Bessel functions, the integral $\Gamma $ is rewritten as

(18)$$\Gamma = \frac{{b{u_2}r_{core}^4}}{{8{\delta ^2}}}({1 - {e^{ - \delta }} - \delta {e^{ - \delta }}} )$$

where $\delta = {{({U_2^2 - 2U_1^2} )} / 3}$, when $\delta $ approaches 0, $\Gamma $ gradually decreases and tends to $b{u_2}r_{core}^4/16$.

When the signal gain along the fiber amplifier is constant, then ${{\partial C} / {\partial z}} = 0$, the semi-analytical solution for the LP₁₁ mode gain ${G_2}$ can be expressed as

(19)$${G_2} = \frac{{d{P_2}}}{{{P_2}dz}} = {g_2} - {\alpha _2} + 2{C_r}{P_1} + 2\sqrt {{{({C_r}{P_1})}^2} - \Delta \beta (\Delta \beta + 2{C_i}{P_1})}$$

where ${C_r}$ and ${C_i}$ are the real and imaginary parts of C, ${C_r}$ obtains its maximum value at $\Omega = {{\kappa |{{b^2}} |} / {\rho {C_0}}}$, and ${C_i}$ is less than 0 and obtains its minimum value at $\Omega = 0$. The LP₁₁ mode gain consists of four components, namely, the pumping gain ${g_2}$, the loss ${\alpha _2}$, the phase matching gain $2{C_r}{P_1}$, and the phase mismatch gain $2\sqrt {{{({C_r}{P_1})}^2} - \Delta \beta (\Delta \beta + 2{C_i}{P_1})}$. Combined with the previous work [27], the phase matching gain is proportional to the LP₀₁ mode power. When the frequency difference is $\Omega = {{\kappa |{{b^2}} |} / {\rho {C_0}}}$, the phase matching gain is maximized. However, the phase mismatch gain has a completely different power characteristic. When LP₀₁ mode power satisfies ${P_1} < {{ - \Delta \beta } / {2{C_i}}}$, the phase mismatch gain is 0, and when LP₀₁ mode power exceeds ${{ - \Delta \beta } / {2{C_i}}}$, the phase mismatch gain increases rapidly and obtains its maximum value at $\Omega = 0$.

3. TMI threshold expression

When the TMI threshold is lower than the excitation power of the phase mismatch gain ${{ - \Delta \beta } / {2{C_i}}}$, the phase mismatch gain is 0. Under this circumstance, the LP₁₁ mode output power and local gain can be expressed as

(20)$${P_2}(L) = {P_2}(0)\textrm{exp} \left( {\int\limits_0^L {{G_2}(z)dz} } \right)$$

(21)$${G_2} = {g_2} - {\alpha _2} + 2{C_r}{P_1}$$

The LP₁₁ mode output power usually has no semi-analytical solution because of the variation of signal gain and gain saturation along the fiber. When the signal gain is stable along the fiber, the LP₁₁ mode output power can be rewritten as

(22)$${P_2}(L) = {P_2}(0)\textrm{exp} ({{g_2}L - {\alpha_{2total}} + 2{C_r}{P_1}(0)G/{g_1}} )$$

where ${\alpha _{2total}} = \int\limits_0^L {{\alpha _2}(z)dz} $ is the total LP₁₁ mode loss, G is the time of signal power amplification.

The condition that the signal gain is constant along the fiber amplifier can be achieved by backward pump and suitable seed power [23]. On this condition, the relationship between pump power and the signal power can be simplified as ${P_s} \approx {\lambda _p}{P_p}/{\lambda _s}$. According to the Eqs. (9) and (10), the amplifier signal gain and fiber length can be approximated as

(23)$${g_s} = \int\!\!\!\int\limits_{core} {\frac{{{g_0}}}{{1 + {I_s}/{I_{sat}}}}\psi _1^2ds} \approx \frac{{({\sigma _{ap}}{\sigma _{es}} - {\sigma _{ep}}{\sigma _{as}})r_{core}^2}}{{({\sigma _{as}} + {\sigma _{es}})r_{clad}^2}}{N_{Yb}}$$

(24)$$L = \frac{{\ln G}}{{{g_s}}} = \frac{{({\sigma _{as}} + {\sigma _{es}})r_{clad}^2\ln G}}{{({\sigma _{ap}}{\sigma _{es}} - {\sigma _{ep}}{\sigma _{as}})r_{core}^2{N_{Yb}}}}$$

The TMI threshold power is usually defined by the ratio $\eta (z)$ of LP₁₁ mode power to LP₀₁ mode power. When the ratio $\eta (L)$ increases to a special value, which could be 5% or other values, the output power at this point is defined as the TMI threshold power [28].

The Eq. (22) can be expressed as:

(25)$$\eta (L) = \eta (0)\textrm{exp} ({{g_2}L - {g_1}L - {\alpha_{2total}} + 2{C_r}{P_1}(L)/{g_1}} )$$

In this paper, $\eta (L) = 5\%$ is chosen as the criteria of the TMI threshold, and the TMI threshold power can be expressed as

(26)$${P_{TMI}} \approx {P_{seed}} + \frac{{16\pi \kappa \tilde{\sigma }\tilde{\alpha }\tilde{N}r_{clad}^2}}{{{k_0}{q_D}{k_T}r_{core}^2}} = \frac{{16\pi \kappa \tilde{\sigma }\tilde{\alpha }\tilde{G}\tilde{N}r_{clad}^2}}{{{k_0}{q_D}{k_T}r_{core}^2}}$$

with

(27)$$\tilde{G} = G/(G - 1)$$

(28)$$\tilde{\sigma } = ({\sigma _{as}} + {\sigma _{es}})/({\sigma _{ap}} + {\sigma _{ep}})$$

(29)$$\tilde{N} = (6U_1^2/U_2^2 - 1)(N_2^2/N_1^2)$$

(30)$$\tilde{\alpha } = {\alpha _{2total}} + \ln \eta (L) - \ln \eta (0)$$

where the TMI threshold expression (26) is made up of some simple fiber parameters and laser parameters. The simple relationship between the TMI threshold and these parameters provides an approach to TMI mitigation. These mitigation strategies can be summarized as: higher seed power (${P_{seed}}$) can enhance the gain saturation effect to achieve higher TMI threshold [29,30], using fiber with greater thermal conductivity ($\kappa $) and lower thermal-optical coefficient (${k_T}$) to directly weaken the heat-induced refractive index grating [31,32], exploiting higher signal absorption (${\sigma _{as}}$) and emission (${\sigma _{es}}$) cross section and lower pump absorption (${\sigma _{ap}}$) and emission (${\sigma _{ep}}$) cross section by shifting the signal and pump wavelength to enhance the gain saturation to suppress the TMI effect [33,34], adopting fiber with a smaller core to cladding ratio (${r_{core}}/{r_{clad}}$) for stronger gain saturation to improve the TMI threshold [35], lowering the quantum defect (${q_D}$) to reduce the heat load [36,37], introducing higher higher-order mode loss (${\alpha _{2total}}$) and lower intensity noise ($\eta (0)$) to suppress higher-order mode transmission to mitigate the TMI effect [38–41], and reducing the V value ($\tilde{N}$) to weaken the grating strength in the core region and increase the TMI threshold [25].

According to the Eq. (26), the threshold expression is mainly applied to the step-index fiber where the gain saturation effect plays a dominant role in TMI mitigation. If the TMI threshold expression is used for the high-power fiber amplifiers, the condition $({\sigma _{as}} + {\sigma _{es}})r_{clad}^2/[({\sigma _{ap}} + {\sigma _{ep}})r_{core}^2] \gg 1$ should be satisfied. When $({\sigma _{as}} + {\sigma _{es}})r_{clad}^2/[({\sigma _{ap}} + {\sigma _{ep}})r_{core}^2] \ge 10$ is met, the TMI threshold expression in this manuscript has a good accuracy. According to this condition $({\sigma _{as}} + {\sigma _{es}})r_{clad}^2/[({\sigma _{ap}} + {\sigma _{ep}})r_{core}^2] \ge 10$, the signal and the pump wavelengths have an important influence on the application range of the threshold expression. For instance, when the cladding diameter is 400 µm, the 976 nm LD-pumped fiber amplifier with a core diameter less than 30 µm can use the TMI threshold expression to calculate the TMI threshold, while when the cladding diameter is 250 µm, the TMI threshold expression can be applied to a 1018 nm tandem-pumped fiber amplifier with a core diameter of 45 µm. The larger the cladding size, the larger the application range of the TMI threshold expression.

In order to evaluate the accuracy of the threshold expression (26), we need to compare the TMI threshold expression with the simulation results of the semi-analytical model without approximation in Section 2. First, the semi-analytical model has been applied to the TMI threshold calculation of a more complex multimode fiber laser, and the simulation results are in good agreement with the experimental results [12]. Then, the semi-analytical model simulates the LP₁₁ mode gain of the forward pumped fiber amplifier according to the parameters of the full numerical model [2]. The simulation results are shown in Fig. 1(a), which are consistent with the full numerical model. Therefore, the semi-analytical model can be used to evaluate the accuracy of the threshold expression (26). Figure 1(b) shows the TMI thresholds for a backward pump fiber amplifier with different core diameters. The main parameters of fiber amplifiers are referenced from Ref. [2] and listed in Table 1, which are used throughout the paper. The red curve represents the calculation results using the TMI threshold expression, and the black curve is the simulation results of the semi-analytical theoretical model. As the core diameter increases from 20 to 30 µm, the fiber length (blue curve) is reduced from 25 m to 11 m due to the constant cladding diameter of 400 µm, the TMI thresholds represented by the red curve and the black curve show a similar trend, where the TMI threshold decreases rapidly. The calculation difference of the two methods is less than 20%, which can come from three aspects. The first is the approximate mode field introduced to solve the heat conduction equation. The second origin is the high gain saturation approximation, which will increase the calculation difference as the core diameter increases. Finally, the thermally induced refractive index grating in the cladding region is ignored, the calculation brought by which becomes larger with the smaller core V value.

Fig. 1. (a) The mode evolution and mode gain in a 30/250 µm fiber amplifier, (b) The TMI threshold power as a function of core diameter, the red curve and the black curve are the calculated results of the expression and the semi-analytical model, respectively.

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Table 1. The main parameters of fiber amplifier

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When the output power is lower than the TMI threshold power but exceeds the trigger power ${{ - \Delta \beta } / {2{C_i}}}$ of phase mismatch gain, the higher-order mode will grow rapidly, which will lead to the early appearance of the TMI phenomenon. According to the previous work [27], the minimum trigger power of phase mismatch gain which is named as the mismatch threshold throughout the paper can be expressed as

(31)$${P_{mismatch}} = \frac{{\Delta \beta }}{{2\max ( - {C_i})}} = \frac{{8\pi \kappa \tilde{\sigma }\tilde{G}\tilde{N}r_{clad}^2\Delta \beta }}{{{q_D}{g_s}{k_0}{k_T}r_{core}^2}} = \frac{{8\pi \kappa \tilde{\sigma }\tilde{G}\tilde{N}r_{clad}^2\Delta \beta L}}{{{q_D}{k_0}{k_T}r_{core}^2\ln G}}$$

where $\Delta \beta $ is approximated as $({U_2^2 - U_1^2} )/2{k_0}{n_{core}}r_{core}^2$ and is positively proportional to V.

See from the exterior form, the mismatch threshold is very similar to the TMI threshold expression (26), for example, they are both positively proportional to the square of cladding radius. On the other hand, the mismatch threshold also shows some new features, such as the mismatch threshold is positively proportional to $\Delta \beta $ and inversely proportional to average signal gain. Therefore, increasing $\Delta \beta $ and decreasing core radius can effectively improve the mismatch threshold.

The ratio of the mismatch threshold to the TMI thereshold can be expressed as:

(32)$$\frac{{{P_{mismatch}}}}{{{P_{TMI}}}} = \frac{{\Delta \beta \max ({C_r})}}{{{g_s}\max ( - {C_r})\ln ({\eta _L} - {\eta _0} + {\alpha _{2total}})}}$$

According to the Eq. (16), the condition $\max ( - {C_i}) = 2\max ({C_r})$ is satisfied, and the Eq. (32) is expressed as:

(33)$$\frac{{{P_{mismatch}}}}{{{P_{TMI}}}} = \frac{{\Delta \beta }}{{2{g_s}\ln ({\eta _L} - {\eta _0} + {\alpha _{2total}})}}$$

Although the phase mismatch gain can lead to the early appearance of the TMI phenomenon, the mismatch threshold is usually much larger than TMI threshold in the small-core long fiber amplifier. As shown in Fig. 2, the $\Delta \beta $ and the ratio of the mismatch threshold to the TMI thereshold for different core diameters are simulated. As the core diameter increases, the signal gain is maintained at 1/m by keeping the core to cladding ratio at 20. The fiber length is maintained at 4.6 m and the total Yb³⁺ concentration is 1.69 × 10²⁶m^-3. The LP₁₁ mode loss is 0. When the core diameter is 20 µm, $\Delta \beta $ is ∼3000 1/m which leads to mismatch threshold being much larger than the TMI threshold. As the core diameter increases, although the ratio of the mismatch threshold to the TMI thereshold decreases rapidly because of $\Delta \beta $, TMI threshold is still less than the mismatch threshold. In addition, actual fiber length is usually much longer than 4.6 m, which will cause the amplifier to have a higher mismatch threshold. These results indicate that the effect of the phase mismatch gain on TMI is almost negligible for the small-core long fiber amplifier.

Fig. 2. The $\Delta \beta $ and the ratio of the mismatch threshold to the TMI thereshold as a function of core diameter. The signal gain is maintained at 1/m by keeping the core to cladding ratio at 20 and the fiber length is 4.6 m.

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4. Simulation results

In the previous section, we propose an analytical TMI threshold expression without the undetermined coefficient, and then theoretically verify that the TMI threshold expression has good accuracy. In addition, the TMI threshold expression has a wide application range, which can analyze the impact of some specific parameters on TMI. Based on the TMI threshold expression, we will demonstrate in this section how the V and higher-order mode loss affect the TMI threshold and the power scaling.

4.1 Influence of V on the TMI threshold

According to the Eqs. (26) and (29), V is implied in the TMI threshold expression and affects the TMI threshold power by changing the variable ${{N_2^2} / {N_1^2}}$. The variable ${{N_2^2} / {N_1^2}}$ is easy to calculate and can be rewritten as

(34)$$\frac{{N_2^2}}{{N_1^2}} = \frac{{S_2^2}}{{S_1^2}}\ast \frac{{S_1^2/N_1^2}}{{S_2^2/N_2^2}}$$

(35)$$S_x^2 = (3 - x)\pi \int\limits_0^{{r_{core}}} {J_{x - 1}^2({u_x}r)rdr}$$

where ${S^2}/{N^2}$ denotes the power proportion of different transverse modes in the core region. Figure 3 shows the relationship between the values of ${{N_2^2} / {N_1^2}}$, ${S^2}/{N^2}$ and normalization constant V. The core diameter is 20 µm, and other parameters are shown in Table 1. As the V value decreases, the power proportions of both LP₀₁ mode and LP₁₁ mode gradually decrease. However, the sensitivity of the power proportions of different transverse modes to the V value is different. Compared with the LP₀₁ mode, the power proportion of the LP₁₁ mode decreases more with decreasing V value. Therefore, ${{N_2^2} / {N_1^2}}$ increases with decreasing V value.

Fig. 3. The ${{N_2^2} / {N_1^2}}$ and ${S^2}/{N^2}$ as a function of V when the core diameter is 20 µm and the cladding diameter is 400 µm. The fiber length is 24 m.

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The calculation results in Fig. 3 are substituted into the TMI threshold expression (26). The results are shown in Fig. 4. As the V value decreases, the TMI threshold gradually increases. When V is less than 3.5, the TMI threshold grows rapidly as the V value decreases. When V exceeds 3.5, the TMI threshold is less sensitive to the change of V value. The influence of V on the TMI threshold matches the simulation results of Tao et al. [42], which shows that the TMI threshold expression has a great application prospect.

Fig. 4. The TMI threshold as a function of V for the 20/400 µm fiber amplifier with a signal wavelength of 1064 nm, V parameter is changed by changing the core NA.

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4.2 Influence of higher-order mode loss on the TMI threshold

The inclusion of the higher-order mode loss in the TMI threshold expression is very necessary because large-mode-area fibers are often required to introduce larger higher-order mode loss in order to achieve the single-mode output with a high TMI threshold in the practical fiber amplifiers. The TMI threshold expression proposed in this paper can help us to simply understand the relationship between the higher-order mode loss and the TMI threshold.

The TMI thresholds of the fiber amplifiers with different higher-order mode losses are simulated in Fig. 5. As the higher-order mode loss continues to increase, the TMI threshold gradually increase. When the higher-order mode loss reaches 100 dB, the TMI threshold is even nearly doubled. Moreover, the amplifier with a smaller core diameter has a higher TMI threshold, which means that the smaller-core fiber amplifier has a better TMI threshold enhancement effect for the same higher-order mode loss. Recently, a 5 kW single-mode fiber laser is achieved by using the Yb-doped fibers with increased higher-order mode loss [43].

Fig. 5. The TMI threshold as a function of higher-order mode loss when the core NA is kept at 0.06. The cladding diameter is 400 µm.

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Bending the fiber is a common way to increase higher-order mode loss. However, as the bending radius decreases, the bending-induced mode distortion becomes more and more serious. Fortunately, some studies have shown that the effect of bending distortion on TMI is weak [44,45]. This also means that the TMI threshold expression in this paper is able to analyze the influence of bending loss on the TMI threshold.

4.3 Power scaling considering SRS and TMI

In the above two subsections, we show the capacity of the TMI threshold expression in modeling the V and higher-order mode loss, which can play an important role in balancing the SRS and TMI effects in the power scaling of high-power fiber lasers. Although there are many theoretical studies on the power scaling, some important experimental conditions are often ignored by these theoretical studies. For example, power scaling predicted by theory almost never takes into account the effect of higher-order mode loss, however, higher-order mode loss is important for high-power fiber lasers to maintain high beam quality and obtain high output power. Besides, V is usually kept constant for convenient calculation of the power scaling [5], but the core NA of the high-power fiber lasers with different core diameters in the experiment is very close, which results in variation of V in practice, leading to deviation between the experimental outcomes and simulation results. Here, we will consider these experimental conditions and calculate the power scaling considering the SRS and TMI effects. The SRS threshold can be expressed as [46]:

(36)$${P_{SRS}} \approx \frac{{16{A_{eff}}\ln G}}{{{g_R}L}}$$

where, ${A_{eff}}$ is the effective mode field area, the SRS gain coefficient ${g_R}$ is 10⁻¹³ m/W, and the total amplifier gain is 13 dB, i.e. G = 20.

According to the TMI and SRS threshold expressions, the power scaling considering TMI and SRS under different higher-order mode losses is simulated in Fig. 6. The core NA is maintained at 0.06. The total Yb³⁺ concentration is 3.25 × 10²⁵m^-3. The fiber length is adjustable by changing the cladding diameter (solid white line). The seed power is changed to ensure the total amplifier gain remains constant. As shown in Fig. 6(a), the TMI and SRS limit the output power growth of large-core short fibers and small-core long fibers, respectively. The large core diameter can effectively suppress the SRS effect, and the long fiber owning a smaller core/cladding diameter ratio can improve the TMI threshold. Therefore, as the core diameter increases from 20 µm to 30 µm, the maximum output power is improved from ∼4.4 kW to ∼5.6 kW, and the corresponding fiber length also increases from ∼30 m to ∼41.5 m. The results shown in Fig. 6(a) are calculated in the way similar to previous theoretical calculation, where higher-order mode loss are absent. In fact, practical high-power fiber lasers usually have a higher output power than those theoretical results because of the higher-order mode loss [38,43]. Therefore, in order to get closer to the practical high-power fiber lasers, the higher-order mode loss should be considered. Figure 6(b) shows the calculation results by introducing higher-order mode loss of 50 dB. Compared with Fig. 6(a), the maximum output power is improved from ∼5.6 kW limited by TMI (fiber length of ∼41.5 m) to ∼7 kW limited by the SRS effect (fiber length of ∼33 m). The reduction of fiber length from ∼41.5 to ∼33 m means a reduction of the TMI threshold and an increase of the SRS threshold, indicating the power scaling brought by the higher-order mode loss is compromised. In this case, the fiber length could be reduced in practical fiber laser system for a higher output power close to the TMI threshold.

Fig. 6. The power scaling as a function of core diameter and fiber length under different higher-order mode loss: (a) without loss, (b) with higher-order mode loss of 50 dB. The core NA is 0.06, and the total Yb³⁺ concentration is 3.25 × 10²⁵m^-3.

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5. Summary

In this paper, we use transverse modes fitted by Gaussian functions to solve the heat conduction equation and the nonlinear coupling equation, and finally obtain an simplified analytical TMI threshold expression without the undetermined coefficient. Compared with other TMI threshold expressions, the TMI threshold expression proposed in this paper contains more fiber parameters and laser parameters. Besides, the TMI threshold expression is consistent with the basic experimental phenomena and simulation results of the theoretical model, which shows that the TMI threshold expression has a great application prospect. Furthermore, the influence of some unique parameters contained in this TMI threshold expression, i.e., V and higher-order mode loss, on the TMI threshold and the power scaling limit is discussed based on the proposed expression. The TMI threshold expression will be helpful to estimate the TMI threshold and assist the mitigation of TMI effects in high-power fiber laser systems.

Funding

Innovative Research Groups of Hunan Province (2019JJ10005); Hunan Provincial Innovation Construct Project (2019RS3018); National Natural Science Foundation of China (62035015, 62305390); National Key Research and Development Program of China (2022YFB3606000).

Acknowledgments

The authors would like to thank Yi An and Xiao Chen for inspiring discussions.

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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Parameter	Value	Parameter	Value
$2 r_{c o r e}$	20∼30 µm	$2 r_{c l a d}$	400 µm
$λ_{p}$	976 nm	$λ_{s}$	1064 nm
$σ_{a p}$	2.47 × 10⁻²⁴ m²	$σ_{e p}$	2.44 × 10⁻²⁴ m²
$σ_{a s}$	5.80 × 10⁻²⁷ m²	$σ_{e s}$	2.71 × 10⁻²⁵ m²
$n_{c l a d}$	1.45	NA	0.06
$k_{T}$	1.2 × 10⁻⁵	$κ$	1.38 W/K
$τ$	901 µs	$ρ C_{0}$	1.55 × 10⁶ J/ K/m³
$G$	100	$η (0)$	1 × 10⁻¹²

Simplified expression for transverse mode instability threshold in high power fiber lasers

Abstract

1. Introduction

2. Theoretical model

3. TMI threshold expression

4. Simulation results

4.1 Influence of V on the TMI threshold

4.2 Influence of higher-order mode loss on the TMI threshold

4.3 Power scaling considering SRS and TMI

5. Summary

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (1)

Equations (36)

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