Comparison of tandem pumping and direct pumping on high-power linearly polarized fiber laser

Yulun Wu; Ping Yan; Dan Li; Lele Wang; Mali Gong; Qirong Xiao

doi:10.1364/OE.500510

1. Introduction

High-power fiber laser is one of the research focuses in the area of laser technology due to the advantages of simple structure, good beam quality and high efficiency [1,2]. In particular, the linearly polarized fiber laser is part of the study because it’s widely used in coherent beam combination [3], nonlinear frequency conversion [4] and coherent detection [5], which demand high polarization extinction ratio (PER) output. This means that, to realize linearly polarized laser output, the methods of conventional fiber laser should be optimized to meet the requirements of polarization control.

In the past decades, linearly polarized fiber lasers have received great development, and many research teams have achieved over one kilowatt linearly polarized fiber laser output. In 2018, Platonov et al. realized 2 kW linearly polarized fiber laser output [6]. The PER was greater than 20 dB and the linewidth was 30 GHz. Wang et al. reported a 1.1 kW all-fiber linearly polarized fiber laser with a PER of 96% and M² of 1.25 [7]. In 2019, Wang et al. increased the laser power to 2.62 kW [8]. The PER was 96.3% and the M² was only slightly increased. In 2021, a 3.96 kW linearly polarized fiber laser of 13.9 dB PER and 0.62 nm linewidth was achieved by Ren et al. [9], and the laser power was further increased to 4.5 kW in 2022 [10]. Although the power of linearly polarized fiber lasers has been widely improved at present, the construction of these lasers is all based on direct pumping scheme.

Tandem pumping, as one of the main pumping schemes in random polarized high-power fiber laser, has the advantages of high quantum efficiency and low heat production [11,12]. At present, in the random polarized fiber laser, 20 kW level output from a single fiber with tandem pumping has been realized [13]. However, in the test of our group in 2021, forward tandem pumping is not suitable in linearly polarized fiber lasers because it demands extremely long fiber length in the main amplifier, which would result in severe polarization deterioration [14]. Besides, the stimulated Raman scattering (SRS) is also easier to happen in this pumping configuration [15]. For direct pumping, SRS and polarization deterioration can be effectively suppressed by applying backward pumping [16], which should also be applicable to tandem pumping. However, the 1018 nm pump source in tandem pumping is extremely sensitive to feedback light [12]. Extremely low power feedback would also lead to a significant reduction in the output power of the 1018 nm laser and could even cause system damage. This problem was solved by Wu et al. in 2022 by applying high isolation backward combiner, which means backward tandem pumping could be used in any fiber laser system [11]. To study the application prospect of tandem pumped linearly polarized fiber laser, it is necessary to establish a model suitable for power amplification of linearly polarized fiber laser.

In this article, we established a laser evolution model considering polarization coupling, multimode gain competition, SRS and SBS process to figure out the influence of pumping scheme on the output parameters of linearly polarized fiber laser. The outputs of linearly polarized fiber laser under different forward and backward power distribution of direct pumping and tandem pumping are calculated. Through theoretical analysis, we find that tandem pumping has advantages over direct pumping in PER and beam quality deterioration, but it suffers from serious SRS effect. Using highly doped fiber and backward pumping at the same time is the best scheme to realize tandem pumped linearly polarized fiber laser. Based on this model, we set up a linearly polarized fiber laser based on backward tandem pumping with an output power of 875 W, which is the first test of backward tandem pumping in linearly polarized fiber laser to the best of our knowledge. It is found that the power improvement of this scheme is limited by the mode field of highly doped fiber.

2. Theoretical model

In our 2021 report, the polarizing coupling coefficient was added into the rate equation, which can calculate the polarization deterioration of different pump conditions during the linearly polarized fiber laser power amplification process [14]. However, the model can only calculate the PER and output power, which is not enough to describe the output condition of the linearly polarized fiber laser. Linearly polarized fiber lasers, compared with random polarized fiber lasers, suffer nonlinear effects more severely [17]. Especially, SRS and SBS processes will greatly affect the output capacity of the laser, so these nonlinear effects must be considered. In addition, according to past reports, the TMI threshold of linearly polarized fiber lasers is lower than random polarized fiber lasers, so the generation of high-order modes will be a barrier to improving the output power of linearly polarized fiber lasers [18]. This indicates that modes analysis needs to be introduced into the model. By calculating the ratio of the fundamental modes during the laser amplification process, the degree of mode deterioration could be estimated. To these ends, we introduce mode coupling, polarization coupling, SRS and SBS to the rate equation. The output conditions of linearly polarized fiber laser under different pump schemes are simulated.

The basic formulae of our model are the rate equations combined with polarization coupling from our previous work, which are listed below [14].

(1a)$$\frac{{dP_p^ + (z )}}{{dz}} ={-} {\mathrm{\Gamma }_p}[{{\sigma_{ap}}N - ({{\sigma_{ap}} + {\sigma_{ep}}} ){N_2}(z )} ]P_p^ + (z )- {\alpha _p}P_p^ + (z )$$

(1b)$$\frac{{dP_p^ - (z )}}{{dz}} = {\mathrm{\Gamma }_p}[{{\sigma_{ap}}N - ({{\sigma_{ap}} + {\sigma_{ep}}} ){N_2}(z )} ]P_p^ - (z )+ {\alpha _p}P_p^ - (z )$$

(1c)$$\frac{{dP_{sx}^ + (z )}}{{dz}} = {\mathrm{\Gamma }_{sx}}[{({{\sigma_{as}} + {\sigma_{es}}} ){N_2}(z )- {\sigma_{as}}N} ]P_{sx}^ + (z )- {\alpha _{sx}}P_{sx}^ + (z )- {\textrm{h}_{xy}}P_{sx}^ + (z )+ {h_{yx}}P_{sy}^ + (z )$$

(1d)$$\frac{{dP_{sy}^ + (z )}}{{dz}} = {\mathrm{\Gamma }_{sy}}[{({{\sigma_{as}} + {\sigma_{es}}} ){N_2}(z )- {\sigma_{as}}N} ]P_{sy}^ + (z )- {\alpha _{sy}}P_{sy}^ + (z )+ {h_{xy}}P_{sx}^ + (z )- {h_{yx}}P_{sy}^ + (z )$$

(1e)$$\frac{{{N_2}(z )}}{N} = \frac{{\frac{{[{P_p^ + (z )+ P_p^ - (z )} ]{\sigma _{ap}}{\mathrm{\Gamma }_p}}}{{h{\nu _p}{A_c}}} + \frac{{{\mathrm{\Gamma }_{sx}}{\sigma _{as}}P_{sx}^ + (z )}}{{h{\nu _s}{A_c}}} + \frac{{{\mathrm{\Gamma }_{sy}}{\sigma _{as}}P_{sy}^ + (z )}}{{h{\nu _s}{A_c}}}}}{{\frac{{[{P_p^ + (z )+ P_p^ - (z )} ]({{\sigma_{ap}} + {\sigma_{ep}}} ){\mathrm{\Gamma }_p}}}{{h{\nu _p}{A_c}}} + \frac{1}{\tau } + \frac{{{\mathrm{\Gamma }_{sx}}({{\sigma_{as}} + {\sigma_{es}}} )P_{sx}^ + (z )}}{{h{\nu _s}{A_c}}} + \frac{{{\mathrm{\Gamma }_{sy}}({{\sigma_{as}} + {\sigma_{es}}} )P_{sy}^ + (z )}}{{h{\nu _s}{A_c}}}}}$$

where $P_p^ + (\textrm{z} )$ and $P_p^ - (\textrm{z} )$ are forward and backward pump power, $P_{sx}^ + (\textrm{z} )$ and $P_{sy}^ + (\textrm{z} )$ are the signal power of two polarization states, ${\sigma _{as}}$ and ${\sigma _{es}}$ are the signal absorption and emission cross section, ${\mathrm{\Gamma }_{sx}}$ and ${\mathrm{\Gamma }_{sy}}$ are the fill factor of two polarization state which are equal to each other under approximate treatment, ${\mathrm{\nu }_\textrm{p}}$ and ${\mathrm{\nu }_\textrm{s}}$ are the frequencies of pump and signal, ${\tau }$ is the photon lifetime, $\textrm{h}$ is the Planck constant. Here, ${h_{xy}}$, ${\textrm{h}_{yx}}$ are the two coupling coefficients that could be calculated as [19]

(2)$${h_{xy}} = {h_{yx}} \approx \frac{{{K_\sigma }}}{{{B^2}}}$$

where ${K_\sigma }$ is a constant related to the influence of internal and external disturbance on the system, which is determined after the fiber is selected to build the system. This parameter can be calculated by testing the polarization degradation after injecting polarized light into one end, which greatly affects the polarization maintenance ability of a system. B is the birefrigence of the fiber and is strongly affected by the temperature inside the fiber core. This model can be used to calculate the polarization deterioration during the amplification stage of linearly polarized fiber laser.

In fiber lasers, mode gain competition and refractive index change are the two main influencing factors of the mode changing process. For the former, we only need to introduce the mode directly into the rate equation [20], while for the latter we need to introduce a new mode coupling coefficient. The coupling coefficient could be derived from Maxwell’s equation as shown in [21], and could be expressed as

(3)$$\begin{array}{{c}} {\frac{{\partial {A_m}}}{{\partial z}} + \frac{{{n_0}}}{c}\frac{{\partial {A_m}}}{{\partial t}} = \frac{i}{2}{k_0}\mathop \sum \limits_n {\kappa _{n \to m}}{A_n}{e^{i({{\beta_m} - {\beta_n}} )z}}} \end{array}$$

(4)$$\begin{array}{{c}} {{\kappa _{n \to m}} = \mathrm{\int\!\!\!\int }\varphi _m^\ast \delta n({x,y,z,t} ){\varphi _n}dxdy\; \; \; \; \; \; \; \; \; } \end{array}$$

In the calculation, the situation of power over the TMI threshold is not considered, and the main purpose is to compare the degree of mode deterioration under different pump schemes. So $\partial {A_m}/\partial t$ is omitted to calculate the steady state. When calculating mode coupling, the thermal-optic effect is mainly considered. During the process, the changes of refractive index are proportional to the temperature change, which could be calculated as

(5)$$\begin{array}{{c}} {\delta n = {k_t}dT\; \; \; \; \; \; \; \; \; } \end{array}$$

where ${P_b}$ is the power of the Brillouin Stokes light and $g_B^{eff}$ is the effective Brillouin gain coefficient which is expressed by [22]

(6)$$\begin{array}{{c}} {g_B^{eff} = \frac{{4{\pi ^2}\gamma _e^2{f_A}}}{{{n_p}c\lambda _s^2{\rho _0}{v_A}{\mathrm{\Gamma }_B}}}\sqrt \pi {e^{{q^2} - {p^2}}}Im[{erfc({q - ip} ){e^{ - 2ipq}}} ]} \end{array}\; $$

(7)$$\begin{array}{{c}} {q = \frac{{{\mathrm{\Gamma }_B}}}{{2\mathrm{\Delta }{\omega _p}}}\; \; \; \; \; \; } \end{array}$$

(8)$$\begin{array}{{c}} {p = \frac{{{\mathrm{\Omega }_B} + {\omega _s} - {\omega _p}}}{{\mathrm{\Delta }{\omega _p}}}\; } \end{array}$$

For SRS effects, the equations for calculation are shown below [23]

(9a)$$\begin{array}{{c}} {\frac{{d{P_s}}}{{dz}} ={-} \frac{{{g_r}}}{{{A_s}}}{P_s}{P_r} - \alpha {P_s}} \end{array}$$

(9b)$$\begin{array}{{c}} {\frac{{d{P_r}}}{{dz}} = \frac{{{g_r}}}{{{A_s}}}\frac{{{\lambda _p}}}{{{\lambda _s}}}{P_s}{P_r} - \alpha {P_s}} \end{array}$$

where ${P_r}$ is the power of Raman Stokes light and ${\lambda _p}$ is the wavelength of Raman light.

In calculation, we consider mode coupling, SBS and SRS in polarization x and y independently owing to the polarization effect of these processes. Then the polarization coupling is considered between the x and y polarization state. The calculation contents are shown in Fig. 1.

Fig. 1. Calculation contents

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In addition, to reduce the complexity of the calculation, we divide the fiber core into six parts, and calculate the temperature, refractive index, particle number and mode power in each area. The division is illustrated in Fig. 2.

Fig. 2. Fiber core division

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The final formula of the model is expressed below.

(10a)$$\begin{array}{{c}} {\frac{{{N_{2,j}}(z )}}{N} = \frac{{{A_{pj}} + {A_{sj}}}}{{{B_{pj}} + \frac{1}{\tau } + {B_{sj}}}}({j = 1,2, \ldots ,6} )} \end{array}$$

(10b)$$\frac{{dP_{sx,i,j}^ + (z )}}{{dz}} = {G_{psx + ,i,j}} + P{C_{psx + ,i,j}} + M{C_{psx + ,i,j}} - SR{S_{psx + ,i,j}} - SB{S_{psx + ,i,j}}$$

(10c)$$\frac{{dP_{sy,i,j}^ + (z )}}{{dz}} = {G_{psy + ,i,j}} + P{C_{psy + ,i,j}} + M{C_{psy + ,i,j}} - SR{S_{psy + ,i,j}} - SB{S_{psy + ,i,j}}$$

(10d)$$\frac{{dP_{srx,i,j}^ + (z )}}{{dz}} = {G_{psrx + ,i,j}} + P{C_{psrx + ,i,j}} + SR{S_{psx + ,i,j}}$$

(10e)$$\frac{{dP_{sry,i,j}^ + (z )}}{{dz}} = {G_{psry + ,i,j}} + P{C_{psry + ,i,j}} + SR{S_{psy + ,i,j}}$$

(10f)$$\frac{{dP_{sbx,i,j}^ - (z )}}{{dz}} = {G_{psbx + ,i,j}} + P{C_{psbx + ,i,j}} + SB{S_{psx + ,i,j}}$$

(10g)$$\frac{{dP_{sby,i,j}^ - (z )}}{{dz}} = {G_{psby + ,i,j}} + P{C_{psby + ,i,j}} + SB{S_{psy + ,i,j}}$$

(10h)$$\frac{{dP_{p,j}^ + (z )}}{{dz}} ={-} [{{\sigma_{ap}}N - ({{\sigma_{ap}} + {\sigma_{ep}}} ){N_{2,j}}(z )} ]P_{p,j}^ + (z )- {\alpha _p}P_{p,j}^ + (z )({\textrm{j} = 1,2,\ldots ,6} )$$

(10i)$$\frac{{dP_{p,j}^ - (z )}}{{dz}} = {\mathrm{\Gamma }_{p,j}}[{{\sigma_{ap}}N - ({{\sigma_{ap}} + {\sigma_{ep}}} ){N_{2,j}}(z )} ]P_{p,j}^ - (z )+ {\alpha _p}P_{p,j}^ - (z )({\textrm{j} = 1,2,\ldots ,6} )$$

where $P_{sx,i,j}^ + $ and $P_{sy,i,j}^ + $ represent the forward signal powers of each transverse mode (denoted by i) under two polarization states (denoted by x,y) in each area (denoted by j). $P_{srx,i,j}^ + $ and $P_{sry,i,j}^ + $ represent the forward SRS powers. $P_{sbx,i,j}^ - $ and $P_{sby,i,j}^ - $ represent the backward SBS powers. $P_{p,j}^ + (z )$ and $P_{p,j}^ - $ represent the pump powers of two directions. ${N_{2,j}}(z )$ represents the inversion particle number. We ignore the backward signal light, SRS light and forward SBS light here, because they are much weaker than those of the other direction.

The changes of the power are devided into multiple parts. G, $PC$, $MC$, $SRS$, $SBS$ represent gain, polarization coupling, mode coupling, SRS process and SBS process respectively. Taking polarization x and LP01 mode (if necessary) as example, they can be calculated as follows

(11a)$${G_{psx,i,j}} = [{({{\sigma_{as}} + {\sigma_{es}}} ){N_{2,j}}(z )- {\sigma_{as}}N} ]P_{sx,i,j}^ + (z )- {\alpha _{sx}}P_{sx,i,j}^ + (z )$$

(11b)$$P{C_{psx,i,j}} ={-} {\textrm{h}_{xy,i,j}}P_{sx,i,j}^ + (z )+ {h_{yx,i,j}}P_{sy,i,j}^ + (z )$$

(11c)$$M{C_{psx,i,j}} = \mathop \sum \limits_{i \ne LP01} {k_{ \to LP01}}P_{sx,i \ne LP01,j}^ + (z )\; - \mathop \sum \limits_{i \ne LP01} {k_{LP01 \to }}P_{sx,i \ne LP01,j}^ + (z )$$

(11d)$$SR{S_{psx,i,j}} = \frac{{{g_r}}}{{{A_{c,j}}}}P_{sx,i,j}^ + (z ){P_{rx,j}}(z )$$

(11e)$$SB{S_{psx,i,j}} = \frac{{{g_B}}}{{{A_{c,j}}}}P_{sx,i,j}^ + (z )P_{bx,j}^ - (z )$$

The power in each area is calculated by total power and overlapping factor, which could be described as

(12a)$$P_{sx,i,j}^ +{=} {\mathrm{\Gamma }_{sx,i,j}}P_{sx,i}^ + ({\textrm{i} = \textrm{LP}01,\textrm{LP}11,\ldots } )({\textrm{j} = 1,2,\ldots ,6} )$$

(12b)$$P_{sy,i,j}^ +{=} {\mathrm{\Gamma }_{sy,i,j}}P_{sy,i}^ + ({\textrm{i} = \textrm{LP}01,\textrm{LP}11,\ldots } )({\textrm{j} = 1,2,\ldots ,6} )$$

(12c)$$P_{p,j}^ +{=} {\mathrm{\Gamma }_{p,j}}P_p^ + ({\textrm{j} = 1,2,\ldots ,6} )$$

(12d)$$P_{p,j}^ -{=} {\mathrm{\Gamma }_{p,j}}P_p^ - ({\textrm{j} = 1,2,\ldots ,6} )$$

(12e)$$P_{srx,j}^ +{=} {\mathrm{\Gamma }_{rx,j}}P_{srx}^ + ({\textrm{j} = 1,2,\ldots ,6} )$$

(12f)$$P_{sbx,j}^ -{=} {\mathrm{\Gamma }_{bx,j}}P_{sbx}^ - ({\textrm{j} = 1,2,\ldots ,6} )$$

(12g)$$P_{sry,j}^ +{=} {\mathrm{\Gamma }_{ry,j}}P_{sry}^ + ({\textrm{j} = 1,2,\ldots ,6} )$$

(12h)$$P_{sby,j}^ -{=} {\mathrm{\Gamma }_{by,j}}P_{sby}^ - ({\textrm{j} = 1,2,\ldots ,6} )$$

And the parameters used to calculate the number of upper level particles are shown as follows

(13a)$${A_{p,j}}= \frac{{[{P_{p,j}^ + (z )+ P_{p,j}^ - (z )} ]{\sigma _{ap}}}}{{h{\nu _p}{A_{c,j}}}}({\textrm{j} = 1,2,\ldots ,6} )$$

(13b)$$\begin{array}{l} {A_{s,j}} = \frac{{\mathop \sum \nolimits_i {\sigma _{as}}P_{sx,i,j}^ + (z )}}{{h{\nu _s}{A_{c,j}}}} + \frac{{\mathop \sum \nolimits_i {\sigma _{as}}P_{sy,i,j}^ + (z )}}{{h{\nu _s}{A_{c,j}}}} + \frac{{\mathop \sum \nolimits_i {\sigma _{ar}}P_{srx,i,j}^ + (z )}}{{h{\nu _r}{A_{c,j}}}} + \frac{{\mathop \sum \nolimits_i {\sigma _{ar}}P_{sry,i,j}^ + (z )}}{{h{\nu _r}{A_{c,j}}}} + \frac{{\mathop \sum \nolimits_i {\sigma _{ab}}P_{sbx,i,j}^ + (z )}}{{h{\nu _b}{A_{c,j}}}}\\ ={+} \frac{{\mathop \sum \nolimits_i {\sigma _{ab}}P_{sby,i,j}^ + (z )}}{{h{\nu _b}{A_{c,j}}}}({i = LP01,LP11 \ldots } )({j = 1,2, \ldots ,6} )\end{array}$$

(13c)$${B_{p,j}} = \frac{{[{P_{p,j}^ + (z )+ P_{p,j}^ - (z )} ]({{\sigma_{ap}} + {\sigma_{ep}}} )}}{{h{\nu _p}{A_{c,j}}}}({\textrm{j} = 1,2,\ldots ,6} )$$

(13d)$$\begin{array}{l} {B_{s,j}} = \frac{{\mathop \sum \nolimits_i ({{\sigma_{as}} + {\sigma_{es}}} )P_{sx,i,j}^ + (z )}}{{h{\nu _s}{A_{c,j}}}} + \frac{{\mathop \sum \nolimits_i ({{\sigma_{as}} + {\sigma_{es}}} )P_{sy,i,j}^ + (z )}}{{h{\nu _s}{A_{c,j}}}} + \frac{{\mathop \sum \nolimits_i ({{\sigma_{ar}} + {\sigma_{er}}} )P_{srx,i,j}^ + (z )}}{{h{\nu _r}{A_{c,j}}}} + \frac{{\mathop \sum \nolimits_i ({{\sigma_{ar}} + {\sigma_{er}}} )P_{sry,i,j}^ + (z )}}{{h{\nu _r}{A_{c,j}}}}\\ + \frac{{\mathop \sum \nolimits_i ({{\sigma_{ab}} + {\sigma_{eb}}} )P_{sbx,i,j}^ + (z )}}{{h{\nu _b}{A_{c,j}}}} + \frac{{\mathop \sum \nolimits_i ({\sigma _{ab}} + {\sigma _{eb}})P_{sby,i,j}^ + (z )}}{{h{\nu _b}{A_{c,j}}}}({i = LP01,LP11\ldots } )({\textrm{j} = 1,2,\ldots ,6} )\end{array}$$

3. Simulation results

At present, the main problems encountered by linearly polarized fiber lasers include the suppression of SRS and SBS, which restrict each other in the process of laser amplification. Therefore, according to different nonlinear situations, the goal of simulation is to study the influence of direct pumping and tandem pumping on the output results of linearly polarized fiber laser. The system diagram used in the simulation is shown in the Fig. 3 and the general parameters are listed in Table 1. To ensure the same pumping absorption ability under different pumping conditions, the total pumping absorption coefficient is fixed at 18 dB, so the type and parameters of active PM fiber can be adjusted according to different pumping wavelengths. The pumping scheme can be forward pumping, backward pumping and bidirectional pumping. Through the simulation of pumping scheme, the selection of pumping method can be further optimized.

Fig. 3. Simulation system

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Table 1. Simulation parameters

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The Brillouin gain coefficient is influenced by factors including fiber material and laser linewidth, etc. According to the calculation formula, we can get the SBS gain coefficient under different linewidths. To distinguish the situations that are dominated by SBS or SRS, Brillouin gain coefficient ${g_b}$ is set in three cases respectively, which can also be regarded as the situation under different linewidth scales. The SBS scenario used for simulation is shown in Fig. 4. In each individual simulation, the forward and backward pump power under different pumping conditions are set from 1 kW to 5 kW, and the resolution is 100 W. Finally, the output parameters such as signal power, SRS power, backward SBS power, signal PER and signal fundamental mode power ratio will be qualitatively analyzed. It is worth noting that when necessary, the SBS threshold power is considered as 10% of the forward signal power. Once SBS power exceeds the threshold, the system is considered limited by SBS regardless of the signal power.

Fig. 4. Brillouin gain coefficient under different laser linewidth and three situations for simulation

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Firstly, the Brillouin gain coefficient is set as ${g_b} = 2.644 \times {10^{ - 13}}/m$ to indicate the condition of low SBS strength. The calculated results are shown in Fig. 5.

Fig. 5. Signal power (a, b), SRS (c, d), PER (e, f) and fundamental mode ratio (g, h) of direct pumping signal output (left part) and tandem pumping (right part) under low SBS strength

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The simulation results are shown in 8 sub-pictures (Fig. 5(a)-(h)). Among them, the four pictures on the left (a,c,e,g) are the results of direct pumping configuration, and the four pictures on the right (b,d,f,h) are the results of tandem pumping configuration. Sub-pictures (a) and (b) represent the signal output power of the two pumping schemes under bidirectional pumping, with the horizontal axis representing the forward pumping power and the vertical axis representing the backward pumping power. In this case, the maximum output power of direct pump and tandem pumping is about 7948W and 6792W respectively. Moreover, the power increase in the case of direct pumping is positively related to the pump power, while tandem pumping configuration has a best pump distribution. To achieve higher signal power output, more pump power should be allocated to the backward direction.

The reason for this result is SRS process. The SRS power at the output of the two pumping configurations is shown in sub-pictures (c) and (d). In this case, due to the small gain coefficient, SBS cannot compete with SRS. No SBS effect is observed under the two pumping schemes, and the intensity of SRS affects the output signal power. When the pumping power reaches the highest, the SRS output power is only about 4W for direct pumping scheme and about 6970W for tnadem pumping scheme. The results show that the SRS intensity under tandem pumping is much higher than that under direct pumping. Although tandem pumping has a lower nonlinear effect, SRS will still be stronger because its fiber length is nearly twice that of direct pumping.

Sub-pictures (e) and (f) show the PER of the output signal under two pumping schemes. With the increase of signal power, PER will deteriorate with the generation of heat. Before SRS occurrs significantly, due to the lower quantum loss, the PER degradation rate of tandem pumping is slower than that of direct pumping, so it has a higher PER. However, when the SRS power rises, the PER of signal light will decrease rapidly due to the heat generation and signal power transfer caused by SRS. In addition, under both pumping schemes and the same output power, the higher the backward pumping power, the higher the PER. This shows that backward pumping is beneficial to improve the polarization of the system. Therefore, if SRS can be controlled, adopting backward tandem pumping scheme will be more conducive to maintaining polarization. It is worth noting that for PER, under normal circumstances, there will be a big drop when the power is just amplified. When the power is rapidly increased to kilowatt level, its deterioration degree is obviously reduced. This is caused by the cooling of the fiber. When there is no gain process, the fiber will hardly generate temperature, so the core temperature is equal to the cooling temperature. However, as the gain process occurs, the fiber begins to generate heat. At this time, the temperature of the fiber surface is limited by the cooling capacity and will be higher than the cooling temperature, so the fiber core at this time not only suffers from heat generation, but also introduces additional outer boundary temperature [14].

Sub-pictures (g) snd (h) show the fundamental mode power ratio of output signal power under two pumping schemes. Similar to PER, with the increase of output power, the fundamental mode power will continue to decrease, which indicates the deterioration of beam quality. Also, backward pumping is more beneficial to improve the beam quality of output laser compared with forward pumping, but the improvement effect is not as good as that of PER. When SRS occurs significantly, the proportion of fundamental mode power of signal light will also decrease rapidly, so suppressing SRS is also helpful to improve beam quality. Because of the large diameter of fiber core, although the heat production is low, the proportion of fundamental mode power of tandem pumping is still slightly lower than that of direct pumping.

Under this circumstance, unless the doping concentration is further increased to shorten the active fiber, tandem pumping has no advantage over direct pumping at present. In addition, the backward pumping scheme is more conducive to the optimization of the output parameters of linearly polarized fiber lasers.

Secondly, the Brillouin gain coefficient is set as ${g_b} = 3.305 \times {10^{ - 13}}/m$ to characterize the condition of moderate SBS strength. The calculated results are shown in Fig. 6.

Fig. 6. Signal power (a, b), SBS or SRS (c, d), PER (e, f) and fundamental mode ratio (g, h) of direct pumping signal output (left part) and tandem pumping (right part) under moderate SBS strength

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The simulation results are shown in 8 sub-pictures (Fig. 6(a)-(h)). Among them, the four pictures on the left (a,c,e,g) are the results of direct pumping configuration, and the four pictures on the right (b,d,f,h) are the results of tandem pumping configuration. Sub-pictures (a) and (b) represent the signal output power of the two pumping schemes under bidirectional pumping. In this case, the maximum output power of direct pump and tandem pumping is about 6241W and 6017W respectively. The output power of direct pumping will first increase and then decrease with the increase of the total pump power, while the output power of tandem pumping is similar to that of Fig. 5(b).

The reason for this difference is the competition between SBS and SRS. In this case, direct pumping is limited by SBS effect, while tandem pumping is limited by SRS effect. As shown in sub-figure (c), with the increase of pump power, the output power of SBS under direct pumping rapidly increases to nearly 3 kW, which makes the signal output power decrease rapidly. Because SBS will damage the system, the output power under direct pumping in the actual experiment will not reach the maximum power obtained by the simulation. In addition, SBS is insensitive to the pumping direction. As shown in sub-figure (d), tandem pumping is still limited by SRS, indicating that SBS in this case still cannot compete with SRS under tandem pumping.

Sub-pictures (e) and (f) show the PER of the output signal under two pumping schemes. With the increase of signal power, PER will deteriorate with the generation of heat. Similarly, tandem pumping has better PER maintenance ability. SBS, like SRS, will also make the output PER drop rapidly. However, because the heat production of SBS process is much lower than that of SRS, its deterioration degree is also much lower than that of SRS. Because SBS is almost insensitive to the pumping direction, it will destroy the power contour of PER.

Sub-pictures (g) snd (h) show the fundamental mode power ratio of output signal power under two pumping schemes. Similarly, with the increase of output power, the fundamental mode power will continue to decrease, which indicates the deterioration of beam quality. Backward pumping is also beneficial to improve the beam quality, and the fundamental mode power ratio of tandem pumping is also slightly lower than that of direct pumping. The degradation of beam quality caused by SBS is significantly more serious than that of PER. The degradation of beam quality caused by SBS is significantly more serious than that caused by PER. Therefore, when SBS occurs in direct pumping, the SRS of tandem pump is not serious at this time, which makes the fundamental mode power ratio of tandem pumping temporarily exceed that of direct pumping.

In this category, the ultimate maximum power under different pumping conditions depends on the relative strength of SBS and SRS. If SBS is weak, the direct pumping is much better, while SBS is relatively strong, the tandem pumping has higher theoretical output power. Regarding PER and beam quality, similar to the simulation of case 1, backward pumping is more advantageous.

Lastly, the Brillouin gain coefficient is set as ${g_b} = 5.663 \times {10^{ - 13}}/m$ to characterize the condition of high SBS strength. The calculated results are shown in Fig. 7.

Fig. 7. Signal power (a, b), PER (c, d), fundamental mode ratio (e, f), SRS (g, h), SBS (i, j) and SBS-free region (k, l) of direct pumping signal output (left part) and tandem pumping (right part) under high SBS strength

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The simulation results are shown in 12 sub-pictures (Fig. 6(a)-(l)). Among them, the six pictures on the left (a,c,e,g,i,k) are the results of direct pumping configuration, and the six pictures on the right (b,d,f,h,j,l) are the results of tandem pumping configuration.

In this case, both pumping schemes are first limited by SBS. However, when the pump power is increased, SRS will gradually gain an advantage in the competition with SBS due to the increase of the gain in the initial stage, and will eventually become the main nonlinear effect. Therefore, no matter the signal output power shown in sub-images (a) and (b), or the PER shown in sub-images (c) and (d), or the fundamental mode power ratio shown in sub-images (e) and (f), there will be a threshold boundary. SBS effect mainly occurs on the left side of the boundary, while SRS effect mainly occurs on the right side, as shown in sub-images (g-j).

However, in the actual experiment, due to the damage characteristics of SBS to the system, the linearly polarized fiber laser can only operate on the left side of the boundary, and SBS needs to be avoided. Therefore, to study the extent to which the two pumping schemes are limited by SBS, we define SBS threshold as when SBS power reaches 1% of output signal power. We can get the area below SBS threshold as shown in the white area of sub-picture (k) and (l).

The white areas of the two pumping schemes are similar in shape and can be basically divided into left and right parts. Among them, the right region is not discussed because it belongs to the part where SRS occurs significantly. The left side represents the area without SBS when the signal power is amplified normally. Compared with direct pumping, the SBS-free area on the left side of tandem pumping is larger. Therefore, it also has a higher maximum signal optical power, which is 5926W, much higher than 3514W of direct pumping, as shown in the sub-picture (a) and (b). Because SBS is insensitive to the pump direction, the optimal pump distribution curve is also significantly different from the first two cases. Due to the weak intensity of SBS, in this case, PER and fundamental mode power ratio of tandem pumping are also higher than that of direct pumping when the signal power reaches a certain level.

Under this configuration, if better output parameters at low power are demanded, direct pumping is more suitable. However, if a breakthrough in power is needed, theoretically, tandem pumping has more potential. Similarly, more backward pumping is beneficial to PER and beam quality.

Finally, we summarize the relationship between beam quality and PER. As shown in Fig. 1, mode coupling will occur between different modes under the same polarization state, so the deterioration or fluctuation of beam quality caused by it will affect the power ratio of different modes, thus affecting the polarization evolution. Due to the design of PANDA fiber, the birefringence of high-order modes and fundamental mode is not the same. Except for the LP11 mode along the axial direction of the stress bar, the birefringence of the other high-order modes is lower than that of the fundamental mode, so its ability to resist disturbance is obviously reduced. When the beam quality deteriorates or fluctuates, the proportion of high-order mode power increases or fluctuates, which will lead to the acceleration or fluctuation of PER deterioration. In the simulations of three situations, the variations of PER and fundamental mode ratio all keep a similar trend, which is consistent with the above theoretical analysis.

4. Experiments

Theoretical simulation shows that compared with direct pumping, tandem pumping has advantages under the maintenance of PER and beam quality. At the same time, the backward pumping scheme can strengthen this advantage.

In order to verify the feasibility of backward tandem pumping scheme as well as further study the effects of the pump method on the output of linearly polarized fiber laser, we set up a backward pumped linearly polarized fiber laser system. The experimental setup is shown in Fig. 8.

Fig. 8. Experiment setup

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The system is a MOPA structure including a fiber Bragg grating (FBG) based seed, a pre-amplifier stage and a main amplifier stage. The FBG-based seed contains a pair of PANDA fiber gratings with a 3 dB linewidth of 0.1 nm and 1 nm for OC-FBG and HR-FBG, 4.5 m polarization maintaining (PM) gain fiber (PLMA-YDF-10/125), a 150 W LD and a cladding power stripper (CPS). The gain fiber is coiled to select the polarization mode of the slow axis and the radius is 2.25 cm. To slightly increase the output power of the seed, the pre-amplification stage also uses a 150 W LD, a 4.5-m-long 10/125 µm Yb-doped PM fiber and a PM CPS. In the main amplifier stage, according to the previous experiment result of forward tandem pumped linearly polarized fiber laser, we have changed the PM gain fiber to special PLMA-YDF-30/250 whose cladding absorption coefficient is risen to about 0.87 dB/m @ 1018 nm. This choice could decrease the length of gain fiber from 42 m to 18 m, which could help suppress SRS and polarization deterioration. The gain fiber is coiled at a radius of 6 cm. The backward tandem pump system consists of a PM (2 + 1)*1 backward pump combiner, two 3*1 signal combiners and five 300 W 1018 nm fiber laser models. Since the 1018 nm pump source is extremely sensitive to the backward light which will cause the output pump power to decrease or even damage the pumping system, we use a two-stage combining system to reduce the single route return light and insert a chirped tilted fiber Bragg grating (CTFBG) in front of the 1018 nm laser to further filter the return signal light. The total pump power output from the combiner is about 1200 W. One CPS is added before the main amplifier stage to prevent residual pump and one CPS is added after the main amplifier stage to filter out the cladding modes. To protect the seed and pre-amplifier, a circular is inserted between the pre-amplifier stage and the main amplifier stage, which could test the backward power.

The output power of the system is 875 W at full pump power, which is shown in Fig. 9. The slope efficiency is only 68.83% mainly caused by two reasons. One is that the fiber coiling radius is small and leads to high pump loss, and the second is that the short length of fiber results in insufficient pump absorption.

Fig. 9. Output power versus pump power

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The output spectrum evolution of the system is shown in Fig. 10. At the highest output power, the SRS suppression ratio is about 36 dB. We further tested the power of Raman light and found the portion is about 1.01%. The spectrum near 1070 nm is shown in Fig. 11. The 3 dB linewidth has arisen from 145.2 pm at seed power (∼44 W) to 254.4 pm at 875 W.

Fig. 10. Spectrum evolution

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Fig. 11. Linewidth evolution

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The PER of output power is shown in Table 2. It could be found that PER also declined significantly when the pump was just added, but was relatively stable when the pump power increased and no significant deterioration occurred.

Table 2. PER evolution

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The beam quality was also tested and M² factor is shown in Table 3. The detail of the beam quality at the highest power is shown in Fig. 12. We can see from the result that the beam quality is much worse than estimated. The reason is from the passive fiber PLMA-GDF-30/250 used in the system (This could be verified by terrible seed beam quality). The core diameter of highly doped gain fiber PLMA-YDF-30/250 is 30µm, but the NA is very high. In order to achieve mode field matching, the core diameter of passive fiber PLMA-GDF-30/250 becomes about 48µm, which excites a large number of high-order modes. It can be confirmed by observing the cross section of the PLMA-GDF-30/250, which is shown in Fig. 13.This is now the main problem of backward tandem pumping when using highly doped PM fibers.

Fig. 12. M² at highest power

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Fig. 13. Cross section of PLMA-GDF-30/250

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Table 3. M² evolution

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Then we compare the laser output parameters of experiments and simulations under forward and backward tandem pumping. We plot the output power, spectrum and PER of the two systems as shown in Fig. 14.

Fig. 14. Comparison of (a)output power versus pump power (b)PER (c)SRS power percentage (d)linewidth change in experiments and simulations under forward and backward tandem pumping

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The experiment results show that compared with forward tandem pumping, backward tandem pumping with PM fiber of higher absorption has better PER maintenance, SRS suppression and linewidth control. The two slope efficiencies are close because both are tandem pumping and have the same total absorptions. It could be concluded that if tandem pumping is applied in a linearly polarized fiber laser, the backward pumping method and highly doped fiber are a better choice.

The variation tendency of simulation results is consistent with those of experiment results. The slope efficiencies of the simulation are higher than that of the experiments mainly because the power loss caused by devices is not considered in the simulation. Similarly, the PER drop caused by splicing points and devices is not considered in the simulations, so the initial values of PERs in static states are quite different (red circles in Fig. 8(b)). In addition, the deterioration rates of PER in the simulations are also lower than those in the experiments, indicating that other factors can affect the deterioration of PER. Because the accurate Raman gain coefficient and Raman noise power cannot be precisely measured, the simulation results of SRS only conform to the experiment in the change tendency. It can be inferred that compared with experiments, the Raman gain coefficient used in the simulations is smaller and the Raman noise power is larger, but the experimental results are still of reference significance. It is worth noting that Raman power ratio fluctuates under forward pumping. This is due to the second-order Raman process. When the second-order Raman occurs, it consumes the first-order Raman power, which causes the total Raman power to decrease. Therefore, when the signal power is low, the power transfered to Raman light is not enough to make up for the loss of this process, and the total Raman light power will decrease. With the increase of signal power, the power obtained by the first-order Raman light exceeds the power loss of the second-order Raman light, and the total Raman power returns to increase. There is no obvious backward power (both less than 1 mW) in the simulation results, which is consistent with the fact that SBS effect has not been observed in the experiment, indicating that special SBS suppression means are not needed under our experimental conditions. Because the mode field of passive fiber in the experiment does not meet the conditions given in our simulation, we can't give a comprehensive corresponding comparison with the beam quality.

5. Conclusion

In this article, a thorough model of linearly polarized fiber laser considering polarization coupling, mode coupling, SBS and SRS effects is established. The output signal power, SRS power, SBS power, signal PER and signal fundamental mode power ratio are analyzed. The simulation results show that tandem pumping suffers from more severe SRS effects due to the demand of longer fiber, while compared with direct pumping, the SBS threshold is higher because of SRS competition. In addition, due to less heat generation, the degradation of beam quality and PER with the increase of power under tandem pumping is weaker than that under direct pumping. This shows that if SRS effect can be suppressed, tandem pumping has the potential to realize high power linearly polarized laser output. The results of theoretical simulation show that the backward pumping scheme can not only suppress SRS, but also help to maintain PER and beam quality, so it can be used for experimental construction.

Based on the theory, a linearly polarized fiber laser using highly doped fiber under backward tandem pumping is built to explore the potential of tandem pumping. Compared with forward pumping, its PER and SRS suppression are greatly improved. The output power, the intensity of SRS and SBS, and the degradation of PER of the experimental system are compared and analyzed with the simulation. The correspondence between theory and experiment is verified. However, due to the design of passive fiber, the output beam quality is disappointing, so if the core diameter of passive fiber designed for highly doped fiber can be controlled, backward tandem pumping can also be applied to linearly polarized fiber lasers.

Funding

National Natural Science Foundation of China (62122040, 61875103, 62075113).

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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Parameters	Symbol	Direct pumping	Tandem pumping
Fiber model	/	PLMA-YDF-20/400	PLMA-YDF-30/250
Fiber core diametre	$d_{c o r e}$	20µm	30µm
Fiber first cladding diametre	$d_{c l a d}$	400µm	250µm
fiber core NA	/	0.065	0.07
Pumping wavelength	$λ_{p}$	976nm	1018nm
Signal wavelength	$λ_{s}$	1070nm	1070nm
Upper energy lifetime	$τ$	0.8µs	0.8µs
Fiber cladding absorption	/	1.8 dB/m	1 dB/m
Fiber length	/	10m	18m
Total pump absorption	/	18dB	18dB
Pump absorption cross section	$σ_{a p}$	$1.767 \times 10^{- 24} m^{2}$	$9.439 \times 10^{- 26} m^{2}$
Pump emission cross section	$σ_{e p}$	$1.713 \times 10^{- 24} m^{2}$	$7.301 \times 10^{- 25} m^{2}$
Signal absorption cross section	$σ_{a s}$	$4.501 \times 10^{- 27} m^{2}$	$4.501 \times 10^{- 27} m^{2}$
Signal emission cross section	$σ_{e s}$	$3.633 \times 10^{- 25} m^{2}$	$3.633 \times 10^{- 25} m^{2}$
Raman gain coefficient	$g_{r}$	$5 \times 10^{- 13} / m$	$5 \times 10^{- 13} / m$
Seed power	$P_{s} (0)$	65W	65W
Seed PER	/	20dB	20dB
Mode number	/	$LP 01, LP 11 a, LP 11 b, LP 21 a, LP 21 b, LP 02, LP 31 a, LP 31 b$
Internal turbulence coefficient	$K_{σ}$	$2.9824 \times 10^{- 12} / m$

Power/W	44	235	394	574	756	875
M²_x	3.258	3.476	3.585	3.061	3.239	3.813
M²_y	2.643	3.834	4.004	4.087	3.227	3.005
M²	3.161	3.664	3.809	3.662	3.325	3.499

Parameters	Symbol	Direct pumping	Tandem pumping
Fiber model	/	PLMA-YDF-20/400	PLMA-YDF-30/250
Fiber core diametre	$d_{c o r e}$	20µm	30µm
Fiber first cladding diametre	$d_{c l a d}$	400µm	250µm
fiber core NA	/	0.065	0.07
Pumping wavelength	$λ_{p}$	976nm	1018nm
Signal wavelength	$λ_{s}$	1070nm	1070nm
Upper energy lifetime	$τ$	0.8µs	0.8µs
Fiber cladding absorption	/	1.8 dB/m	1 dB/m
Fiber length	/	10m	18m
Total pump absorption	/	18dB	18dB
Pump absorption cross section	$σ_{a p}$	$1.767 \times 10^{- 24} m^{2}$	$9.439 \times 10^{- 26} m^{2}$
Pump emission cross section	$σ_{e p}$	$1.713 \times 10^{- 24} m^{2}$	$7.301 \times 10^{- 25} m^{2}$
Signal absorption cross section	$σ_{a s}$	$4.501 \times 10^{- 27} m^{2}$	$4.501 \times 10^{- 27} m^{2}$
Signal emission cross section	$σ_{e s}$	$3.633 \times 10^{- 25} m^{2}$	$3.633 \times 10^{- 25} m^{2}$
Raman gain coefficient	$g_{r}$	$5 \times 10^{- 13} / m$	$5 \times 10^{- 13} / m$
Seed power	$P_{s} (0)$	65W	65W
Seed PER	/	20dB	20dB
Mode number	/	$LP 01, LP 11 a, LP 11 b, LP 21 a, LP 21 b, LP 02, LP 31 a, LP 31 b$
Internal turbulence coefficient	$K_{σ}$	$2.9824 \times 10^{- 12} / m$

Power/W	44	235	394	574	756	875
M²_x	3.258	3.476	3.585	3.061	3.239	3.813
M²_y	2.643	3.834	4.004	4.087	3.227	3.005
M²	3.161	3.664	3.809	3.662	3.325	3.499

Comparison of tandem pumping and direct pumping on high-power linearly polarized fiber laser

Abstract

1. Introduction

2. Theoretical model

3. Simulation results

4. Experiments

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (14)

Tables (3)

Equations (40)

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