1. Introduction

Among high-power fiber lasers (HPFLs), single-mode fiber-based lasers are limited by nonlinear and thermo-optic effects, power saturation, and optical damage, which make it difficult to significantly increase their output power [1,2]. In addition, the presence of transverse mode instability (TMI) in single-mode fibers (SMFs) fundamentally limits the output power [3]. The application of multimode fibers (MMFs) alleviates the limitations of thermo-optic effects and optical damage, thereby effectively improving the output power of fiber lasers [4]. In MMF-based fiber lasers, nonlinear effects are the main factors that hinder the improvement in their output beam characteristics (power, beam quality, spectral characteristics, etc.). The nonlinear effects of high-power near-single-mode continuous-wave fiber lasers (NSM-CWHPFLs) based on MMFs include intramodal nonlinear effects such as self-phase modulation (SPM), cross-phase modulation (XPM), stimulated Raman scattering (SRS), and four-wave mixing (FWM) [5–9] as well as intermodal nonlinear effects such as intermodal four-wave mixing (IMFWM) and intermodal Raman scattering (IMRS) [10,11]. These intermodal nonlinear effects result in the generation of light waves at new frequencies, which seriously affect the purity of the output spectra and cause significant energy dispersion and power fluctuation of the output laser. Consequently, the maximum obtainable output power and its application in the field of spectral-beam combination are limited. In addition, intermodal nonlinear effects deteriorate the output beam quality and reduce the energy concentration of the laser beam, which severely hinders laser processing and laser damage applications [4]. Therefore, it is important to study intermodal nonlinear effects in high-power near-single-mode CW fiber lasers.

The current related studies on NSM-CWHPFLs mainly focus on intramodal nonlinear effects, while intermodal nonlinear effects such as IMFWM and IMRS have gradually attracted the attention of researchers in recent years. In some studies, Stokes and anti-Stokes peaks caused by IMFWM were observed in the output spectra of fiber lasers; however, no detailed explanation regarding their generation mechanism was provided [12,13]. In 2017, Fang et al. observed peaks at approximately 1060 and 1100 nm in the output spectrum and indicated that they were caused by FWM in the fiber amplifier [14]. In 2018, we had conducted a preliminary study on the physical mechanism and suppression method of the IMFWM effect in HPFLs and established a phase-matching frequency shift and coherence length calculation model. Moreover, we had previously proposed an IMFWM suppression method based on fiber coiling and achieved effective suppression of IMFWM products in HPFLs [15]. In 2020, Chen et al. experimentally studied the influence of the pump direction on the IMFWM in HPFLs and indicated that the counter-pumping scheme is beneficial for suppressing the IMFWM compared with the co-pump [16]. Cao et al. established a power variation model of each mode involving IMFWM via the addition of an IMFWM-induced power coupling term to analyze the power variation of each mode caused by the IMFWM [17]. However, this model is a simplified power model and cannot simulate the spectral evolution process involving IMFWM.

The IMRS effect in NSM-CWHPFLs usually refers to the intermodal wave mixing process involving Raman-Stokes light, including SRS-induced intermodal wave mixing (SRS-IMWM) and IMFWM based on multimode Raman-Stokes light (MMRS-IMFWM). Among them, SRS-IMWM mainly refers to the process of converting the signal light in the fundamental mode (FM) into Raman-Stokes light in high-order modes (HOMs) in NSM-CWHPFLs. In 2018, Liu et al. simulated the spectral evolution process of the light in the LP₀₁ and LP₁₁ modes including SRS-IMWM based on the two-mode nonlinear Schrödinger equations (NLSEs). Subsequently, they theoretically demonstrated that the SRS leads the signal light in the FM to convert to the Raman-Stokes light in the LP₁₁ mode [18]. In 2020, based on the wavelength of IMFWM peaks observed in the output spectrum, Chu et al. indicated that the HOM light caused by SRS was mainly in the LP₂₁ mode [19]. MMRS-IMFWM is an IMFWM process wherein the second-order Stokes light in the HOM is generated from the Raman-Stokes lights of the FM and HOM. In 2020, Lin et al. simulated the SRS-IMWM between the LP₀₁ and LP₁₁ modes based on two-mode NLSEs. Their simulation results indicated that the second-order Stokes light in the LP₁₁ mode at 1178 nm was generated by the MMRS-IMFWM process [20].

Existing studies on the intermodal nonlinear effects of NSM-CWHPFLs, such as IMFWM, SRS-IMWM, and MMRS-IMFWM, remain insufficient. First, the lack of a theoretical model for the spectral evolution of the intermodal nonlinear effects of the NSM-CWHPFLs makes it impossible to determine the generation mechanism and evolution process of new spectral sidebands caused by IMFWM and MMRS-IMFWM. Second, the lack of theoretical models of mode degradation caused by intermodal nonlinear effects results in difficulties in clarifying the physical mechanism by which intermodal nonlinearities affect the output beam quality. Furthermore, no detailed theoretical and experimental research on the influence of various laser parameters on intermodal nonlinear effects has been performed to date; thus, there exists no comprehensive and accurate theoretical reference for intermodal nonlinearity suppression. Therefore, in this study, we proposed an intermodal-nonlinearity-induced time-frequency evolution model of NSM-CWHPFLs and simulated the evolution process of spectral characteristics and output beam quality factor M² under the combined action of intermodal and intramodal nonlinear effects. The aim was to obtain an in-depth understanding of the generation mechanism of various spectral sidebands and the comprehensive optimization of spectral distortion and mode degradation caused by intermodal nonlinear effects.

2. Theoretical model

To study the intermodal nonlinear effects in NSM-CWHPFLs, an intermodal-nonlinearity-induced time-frequency characteristic evolution model must be established to analyze the spectral distortion and mode degradation caused by intermodal nonlinear effects. Currently, the time-frequency characteristic evolution model of HPFLs is usually established using the rate equation combined with single-mode NLSEs. However, this model cannot be used to analyze the impact of mode competition and intermodal nonlinearities. Therefore, it is necessary to establish a time-frequency evolution model of NSM-CWHPFLs, including intermodal nonlinear effects based on the multimode-pulse propagation equations and the high-power CW fiber laser model.

2.1 Multimode pulse propagation equation including polarization characteristic correction and Raman response function optimization

In NSM-CWHPFLs, the master oscillator power amplifier (MOPA) scheme, based on a multi-longitudinal-mode seed oscillator achieved through the use of a pair of narrow-band fiber Bragg gratings (FBGs), is the most economical and effective method to achieve high-power, narrow-spectral-width (sub-nanometer), and near-single-mode laser output. For the narrow-band FBG-based MOPA, the complex interaction between nonlinear effects, gain characteristics, and dispersion characteristics causes it to operate in a state between an ideal continuous wave and a pulse [21]. It includes turbulent-like pulses on short time scales, reflecting the superposition state of laser pulses at different frequencies with rapidly changing amplitudes and phases. Moreover, it can be considered continuous on millisecond time scales and pulsed on sub-nanosecond time scales [22]. Therefore, the time-frequency characteristic evolution of the NSM-CWHPFL must be described based on the pulse propagation equation in the fibers. Commonly used pulse propagation equations include the nonlinear Schrödinger equation (NLSE), generalized NLSE (GNLSE), coupled NLSE, and frequency-band NLSEs [23]. GNLSE can be used in cases involving the combined action of multiple nonlinear effects, and it is widely used in modeling the evolution process of the electric field amplitude envelope of laser pulses in single-mode fibers [23]. However, for multimode fiber lasers, multimode GNLSEs based on the spatial distribution characteristics of transverse modes must be applied to analyze the pulse propagation in MMFs. They are typically referred to as generalized multimode nonlinear Schrödinger equations (GMMNLSEs).

Based on the commonly used GMMNLSEs derived from Maxwell’s equations proposed by Poletti et al. [24] and several approximate conditions associated with HPFLs, this study further derived the GMMNLSEs. The modes of the four waves participating in nonlinear effects were defined as ${m_g} + {m_h} \to {m_k} + {m_l}$ and abbreviated as $kghl$. As the fiber mode can be described by the propagation constant $\beta _{}^{({m_k})}(\omega )$ and mode function ${\mathbf{F}_{{m_k}}}(x,y,\omega )$, the electric field of a multimode pulse is decomposed into a sum of mode functions with envelopes. The GMMNLSEs were expressed as a set of partial differential equations that described the dynamics of the mode envelope A_mk(z,t). For simplicity, the following assumptions were considered in the further derivation of the GMMNLSEs: (1) Assuming that the Raman response function and pulse envelope function A_mk(z,t) vary slowly on the time scale, a rapidly oscillating term (∂/∂t)ⁿA_mk(z,t) was ignored [20]. (2) The frequency dependence of the mode function ${\mathbf{F}_{{m_k}}}(x,y,\omega )$ and of the corresponding normalization constant was neglected and then finally truncated to the linear term around a central frequency ω₀ [24]. (3) Only positive frequency components were considered [24]. (4) The non-phase-matched terms at 3ω₀ responsible for third-harmonic generation were discarded [25]. (5) The self-steepening terms $\tau _{kghl}^{(1,2)} = 1/{\omega _0} + \partial \ln [Q_{kghl}^{(1,2)}(\omega )]/\partial \omega$ were different for every set of the four mode indices. The differences were neglected, and the approximation $\tau _{kghl}^{(1,2)} \approx 1/{\omega _0}$ was used [26]. (6) The Raman contribution at 2ω₀ was sufficiently small to be neglected [25].

Based on the above assumptions, GMMNLSE for the pulse envelope ${A_{{m_k}}}(z,t)$ of mode m_k was expressed, with ${G^{({m_k})}}(z,t)$, ${\Theta ^{({m_k})}}(z,t)$, ${D^{({m_k})}}(z,t)$, and ${N^{({m_k})}}(z,t)$ indicating the gain, loss, dispersion, and nonlinear terms for mode m_k, respectively, where ${g_{{m_k}}}(z,t)$ and ${\alpha _{{m_k}}}(z)$ represent the gain and loss coefficients of mode m_k, respectively. The nonlinear term ${N^{({m_k})}}(z,t)$ represents the coupling process between mode m_k and every combination of modes m_g, m_h, and m_l. $\beta _n^{({m_k})} = \partial \beta _{}^{({m_k})}/\partial {\omega ^n}$ is the n-order dispersion parameters of mode m_k, where m₀ represents the fundamental mode. Further, ${n_2}$ is the nonlinear refractive index and c is the speed of light in a vacuum. In addition, f_R is the fractional contribution of the Raman response function to the overall nonlinearity, h(t) is the delayed Raman response function, and $Q_{kghl}^{(1,2)}(\omega )$ represents the mode overlap coefficients.

For the Raman contribution, the imaginary part of the delayed Raman response function h(t) is related to the Raman gain spectrum [27]. To simulate the Raman contribution more accurately, the multiple-vibration-mode-based Raman response function was expressed as [28]

Table 1. IMFWM and MMRS-IMFWM frequency shifts for the 25/400 fiber

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According to Eq. (4), the mode overlap coefficients differ for every set of four mode indices, and several coefficients vanish exactly or approximately because of the mode symmetries. Consider two linearly x-polarized modes LP₀₁ and LP_11e as examples, that is, ${m_1} \to L{P_{01}}\& {\sigma _1} \to {\mathbf{e}_x}$ and ${m_2} \to L{P_{11e}}\& {\sigma _2} \to {\mathbf{e}_x}$. The mode overlap coefficients corresponding to the four mode combinations that contribute to multimode nonlinearities primarily included $Q_{1111}^{(1)}\& Q_{1122}^{(1)}\& Q_{1212}^{(1)}\& Q_{1221}^{(1)}$ and $Q_{2222}^{(1)}\& Q_{2211}^{(1)}\& Q_{2121}^{(1)}\& Q_{2112}^{(1)}$. Moreover, owing to the mode symmetries of the multimode step-index fibers, the coefficients of mode combinations involving the same fiber modes were identical, that is, $Q_{1122}^{(1)} = Q_{1212}^{(1)} = Q_{1221}^{(1)} = Q_{2211}^{(1)} = Q_{2121}^{(1)} = Q_{2112}^{(1)}$. However, for orthogonal polarization situations, that is, ${\sigma _2} \to {\mathbf{e}_y}$, $Q_{1212}^{(1)} = Q_{1221}^{(1)} = Q_{2121}^{(1)} = Q_{2112}^{(1)} = 0$ was obtained. To incorporate the influence of polarization characteristics in a more convenient manner, polarization-related correction coefficients $f_{pol}^K$ and $f_{pol}^R$ were introduced. Consequently, the nonlinear term of the GMMNLSEs was then modified to

Considering the mode degeneration in the weakly guiding limit, the fiber modes fall into groups of LP_mn modes containing either two (m = 0) or four (m > 0) degenerate modes. Within each group, degenerate modes can be combined into either ${\mathbf{e}_ + }$ or ${\mathbf{e}_ - }$ circularly polarized modes, where ${\mathbf{e}_ \pm } = ({\mathbf{e}_x} \pm i{\mathbf{e}_y})/\sqrt 2 $. Using the circular polarization properties, $Q_{kghl}^{(2)}(\omega ) = 0$ was obtained, and consequently, the polarization-related correction coefficients were set to $f_{pol}^K = 2/3\& f_{pol}^R = 1$.

In addition, in the derivations of GMMNLSEs, the optical field was assumed to maintain its polarization along the fiber length, such that a scalar approach was valid. However, in practical situations, unless polarization-maintaining (PM) fibers are used, the residual birefringence in the fiber can continuously change the polarization state of light in a random manner [24]. Thus, for non-polarization-maintaining cases, the effective value of the nonlinear term should be reduced by setting the polarization-related correction coefficients to be $f_{pol}^K = 1\& f_{pol}^R = 2/3$, thereby avoiding a complicated vector algorithm process [29]. In NSM-CWHPFLs, without the PM fiber devices to maintain the polarization, the polarization states vary in a complicated manner; therefore, they are generally considered to be non-polarization- or polarization-maintaining on a very short time scale. Therefore, combined with the degeneration of the LP_mn modes presented above, the polarization-related correction coefficients were set as $f_{pol}^K = 2/3\& f_{pol}^R = 2/3$.

The split-step Fourier method (SSFM) is the most classical algorithm for the numerical computation of GMMNLSEs. It comprises a set of partial differential equations describing the dynamics of the mode envelopes. However, the computational complexity of the SSFM increases rapidly with increase in the number of fiber modes. Moreover, simulations of intermodal nonlinear effects should be performed in a large spectral range to satisfy the requirement of multimode-nonlinear frequency shifts, which requires a long computing time. Thus, to improve computational efficiency, the massively parallel algorithm (MPA) [30–32] combined with the SSFM was used to accelerate the computing process through parallel computing [33]. Using SSFM, the GMMNLSE was separated into linear and nonlinear terms, and these effects were assumed to be approximately independent of each small step. The linear term $\widetilde {{L^{({m_k})}}}(z,\omega )$ included the gain, loss, and dispersion terms, which were evaluated as multiplications in the spectral domain. In contrast, the nonlinear term was integrated in the time domain. Subsequently, through the application of the MPA, the integrals were computed in parallel with multiple iterations until the error of the approximate solution was smaller than a given tolerance. Because the number of iterations is always smaller than the extent of parallelization, the massively parallel split-step algorithm (MPSSA) is faster than the traditional serial SSFM. Subsequently, solving the GMMNLSEs requires the gain coefficients of different modes. Therefore, the NSM-CWHPFL model to obtain the gain coefficients were established.

2.2 NSM-CWHPFL Model including mode competition and fiber coiling

Considering the multimode aspects of the large-mode-area fibers used in NSM-CWHPFLs, the classical single-mode fiber laser model, which considers only the fundamental mode, cannot be applied. Thus, the theoretical model of NSM-CWHPFLs were established based on the mode competition in the multimode fiber, the interaction process between the laser and the Yb³⁺ ion, and the structural properties of the laser system [34]. In addition, the influence of fiber coiling, which is widely used in NSM-CWHPFLs to improve the output beam quality and heat dissipation performance of the fiber, was considered. Fiber coiling results in the introduction of mode-dependent bend losses. The bend losses of the HOMs are greater than those of the fundamental mode. The theoretical model of NSM-CWHPFLs, including mode competition and fiber coiling can be found in Ref. [35]. In this model, the beam propagation method was applied to simulate the bend loss of each mode, thereby improving the accuracy of the NSM-CWHPFL model. According to Eqs. (3)–(6) in Ref. [35], the gain coefficient of mode m_k can be expressed as

2.3 Seed simulation model based on the polarization thermal source theory

Prior to solving the GMMNLSEs and NSM-CWHPFL models, the incident multimode seed light must be determined. The NSM-CWHPFL configured with the MOPA scheme comprises a multilongitudinal-mode seed oscillator based on narrow-band FBGs and a multimode fiber amplifier. The narrow-linewidth seed oscillator with low power includes thousands of longitudinal modes that are not correlated with each other and should be simulated by the polarization thermal source model (PTSM) [36]. Based on the PTSM, the output laser of the multilongitudinal-mode seed oscillator can be regarded as a superposition of several longitudinal modes, which can be approximately expressed as follows:

In addition, the seed light may also include the Raman-Stokes light with the center wavelength of $\lambda _0^{SRS}$ or other frequency components. These can also be simulated based on the PTSM and then added to the seed signal pulse. Furthermore, amplified spontaneous emission (ASE) noise must also be considered, which can be approximated as Gaussian white noise with an average power of P_noise. Based on the incident seed pulse of each mode with a specific spectral width, average power, and central wavelength obtained via the PTSM, first, the gain coefficient of each mode was obtained using the NSM-CWHPFL model, including mode competition and fiber coiling. Subsequently, through the introduction of the incident seed pulse of each mode and the corresponding gain coefficients into the GMMNLSEs, the evolution of the time-frequency characteristics for each mode could be solved to obtain the output spectrum.

2.4 Mode mixing model in multimode fiber

In addition to the spectral characteristics, the output beam quality has attracted considerable attention. For the high-power regime, we chose the commonly used factor M² to evaluate beam quality, which is defined as the ratio of the divergence of the beam to that of an imaginary fundamental Gaussian beam such that the two beams have the same second-order intensity moment at the waist [38]. For MMF-based fiber lasers, the HOM content is inevitably present in the output laser beam. The degradation of the beam quality can be analyzed based on the mode-mixing theory of multimode fibers [38].

Based on the mode mixing theory, the normalized electric field amplitude distribution of mode mixing at the exit fiber facet can be expressed as $E(x,y) = \sum {\sqrt {{\eta _{{m_k}}}} } {E_{{m_k}}}(x,y) \cdot {e^{i{\Psi _{{m_k}}}}}$, where ${\eta _{{m_k}}}$, ${E_{{m_k}}}(x,y)$, and ${\Psi _{{m_k}}}$ are the power ratio, electric field distribution and phase for mode m_k, respectively. Using a scanning-slit device to analyze the mixing field diverging of LP_mn modes in free space, with the silt parallel to the y-axis, the $M_x^2$ calculation formula can be expressed as follow. Similarly, $M_y^2$ can be obtained and M² can be calculated using the relationship ${M^2} = \sqrt {M_x^2M_y^2}$.

2.5 Intermodal-nonlinearity-induced time-frequency evolution model of NSM-CWHPFLs

Based on the models presented in the previous Section, an intermodal-nonlinearity-induced time-frequency evolution model was proposed for the NSM-CWHPFLs. The model incorporated the seed simulation model based on the polarization thermal source theory, the NSM-CWHPFL model including mode competition and fiber coiling, the GMMNLSE model including polarization characteristic correction and Raman response function optimization, and the linear polarization mode mixing model, as shown in Fig. 1.

 

Fig. 1. Schematic of the intermodal-nonlinearity-induced time-frequency evolution model of NSM-CWHPFLs.

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First, the incident seed pulse $\widetilde {A_{{m_k}}^{in}}(\omega )$ with a specific spectral width, average power, and central wavelength was obtained using the PTSM seed simulation model. Thereafter, the corresponding gain coefficient ${g_{{m_k}}}(z,\omega )$ was calculated based on the NSM-CWHPFL model. Subsequently, through the introduction of $\widetilde {A_{{m_k}}^{in}}(\omega )$ and ${g_{{m_k}}}(z,\omega )$ into the GMMNLSE model, the evolution of the time-frequency characteristics for mode m_k were simulated. Through the application of the MPSSA algorithm presented above within the Yb-doped fiber (YDF) with a length of ${L_{MM - YDF}}$, the output pulse $\widetilde {A_{{m_k}}^{}}({L_{MM - YDF}},\omega )$ of mode m_k was obtained at the end of the YDF. Similarly, for a passive multimode fiber (PMMF) with a length of ${L_{PMMF}}$, while ignoring the gain term of the GMMNLSEs and considering only the loss term ${\Theta ^{({m_k})}}(z,t)$, the MPSSA algorithm was applied to obtain the frequency-domain amplitude of the output pulse $\widetilde {A_{{m_k}}^{out}}(\omega )$ of mode m_k. Thereafter, based on the time-domain amplitude $A_{{m_k}}^{out}(t) = {{{\cal {\boldsymbol F}}}^{ - 1}}[{\widetilde {A_{{m_k}}^{out}}(\omega )} ]$, the power propagating in mode m_k was calculated using the relationship ${P_{{m_k}}}(t) = {|{A_{{m_k}}^{out}(t)} |^2}$, in units of W. Finally, through the introduction of the output pulse of each mode into the linear polarization mode mixing model, the speckle pattern and M² of the mixing field of the LP_mn modes were obtained.

Moreover, to analyze the spectral characteristics, first, the output spectrum of mode m_k by ${P_{{m_k}}}(\lambda ) = {|{\widetilde {A_{{m_k}}^{out}}(\lambda )} |^2}$ and the total spectrum $P(\lambda )$ were obtained via the numerical summation of the spectra of different modes. Then, taking the IMFWM as an example, based on the Stokes and anti-Stokes sidebands induced by the IMFWM, the nonlinear power ratio can be obtained by $IMFWM\;ratio = 10 \cdot {\log _{10}}\left\{ {{{\int_{{\lambda_{IMFWM}}} {P(\lambda )d\lambda } } / {\int_{{\lambda_{\min }}}^{{\lambda_{\max }}} {P(\lambda )d\lambda } \int {} }}} \right\}$, where $\lambda _{IMFWM}^{}$ represents the wavelength ranges of the IMFWM-Stokes and IMFWM-anti-Stokes bands, and $\lambda _{\min }^{}$ and $\lambda _{\max }^{}$ are the wavelength ranges in the simulation, which are determined by the time extent of the time window ${T_{windows}}$ and the number of time grid points N. Similarly, the power ratios of other nonlinear sidebands were obtained, such as SRS and MMRS-IMFWM.t.

In the MPSSA algorithm, two longitudinal step sizes were defined. A large step size h was divided into M small steps Δh to compute the integrals in parallel, that is, h = M·Δh, where M represents the extent of parallelization. The computation process of each integral in parallel and the summation over a small step via the trapezoid rule were repeated iteratively to obtain a more accurate approximation solution. When the error was smaller than a particular tolerance, the process converged. In addition to the number of iterations n_ite, the computational accuracy is also dependent on the step size h, which is determined by the nonlinear length $h \ll {L_{NL}} = c{A_{eff}}/{n_2}{\omega _0}{P_{peak0}}$, where ${P_{peak0}}$ is the peak power of the initial pulse and ${A_{eff}}$ is the effective mode area. In practice, $h \approx 0.1 \cdot {L_{NL}}$ and ${n_{ite}} = 4$ are sufficient to match the relative error requirement, which is less than 1e^-5. To achieve a high degree of parallelization, n_ite should be significantly smaller than M. In the following simulations for high-power applications, $M = 10$, $h = 0.05$, and ${n_{ite}} = 4$ were sufficient to satisfy the requirements of computational accuracy and parallel speedup of the MPSSA algorithm. Based on the time-frequency evolution model of the NSM-CWHPFLs, the pulses of different modes and fiber positions were saved to simulate the evolution process of the spectral characteristics and M² under the combined action of nonlinear effects. Consequently, the influence of intermodal nonlinear effects on high-power fiber lasers was analyzed.

Moreover, in practical HPFL systems, the phenomenon of transverse mode instability (TMI) occurs beyond a certain threshold pump power, resulting in a sharp degradation of the output beam quality and a significant increase in the M², thus leading to a significant difference between the theoretical simulation and experimental results at high power levels [3,40]. To incorporate the influence of TMI-induced mode degradation in the simulation and further improve the accuracy of the theoretical model at high powers, the mode-coupling coefficients d_ij in Eq. (6) of Ref. [35] of the NSM-CWHPFL model need to be modified as the pump power varies, to approximately characterize the power transfer from the FM to the HOM caused by TMI.

3. Simulation and analysis

Prior to performing the simulation of intermodal nonlinear effects based on the proposed theoretical model, the frequency-shift calculation formulae for intermodal nonlinearities, such as IMFWM and MMRS-IMFWM were derived. They were then used as supporting evidence combined with the simulation results of different mode combinations to determine the positions and generation mechanisms of various intermodal nonlinear spectral sidebands. Subsequently, the influence of the fiber laser parameters, such as fiber length, pump power, pump direction, and seed spectral width, on the intermodal nonlinear effects in the CWHPFLs was analyzed. In addition, considering the influence of mode control, which is achieved through the combination of fiber coiling and seed mode characteristic optimization in the NSM-CWHPFLs to ensure a near-single-mode laser output, the influence of the HOM power ratios of the seed light and bend radii of the YDF in the amplifier on the nonlinear power ratio and M² were simulated.

3.1 IMFWM and MMRS-IMFWM frequency shifts

The IMFWM process is a four-wave mixing process involving different modes, wherein two photons are annihilated to produce a low-frequency Stokes photon and high-frequency anti-Stokes photon. The occurrence of the IMFWM process must satisfy ${\omega _c} + {\omega _d} = {\omega _a} + {\omega _b}$ and the phase-matching condition $\Delta \beta _{abcd}^{}\textrm{ = }0$. The fiber mode combinations of the IMFWM can be separated into two cases: the two incident waves in different fiber modes (Case 1) and those in the same fiber modes (Case 2), corresponding to different mode overlap coefficients. Considering the coefficients between the two modes, denoted as m_g and m_h, $Q_{1122}^{(1)}$ and $Q_{1212}^{(1)}$ correspond to Case 1 and $Q_{1221}^{(1)}$ corresponds to Case 2. Considering $Q_{1122}^{(1)}$ as an example, the IMFWM phase-matching condition is $\beta _{}^{({m_g})}({\omega _c}) + \beta _{}^{({m_h})}({\omega _d}) - \beta _{}^{({m_g})}({\omega _a}) - \beta _{}^{({m_h})}({\omega _b}) = 0$. Assuming $\Delta \omega = {\omega _a} - {\omega _c} = {\omega _d} - {\omega _b}$ and expanding the propagation constants $\beta _{}^{({m_g})}({\omega _c})$ and $\beta _{}^{({m_h})}({\omega _d})$ in Taylor series around the frequencies ω_a and ω_b while ignoring high-order terms, the approximate calculation formula of the IMFWM frequency shift is expressed as follows:

Considering a 25/400 fiber with a numerical aperture (NA) of 0.06/0.46 as an example, the fiber supports four modes (LP₀₁, LP₁₁, LP₂₁, and LP₀₂), wherein LP₁₁ includes LP_11e and LP_11o, and LP₂₁ includes LP_21e and LP_21o. When the light wavelength was 1080 nm, the first-order dispersion parameters of four modes for 25/400 fibers were 0, 0.47, 0.86, and −0.21 fs/mm, respectively, and the second-order dispersion parameters are 15.09, 15.33, 17.05, and 23.86 fs²/mm, respectively. Using Eq. (10), as in $\beta _1^{11}({\omega _0}) > \beta _1^{01}({\omega _0})$, the IMFWM frequency shift for Case 1 was negative; that is, the IMFWM-Stokes wave at the low frequency was in the LP₁₁ mode, and the IMFWM-anti-Stokes wave at the high frequency was in the LP₀₁ mode. This verified the significant mode differences of the intermodal-nonlinearity spectral sidebands. Moreover, the IMFWM corresponding to Case 2 was not observed in high-power fiber lasers [15], which is neglected in the following simulation. The IMFWM frequency shifts corresponding to different fiber mode combinations are listed in Table 1, and IMFWM Stokes wavelength λ₃ and anti-Stokes wavelength λ₄ were calculated according to the frequency shift.

Similarly, the MMRS-IMFWM refers to an IMFWM process between Raman-Stokes waves in different modes, and thus must satisfy the phase matching condition. Using Eq. (10), the center frequency ω₀ was changed to the Raman-Stokes frequency ω_SRS, which was downshifted from ω₀ by approximately 13.2 THz, to obtain the MMRS-IMFWM frequency shifts. For example, when the light wavelength was 1080 nm, the Raman-Stokes wavelength was approximately 1135 nm. According to the dispersion parameters of the 25/400 fiber at 1135 nm, the MMRS-IMFWM frequency shifts are also listed in Table 1. As $\beta _1^{11}({\omega _{SRS}}) > \beta _1^{01}({\omega _{SRS}})$, the Stokes and anti-Stokes waves of the MMRS-IMFWM were in the LP₁₁ and LP₀₁ mode, respectively, which verified that significant mode differences were also found in the MMRS-IMFWM. In addition, the occurrence of the intermodal nonlinearities must also satisfy the condition that the nonlinear length should be smaller than the walk-off length. The walk-off lengths between two modes were calculated using the relationship $L_W^{gh} = {\tau _0}/|{\beta_1^{({m_g})} - \beta_1^{({m_h})}} |$. The results are presented in Table 1. In practice, the nonlinear length is approximately 0.5 m and is greater than the walk-off lengths of different mode combinations; thus, these nonlinearities can be observed in high-power fiber lasers.

3.2 Intermodal nonlinear effects of different mode combinations

To verify the accuracy of the frequency shift calculation formula of intermodal nonlinear effects, considering the IMFWM as an example, the IMFWM corresponding to different fiber mode combinations was simulated based on the proposed theoretical model. The parameters used in the simulations are listed in Table 2.

Table 2. Parameters used in the simulation of intermodal nonlinear effects

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It was assumed that the total seed light power was 100 W with a power proportion of the two modes of 1:1. For example, for the IMFWM between the LP₀₁ and LP₁₁ modes, denoted as IMFWM@LP₀₁+ LP₁₁, the power of LP₀₁ mode was 50 W, that of the LP_11e and LP_11o modes were 25 W, and that of other modes were 0.1 W. Subsequently, the IMFWM corresponding to different mode combinations were simulated by changing the mode power proportion of the incident seed and choosing the corresponding mode overlap coefficients. Figure 2 shows the simulation results for the output spectra and speckle patterns. For the six-modes mixing simulations, the phase values of the LP₀₁, LP_11e, LP_11o, LP_21e, LP_21o, and LP₀₂ modes were specified as 0, 0.16π, 0.57π, 0, 0.16π, and 0.56π, respectively. To maintain consistency and mitigate potential confounding effects that may arise from changes in phases, these phase values were kept constant throughout subsequent six-modes mixing simulations.

 

Fig. 2. Theoretical model simulation results of the IMFWM of different mode combinations: (a) LP₀₁+ LP₁₁; (b) LP₀₁+ LP₂₁; (c) LP₁₁+ LP₂₁; (d) LP₀₁+ LP₀₂; (e) LP₁₁+ LP₀₂; (f) LP₂₁+ LP₀₂.

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These simulation results indicate that the IMFWM frequency shifts of different mode combinations are different, and the IMFWM-Stokes and IMFWM-anti-Stokes peaks at different wavelengths were observed in the output spectra. The consistency between the wavelengths of the IMFWM peaks shown in Fig. 2 and the calculation results in Table 3 verified the accuracy of Eq. (10) and clarified the positions of various IMFWM sidebands. This provided a theoretical basis for determining the generation mechanism of the intermodal nonlinear sidebands. In addition, the IMFWM power ratios of the different mode combinations were different, and the power ratio of IMFWM@LP₀₁+ LP₁₁ was the highest.

3.3 Intermodal nonlinear effects in NSM-CWHPFLs

Several signal lights in different modes are transmitted and amplified in the NSM-CWHPFLs; therefore, the combined action of intermodal nonlinear effects along with intramodal nonlinearities such as SRS, SPM, XPM, and FWM must be considered. Based on the proposed theoretical model, the evolution of the spectral characteristics and M² of the 25/400 fiber-based NSM-CWHPFL was simulated. It was assumed that the seed light powers of the six modes (LP₀₁, LP_11e, LP_11o, LP_21e, LP_21o, and LP₀₂) were 50, 25, 25, 25, 25, and 50 W, respectively. When the lengths of the MM-YDF and PMMF were 15 and 5 m, respectively, the simulation results for pump powers of 1000, 1500, 2000, 2500, 3000, and 3500 W, are shown in Fig. 3. Considering strong fluctuations in the simulated spectra, we smoothed the simulation results based on the moving filter to be better observed.

 

Fig. 3. Simulation results of the combined action of intermodal and intra-mode nonlinear effects in the 25/400 fiber-based NSM-CWHPFL: (a) output spectra of each mode at 3500 W; (b) spectral evolution diagram at 3500 W; (c) output spectrum vs pump power; (d) nonlinear power ratio vs pump power & fiber length.

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Figure 3(a) shows the output spectra of each mode when the pump power was 3500 W. The intermodal nonlinear effects included the IMFWM, SRS-IMWM, and MMRS-IMFWM. The new spectral sidebands caused by the IMFWM included IMFWM sidebands of LP₀₁+ LP₁₁ at 1098.6 and 1061.9 nm, those of LP₀₁+ LP₂₁ at 1112.4 and 1048.4 nm, and those of LP₁₁+ LP₂₁ at 1092.5 and 1065.6 nm. Among the IMFWM sidebands, the power ratio of IMFWM@LP₀₁+ LP₁₁ was the highest. For each mode, the SRS-IMWM effect contributed to the Raman-Stokes light at 1135 nm; however, the contribution of the SRS-IMWM was much smaller than that of the intramodal SRS. Thus, the Raman-Stokes light at 1135 nm was primarily generated via intramodal SRS. In addition, with increase in the pump power, the MMRS-IMFWM effect resulted in the generation of new sidebands at 1160 and 1150 nm. The sideband at 1160 nm corresponded to second-order Stokes light in the LP₁₁ mode generated by the IMFWM process of the Raman Stokes light in the LP₀₁ and LP₁₁ modes. In contrast, that at 1150 nm corresponded to the second-order Stokes light in the LP₂₁ mode generated via MMRS-IMFWM@LP₀₁+ LP₂₁. Furthermore, with an increase in the fiber length and pump power, the intermodal and intramodal nonlinear effects were enhanced, and the power ratios of the IMFWM, SRS, and MMRS-IMFWM sidebands increased significantly.

According to the theoretical model, the pump direction and seed spectral width also affect nonlinear effects. Figure 4(a) shows the simulation results of the bidirectional pump, assuming that the co-pump and counter-pump power ratios were 1:1. Compared with the co-pump, the power distribution along the fiber axis was more uniform for the bi-pump, resulting in a significant decrease in the power ratios of the IMFWM, SRS, and MMRS-IMFWM sidebands. Figures 4(b) and 4(c) show the simulation results when the seed spectral widths were 0.1, 0.2, 0.4, 0.8, 1.6, and 3.2 nm, respectively. An increase in the seed spectral width resulted in an increase in the time-domain power fluctuation, thereby shortening the pulse spikes. Because of the walk-off caused by the group velocity mismatch between the signal and nonlinear lights, maintaining the temporal overlap for short pulses is more challenging than for long pulses. Therefore, an increase in the seed spectral width weakens the nonlinear effects, thereby resulting in a decrease in the power ratios of these nonlinear sidebands.

 

Fig. 4. Influence of pump direction and seed spectral width on intermodal and intramodal nonlinear effects: (a) comparison of nonlinear power ratio between bi-pump and co-pump; (b) output spectrum vs seed spectral width; (c) nonlinear power ratio vs seed spectral width.

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3.4 Influence of mode control on intermodal nonlinear effects

Mode control technology is commonly used in NSM-CWHPFLs to achieve near-single-mode laser output. The mode control method mainly includes fiber coiling and seed mode characteristic optimization. Fiber coiling suppresses HOMs by increasing bend losses. Seed-mode characteristic optimization reduces the HOM power ratio of the seed light through mode control of the seed oscillator or fiber coiling of the incident passive fiber in fiber amplifiers. Because intermodal nonlinearities involve different modes, mode control improves the output beam quality while affecting the nonlinear effects. Therefore, the influence of the HOM power ratios of the seed light and bend radii of the YDF in the amplifier on the nonlinear power ratio and M² were analyzed. Considering the intermodal nonlinear effects between the LP₀₁ and LP₁₁ modes as an example, the simulation results when the HOM ratios of the seed light were 10, 25, 50, 75, and 90% are shown in Fig. 5.

 

Fig. 5. Influence of HOM ratio of the seed light on intermodal nonlinear effects @LP₀₁+ LP₁₁: (a) IMFWM ratio; (b) M²; (c) SRS ratio; (d) MMRS-IMFWM ratio.

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The IMFWM power ratio was highest when the HOM ratio was 50%. The increase and decrease in the HOM ratio both resulted in a decrease in the IMFWM power ratio. In contrast, the SRS power ratio increased slightly. Similar to the IMFWM, the MMRS-IMFWM power ratio was the highest when the HOM power ratio was 50%, and an increase in the HOM power ratio resulted in a significant decrease in the MMRS-IMFWM power ratio. In addition, an increase in the HOM power ratio resulted in a significant increase in M². Figure 6 shows the simulation results when the bend radii of the YDF in the amplifier were 35, 40, 45, 50, 75, and 100 mm. With the gradual decrease in the bend radius, the IMFWM power ratio and M² gradually decreased. However, for the 25/400 fiber, the bend radius must be reduced to 35 mm to effectively suppress IMFWM@LP₀₁+ LP₁₁ at the cost of reducing the total power and system stability. Therefore, for the 25/400 fiber-based MOPA, the intermodal nonlinear effects cannot be completely suppressed by fiber coiling and must be further optimized in combination with other methods.

 

Fig. 6. Influence of bend radius on intermodal nonlinear effects @LP₀₁+ LP₁₁: (a) output spectra; (b) IMFWM ratio; (c) M².

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The simulation results for the intermodal nonlinearities @LP₀₁+ LP₂₁ are shown in Figs. 7 and 8. The IMFWM and MMRS-IMFWM were the strongest when the HOM ratio was 50%. An increase in the HOM ratio resulted in a decrease in the SRS ratio and an increase in M². With decrease in the bend radius, the power ratio of IMFWM@LP₀₁+ LP₂₁ gradually decreased, and that of M² decreased significantly.

 

Fig. 7. Influence of HOM ratio of the seed light on intermodal nonlinear effects @LP₀₁+ LP₂₁: (a) IMFWM ratio; (b) M²; (c) SRS ratio; (d) MMRS-IMFWM ratio.

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Fig. 8. Influence of bend radius on intermodal nonlinear effects @LP₀₁+ LP₂₁: (a) output spectra; (b) IMFWM ratio; (c) M².

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These simulation results theoretically validated the effectiveness of suppressing intermodal nonlinear effects based on fiber coiling and seed mode characteristic optimization to provide theoretical guidelines for the comprehensive optimization of spectral distortion and mode degradation caused by intermodal nonlinear effects. In addition, the simulation results for LP₀₁+ LP₂₁ were different from those for LP₀₁+ LP₁₁. With an increase in the pump power, M²@LP₀₁+ LP₁₁ increased, whereas M²@LP₀₁+ LP₂₁ decreased, owing to the difference in the combined action of mode competition and intermodal nonlinearities. IMFWM@LP₀₁+ LP₁₁ was obviously stronger than IMFWM@LP₀₁+ LP₂₁, and the gain of LP₁₁ was smaller than that of LP₂₁, thus enhancing the IMFWM-induced mode degradation and increasing M².

Additionally, based on the comparison between Figs. 6(c) and 8(c), it is observed that the beam quality of LP₀₁+ LP₂₁ mode is superior to that of LP₀₁+ LP₁₁ mode when the bending radius is set to 100 mm, which is due to the fact that the bending loss of LP₂₁ mode is significantly higher than that of the LP₁₁ mode at the same bending radius. Consequently, the LP₂₁ mode experiences a greater reduction in gain coefficient due to the higher bending loss, causing the output mode component to be predominantly in the fundamental mode.

4. Experiments

To verify the accuracy of the theoretical model, MOPA systems based on 20/400, 25/400, and 30/600 fibers, respectively were established for the verification experiments.

4.1 Experiment on intermodal nonlinear effects in 20/400 fiber-based MOPA system

A schematic of the fiber laser configured using the MOPA scheme based on the 20/400 fiber and the parameters of the fiber devices we had provided in Ref. [15]. When the pump power was ∼170 W, the output power of the seed laser was ∼90 W with a spectral width of ∼0.0775 nm and M² of ∼1.25. As the light wavelength was 1090 nm, the 20/400 fiber supported two modes (LP₀₁ and LP₁₁), wherein LP₁₁ included LP_11e and LP_11o, with the second-order dispersion parameters of two modes being 14.36 and 15.3 fs²/mm. Using the mode-mixing model, the seed light powers of the LP₀₁, LP_11e, and LP_11o mode were estimated to be 80, 5, and 5 W, respectively. Based on the experimental system parameters, the evolution processes of the spectral characteristics and beam quality under different pump powers were obtained using the theoretical model and then compared with the experimental results. Further, the output spectrum was measured using an optical spectrum analyzer (OSA; YOKOGAWA, AQ6370C) and M² was measured using a beam quality analyzer (Ophir Photonics, M²-200s). Figure 9 shows the simulation and experimental results for the output spectra at different pump powers. For the three-modes mixing simulation of the 20/400 fiber, the phase values of the LP₀₁, LP_11e, and LP_11o modes were specified as 0, 0.48π, and 0.16π, respectively.

 

Fig. 9. Output spectra under different pump powers in the 20/400 fiber-based MOPA system: (a) pump power of 1900 W; (b) pump power of 2280 W; (c) pump power of 2660 W.

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Considering the influence of the measurement error and the approximation of the theoretical model, it was concluded that the experimental results of the output spectra were consistent with the simulation results obtained using the theoretical model, thus verifying its accuracy. In addition, the physical mechanism of the various nonlinear spectral sidebands observed in the experimental spectra were determined according to the simulation results. As shown in Fig. 10(a), the sidebands at 1109 and 1070 nm corresponded to IMFWM@LP₀₁+ LP₁₁. Using Eq. (10), the calculated IMFWM frequency shift for the 20/400 fiber was 5.03 THz, which is consistent with the experimental and simulation results. Further, the wide-bandwidth sideband near 1145 nm was the Raman-Stokes sideband, which primarily included the contribution of the intramodal SRS effect with a small SRS-IMWM contribution. When the pump power was further increased, a new sideband at 1169 nm was observed in the output spectrum, which was the second-order Stokes sideband in LP₁₁ generated by MMRS-IMFWM@LP₀₁+ LP₁₁. The calculation result of the MMRS-IMFWM frequency shift was 5.42 THz, which is consistent with the experimental and simulation results.

 

Fig. 10. Comparison of experimental and simulation results in the 20/400 fiber-based MOPA system: (a) output spectra of three mode at pump power of 2660 W; (b) comparison of nonlinear power ratio; (c) comparison of M².

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Figures 10(b) and 10(c) show a comparison of the experimental and simulation results for the IMFWM, SRS, MMRS-IMFWM power ratio, and M². When the pump power exceeded 2000 W, M² increased, and the beam quality deteriorated significantly, which is not completely consistent with the simulation results. This is because in an actual laser system, TMI occurs when the pump power exceeds a threshold [3,40]. Therefore, to simulate an actual laser system, the mode-coupling coefficient d_ij in Eq. (6) of Ref. [35] of the NSM-CWHPFL model must be modified to characterize the power transfer from the FM to the HOM caused by TMI. The mode coupling coefficient is related to many factors such as the stress and temperature [35,39]. Assuming that the mode coupling coefficient d_ij between the adjacent modes is the average mode coupling coefficient d, it is difficult to determine the coefficient d, and it can only be obtained by an experimental measurement. Generally, d = 0.001 or 0.01 m⁻¹ [35,43]. When the pump powers were 1900, 2280, and 2660 W, the TMI-correction mode coupling coefficient d_ij were set to 0.005, 0.01, and 0.025, respectively, thus incorporating the influence of TMI-induced mode degradation in the simulation.

These simulation and experimental results indicate that the theoretical model simulation, frequency shift calculation, and experimental (spectral sideband positions, IMFWM, SRS, MMRS-IMFWM power ratio, and M²) results were consistent. Considering the M² measurement error and the theoretical model approximation, this consistency verified the accuracy of the theoretical model and frequency-shift calculation formula. Further, the physical mechanisms of various nonlinear spectral sidebands observed in the experimental spectra were clarified, thus providing a more comprehensive theoretical analysis of the spectral distortion and mode degradation caused by intermodal nonlinear effects.

4.2 Experiment on intermodal nonlinear effects in 25/400 fiber-based MOPA system

The schematic and fiber device parameters of the 25/400 fiber-based MOPA system we had provided in Ref. [35]. When the pump power was ∼470 W, the output power of the seed laser was ∼300 W. The seed light was transmitted through the 25/400 passive fiber into the amplifier stage and excited HOMs with an M² of ∼1.5. Using the mode mixing model, the seed light powers of six modes were estimated as 170, 40, 60, 15, 15, and 0.1 W. Figure 11 shows the experimental and simulation results of output spectra at the pump power of 3265 W.

 

Fig. 11. Experimental and simulation results of output spectra in the 25/400 MOPA system at 3265 W: (a) comparison of experimental and simulation results; (b) output spectra of six mode.

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As shown in Fig. 11(b), the sidebands at 1099.1 and 1061.6 nm were caused by IMFWM @LP₀₁+ LP₁₁, those at 1092.1 and 1065.7 nm were caused by IMFWM@LP₁₁+ LP₂₁, and those at 1112.7 and 1048.2 nm corresponded to IMFWM@LP₀₁+ LP₂₁. Among the IMFWM sidebands, the IMFWM power ratio of LP₀₁+ LP₁₁ was the highest. The wide-bandwidth sideband near 1135 nm was the Raman-Stokes sideband with contributions from SRS and SRS-IMWM. The sideband at approximately 1160 nm was the second-order Stokes sideband in LP₁₁ generated by MMRS-IMFWM@LP₀₁+ LP₁₁, and that at approximately 1170 nm was the second-order Stokes sideband in LP₂₁ generated by MMRS-IMFWM@LP₀₁+ LP₂₁.

Figure 12 shows a comparison of the experimental and simulation results for the nonlinear power ratio and M² under different pump powers in the 25/400 fiber-based MOPA system. Considering the TMI-induced mode degradation in the simulation, the TMI-correction mode coupling coefficient d_ij was set to 0.0025, 0.005, 0.01, 0.02, and 0.03. The consistency between the simulation, calculation, and experimental results verified the accuracy of the theoretical model. In addition, the experimental results of M² are slightly larger than the simulation results because of the mode distortion caused by fiber coiling, stress, temperature, etc. Thus, considering the influence of the mode distortion, the simulation results of M² must be multiplied by a mode-distortion correction factor, as indicated by the red dotted line in Fig. 12(b).

 

Fig. 12. Comparison of experimental and simulation results in the 25/400 fiber-based MOPA system: (b) nonlinear power ratio; (c) M².

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Subsequently, the mode control method was applied to verify the effectiveness of the intermodal nonlinear effect suppression based on fiber coiling and seed mode characteristic optimization. The bend radius of the YDF in the amplifier stage was reduced from 100 to 50 mm, and the input signal PMMF of the pump-signal combiner (PSC) was coiled with a bend radius of ∼40 mm to optimize the mode characteristics of the seed light incident on the amplifier. Figure 13 shows a comparison of the simulation and experimental results before and after the fiber coiling. The IMFWM power ratio and M² were significantly reduced, thereby verifying the effectiveness of the comprehensive optimization of the spectral distortion and mode degradation caused by intermodal nonlinear effects based on fiber coiling combined with seed mode characteristic optimization.

 

Fig. 13. Comparison of experimental and simulation results before and after fiber coiling in the 25/400 fiber-based MOPA system: (a) IMFWM power ratio; (b) M².

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Finally, the gain fiber in the amplifier stage was changed to 30/600 YDF, with a numerical aperture of 0.06/0.46, to analyze the influence of the fiber core diameter on the intermodal nonlinear effects. Considering the higher cost and increased fiber coiling difficulty resulting from its thicker cladding of the 30/600 fiber compared to the 25/400 fiber, the 30/600 gain fibers were coiled on a water-cooled plate engraved with a circular spiral groove [44]. Figure 14 shows a comparison of the simulation and experimental results for 25/400 and 30/600 fibers. The simulation results of the 30/600 fiber were also obtained via the proposed theoretical model under the identical simulation parameter configurations. The IMFWM and SRS power ratios of the 30/600 fibers were significantly lower than those of the 25/400 fibers. According to the theoretical model, an increase in the fiber core diameter leads to an increase in the mode field area and a decrease in the mode overlap coefficient, thereby resulting in a decrease in the nonlinear power ratio. Therefore, the use of fibers with larger core diameters can effectively suppress the intermodal and intramodal nonlinear effects.

 

Fig. 14. Comparison of experimental and simulation results between the 25/400 and 30/600 fibers: (a) IMFWM power ratio; (b) SRS power ratio; (c) M².

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Furthermore, the differences in the M² factor between the experimental and simulation results at high power levels were more significant for the 30/600 fiber compared to the 25/400 fiber. This disparity was attributed to the increasing bending radius of the 30/600 gain fibers caused by circular spiral groove coiling, leading to a decrease in mode control effectiveness when compared to the coiling method with a constant bending radius [35,44]. In addition, as the core radius of gain fibers increases, the HOM ratio and the number of modes further increase, which can exacerbate the errors caused by both the mode mixing model and stress-induced mode distortions. Consequently, to improve the accuracy of the theoretical model, we plan to incorporate mode spatial distribution characteristics into the framework, aiming at refining the simulation accuracy of beam profile evolutions.

5. Conclusions

In this study, we proposed an intermodal-nonlinearity-induced time-frequency evolution model of NSM-CWHPFLs, which combined the NSM-CWHPFL model including mode competition and fiber coiling, the GMMNLSE model including polarization characteristic correction and Raman response function optimization, and the linear polarization mode mixing model. Based on the theoretical model, the evolution of the spectral characteristics and M² under the combined action of intermodal nonlinear effects (IMFWM, SRS-IMWM, and MMRS-IMFWM) and intramodal nonlinear effects (SRS, SPM, XPM, and FWM) in the NSM-CWHPFLs were simulated for the first time to the best of the authors’ knowledge. Further, the frequency-shift calculation formulae of the intermodal nonlinearities were derived and then used as supporting evidence combined with the simulation results to determine the positions and generation mechanisms of various intermodal nonlinear spectral sidebands. Moreover, the influence of fiber laser parameters, such as fiber length, pump power, pump direction, seed spectral width, HOM power ratio of the seed light, and bend radii of the YDF in the amplifier on the nonlinear power ratio and M² were simulated. Consequently, a suppression method for the intermodal nonlinear effects based on fiber coiling and seed mode characteristic optimization was proposed. Furthermore, verification experiments were conducted using 20/400, 25/400, and 30/600 fiber-based MOPA systems. Under different pump powers and fiber core diameters, the consistency of the theoretical model simulation, frequency-shift calculation, and experimental results verified the accuracy of the theoretical model and frequency-shift calculation formula. The physical mechanisms of various observed nonlinear spectral sidebands were clarified based on these results, thereby providing a more comprehensive and in-depth understanding of the intermodal nonlinearities. Finally, the mode control method was applied to the 25/400 fiber-based MOPA to verify the suppression of intermodal nonlinear effects based on fiber coiling and seed mode characteristic optimization. Consequently, a comprehensive optimization of intermodal-nonlinearity-induced spectral distortion and mode degradation was realized.

In addition, based on the physical mechanism of intermodal-nonlinearity spectral sidebands and the significant mode differences of the sidebands, a new method of realizing laser sources at special wavelengths in pure FM or specific HOM via the application of intermodal nonlinearities was determined. In addition, the proposed theoretical model established a relationship between the nonlinear power ratio and M², thus theoretically verifying the possibility of determining the HOM ratio based on the intermodal nonlinear effects. Consequently, this method can be used for mode decomposition. In future studies, considering the differences between the experimental and simulation results, we plan to further study the improvement of the theoretical model, such as combining the TMI model based on stimulated thermal Rayleigh scattering [41,42], incorporating mode spatial distribution characteristics, etc., thereby further improving its accuracy. Furthermore, the application of new spectral sidebands caused by intermodal nonlinear effects will be explored, and the mode decomposition based on intermodal nonlinearities will be investigated and validated in comparison with conventional mode decomposition methods such as the spatially and spectrally resolved imaging (S²), the stochastic parallel gradient descent (SPGD) algorithm, and deep learning method [45–50].