Mode instabilities in Yb:YAG crystalline fiber amplifiers

Shicheng Zhu; Shicheng Zhu; Jinyan Li; Li Li; Kexiong Sun; Chang Hu; Xinyu Shao; Xiuquan Ma; Xiuquan Ma

doi:10.1364/OE.27.035065

1. Introduction

High average power fiber laser with near-diffraction-limited beam quality has a wide range of applications in scientific research and industrial processing [1]. In the past two decades, with the progress of several key technologies such as the fabrication of high-quality large-mode-area (LMA) fibers and high power passive fiber components, multi-kilowatt average output power with near-diffraction-limited beam quality has been achieved in silica-glass fiber lasers [2,3]. However, the newly discovered nonlinear effect called mode instability (MI) has caused widespread concern about the continuation of further power scaling in the silica-glass fiber [4]. MI is characterized by a threshold-like onset of output beam quality degradation when a certain average power is reached. Reported MI thresholds range from hundreds of watts to several kilowatts, which is found to be a worse problem when increasing the fiber core size [5–7]. On the other hand, increasing the fiber core size is the main approach to increase the average power for mitigating conventional nonlinear effects such as stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS) and so on. Therefore, MI has been becoming the primary obstacle for continuously increasing power output of single-cavity fiber lasers.

It is now widely recognized that the thermal effect is the origin of MI in the high power fiber laser system. The interference pattern between the fundamental mode and the higher order mode (HOM) makes the quasi-periodic heat distribution through the quantum defect. Such a heat distribution will cause a refractive index grating in the fiber core through the thermo-optic effect. Once there is a phase shift between the refractive index grating and the modal interference pattern, the power of higher order mode will increase exponentially and then, leads to the MI. Based on this concept, several mitigation strategies have been proposed and some of those have been experimentally demonstrated to be effective such as tailoring the doping distribution [8], increasing the ratio of pump cladding diameter [9], increasing the loss of HOMs [10], washing out the thermally-induced index grating through pump power modulation [11]. However, considering the physical origin of MI, there are several problems that are fundamentally difficult to solve: First, according to the theoretical analysis, the unwanted HOM power is increasing exponentially [4], which means that those mitigation strategies based on increasing the loss of HOMs would not be effective enough to give an order of magnitude improvement [12]; Second, both experimental and theoretical studies indicate that the photodarkening (PD) effect greatly reduces the MI threshold [13,14], which has been and will be a problem always existing in silica-glass fibers. Therefore, although recent research suggests that MI thresholds for diode-pumped silica-glass fiber lasers can exceed 10 kW by properly combing the mitigation methods [15], further improvement to multi-ten-kilowatts level seems quite difficult.

Due to the poor optical and thermal properties of silica-glass (mainly the PD degradation effect and the low thermal conductivity), researchers have been exploring new materials for further power scaling for quite a while. Recently, YAG-based full crystalline fiber lasers have been making great progress [16–18], since they have the potential for achieving multi-ten-kilowatts output power out of a single fiber [19]. Comparing with the silica-glass fiber, full crystalline YAG fibers can offer quite a few important advantages including the complete elimination of PD effects [17], the much higher thermal conductivity (10 W/m/K vs 1.38 W/m/K), the much lower stimulated Brillouin cross-section (more than two orders of magnitude lower), the higher melting temperature, and much higher doping concentrations.

Up to date, C4 (crystalline-core/crystalline-clad) fibers have been successfully fabricated, and the corresponding laser operation with the optical-to-optical laser efficiency approaching 70% was reported [17]. Recent MI thresholds estimated by other researchers indicate that MI thresholds of full crystalline fiber lasers should be 10 times higher than that of the silica-glass fiber lasers [20]. However, considering that Yb:YAG is a quasi-three-level laser at room temperature while Yb-doped silica-glass is a four-level laser, accurate calculations require a full numerical model. In this paper, by constructing a full numerical model, we simulated the MI behavior of Yb:YAG crystalline fiber amplifiers. The temperature-dependent laser performance of Yb:YAG crystalline fiber amplifiers is obtained by the iterative method. A comparison of MI thresholds between Yb:YAG crystalline and Yb-doped silica-glass fibers is presented. The found MI thresholds in the crystalline fibers are even higher than expected, up to 36 times that of silica-glass fibers.

2. Numerical model and implementation

There are several kinds of MI approaches, such as full time-dependent approaches [5,21,22], steady-periodic numerical approaches [4,23] and semi-analytical approaches [11,24,25]. Full time-dependent approaches give the most accurate modeling of the temporal dynamics of MI, but require too much computation time. Semi-analytical approaches are satisfactory in terms of simulation speed, but the thermal lensing effect is missing. The beam-propagation-method (BPM) based steady-periodic approach, which is the one we use, provides acceptable simulation speed and automatically includes the thermal lensing effect, the gain saturation effect and all transverse modes.

However, while the steady-periodic approach is widely used in the modeling of MI thresholds of silica-glass fibers, it cannot be directly applied in the simulation of Yb:YAG crystalline fibers. The steady-periodic approach usually has a small thermal simulation window (two or three core diameters) with a constant temperature boundary condition. The simulation speed benefits from this, but the missing of real boundary condition means that the uniform temperature rise within the core is ignored. For silica-glass fibers, the influence of such a uniform rise would be small because the MI behavior only depends on the local temperature rise within the core and the laser performance of Yb-doped silica-glass changes slightly with temperature. From 300 to 500 K, reductions of emission and absorption cross-sections at the signal (1064 nm) and pump (976 nm) wavelengths for the Yb-doped silica-glass are less than 10% [26]. But for the Yb:YAG, those two values are 61% and 51% (signal and pump wavelengths at 1030 nm and 941 nm) and we will study them later [27–29]. Therefore, the laser performance of Yb:YAG may change with temperature and the uniform temperature rise should not be ignored. To solve this problem, we divide the MI simulation of Yb:YAG crystalline fiber amplifiers into two steps: first obtain the steady state with given signal and pump powers, and then use the steady-periodic MI approach to model the MI behavior.

In the first step, the iterative method is employed because the temperature-dependent cross-sections make the steady state population equations implicit. According to the heat generation mechanism of Yb:YAG, the upper manifold populations are depleted by stimulated emission, spontaneous relaxation and non-radiative relaxation [30], so the rate equations for the upper and lower manifold are

(1)$$\frac{{\partial {n_u}(x,\;y,\;z,\;t)}}{{\partial t}} ={-} \frac{{\partial {n_l}(x,\;y,\;z,\;t)}}{{\partial t}} = {\sigma _P}(f_{01}^{}{n_l} - f_{12}^{}{n_u})\frac{{I_P^ +{+} I_P^ - }}{{h{v_P}}} + {\sigma _S}(f_{03}^{}{n_l} - f_{11}^{}{n_u})\frac{{I_S^{}}}{{h{v_S}}} - \frac{{{n_u}}}{\tau },$$

where n_u and n_l are the population of the upper and lower manifold, ${{\sigma }_P}$ and ${{\sigma }_S}$ are spectroscopic cross-sections at the pump and signal wavelengths, I_S is the signal intensity, $\; I_P^ + $ is the co-propagating pump intensity, $\; I_P^ - $ is the counter-propagating pump intensity, hv_p and hv_s are the pump and signal photon energies, τ is the effective lifetime, and f₀₁, f₀₃, f₁₁ and f₁₂ are the Boltzmann occupation factors for each state manifold.

Solving Eq. (1) for the steady-state laser operation, we can get the population of the upper manifold

(2)$${n_u}(x,\;y,\;z) = \frac{{{\sigma _P}f_{01}^{}\frac{{I_P^ +{+} I_P^ - }}{{h{v_P}}} + {\sigma _S}f_{03}^{}\frac{{I_S^{}}}{{h{v_S}}}}}{{{\sigma _P}(f_{01}^{} + f_{12}^{})\frac{{I_P^ +{+} I_P^ - }}{{h{v_P}}} + {\sigma _S}(f_{03}^{} + f_{11}^{})\frac{{I_S^{}}}{{h{v_S}}} + \frac{1}{\tau }}}{n_t},$$

where n_t is the ion concentration. The gain of signal and pump lights are given by

(3)$${g_s}(x,\;y,\;z) ={-} {\sigma _s}[f_{03}^{}{n_l}(x,\;y,\;z) - f_{11}^{}{n_u}(x,\;y,\;z)],$$

(4)$$g_p^ \mp (x,\;y,\;z) ={\pm} {\sigma _P}[f_{01}^{}{n_l}(x,\;y,\;z) - f_{12}^{}{n_u}(x,\;y,\;z)],$$

where n_l(x, y, z) = n_t – n_u(x, y, z) and the heat load is given by [30]

(5)$$Q(x,\;y,\;z)\textrm{ = \{ 1 - }{\eta _p}[(1 - {\eta _l}){\eta _r}\frac{{{\lambda _p}}}{{{{\bar{\lambda }}_f}}} + {\eta _l}\frac{{{\lambda _p}}}{{{\lambda _s}}}]\} \frac{{I_P^ + (z)\textrm{ + }I_P^\textrm{ - }(z)}}{{{A_p}}},$$

(6)$${\eta _l}(x,\;y,\;z) = \frac{{\frac{{I_S^{}}}{{h{v_S}}}{g_s}(x,\;y,\;z)}}{{\frac{{I_S^{}}}{{h{v_S}}}{g_s}(x,\;y,\;z) + \frac{{{n_u}}}{\tau }}},$$

where ${{\lambda }_s}$ and ${{\lambda }_p}$ are the signal and pump wavelengths, A_p is the area of the pump cladding, ${{\bar{\lambda }}_f}$ is the average fluorescence wavelength which is 1009.1 nm at room temperature and changes slightly with temperature, ${\eta _p}$ is the pump efficiency which refers to the fraction of absorbed pump photons contributing to inversion and is usually set to 1, ${\eta _r}$ is the radiative quantum efficiency for the upper manifold and we set it to be 0.97 [29] (The non-unity radiative quantum efficiency can be related to concentration quenching or multiphonon relaxation), ${\eta _l}$ is the laser extraction efficiency which is the fraction of excited ions that are extracted by the laser.

Among parameters used in Eqs. (2)–(6), spectroscopic cross-sections at pump [27] and signal [28] wavelengths are given by

(7)$${\sigma _P}(941,\;T) = [0.207 + 0.637\exp( - \frac{{T - 273}}{{288}})] \times {10^{ - 20}}c{m^2},$$

(8)$${\sigma _S}(1030,\;T) = [0.95334 + 33.608\exp( - \frac{T}{{92.82465}})] \times {10^{ - 20}}c{m^2},$$

and the Boltzmann occupation factors for each state manifold are

(9)$${f_{ij}} = \frac{{{e^{ - \frac{{{E_{ij}}}}{{kT}}}}}}{{\sum\limits_{i = 1}^4 {{e^{ - \frac{{{E_{ij}}}}{{kT}}}}} }},$$

where k is the Boltzmann constant, T is the absolute temperature, and E_ij is the energy for each Stark level of the upper and lower manifolds and has been shown in Table 1 [31]. The temperature-dependent emission and absorption cross-sections (${{\sigma }_S}$ f₁₁ and ${{\sigma }_P}$ f₀₁) are shown in Fig. 1.

Fig. 1. Temperature-dependent emission and absorption cross-sections of the Yb:YAG. The signal wavelength is 1030 nm and the pump wavelength is 941 nm. As we can see, when the temperature rises by 200 K, the emission cross-section reduced by 61%, from 1.76 × 10⁻²⁰ cm² to 0.69 × 10⁻²⁰ cm², and the absorption cross-section reduced by 51%, from 0.69 × 10⁻²⁰ cm² to 0.34 × 10⁻²⁰ cm². So Yb:YAG’s emission and absorption cross-section values are strongly dependent on temperature, which means one has to use the iterative method to solve the implicit rate equations for Yb:YAG.

Download Full Size | PDF

Table 1. Yb:YAG energy level values

View Table | View all tables in this article

One can see that, when temperature-dependent parameters are included in Eqs. (2)–(6), their solutions cannot be directly obtained because the temperature T is both the parameter we used and the solution we will get. Thus, the iterative method is required. To start the iterative process, parameters given by Eqs. (7)–(9) are calculated at room temperature. We use finite-difference beam propagation method (FD-BPM) [32] combined with Eqs. (2)–(4) to model the laser performance along the whole fiber. The heat load given by Eq. (6) is stored to update material parameters used in the next iterative process. The steady state is defined when the change of temperature at every position between two adjacent iterative processes is less than 0.1 K.

In this part, because the uniform temperature rise within the core is important, we use the average temperature of the fiber core to calculate ${{\sigma }_P}$, ${{\sigma }_S}$, and f_ij . The temperature T used in Eqs. (7)–(9) is given by [33]

(10)$$T(z) = \frac{{\int_0^{{R_{core}}} {T(r,\;z)dr} }}{{\int_0^{{R_{core}}} {dr} }},$$

(11)$$T(r,\;z) = {T_c} + \frac{{{Q_0}(z){R_{core}}^2}}{{4\kappa }}[1 + 2\ln (\frac{{{R_{core}}}}{{{R_{outer}}}}) + \frac{{2\kappa }}{{{R_{outer}}{h_q}}}] + \frac{{{Q_0}(z){R_{core}}^2}}{{4\kappa }}[1 - {(\frac{r}{{{R_{core}}}})^2}],$$

where T_c is the coolant temperature at the boundary, Q₀(z) is the average heat density within the core for each longitudinal position, R_core is the radius of core, R_outer is the radius of outer cladding, κ is the thermal conductivity (here we assume that the core, inner cladding and outer cladding have the same thermal conductivity), and h_q is the convective coefficient for Newton’s law of cooling.

When the steady state in the Yb:YAG crystalline fiber is obtained, we use stored material parameters along the fiber length in the steady-periodic MI approach. In this step, we use FD-BPM to simulate the optical propagation and alternating-direction-implicit (ADI) method [34] to solve the heat equations. As the heat solver is well presented in [23], we briefly show how gains and thermal-induced refractive changes are included in our model. Here, the simulation starts with the input signal field

(12)$$E(x,\;y,\;0,\;t) = \sqrt {{P_{01}}} {E_{01}}(x,\;y) + \sqrt {{P_{11}}} {E_{11}}(x,\;y){e^{ - i\Delta \omega t}},$$

where P₀₁ and P₁₁ are the powers of the LP₀₁ and LP₁₁ mode, E₀₁ and E₁₁ are the normalized mode amplitudes for each mode, $\Delta {\omega }$ is the frequency difference between modes. It is worth noting that, both in the iterative process and MI simulation, E₀₁ and E₁₁ are thermal lensed mode amplitudes.

The wave equation that governs the beam propagation is given by

(13)$$2i{n_0}{k_0}\frac{{\partial E}}{{\partial z}} = (\frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}})E(x,\;y,\;z) + {k_0}[{n^2}(x,\;y,\;z) - n_0^2]E(x,\;y,\;z),$$

where n₀ is the reference refractive index, k₀ is the wavenumber in free space. Here, the waveguide index profiles, gains and thermal-induced refractive changes are incorporated in

(14)$${n^2}(x,\;y,\;z) = {n_{wg}}(x,\;y) + i{n_0}{g_s}(x,\;y,\;z) + 2{n_0}\frac{{dn}}{{dT}}\Delta T(x,\;y,\;z),$$

where n_wg is the waveguide index profile, g_s is the signal gain given by Eq. (3), dn/dT is the thermo-optic coefficient, $\Delta T$ is the local temperature rise calculated by the ADI method with the heat source in Eq. (5).

We integrate optical fields and temperature distributions for each z position over one period from the input end to the output end. Figure 2 shows the flow diagram in our model. Because the ADI method is sequential by nature and then cannot be parallelized, the simulation speed of our model is not fast. With the desktop computer based on an Intel Core E5-2683 (2.1 GHz) processor, the run time in the MI simulation step is about 2-3 hours per meter. Meanwhile, depending on the operation power and fiber length, the convergence time required in the iterative step is between a few minutes and two hours. However, because both the FD-BPM and the ADI method are unconditionally stable, our model allows coarser grids (e.g. dx = 3 µm and dz = 40 µm) to preliminary locate the MI threshold and then use fine grids to make the precise simulation.

Fig. 2. Flow diagram of the numerical model. Because of the temperature-dependent material parameters, simulations of Yb:YAG crystalline fiber amplifiers are divided into two steps: first obtain the steady state with given signal and pump powers, and then use the steady-periodic MI approach to model the MI behavior.

Download Full Size | PDF

The implementation of our MI approach is similar to that in [35] except that they use the split-step fast-Fourier-transform (FFT) BPM as the optical solver. We have verified our MI model by comparing our modeling results with reported data. According to [35], the MI threshold for a counter-pumped 0.8 m long 80/170 fiber amplifier is 353 W. The total output signal power is 356 W with 8.4 W LP₁₁ mode power. In our model, with all the same parameters, the found MI threshold is 355 W and the total output signal power is 360.1 W with 7.7 W in LP₁₁ mode.

3. Modeling results

Before the MI behavior is presented, it is of interest to see how the laser performance of the Yb:YAG crystalline fiber is affected by the uniform temperature rise. We simulate a Yb:YAG crystalline fiber with difference convective coefficients. Fiber parameters are: core diameter 50 µm, inner cladding diameter 200 µm, outer cladding diameter 500 µm, fiber length 1.6 m. Other material parameters of Yb:YAG used in this paper are presented in Table 2.

Table 2. Optical material parameters of Yb:YAG used for the simulation

View Table | View all tables in this article

The laser performance is simulated with convective coefficients at 0.1 W/cm²/K and 1 W/cm²/K, and the coolant temperature at the boundary is fixed at 300 K. The steady state obtained by the iterative method is shown in Fig. 3. One can see that, the laser performance of Yb:YAG crystalline fiber is indeed influenced by the temperature rise. When the convective coefficient changes from 1 W/cm²/K to 0.1 W /cm²/K, the maximum temperature change along the fiber is 98.5 K and the average change is 70.5 K. The output signal power reduced from 4850 W to 4620 W and the signal gain at the input end reduced by 36%, from 66 dB/m to 42.4 dB/m.

Fig. 3. Laser performances of a 1.6 m long 50/200 Yb:YAG crystalline fiber amplifier with different convective coefficients. The fiber is co-pumped with pump power at 5000W. The input signal power is 400W with only the fundamental mode. (a) shows the power evolutions of signal and pump powers along the fiber and the output signal powers are 4620 W and 4850 W. (b) shows the corresponding gains and average core temperatures. When the convective coefficient changes from 1 W/cm²/K to 0.1 W/cm²/K, the maximum temperature change along the fiber is 98.5 K and the average change is 70.5 K. The output signal power reduced by 230 W. The signal gain at the input end reduced by 36%, from 66 dB/m to 42.4 dB/m.

Download Full Size | PDF

Also, the relatively low temperature risen shows the great thermal management in the crystalline fiber. When the convective coefficient is 1 W/cm²/K, which corresponding to the efficient water cooling, the maximum temperature rise is about 50 K. Accordingly, the maximum temperature rise along the length of such a fiber can be controlled to about 100 K even with the operation power at 10 kW. Actually, here we assume that the outer cladding has the same thermal conductivity of un-doped YAG (10 W/m/K). The actual material can be sapphire and the thermal conductivity is higher than 30 W/m/K. Therefore, the temperature rise can be further reduced. Note that the thicker temperature line in the front end of the fiber is due to the fast mode area oscillation brought by the thermal lensing effect.

Then we make a comparison of MI behaviors between the Yb:YAG crystalline fiber and the Yb-doped silica-glass fiber. Structure parameters for both fibers are the same as used in Fig. 3. Material parameters of Yb-doped silica-glass are: mass density 2201 kg/m³, heat capacity 703 J/kg/K, thermal conductivity 1.38 W/m/K, thermo-optic coefficient 1.2 × 10⁻⁵ K⁻¹, effective lifetime and ion concentration of ytterbium 901 µs and 3 × 10²⁵ m⁻³. Spectroscopic cross-sections for Yb-doped silica-glass at signal (1064 nm) and pump (976 nm) wavelengths are: signal and pump absorption cross-sections ${\sigma }_\textrm{S}^\textrm{a}$ = 5.8 × 10⁻²³ cm² and ${\sigma }_\textrm{P}^\textrm{a}$ = 2.47 × 10⁻²⁰ cm², signal and pump emission cross-section ${\sigma }_\textrm{S}^\textrm{e}\; $ = 5 × 10⁻²¹ cm² and ${\sigma }_\textrm{P}^\textrm{e}$ = 2.44 × 10⁻²⁰ cm². The convective coefficient is 1 W/cm²/K.

The seeded fundamental mode power in the crystalline fiber is 400 W and in the silica-glass fiber is 15 W as we try to keep the net gain of both fibers close. The input LP₁₁ mode power can be estimated for spontaneous thermal Rayleigh seeding [46] or from pump or signal amplitude modulation [35], and we believe 10⁻⁷ W is a moderate value. The frequency difference between the LP₁₁ mode and the fundamental mode is estimated from the thermal diffusion time across the core radius (R_core²Cρ/κ) [4,47]. With given structure and material parameters, frequency offsets for the crystalline fiber and the silica-glass fiber are about 5950 Hz and 1450 Hz. The MI threshold here is defined when the power of LP₁₁ mode reaches 5% of the output signal power.

Figure 4 shows the time average powers along the fiber length for the two fibers in the co-pumped scheme at MI thresholds. More details and simulation results in the counter-pumped scheme are shown in Table 3. The computational windows are L_x = L_y = 150 µm. The discretization in the simulation is set to dx = dy = 2 µm, dz = 20 µm and the time grid is set to be 64 points for one beat cycle (given by the inversion of thermal diffusion time).

Fig. 4. Time average powers along the fiber length for the co-pumped 1.6 m long 50/200 (a) Yb:YAG crystalline fiber amplifier and (b) Yb-doped silica-glass fiber amplifier at the MI threshold. It is shown that MI phenomenon will also occur in the Yb:YAG crystalline fiber amplifier at high power laser operation. The found MI threshold of the crystalline fiber is 7 kW, 28 times that of the silica-glass fiber (248 W). The result is higher than previously expected (Ten times, estimated from the higher thermal conductivity and lower thermo-optic coefficient), indicates that there are more factors affecting the MI threshold in the crystalline fiber amplifier.

Download Full Size | PDF

Table 3. Signal and pump powers at the MI threshold for two fibers in the co- and counter-pumped scheme

View Table | View all tables in this article

The simulation results show that, as expected, the MI phenomenon occurs in the Yb:YAG crystalline fiber amplifier, and the MI threshold is higher than that in the Yb-doped silica-glass fiber amplifier. In the co-pumped scheme, the MI threshold of the Yb:YAG crystalline fiber is found to be 7 kW, 28 times that of the silica-glass fiber (248 W). For the counter-pumped case, the MI threshold of the crystalline fiber has been improved to 9.17 kW, which is 36 times that of the silica-glass fiber (253 W). Such a result is significantly higher than previously expected. Researches on the MI behavior of silica-glass fiber have shown that MI threshold is proportional to the thermal conductivity and inversely proportional to the thermo-optic coefficient [20]. In this simulation, the thermal conductivity of crystalline is about 7 times higher than that of silica-glass fiber (10 W/m/K vs 1.38 W/m/K), and the thermo-optic coefficient is 0.75 times (0.9 × 10⁻⁵ K⁻¹ vs 1.2 × 10⁻⁵ K⁻¹). Thus, those two factors account for a factor of approximately ten in thresholds, leaving a factor of about three unexplained.

It has already been demonstrated that the degree of gain saturation will strongly influence the MI threshold [9,48]. Because the saturation is strong at the center of the core but moderate near the core boundary, as the heat responsible for MI derives from the quantum defect, strong gain saturation will flatten the heat profile across the fiber core. Then the strength of the refractive index grating will be reduced and the MI threshold is improved. The degree of gain saturation is determined by the competition between the populations pumped into upper state level and stimulated out of that level. For the Yb:YAG at room temperature, the emission cross-section is more than three times that of Yb-doped silica-glass (1.76 × 10⁻²⁰ cm² vs 5 × 10⁻²¹ cm²), and the absorption cross-section is less than one-third (6.89 × 10⁻²¹ cm² vs 2.47 × 10⁻²⁰ cm²). Such a difference determines that the degree of gain saturation in the Yb:YAG crystalline fiber amplifier must be higher than in the Yb-doped silica-glass fiber amplifier. Figure 5 show heat profiles of the Yb:YAG crystalline fiber and the Yb-doped silica-glass fiber. All parameters are consistent with those used in Fig. 4 except that pump powers are set to 6000 W and 220 W, slightly below the MI threshold. We can see that the heat profile of the crystalline fiber at all positions is significantly flatter than that of the silica-glass fiber.

Fig. 5. Heat profiles along the fiber length of the (a) Yb:YAG crystalline fiber and (b) silica-glass fiber. All parameters are consistent with those used in Fig. 4 except that pump powers are set to be 6000 W and 220 W (slightly below the MI threshold). Due to the larger emission cross-section (1.76 × 10⁻²⁰ cm² vs 5 × 10⁻²¹ cm²) and smaller pump absorption cross-section (6.89 × 10⁻²¹ cm² vs 2.47 × 10⁻²⁰ cm²), gain saturation effect in the Yb:YAG crystalline fiber is much stronger than in the Yb-doped silica-glass fiber. The higher degree of gain saturation in the Yb:YAG crystalline fiber flattens the heat profile and then leads to the improvement of MI thresholds.

Download Full Size | PDF

4. Power scalability of Yb:YAG crystalline fiber amplifiers

Since we have shown the high MI threshold of Yb:YAG crystalline fiber in the 1.6 m long 50/200 fiber, one may wonder whether the MI is still the major obstacle of power scaling in such fibers. We first simulate MI thresholds of the Yb:YAG crystalline fiber with different core and inner-cladding sizes and results are shown in Fig. 6. Here, core diameters range from 40 µm to 60 µm and inner cladding diameters range from 150 µm to 300 µm. Fiber lengths vary with core and inner cladding diameters to ensure that most of the pump power is absorbed (not all, which is not suitable for the three-level laser), ranging from 0.8 m to 4 m. The input signal powers are fixed at 400 W LP₀₁ mode and 10⁻⁷ W frequency-shifted LP₁₁ mode. Other parameters are consistent with those used in Fig. 4. The highest MI threshold we obtained is 16.7 kW, in the 40/300 fiber whose fiber length is 4 meters.

Fig. 6. MI thresholds for the co-pumped Yb:YAG crystalline fiber amplifiers with different core and inner cladding sizes. Core and inner diameters range from 40 µm to 60 µm and 150 µm to 300 µm. Fiber lengths range from 0.8 m to 4 m, defined as the pump absorption > 90%. The input power of LP₀₁ mode is fixed at 400 W, and the input power of LP₁₁ mode is 10⁻⁷ W with the frequency shift estimated through the thermal diffusive time. The found highest MI threshold is 16.7 kW with the 40/300 fiber whose fiber length is 4 m. It is shown that Yb:YAG crystalline fibers can achieve multi-ten-kilowatts output power out of a single fiber without any mitigation strategy of MI effect.

Download Full Size | PDF

Then we calculate conventional power limits include the pump brightness, SRS, SBS, and thermal lensing limitations [40]

(15)$${P_{\textrm{ lens }}} = \frac{{{\eta _{\textrm{laser }}}}}{{{\eta _{\textrm{heat }}}}}\frac{{\pi \kappa {\lambda _s}^2}}{{2\frac{{dn}}{{dT}}{R_{core}}^2}}L,$$

(16)$${P_{\textrm{SRS}}} = \frac{{{f_{\textrm{SRS}}}\pi {R_{core}}^2}}{{{g_R}L}}{\Gamma ^2}\ln (G),$$

(17)$${P_{\textrm{SBS}}} = \frac{{{f_{\textrm{SBS}}}\pi {R_{core}}^2}}{{{g_B}L}}{\Gamma ^2}\ln (G),$$

(18)$${P_{\textrm{ pump}}} = {\eta _{\textrm{laser }}}{I_{\textrm{pump }}}({{\pi^2}\textrm{NA}_{clad}^2} )\frac{{{\alpha _{\textrm{core }}}}}{{{A_{pump}}}}L{R_{core}}^2,$$

where ${\eta _{laser}}$ is the laser efficiency, ${\eta _{heat}}$ is the fraction of power turned into heat, ${{\alpha }_{core}}$ is the pump core absorption coefficient, L is the fiber length. f_SRS is the SRS numerical factor and ${g_R}$ is the SRS gain coefficient, f_SBS is the SBS numerical factor and ${g_B}$ is the SBS gain coefficient, ${\Gamma }$ is the mode field radius to core radius, G is the assumed laser gain, A_pump is the small signal total pump absorption, I_pump is the pump brightness. Parameters used here are shown in Table 2 and results are shown in Fig. 7.

Fig. 7. Contour plots of power limits (in kW) of Yb:YAG crystalline fiber amplifiers in (a) the single frequency case and (b) the broad bandwidth case. Combing with Fig. 6, one can see that MI is still the main limitation of power scaling in Yb:YAG crystalline fiber amplifiers. Only in the broad bandwidth case, for the 4 m long 40/300 fiber, SRS limits start to be lower than MI thresholds.

Download Full Size | PDF

As we can see, in most areas, calculated limits are largely higher than MI thresholds in Fig. 6. Note that the power shown here is the output signal power and MI thresholds we showed in this paper are the input pump power. Whatever, in the single frequency case, MI will be the only limitation. In the broad bandwidth case, for the 4 m long 40/300 fiber, the SRS limit starts to be lower than the MI threshold. As we know that with smaller core diameters and longer fiber lengths, SRS power limits will be reduced and MI thresholds will be improved, it suggests that the SRS and MI will finally limit the power scaling of broad bandwidth Yb:YAG crystalline fiber amplifiers.

5. Conclusion

In this paper, MI thresholds of Yb:YAG crystalline fiber amplifiers are investigated through full numerical simulations. Temperature-dependent physical parameters of Yb:YAG are accommodated into the steady-periodic MI approach through an iterative process before the MI simulation is started. A comparison of MI behaviors in the Yb:YAG crystalline and Yb-doped silica-glass fiber amplifiers is presented. Simulation results show that, the same as the Yb-doped silica-glass fiber, MI will occur in Yb:YAG crystalline fiber amplifiers under high power laser operation. The simulated MI threshold of a 50/200 Yb:YAG crystalline fiber is 28 times that of the Yb-doped silica-glass fiber in the co-pumped scheme and 36 times that in the counter-pumped scheme. We show that, except the higher thermal conductivity and the lower thermo-optic coefficient, the much higher degree of gain saturation caused by the larger emission cross-section and smaller absorption cross-section of Yb:YAG plays an important role in the improvement of MI thresholds.

We investigated MI thresholds of Yb:YAG crystalline fibers with various core and inner cladding sizes in the co-pumped scheme and compared them with calculated other power limitations. We find that, although the MI threshold can easily break through 10 kW, MI is still the one of the main limitations for power scaling of Yb:YAG crystalline fiber amplifiers.

Funding

National Natural Science Foundation of China (61735007); Science and Technology Planning Project of Guangdong Province (2017B090913001).

Disclosures

The authors declare no conflicts of interest.

References

1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]

2. C. X. Yu, O. Shatrovoy, T. Y. Fan, and T. F. Taunay, “Diode-pumped narrow linewidth multi-kilowatt metalized Yb fiber amplifier,” Opt. Lett. 41(22), 5202–5205 (2016). [CrossRef]

3. Q. Fang, J. Li, W. Shi, Y. Qin, Y. Xu, X. Meng, R. A. Norwood, and N. Peyghambarian, “5 kW near-Diffraction-limited and 8 kW High-Brightness Monolithic Continuous Wave Fiber lasers Directly Pumped by laser Diodes,” IEEE Photonics J. 9(5), 1–7 (2017). [CrossRef]

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

5. 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]

6. C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE 7580, 75801H (2010). [CrossRef]

7. H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20(14), 15710–15722 (2012). [CrossRef]

8. B. G. Ward, “Maximizing power output from continuous-wave single-frequency fiber amplifiers,” Opt. Lett. 40(4), 542–545 (2015). [CrossRef]

9. A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21(13), 15168–15182 (2013). [CrossRef]

10. C. Jauregui, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Passive mitigation strategies for mode instabilities in high-power fiber laser systems,” Opt. Express 21(16), 19375–19386 (2013). [CrossRef]

11. C. Jauregui, C. Stihler, A. Tünnermann, and J. Limpert, “Pump-modulation-induced beam stabilization in high-power fiber laser systems above the mode instability threshold,” Opt. Express 26(8), 10691–10704 (2018). [CrossRef]

12. A. V. Smith and J. J. Smith, “Overview of a steady-periodic model of modal instability in fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 20(5), 472–483 (2014). [CrossRef]

13. H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tünnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23(12), 15265–15277 (2015). [CrossRef]

14. C. Jauregui, H.-J. Otto, F. Stutzki, J. Limpert, and A. Tünnermann, “Simplified modelling the mode instability threshold of high power fiber amplifiers in the presence of photodarkening,” Opt. Express 23(16), 20203–20218 (2015). [CrossRef]

15. R. Tao, X. Wang, and P. Zhou, “Comprehensive Theoretical Study of Mode Instability in High-Power Fiber Lasers by Employing a Universal Model and Its Implications,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–19 (2018). [CrossRef]

16. J. Zhang, Y. Chen, S. Yin, C. Luo, and M. Dubinskii, “High-Power Diode-Cladding-Pumped Yb:YAG Laser Based on True Double-Clad Fully Crystalline Fiber,” in (2016), p. ATu6A.4

17. M. Dubinskii, J. Zhang, V. Fromzel, Y. Chen, S. Yin, and C. Luo, “Low-loss ‘crystalline-core/crystalline-clad’ (C4) fibers for highly power scalable high efficiency fiber lasers,” Opt. Express 26(4), 5092–5101 (2018). [CrossRef]

18. L. B. Shaw, S. Bayya, W. Kim, J. Myers, D. Rhonehouse, S. N. Qadri, C. Askins, J. Peele, R. Thapa, D. Gibson, R. R. Gattass, J. Kolis, B. Stadleman, and J. S. Sanghera, “Cladded Single Crystal Fibers for All-Crystalline Fiber Lasers,” in 2018 Conference on Lasers and Electro-Optics, CLEO 2018 - Proceedings (2018), pp. SF3I.3.

19. T. A. Parthasarathy, “Predicted performance limits of yttrium aluminum garnet fiber lasers,” Opt. Eng. 49(9), 094302 (2010). [CrossRef]

20. M. N. Zervas, “Transverse mode instability analysis in fiber amplifiers,” Proc. SPIE 10083, 100830M (2017). [CrossRef]

21. S. Naderi, I. Dajani, T. Madden, and C. Robin, “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21(13), 16111–16129 (2013). [CrossRef]

22. B. G. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21(10), 12053–12067 (2013). [CrossRef]

23. A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21(3), 2606–2623 (2013). [CrossRef]

24. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37(12), 2382–2384 (2012). [CrossRef]

25. L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013). [CrossRef]

26. S. W. Moore, T. Barnett, T. A. Reichardt, and R. L. Farrow, “Optical properties of Yb + 3-doped fibers and fiber lasers at high temperature,” Opt. Commun. 284(24), 5774–5780 (2011). [CrossRef]

27. Q. Liu, X. Fu, M. Gong, and L. Huang, “Effects of the temperature dependence of absorption coefficients in edge-pumped Yb:YAG slab lasers,” J. Opt. Soc. Am. B 24(9), 2081 (2007). [CrossRef]

28. B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. A. Bass, “Modeling of high-power solid-state slab lasers,” Proc. SPIE 4968, 1 (2003). [CrossRef]

29. F. D. Patel, E. C. Honea, J. Speth, S. A. Payne, R. Hutcheson, and R. Equall, “Laser demonstration of Yb3Al5O12 (YbAG) and materials properties of highly doped Yb:YAG,” IEEE J. Quantum Electron. 37(1), 135–144 (2001). [CrossRef]

30. S. Chénais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers-Part I: Theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. 40(9), 1217–1234 (2004). [CrossRef]

31. A. A. Kaminskiǐ, Crystalline Lasers: Physical Processes and Operating Schemes (CRC, 1996), 12.

32. J. Yamauchi, T. Ando, and H. Nakano, “Beam-propagation analysis of optical fibres by alternating direction implicit method,” Electron. Lett. 27(18), 1663–1665 (1991). [CrossRef]

33. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001). [CrossRef]

34. G. B. Ermentrout, Numerical Recipes in C (Cambridge university, 1989), 93(1).

35. A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012). [CrossRef]

36. Y. Sato, J. Akiyama, and T. Taira, “Effects of rare-earth doping on thermal conductivity in Y3Al5O12 crystals,” Opt. Mater. 31(5), 720–724 (2009). [CrossRef]

37. R. Wynne, J. L. Daneu, and T. Y. Fan, “Thermal coefficients of the expansion and refractive index in YAG,” Appl. Opt. 38(15), 3282–3284 (1999). [CrossRef]

38. A. A. Kaminskiǐ, H. J. Eichler, K. Ueda, S. N. Bagaev, G. M. A. Gad, J. Lu, T. Murai, H. Yagi, and T. Yanagitani, “Observation of stimulated Raman scattering in Y 3 Al 5 O 12 single crystals and nanocrystalline ceramics and in these materials activated with laser ions Nd 3+ and Yb 3+,” JETP Lett. 72(10), 499–502 (2000). [CrossRef]

39. B. T. Do and A. V. Smith, “Bulk optical damage thresholds for doped and undoped, crystalline and ceramic yttrium aluminum garnet,” Appl. Opt. 48(18), 3509–3514 (2009). [CrossRef]

40. 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]

41. Y. Zhou, Q. Thai, Y. C. Chen, and S. Zhou, “Monolithic Q-switched Cr, Yb:YAG laser,” Opt. Commun. 219(1-6), 365–367 (2003). [CrossRef]

42. A. Giesen, “High-power thin-disk lasers,” in Advanced Solid-State Photonics (2007), p. MA1.

43. D. S. Sumida and T. Y. Fan, “Emission Spectra and Fluorescence Lifetime Measurements of Yb:YAG as a Function of Temperature,” in Advanced Solid State Lasers (2017), p. YL4.

44. J. Petit, B. Viana, P. Goldner, J. P. Roger, and D. Fournier, “Thermomechanical properties of Yb3+ doped laser crystals: Experiments and modeling,” J. Appl. Phys. 108(12), 123108 (2010). [CrossRef]

45. H. Qiu, P. Yang, J. Dong, P. Deng, J. Xu, and W. Chen, “The influence of Yb concentration on laser crystal Yb:YAG,” Mater. Lett. 55(1-2), 1–7 (2002). [CrossRef]

46. A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5(5), 7100807 (2013). [CrossRef]

47. A. V. Smith and J. J. Smith, “Frequency dependence of mode coupling gain in Yb doped fiber amplifiers due to stimulated thermal Rayleigh scattering,” arXiv Prepr. arXiv:1301.4277 (2013).

48. K. R. Hansen and J. Lægsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22(9), 11267–11278 (2014). [CrossRef]

Level assignment	Energy (cm⁻¹)	Level assignment	Energy (cm⁻¹)
(1,3)	10941	(0,4)	785
(1,2)	10624	(0,3)	621
(1,1)	10327	(0,2)	565
		(0,1)	0

Parameter	Symbol	Value	Reference
Pump brightness	I_pump	0.1 W/µm²/sr	[19]
Pump clad NA	NA_clad	0.5	[19]
Thermal conductivity	κ	10 W/m/K	[36]
Thermo-optic coefficient	dn/dT	9 × 10⁻⁶	[37]
Raman gain coefficient	$g_{R}$	1 × 10⁻¹²	[38]
Brouillon gain coefficient	$g_{B}$	5 × 10⁻¹⁴	[39]
Numerical factor (SRS)	f_SRS	25 at 1 kW	[40]
Numerical factor (SBS)	f_SBS	26 at 1 kW	[40]
Small-signal total pump absorption	A_pump	20 dB	[40]
Laser gain (assumed)	G	10	[40]
Model field radius/core radius	$Γ$	0.8	[40]
Small-signal pump core absorption	$α_{c o r e}$	3250 dB/m	[41]
Laser efficiency	$η_{l a s e r}$	0.65	[42]
Heat fraction	$η_{h e a t}$	0.1	[43]
Effective lifetime	τ	950 µs	[44]
Mass density	ρ	4560 kg/m³	[44]
Heat capacity	C	590 J/kg/K	[45]
Ion concentration	n_t	6.9 × 10²⁵ m⁻³
Signal wavelength	$λ_{s}$	1030 nm
Pump wavelength	$λ_{p}$	941 nm

Pumping scheme	Fiber	Pump (in)	Pump (out)	Signal (in)	Signal (out)	Output LP₁₁ mode power
Co-	YAG	7000 W	195 W	400 W	6604 W	480.6 W (7.26%)
Co-	Silica-glass	248 W	2.4 W	15 W	242.8 W	16.7 W (6.88%)
Counter-	YAG	9171 W	310 W	400 W	8483 W	479.6 W (5.65%)
Counter-	Silica-glass	253 W	3.2 W	15 W	241.9 W	19.0 W (7.87%)

Level assignment	Energy (cm⁻¹)	Level assignment	Energy (cm⁻¹)
(1,3)	10941	(0,4)	785
(1,2)	10624	(0,3)	621
(1,1)	10327	(0,2)	565
		(0,1)	0

Parameter	Symbol	Value	Reference
Pump brightness	I_pump	0.1 W/µm²/sr	[19]
Pump clad NA	NA_clad	0.5	[19]
Thermal conductivity	κ	10 W/m/K	[36]
Thermo-optic coefficient	dn/dT	9 × 10⁻⁶	[37]
Raman gain coefficient	$g_{R}$	1 × 10⁻¹²	[38]
Brouillon gain coefficient	$g_{B}$	5 × 10⁻¹⁴	[39]
Numerical factor (SRS)	f_SRS	25 at 1 kW	[40]
Numerical factor (SBS)	f_SBS	26 at 1 kW	[40]
Small-signal total pump absorption	A_pump	20 dB	[40]
Laser gain (assumed)	G	10	[40]
Model field radius/core radius	$Γ$	0.8	[40]
Small-signal pump core absorption	$α_{c o r e}$	3250 dB/m	[41]
Laser efficiency	$η_{l a s e r}$	0.65	[42]
Heat fraction	$η_{h e a t}$	0.1	[43]
Effective lifetime	τ	950 µs	[44]
Mass density	ρ	4560 kg/m³	[44]
Heat capacity	C	590 J/kg/K	[45]
Ion concentration	n_t	6.9 × 10²⁵ m⁻³
Signal wavelength	$λ_{s}$	1030 nm
Pump wavelength	$λ_{p}$	941 nm

Mode instabilities in Yb:YAG crystalline fiber amplifiers

Abstract

1. Introduction

2. Numerical model and implementation

3. Modeling results

4. Power scalability of Yb:YAG crystalline fiber amplifiers

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (7)

Tables (3)

Equations (18)

Optics Express