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Accurate modeling of ultrafast nonlinear pulse propagation in multimode gain fiber

Yi-Hao Chen, Henry Haig, Yuhang Wu, Zachary Ziegler, and Frank Wise

Open Access Open Access

Abstract

The nonlinear propagation of picosecond or femtosecond optical pulses in multimode fiber amplifiers underlies a variety of intriguing physical phenomena as well as the potential for scaling sources of ultrashort pulses to higher powers. However, existing theoretical models of ultrashort-pulse amplification do not include some critical processes, and, as a result, they fail to capture basic features of experiments. We introduce a numerical model that combines steady-state rate equations with the unidirectional pulse propagation equation, incorporating dispersion, Kerr and Raman nonlinearities, and gain/loss-spectral effects in a mode-resolved treatment that is computationally efficient. This model allows investigation of spatiotemporal processes that are strongly affected by gain dynamics. Its capabilities are illustrated through examinations of amplification in few-mode gain fiber, multimode nonlinear amplification, and beam cleaning in a multimode fiber amplifier.

© 2023 Optica Publishing Group

1. INTRODUCTION

Optical fibers are building blocks of various applications, including telecommunications [1–4], remote sensing [5–7], and fiber lasers [8–11]. Among these, multimode fibers serve as a platform for studying the spatiotemporal dynamics of optical waves, showcasing various intriguing nonlinear effects, such as spatiotemporal mode-locking [12–20] and instability [21,22], spatial beam cleaning [23–28], multimode soliton formation [29,30] and its self-frequency shift [31], and multimode dispersive-wave [32,33] and supercontinuum generation [34–37], to name just a few.

Despite the abundant physics accessible with multimode fibers, research on nonlinear multimodal effects primarily focuses on passive fibers. Only a few investigations have been conducted on multimode gain fibers. One particular area of interest is modal gain equalization for telecommunications, which relies on the control of signal or pump modal distributions to reduce differential modal gain [38–40]. In addition, studies have been carried out to examine mode competition in large-core gain fibers with index anti-guiding [41]. By employing gain guidance and appropriate core sizing, it becomes possible to selectively amplify only the fundamental mode. However, multimodal nonlinear effects are neglected in these studies. Nonlinear modal coupling in a gain fiber can play an important role in several physical phenomena, such as spatiotemporal mode-locking and beam cleaning. Multimode gain fiber in a spatiotemporal laser enables the generation of multimode dissipative solitons [12,13] and multimode soliton molecules [15]. In addition, the amplification process facilitates beam self-cleaning by increasing the peak power [24] and enables the generation of multimode supercontinuum [42] and amplified ultrashort pulses [28].

Modeling the amplification process in a multimode gain fiber plays a crucial role in the comprehensive understanding of underlying nonlinear dynamics. Because the small-signal gain in a rare-earth-doped fiber has a bell-shaped frequency dependence near the gain peak, it is customary to use a Gaussian function with an additional gain-saturation term to model the amplification. The single-mode version of this model was first introduced for Er-doped fiber amplifiers [43]. It was later generalized to the multimode version by considering the gain at each spatial point of an optical field [12–14]. However, this simplified model fails to capture several phenomena that can be essential in spatiotemporal dynamics, such as pump depletion, spatial gain saturation, and the variation of the gain spectrum with saturation level [44]. In fact, all of these processes in a rare-earth-doped fiber amplifier can be accurately described by rate equations that govern the population inversion, signal and pump fields, and amplified spontaneous emission (ASE). Both single-mode [45–51] and multimode models that incorporate spatial gain saturation [39,52] have been developed. Although modal coupling has already been introduced in the Gaussian-gain framework, only a few works considered modal coupling in a multimode gain fiber in the rate-equation framework. Modal coupling was first considered by Gong et al. with a linear coupling term, similar to the random-mode-coupling approach in our unidirectional pulse propagation equation (UPPE) [52]. But not until very recently was nonlinear modal coupling introduced by Jima et al. in their $({3 + 1}){\rm D}$ gain model to simulate Kerr beam cleaning [53]. However, their work assumes quasi-continuous-wave scenarios and thus neglects frequency-dependent features such as dispersion and gain that are essential in ultrafast nonlinear dynamics. Consequently, an accurate modeling framework for ultrafast nonlinear pulse propagation in a multimode gain fiber is still lacking.

There are various nonlinear pulse propagation models [54], among which the UPPE has been widely used in nonlinear fiber research [55]. In this paper, we generalize the UPPE and obtain its multimode mode-resolved form [56]:

$$\begin{split}&{\partial _z}{A_p}(z,\Omega)\\&\quad = \left({\hat {\cal D} + \hat {\cal G} + \hat {\cal R} + \hat {\cal N}} \right){A_p}(z,\Omega)\\&\quad = i\Big[{{\beta _p}(\omega) - \left({{\beta _{(0)}} + {\beta _{(1)}}\Omega} \right)} \Big]{A_p}(z,\Omega) + {g_p}(z,\Omega){A_p}(z,\Omega)\\&\qquad + i\sum\limits_\ell {Q_{{p\ell}}}(z){A_\ell}(z,\Omega) \\&\qquad+ i\frac{{\omega {n_2}}}{c}\sum\limits_{\ell mn} \Bigg\{\vphantom{\left. {{f_b}S_{p\ell mn}^{{R_b}}\mathfrak{F}\left[{{A_\ell}\left[{{h_b} * \left({{A_m}A_n^ *} \right)} \right]} \right]} \right\}}{({1 - {f_R}} )S_{p\ell mn}^K\mathfrak{F}\left[{{A_\ell}{A_m}A_n^ *} \right] } \\&\qquad\qquad\qquad\qquad +{f_R}\bigg\{{f_a}S_{p\ell mn}^{{R_a}}\mathfrak{F}\Big[{A_\ell}\big[{{h_a} * \left({{A_m}A_n^ *} \right)} \big] \Big] \\& \qquad\qquad\qquad\qquad\quad\;\;\,\,+ {{f_b}S_{p\ell mn}^{{R_b}}\mathfrak{F}\Big[{{A_\ell}\big[{{h_b} * \left({{A_m}A_n^ *} \right)} \big]} \Big]} \bigg\} \Bigg\},\end{split}$$
where ${A_p}(z,t)$ is the electric field ($\sqrt {\rm W}$) of mode $p$, whose Fourier transform is ${A_p}(z,\Omega) = \mathfrak{F}[{A_p}(z,t)]$. $z$ is the propagation distance. $\hat {\cal D}$, $\hat {\cal G}$, $\hat {\cal R}$, and $\hat {\cal N}$ are dispersion, gain, random-mode-coupling/disorder, and nonlinear operators, respectively. The Fourier transform is applied with respect to angular frequency $\Omega = \omega - {\omega _0}$. ${\beta _p}$ is the propagation constant of mode $p$; ${\beta _{(0)}}$ is a free parameter to reduce the global-phase increment during propagation and thus allows a larger step size for faster simulations; ${\beta _{(1)}}$ is the inverse group velocity of the moving frame. ${Q_{{p\ell}}}$ is responsible for the random linear mode coupling, caused by fiber disorder, during pulse propagation [57–59]. ${n_2}$ is the nonlinear refractive index. $c$ is the speed of light. ${f_R}$ is the Raman fraction representing the contribution of the Raman response of all nonlinearities, where ${f_a}$ and ${f_b}$ are Raman fractions of the total Raman response for isotropic and anisotropic Raman responses, respectively (${f_a} + {f_b} = 1$); ${h_a}$ and ${h_b}$ are isotropic and anisotropic Raman response functions [60,61], respectively. $p$, $\ell$, $m$, and $n$ are eigenmode indices; $S_{p\ell mn}^K$, $S_{p\ell mn}^{{R_a}}$, and $S_{p\ell mn}^{{R_b}}$ are overlap integrals of eigenmode fields ${\vec F_j}({\vec r_ \bot})$ (1/m) [62], where ${\vec r_ \bot} = (x,y)$:
$$S_{p\ell mn}^K \def\LDeqtab{} = \frac{2}{3}S_{p\ell mn}^{{R_a}} + \frac{1}{3}S_{p\ell mn}^k,$$
$$S_{p\ell mn}^k \def\LDeqtab{} = \int \left({\vec F_p^ * \cdot \vec F_n^ *} \right)\left({{{\vec F}_\ell} \cdot {{\vec F}_m}} \right){{\rm d}^2}x,$$
$$S_{p\ell mn}^{{R_a}} \def\LDeqtab{} = \int \left({\vec F_p^ * \cdot {{\vec F}_\ell}} \right)\left({{{\vec F}_m} \cdot \vec F_n^ *} \right){{\rm d}^2}x,$$
$$S_{p\ell mn}^{{R_b}}\def\LDeqtab{} = \frac{1}{2}\left[{\int \left({\vec F_p^ * \cdot {{\vec F}_m}} \right)\left({{{\vec F}_\ell} \cdot \vec F_n^ *} \right){{\rm d}^2}x + S_{p\ell mn}^k} \right]\!,$$
where ${{\rm d}^2}x = {\rm d}x{\rm d}y$ represents the integral over the spatial domain. The derivation of the UPPE is partially described in [63] and its supplemental document. The main focus of the present paper is to derive ${g_p}(z,\Omega)$, the gain (or loss), which can be incorporated in the UPPE [Eq. (1)].

We propose a gain model based on rate equations to describe the nonlinear evolution in a multimode gain fiber. It incorporates important aspects, such as nonlinear modal coupling and spatial gain saturation, as well as frequency-dependent features including dispersion, gain spectrum, and self-steepening. Unlike prior works that rely on the computationally intensive $({3 + 1}){\rm D}$ scheme, we employ a mode-resolved representation, which naturally facilitates modal analysis. By combining the gain model with the UPPE, investigation of the underlying nonlinear dynamics in the mode-resolved picture becomes possible.

To demonstrate the applicability of our model, we initially apply it to analyze the pulse propagation in a few-mode large-mode-area (LMA) gain fiber. We compare it with a widely used Gaussian model and show that few-mode gain fibers exhibit nonlinear amplification that can be captured only by using the rate-equation model. Recently, gain-managed nonlinear (GMN) amplification has been discovered and exploited to generate microjoule-energy and sub-40-fs pulses [64]. It will be valuable to explore the potential of this approach beyond the current single-mode scenario and investigate its scalability with a multimode fiber. Last, we investigate Kerr beam cleaning in an amplifier. New nonlinear dynamics, including the role of loss-enhanced Kerr beam cleaning, are revealed by use of the model.

2. MODE-RESOLVED GAIN MODELS

A. Gaussian-Gain Model

Before developing the model that includes the rate equations, we first introduce multimode amplification with a simple Gaussian-gain spectrum. This will help demonstrate the necessity of using the rate equations to capture certain nonlinear dynamics. Although a $({3 + 1}){\rm D}$ model with a Gaussian-gain spectrum has been employed in the study of multimode pulse propagation [12–14], it remains in its full-field form; a mode-resolved formulation is lacking.

To simulate the UPPE, all operators are computed independently using methods such as the split-step [65] or Runge–Kutta in the interaction picture [66,67]. The gain component of the $({3 + 1}){\rm D}$ pulse evolution is modeled with

$${\partial _z}{\vec {\mathbb A}}(\vec r,\Omega) = \frac{{{g_0}/2}}{{1 + \frac{{\int I(\vec r,t){\rm d}t}}{{{I_{{\rm sat}}}}}}}f(\Omega){P_d}({\vec r_ \bot}){\vec {\mathbb A}}(\vec r,\Omega),$$
where the field ${\vec {\mathbb A}}(\vec r,\Omega) = \sum\nolimits_p {\vec F_p}({\vec r_ \bot}){A_p}(z,\Omega) = \mathfrak{F}[\mathbb{A}(\vec r,t)]$, and its intensity $I(\vec r,t) = {| {{\vec {\mathbb A}}(\vec r,t)} |^2}$ [in units of ${{\rm W/m}^2}$; $\vec r = ({\vec r_ \bot},z)$]. ${g_0}$ is the peak gain. ${I_{{\rm sat}}}$ is the saturation energy per area (${{\rm J/m}^2}$). $f(\Omega) = {{\rm e}^{- {\Omega ^2}/\Delta {\Omega ^2}}}$ represents the frequency dependence of the gain spectrum, with $\Delta \Omega$ the gain bandwidth. ${P_d}({\vec r_ \bot})$ is the doping profile of the gain medium. Next, we transform Eq. (3) to its mode-resolved counterpart by multiplying both sides of the equation by the denominator of the Gaussian-gain factor, followed by taking an inner product with $\vec F_p^ * ({\vec r_ \bot})$. Eq. (3) subsequently becomes
$$\begin{split}\!\!\!\!\!\!\!\!\!\!\!\! {\partial _z}{A_p}(z,\Omega) & + \sum\limits_{\ell mn} S_{p\ell mn}^{{R_a}}{B_{{mn}}}(z)\left[{{\partial _z}{A_\ell}(z,\Omega)} \right]\\ & \quad \quad \quad \quad\approx \frac{{{g_0}}}{2}f(\Omega){\Gamma _p}{A_p}(z,\Omega),\end{split}$$
where ${B_{{mn}}}(z) = [{\int {A_m}(z,t)A_n^ * (z,t){\rm d}t}]/{I_{{\rm sat}}}$ represents gain saturation. ${\Gamma _p} = \int {P_d}({\vec r_ \bot}){| {{{\vec F}_p}({{\vec r}_ \bot})} |^2}{{\rm d}^2}x$ is the doping overlap integral. The approximation in Eq. (4) results from $\int \vec F_p^ * ({\vec r_ \bot}) \cdot {\vec F_\ell}({\vec r_ \bot}){P_d}({\vec r_ \bot}){{\rm d}^2}x \approx {\delta _{{p\ell}}}{\Gamma _p}$ assuming that all eigenmodes stay mostly within the doped region to apply the orthogonality relation. Further simplification leads to
$$\sum\limits_\ell {T_{{p\ell}}}(z)\left[{{\partial _z}{A_\ell}(z,\Omega)} \right] = \frac{{{g_0}}}{2}f(\Omega){A_p}(z,\Omega),$$
where we define a transfer matrix ${\textbf T}(z)$ with entries ${T_{{p\ell}}}(z)$:
$${T_{{p\ell}}}(z) = \left[{{\delta _{{p\ell}}} + \sum\limits_{{mn}} S_{p\ell mn}^{{R_a}}{B_{{mn}}}(z)} \right]\Gamma _p^{- 1}.$$
Equation (5) can be written in matrix form by treating the field ${A_p}(z,\Omega)$ as a column vector ${\textbf A}(z,\Omega)$. As a result, the mode-resolved gain term [Eq. (1)] is
$${g_p}{A_p} = {\left[{{{\textbf T}^{- 1}}(z)\frac{{{g_0}}}{2}f(\Omega){\textbf A}(z,\Omega)} \right]_p}.$$
With this gain term, mode-resolved evolution in a multimode gain fiber can be numerically calculated.

B. Rate-Equation-Gain Model

Due to the frequency dependence of the gain [44], the Gaussian model is valid only for narrowband pulses, negligible gain saturation, and operation around the gain peak. To overcome these limitations, rate equations for the populations can accurately simulate the amplification process. These equations have been applied to amplifiers based on various ions, such as Er [39,45,46], Yb [48–52], and Nd [68]. In general, describing the dynamics requires considering the populations of all relevant energy levels, especially for Er [46] and Nd [68]. However, Yb exhibits a simple two-multiplet energy level scheme, which allows for simulations using a convenient two-multiplet system [44]. Although many prior studies have employed rate equations, their integration with a nonlinear pulse propagation equation is not trivial. Inclusion of the gain was described by either a susceptibility for the gain medium [47] or a signal-gain term determined with rate equations in each propagation step [49–51]. For simplicity, in this paper, we solve the two-multiplet system of Yb to demonstrate the capability of our model. We apply the approach of jointly solving the signal-gain term and the UPPE to correctly include the interplay of dispersion, nonlinearity, and gain. Moreover, we ignore temporal gain saturation [69], which causes the trailing edge of a pulse to experience less gain than the leading edge, since it is a small effect for the picosecond or femtosecond pulses of interest here.

Unlike the rate equations based on the power in each mode [39,52], we use intensities for the signal and the ASE to include both nonlinear-coupling and field-interference effects in the UPPE. Furthermore, we assume that the pulse repetition rate is sufficiently high that the population inversion almost reaches the steady state [49,51]. Thus, the rate equations become

$$\begin{split}\frac{{\rm d}{I^{\pm ,\hat\mu}}}{{\rm d}z}(\vec r,\nu)& = \mathfrak{g}(\vec r,\nu){I^{\pm ,\hat\mu}}(\vec r,\nu)\\&\quad + {\sigma _e}(\nu){N_2}(\vec r)h\nu \sum\limits_{p = 1}^{{N_m}} {\left| {F_p^{{\hat\mu}}({{\vec r}_ \bot})} \right|^2},\end{split}$$
where the second term represents spontaneous emission, and ${N_m}$ is the total number of supported spatial modes. ASE is generated in all directions, but only those in the supported modes propagate in the fiber. ${\pm}$ represents forward or backward propagation, and $\hat\mu$ represents one of the orthogonal polarizations in the polarization basis, such as the linear basis $\{{\hat x,\hat y} \}$ or the circular basis $\{{\hat + ,\hat -} \}$. ${I^{{\hat\mu}}}(\vec r,\nu)$ is the averaged spectral intensity in the $\mu$ polarization (${\rm W}/{\rm Hz}/{{\rm m}^2}$) because the gain medium reacts to the average power, not the pulse energy. The total intensity ${I^ \pm} = \sum\nolimits_{\hat\mu= \hat x,\hat y} {I^{\pm ,\hat\mu}}$. For a discrete Fourier transform with $\mathfrak{F}[A](\Omega) = \frac{1}{N}\sum\nolimits_{n = 1}^N A({t_n}){{\rm e}^{i\Omega {t_n}}}$, where $N$ is the number of discrete points, the signal spectral intensity is given by
$$I_s^{\pm ,\hat\mu}\left({\vec r,\nu = \frac{{\Omega + {\omega _0}}}{{2\pi}}} \right) = {\left| {\mathfrak{F}[{\mathbb{A}^{\pm ,\hat\mu}}(\vec r,t)]} \right|^2}{R_r}{\left({N\Delta t} \right)^2},$$
where ${R_r}$ is the pulse repetition rate, and $\Delta t$ is the temporal spacing. [See Supplement 1 regarding how to obtain the spectral density with correct units in Eq. (9).] $\mathfrak{g}(\vec r,\nu) = {\sigma _e}(\nu){N_2}(\vec r) \,-\def\LDeqbreak{} {\sigma _a}(\nu){N_1}(\vec r)$ is polarization independent. ${\sigma _e}$ and ${\sigma _a}$ are emission and absorption cross sections (${{\rm m}^2}$), respectively. ${N_1}$ and ${N_2}$ are the populations of the lower and upper multiplets, respectively, which satisfy
$${N_2}(\vec r) = \frac{{{R_a}(\vec r)}}{{{R_a}(\vec r) + {R_e}(\vec r) + \frac{1}{\tau}}}{N_T}(\vec r),$$
where $\tau$ is the lifetime of the upper multiplet, ${N_T}$ is the total volume density of rare-earth ions ($1/{{\rm m}^3}$), ${N_1} + {N_2} = {N_T}$, and
$${R_i}(\vec r) = \sum\limits_{\hat\mu= \hat x,\hat y} \sum\limits_ \pm \int \frac{1}{{h\nu}}{\sigma _i}(\nu){I^{\pm ,\hat\mu}}(\vec r,\nu){\rm d}\nu ,$$
where $i = a,e$.

To obtain the intensity evolutions of the pump, signal, and ASE, we observe that the total intensity is composed of three parts:

$${I^{\pm ,\hat\mu}}(\vec r,\nu) = \frac{{P_p^{\pm ,\hat\mu}(z,{\nu _p})}}{{{A_{{\rm cladding}}}}} + \int \left[{I_s^{\pm ,\hat\mu}(\vec r,\nu) + I_{{\rm ASE}}^{\pm ,\hat\mu}(\vec r,\nu)} \right]{\rm d}\nu ,$$
where $p$ and $s$ represent pump and signal fields, respectively. Since the pump is continuously absorbed by the core in a double-clad fiber, pump power (W) is used instead. By inserting it into Eq. (8), we obtain the following governing equations:
$$\frac{{\rm d}P_p^ \pm}{ {\rm d}z}(z,{\nu _p}) \def\LDeqtab{}= \frac{{P_p^ \pm (z,{\nu _p})}}{{{A_{{\rm cladding}}}}}\int_{{A_{{\rm core}}}} \mathfrak{g}(\vec r,{\nu _p}){{\rm d}^2}x,$$
$$\frac{{\rm d}I_s^{\pm ,\hat\mu}}{{\rm d}z}(\vec r,\nu) \def\LDeqtab{}= \mathfrak{g}(\vec r,\nu)I_s^{\pm ,\hat\mu}(\vec r,\nu),$$
$$\begin{split} \frac{{\rm d}I_{{\rm ASE}}^{\pm ,\hat\mu}}{{\rm d}z}(\vec r,\nu) & = \mathfrak{g}(\vec r,\nu)I_{{\rm ASE}}^{\pm ,\hat\mu}(\vec r,\nu)\\[-3pt]&\quad + {\sigma _e}(\nu){N_2}(\vec r)\textit{sh}\nu \sum\limits_{p = 1}^{{N_m}} {\left| {F_p^{{\hat\mu}}({{\vec r}_ \bot})} \right|^2},\end{split}$$
where the total pump power $P_p^ \pm = \sum\nolimits_{\hat\mu= \hat x,\hat y} P_p^{\pm ,\hat\mu}$. Because ASE is always generated unpolarized, $s$ is one in polarized simulations or two in scalar ones. In addition, Eq. (11) becomes
$$\begin{split} {R_i}(\vec r) & = \sum\limits_ \pm \frac{1}{{h{\nu _p}}}{\sigma _i}({\nu _p})\frac{{P_p^ \pm (z,{\nu _p})}}{{{A_{{\rm cladding}}}}}\\&\quad + \sum\limits_{\hat\mu= \hat x,\hat y} \sum\limits_ \pm \int \frac{1}{{h\nu}}{\sigma _i}(\nu)\left[{I_{{\rm ASE}}^{\pm ,\hat\mu}(\vec r,\nu) + I_s^{\pm ,\hat\mu}(\vec r,\nu)} \right]{\rm d}\nu .\end{split}$$

Next, we proceed to transform Eq. (13) into the mode-resolved framework. Due to the spectral incoherence of ASE, its intensity follows the eigenmode decomposition ${P_{{\rm ASE}(p)}}$ (W/Hz): $I_{{\rm ASE}}^{\pm ,\hat\mu}(\vec r,\nu) = \sum\nolimits_{p = 1}^{{N_m}} {| {F_p^{{\hat\mu}}({{\vec r}_ \bot})} |^2}{}P_{{\rm ASE}(p)}^{\pm ,\hat\mu}(z,\nu)$. If the eigenmode fields are represented in the polarization basis of a fiber, such as LP modes rather than HE modes in a step-index fiber, they satisfy $\int {| {F_p^{{\hat\mu}}({{\vec r}_ \bot})} |^2}{{\rm d}^2}x = \int {| {{{\vec F}_p}({{\vec r}_ \bot})} |^2}{{\rm d}^2}x = 1$ due to the normalization relation. As a result, Eq. (13c) can be derived into the following mode-resolved form:

$$\begin{split} \frac{{\rm d}P_{{\rm ASE}(p)}^{\pm ,\hat\mu}}{{\rm d}z} (z,\nu) & = P_{{\rm ASE}(p)}^{\pm ,\hat\mu}(z,\nu)\int_{{A_{{\rm core}}}} \mathfrak{g}(\vec r,\nu){\left| {F_p^{{\hat\mu}}({{\vec r}_ \bot})} \right|^2}{{\rm d}^2}x\\&\quad + {\sigma _e}(\nu)\textit{sh}\nu \int_{{A_{{\rm core}}}} {\left| {F_p^{{\hat\mu}}({{\vec r}_ \bot})} \right|^2}{N_2}(\vec r){{\rm d}^2}x.\end{split}$$
On the other hand, to solve for the mode-resolved signal field, we derive its mode-resolved gain for the UPPE. By assuming that the full field ${{\vec {\mathbb A}}^ \pm}(\vec r \pm \Delta z\hat z,\Omega) = {{\rm e}^{\pm g(\vec r,\nu)\Delta z}}{{\vec {\mathbb A}}^ \pm}(\vec r,\Omega)$, where $g(\vec r,\nu)$ is the gain of each spatial point $\vec r$, the mode-resolved field $A_p^ \pm (z \pm \Delta z,\Omega) = \sum\nolimits_{n = 1}^{{N_m}} A_n^ \pm (z,\Omega)\def\LDeqbreak{}\int {{\rm e}^{\pm g(\vec r,\nu)\Delta z}}\vec F_p^ * ({\vec r_ \bot}) \cdot {\vec F_n}({\vec r_ \bot}){{\rm d}^2}x$, which leads to the mode-resolved field amplification ${G_p}(z,\Omega)$:
$$\begin{split} {G_p}(z,\Omega) & = \frac{{A_p^ \pm (z \pm \Delta z,\Omega)}}{{A_p^ \pm (z,\Omega)}}\\& = \sum\limits_{n = 1}^{{N_m}} \frac{{A_n^ \pm (z,\Omega)}}{{A_p^ \pm (z,\Omega)}}\int {{\rm e}^{\pm g(\vec r,\nu)\Delta z}}\vec F_p^ * ({{\vec r}_ \bot}) \cdot {{\vec F}_n}({{\vec r}_ \bot}){{\rm d}^2}x\\& \approx 1 \pm \Delta z\sum\limits_{n = 1}^{{N_m}} \frac{{A_n^ \pm (z,\Omega)}}{{A_p^ \pm (z,\Omega)}}\int g(\vec r,\nu)\vec F_p^ * ({{\vec r}_ \bot}) \cdot {{\vec F}_n}({{\vec r}_ \bot}){{\rm d}^2}x,\end{split}$$
where the relation ${{\rm e}^{\pm g\Delta z}} \approx 1 \pm g\Delta z$ is employed. In addition, ${G_p} = {{\rm e}^{\pm {g_p}\Delta z}} \approx 1 \pm {g_p}\Delta z$ [Eq. (1)]. The spectral-intensity amplification, with respect to $I_s^{\pm ,\hat\mu}(\vec r,\nu) \propto {| {{\mathbb{A}^{\pm ,\hat\mu}}(\vec r,\Omega)} |^2}$ [Eq. (9)], is ${{\rm e}^{\pm 2g\Delta z}} \approx 1 \pm 2g\Delta z$, while it is ${{\rm e}^{\pm \mathfrak{g}\Delta z}} \approx 1 \pm \mathfrak{g}\Delta z$ from Eq. (13b); as a result, $g = \mathfrak{g}/2$. Equation (16) eventually leads to the following mode-resolved gain term that can be jointly solved with the UPPE [Eq. (1)]:
$${g_p}A_p^ \pm \approx \frac{1}{2}\sum\limits_{n = 1}^{{N_m}} A_n^ \pm (z,\Omega)\int_{{A_{{\rm core}}}} \mathfrak{g}(\vec r,\nu)\vec F_p^ * ({\vec r_ \bot}) \cdot {\vec F_n}({\vec r_ \bot}){{\rm d}^2}x.$$
Accurate modeling of nonlinear processes can be performed with Eqs. (13a), (14), (15), (17), and the UPPE [Eq. (1)], but simulations are computationally expensive due to their 2D integrals. This problem can be easily mitigated by minimizing the number of 2D computations during pulse propagation through several pre-computations, leading to comparable computational times for the optimized rate-equation-gain model and the aforementioned Gaussian-gain model. With the approach proposed by Lindberg et al. [51], solving various pumping schemes becomes possible. Moreover, the background ASE is added to the signal pulse during pulse propagation to include the effect of ASE-induced instability. For details, see the section on “Rate-equation-gain modeling” in Supplement 1.

3. STUDIES OF MULTIMODE FIBER AMPLIFIERS

In this section, we explore three ultrafast nonlinear pulse evolutions in multimode fiber amplifiers. Spatial gain saturation, evolution of the gain spectrum, and nonlinear intermodal interactions play significant roles in these examples. We offer these as example applications of the multimode gain model. The results satisfy basic expectations and thus serve as first checks of the model, but they are not intended to constitute careful or thorough bench-marking of the code.

Simulation of multimode processes is challenging due to the strong beating between modes, which often necessitates a small step size. To overcome this challenge, we employ a massively parallel algorithm designed to handle multimode propagation [70]. In addition, we use the Adams–Moulton method [71] to reduce the truncation error without requiring additional computations. Furthermore, we have incorporated adaptive step-size control, which improves the computational performance, especially with strong amplification. Details are discussed in Supplement 1.

A. Differential Modal Gain in a Few-Mode Gain Fiber

LMA gain fibers are widely used to achieve high-power output. However, they often require careful mode-management techniques, such as fiber coiling [72], to suppress higher-order modes. With the recent resurgence of interest in multimode fibers, LMA fibers have emerged as one of the preferred choices in spatiotemporal studies due to their commercial availability [14–17]. Depending on the fiber parameters, such as core size and numerical aperture (NA), modes can experience significantly different gain.

To study the differential modal gain, we launch a pulse at 1030 nm into a Yb-doped polarization-maintaining (PM) fiber with 10-µm core diameter (Thorlabs YB1200-10/125DC-PM), which supports the three modes illustrated at the top of Fig. 1(a). The input pulse energy is evenly distributed among the modes. We adjusted the Gaussian-gain parameters to match the energy evolution as closely as possible to the solution obtained with the rate equations. Due to the low NA, the ${{\rm LP}_{11{\rm a}}}$ and ${{\rm LP}_{11{\rm b}}}$ modes extend beyond the core, resulting in smaller overlaps with the doped region compared to the fundamental ${{\rm LP}_{01}}$ mode (${\Gamma _1} = 0.88$, ${\Gamma _2} = {\Gamma _3} = 0.61$). Therefore, the fundamental mode is preferentially amplified [Fig. 1(a)]. Although the two models show decent agreement in terms of energy evolution, there is inconsistency in the output spectrum. Because the Yb gain spectrum shifts to longer wavelengths at higher saturation levels [44], the pulse spectrum shifts to the red [Fig. 1(b)]. This phenomenon is accurately captured by the rate-equation model, while the Gaussian model yields a spectrum centered around the seed wavelength. Unless higher-order modes are intentionally excited, the fundamental mode predominates after amplification. In practical scenarios, decent fiber coiling can induce high bend loss for the two weakly confined ${{\rm LP}_{11}}$ modes; hence, the fundamental mode dominates almost all the time.

 figure: Fig. 1.

Fig. 1. Pulse-energy evolutions of three modes supported in the LMA gain fiber. The modes are shown at the top, where black circles represent the fiber core. Pulses with 1-nJ energy, 1-ps duration, and 15-MHz repetition rate are launched into a 2-m-long Yb-doped fiber with a 0.08 NA. The fiber is co-pumped by a 3-W multimode diode at 976 nm. (a) Left: results of Gaussian model. Right: results of rate-equation model. (b) Output spectra produced by the (top) Gaussian model and (bottom) rate-equation model. PSD, power spectral density. The slight deviation between the ${{\rm LP}_{11{\rm a}}}$ and ${{\rm LP}_{11{\rm b}}}$ modes results from the numerical error of the finite-difference mode solver [73].

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B. Gain-Managed Nonlinear Evolution in a Multimode Fiber

Recently, GMN amplification has emerged as a simple way to generate sub-40-fs pulses from Yb fiber amplifiers [64]. In this technique, a narrowband seed pulse is amplified while simultaneously undergoing strong nonlinear spectral broadening, with a phase that can be compensated for by a standard grating-pair compressor. The amplification process relies on the interaction between the gain spectrum and the spectral broadening. The middle- and long-wavelength parts of the spectrum experience gain and develop a linear frequency sweep with the corresponding parabolic temporal leading edge, despite the accumulation of a huge nonlinear phase. The short-wavelength part of the spectrum exhibits a (possibly nonlinear) frequency sweep expected from self-phase modulation, but this part of the spectrum is later absorbed by the saturated gain medium. As a result, a linear frequency sweep is generated over a bandwidth much larger than the gain-narrowing limit, which subsequently produces a sub-40-fs transform-limited pulse after being dechirped. Pulses with energy over 1 µJ and duration of 40 fs can be generated using 30-µm LMA Yb-doped PM fiber [74]. Because of limited mode areas of commercial LMA fibers (e.g., $3300\;{{\unicode{x00B5}{\rm m}}^2}$ from NKT aeroGAIN-ROD), further energy scaling in the single-mode scenario is restricted. Multimode gain fiber offers another potential approach to energy scaling, through both the larger area of the fundamental mode and the possibility of multimode evolution (if it can be controlled).

Before investigating the gain-managed evolution in a multimode gain fiber, we explored the effect of polarization modes on the GMN dynamics in a single-mode non-PM fiber. We launch a pulse with a single polarization ${\hat e_1} = \frac{{\hat x + ie\hat y}}{{\sqrt {1 + {e^2}}}}$ of various ellipticities $e$ in the non-PM counterpart of the previously studied 10-µm Yb-doped fiber (Thorlabs YB1200-10/125DC), while introducing noise in the other orthogonal polarization ${\hat e_2} = \frac{{e\hat x - i\hat y}}{{\sqrt {1 + {e^2}}}}$. Although both linear and circular polarizations can serve as linearly independent bases of a non-PM fiber, only circular polarizations form a stable basis with minimal energy coupling to the other orthogonal polarization. This is due to the presence of the four-wave-mixing term, ${\cal O}(A_x^ * A_y^2)$, in the linear basis [Eq. (18a)], which results in strong energy exchange between two polarization modes. (The use of ${\cal O}$ notation stands for terms with the same form.) By contrast, in the circular basis [Eq. (18b)], there are only phase-modulation terms. Since GMN pulses evolve with significant nonlinearity, only a seed with a single circular polarization can enter the GMN regime; any polarization coupling will interfere with the nonlinear dynamics and prevent the pulse from reaching the GMN regime (Fig. 2). In fact, even in the circular basis, the noise in the other polarization must be kept below 1% of the seed energy to achieve a clean and smooth GMN spectrum. In practical scenarios, random polarization-mode coupling, resulting from fiber disorder, may be significant and could disrupt the GMN evolution. Therefore, even the GMN evolution of a single circularly polarized field in a single-mode non-PM fiber cannot be easily realized. This is verified both numerically and experimentally in Supplement 1:

$${\hat {\cal N}_{{\rm linear}}}{A_x}\def\LDeqtab{} \propto i \mathfrak{F}\left[{\left({{{\left| {{A_x}} \right|}^2} + \frac{2}{3}{{\left| {{A_y}} \right|}^2}} \right){A_x} + {\cal O}(A_x^ * A_y^2)} \right],$$
$${\hat {\cal N}_{{\rm circular}}}{A_ +}\def\LDeqtab{} \propto i \mathfrak{F}\left[{{\cal O}\left({{{\left| {{A_ +}} \right|}^2}{A_ +}} \right) + {\cal O}\left({{{\left| {{A_ -}} \right|}^2}{A_ +}} \right)} \right].$$
 figure: Fig. 2.

Fig. 2. Peak power achieved through nonlinear amplification of pulses with varying ellipticity. A pulse of 1-nJ energy, 1-ps duration, and 15-MHz repetition rate at 1030 nm is launched into a 2-m-long gain fiber with polarization ${\hat e_1}$. Noise is introduced in the orthogonal polarization (${\hat e_2}$). The fiber is co-pumped by a 3-W multimode diode at 976 nm. $e = 0\;{\rm and}\;1$ represent linearly and circularly polarized seed pulses, respectively. The insets are the output spectra corresponding to $e = 0,0.2,0.6$, and 1. The blue and red lines indicate the output spectra of the same (${\hat e_1}$) and orthogonal (${\hat e_2}$) polarizations as the input pulse, respectively. Results of 10 simulations are shown to demonstrate the mean values (center line) and standard deviations (shaded area).

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Next, we explore the feasibility of a multimode GMN amplifier. A pulse at 1030 nm is launched into a Yb-doped PM fiber with a 30-µm core and 0.06 NA (parameters of Coherent PLMA-YDF-30/400-VIII). This fiber supports 10 modes, with the last few modes extending beyond the core and exhibiting lower gain. However, the more-tightly-confined lower-order modes can potentially be coupled significantly by nonlinearity. As the pulse propagates, it experiences strong nonlinear intermodal coupling despite the dominant fundamental mode in the seed, which prevents the pulse from reaching the GMN regime [“MM” (multimode) scenario (solid lines) in Fig. 3]. In comparison, we considered a scenario involving only the fundamental mode, which can generate a GMN pulse up to 1 µJ [“SM” (single-mode) scenario (dashed line) in Fig. 3], as demonstrated in [74] by tightly coiling this fiber to realize single-mode operation.

 figure: Fig. 3.

Fig. 3. Pulse-energy evolutions of single-mode (SM; top dashed line) and multimode (MM; bottom solid lines) scenarios aiming for a GMN state. A pulse of 5-nJ energy, 0.9-ps duration, and 5-MHz repetition rate is launched into the fundamental mode of each fiber. In the MM scenario, each higher-order mode of the seed has 0.01% of the total energy. The SM and MM fibers are co-pumped with 15 W and 10 W, respectively. Output spectra are shown on the right, where the black lines represent the fundamental mode, while other colors represent higher-order modes.

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Overall, despite the remarkable pulse parameters achieved by a single stage of GMN amplification, GMN evolution will be limited by the requirement of scalar-field behavior, involving only a single polarization in a single spatial mode throughout the evolution. As a result, GMN evolution is currently possible only in single-mode PM fibers.

C. Kerr Beam Cleaning in a Multimode Amplifier

Beam cleaning in multimode fibers has long been a topic of interest, owing to its scientific significance and potential applications. A speckled multimode beam can “condense” into a bell-shaped profile that is dominated by the fundamental mode by means of nonlinear modal coupling, through Raman scattering [75,76], Brillouin scattering [77], or the Kerr effect [23]. The intricate interplay between linear and nonlinear modal coupling in Kerr beam cleaning has sparked active discussions of mechanisms understood through wave turbulence theories [78,79] or statistical mechanics [27,80–83]. This effect can effectively mitigate the degradation of beam quality that ordinarily occurs on transmission through a multimode fiber, thereby extending its practical utility beyond research laboratories and to applications in areas such as nonlinear microscopy [84] or delivery of high-power beams.

Only a limited number of investigations have considered beam cleaning in active fibers [24,28,42,53,85]. The current understanding of beam cleaning in such systems mirrors that for conservative systems: the gain simply facilitates reaching the peak powers that underlie the beam cleaning. In this study, we simulate beam cleaning in a multimode amplifier with the rate-equation model. The results reveal that Kerr beam cleaning can be enhanced by loss and account for frequency-dependent phenomena observed in experiments performed with femtosecond pulses [28].

To establish consistency between beam-cleaning experiments and simulations, especially in fibers with modest core diameters, e.g., 15–50 µm, it is crucial to incorporate weak disorder, which dramatically accelerates the optical thermalization process [86,87]. However, inclusion of disorder introduces substantial computational complexity. Without disorder, observation of beam cleaning in simulations can require careful adjustment of the parameters of the input field [87,88]. To circumvent these challenges and ensure reliable repeatability of beam-cleaning results, we choose to model beam cleaning in a step-index multimode fiber with a core diameter of 168 µm and 0.2 NA, as was done in [25]. The large core of this fiber results in small differences in propagation constants among the first few modes, which facilitates the strong intermodal four-wave mixing that underlies beam cleaning. Assuming these fiber parameters, we can reliably observe beam cleaning without including disorder and in reasonable computational time. We expect that the observed physics will be similar in a realistic smaller-core fiber with the inclusion of disorder.

To begin with, we study the quasi-continuous-wave scenario by injecting a 10-ns pulse into a Yb-doped counterpart of the aforementioned fiber. We assume a pump absorption of 40 dB/m at 975 nm and a cladding diameter of 400 µm, parameters similar to those of a commercial 20-µm gain fiber (Thorlabs YB1200-20/400DC). The doped region is assumed to have a diameter of 84 µm, which is smaller than the fiber core to suppress ASE while still larger than all the modes considered in the simulations. With 1-kW initial peak power, the pulse does not trigger beam cleaning. Interestingly, even during the amplification process, there is no indication of beam cleaning [Fig. 4(a)]. While Kerr beam cleaning tries to drive the energy toward the fundamental mode, the pulse also encounters spatial gain saturation, which can be the strongest at the beam center, particularly in the presence of some cleaning, resulting in stronger gain for higher-order modes. These two effects, Kerr beam cleaning and gain saturation, compete with each other, with the former governed by peak power and the latter by pulse energy. Because of the long pulse duration, gain saturation dominates, and beam cleaning is impeded during the amplification stage. However, once the system transitions into the loss regime [0.8 m in Fig. 4(b)], the competition from gain saturation ceases, which allows Kerr beam cleaning to prevail. Surprisingly, the fundamental-mode occupancy nearly reaches unity, in contrast with the typical values of 0.3–0.5 observed in passive fibers. (See Supplement 1 for simulations with passive fiber.) This exceptional cleaning performance is attributed to lower loss at the center of the beam caused by gain saturation, despite the absence of actual “gain” [Fig. 4(b)]. Since Yb absorbs at 1030 nm, the central portion of the gain medium is slightly more inversely populated than the outer part, leading to greater loss for the higher-order modes [Fig. 4(b) and insets of Fig. 4(c)]. Ultimately, only the fundamental mode survives. Such “loss-enhanced” beam cleaning further drives the fundamental-mode occupancy beyond what can be achieved in conservative systems, following the trigger by Kerr beam cleaning.

 figure: Fig. 4.

Fig. 4. Dissipative beam-cleaning process with 10-ns pulses. Simulations include 21 modes, with equal seed energy in the first six modes with random phases, and noise in the other modes. The pulse repetition rate is 10 kHz and the co-pumping power is 50 W. Evolutions of (a) mode occupations (energy ratios of each mode), (b) loss at 1030 nm of a cut-line through the fiber core, and (c) peak power during the process are shown. Deeply and lightly colored lines represent mean values and results from 20 simulations, respectively. White dashed line in (b) represents the transition from amplification to absorption of the signal. Intensity (left; $I$) and inversion (right; ${N_2}$) profiles are displayed with normalized color maps at several propagation distances in (c), accompanied by doped cores (black circles).

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It is worth noting that the interpretation of this loss-enhanced effect cannot rely solely on the larger mode areas of higher-order modes, as it neglects the non-uniform spatial gain-saturation effect caused by signal absorption. Without Kerr beam cleaning triggering the nonlinear process, e.g., at low input power, the loss region from gain saturation will not exhibit the circular profile at the core center that underlies preferential elimination of higher-order modes. (See Supplement 1 for validation of this statement through simulations without Kerr nonlinearity.) Additionally, in our simulations, the fundamental-mode occupation reaches a plateau once the loss becomes more uniform across the doped region due to weak saturation from a weak signal [after 9-m propagation in Figs. 4(a) and 4(b)]. (For more details about loss-enhanced Kerr beam cleaning in the nanosecond regime and another demonstration of cleaning in the ${{\rm LP}_{{11}}}$ modes, see Supplement 1.)

Now we turn to beam cleaning in the ultrafast regime. In contrast to the behavior of nanosecond pulses, picosecond pulses undergo significant spectral broadening and experience strong effects of modal dispersion, as well as temporal stretching caused by chromatic dispersion. These factors raise the question of whether beam cleaning operates under the same mechanism in the picosecond regime as it does in the nanosecond regime. In simulations, we launched a 1-ps pulse into the same gain fiber considered above. The repetition rate and the peak power of the input pulse were increased to 10 MHz and 10 kW, respectively, to avoid excessive ASE and enhance the nonlinear interactions. (For the estimate of ASE, see Supplement 1.) During the amplification, beam cleaning does not occur because of spatial gain saturation. Once the pulse enters the loss regime, cleaning begins; however, the fundamental-mode occupancy increases rapidly only initially, and then reaches a plateau around 0.5 [Fig. 5(a)]. Since Yb absorbs light below ${\sim}1050\;{\rm nm}$, the modes that are less spectrally broadened are absorbed, leaving primarily the most spectrally broadened fundamental mode. In addition, the spectrum tends to redshift due to the saturating gain spectrum, leading to rapid completion of Yb absorption [Fig. 5(b)]. The spectral discontinuity observed in the higher-order modes during pulse propagation (2–4-m spectral evolution of higher-order modes in Fig. 6) indicates the total absorption of the initial higher-order modes, followed by the generation of new higher-order modes from the surviving fundamental mode through the intermodal four-wave mixing. Since Yb absorption does not affect long wavelengths, the pulse evolves in a quasi-conservative manner as it propagates further [Fig. 5(c)]. Due to the broad spectrum, the pulse in this regime undergoes significant dispersive temporal stretching, resulting in no further beam-cleaning enhancement. Consequently, the initial beam cleaning alone falls far short of the level achieved in nanosecond scenarios. Nonetheless, it still out-performs those from a passive fiber with a picosecond pulse of its maximum achievable peak power during propagation [300 kW from Fig. 5(c)], leading to the loss-enhanced Kerr beam cleaning in the picosecond regime. (See Supplement 1 for the simulation of a 300-kW-peak-power picosecond pulse in passive fiber.)

 figure: Fig. 5.

Fig. 5. Dissipative beam-cleaning process with 1-ps pulses. The co-pumping power is 20 W. Evolutions of (a) mode occupations (energy ratios of each mode), (b) loss at 1030 nm of a cut-line through the fiber core, and (c) total pulse energy and peak power with propagation.

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 figure: Fig. 6.

Fig. 6. Spectral evolutions of the (top) fundamental ${{\rm LP}_{{01}}}$ mode and (bottom) summation of all higher-order modes (HOMs) in picosecond beam cleaning (Fig. 5). All spectra are normalized to one in the linear scale. White dashed line represents the transition from the amplification to the absorption regime.

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Loss-enhanced Kerr beam cleaning is achieved by different mechanisms in the two regimes. With nanosecond pulses, spatial gain saturation with a circular profile at the core center, induced by Kerr beam cleaning, influences the loss, leading to higher loss for higher-order modes. On the other hand, in the picosecond regime, higher loss for higher-order modes results from differential spectral broadening between the fundamental mode and the higher-order modes. Kerr beam cleaning drives energy toward the fundamental mode, which subsequently generates the most nonlinear spectral broadening. It thus remains the dominant mode after Yb absorption eliminates the less-broadened higher-order modes. In these two scenarios, loss-enhanced Kerr beam cleaning, triggered by Kerr beam cleaning, induces a stronger preference for the fundamental mode, further stabilizing the fundamental-mode evolution.

In addition to loss-enhanced Kerr beam cleaning, Kerr beam cleaning in active fibers can also be achieved without the loss-enhanced effect by reducing gain saturation, while driving the desired Kerr beam cleaning solely by the amplified peak power. This regime of Kerr beam cleaning aligns well with the interpretation obtained from prior works [24,28,42,53,85]; however, further work is required to conclude which regime they are in. For the highly doped fibers used in [24,53], the loss-enhanced effect should be significant, whereas beam cleaning with femtosecond pulses might be dominated by the peak-power-induced Kerr nonlinearity, rather than gain saturation, resulting in weak loss enhancement [28]. More details are presented in Supplement 1.

4. CONCLUSION

In conclusion, we have developed a mode-resolved gain model based on rate equations to accurately describe the gain dynamics. This model, coupled with the mode-resolved UPPE, provides a comprehensive framework for analyzing nonlinear pulse-propagation dynamics in multimode gain fibers. Through the application of this model, we have explored three distinct multimode nonlinear processes within a gain fiber. The results illustrate the intricate interplay of dispersion, nonlinear intermodal four-wave mixing, and spatially and spectrally dependent gain saturation, among others.

The presented model and findings hold significant potential for their application in complex nonlinear dissipative systems. Refined understanding of multimode nonlinear dynamics may allow unlocking of new possibilities, to harness the capabilities of these systems to their full extent.

Funding

Office of Naval Research (N00014-19-1-2592, N00014-20-1-351 2789); National Science Foundation (ECCS-1912742).

Disclosures

The authors declare no conflicts of interest.

Data availability

The code used in this work has been made publicly available at [89]. It allows modeling not only amplifiers as in this paper, but also passive fibers, with both single-mode and multimode scenarios.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Supplemental document.

Data availability

The code used in this work has been made publicly available at [89]. It allows modeling not only amplifiers as in this paper, but also passive fibers, with both single-mode and multimode scenarios.

89. Y.-H. Chen, “MMTools,” GitHub (2023), https://github.com/AaHaHaa/MMTools.

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Figures (6)
Fig. 1.
Fig. 1. Pulse-energy evolutions of three modes supported in the LMA gain fiber. The modes are shown at the top, where black circles represent the fiber core. Pulses with 1-nJ energy, 1-ps duration, and 15-MHz repetition rate are launched into a 2-m-long Yb-doped fiber with a 0.08 NA. The fiber is co-pumped by a 3-W multimode diode at 976 nm. (a) Left: results of Gaussian model. Right: results of rate-equation model. (b) Output spectra produced by the (top) Gaussian model and (bottom) rate-equation model. PSD, power spectral density. The slight deviation between the ${{\rm LP}_{11{\rm a}}}$ and ${{\rm LP}_{11{\rm b}}}$ modes results from the numerical error of the finite-difference mode solver [73].

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Fig. 2.
Fig. 2. Peak power achieved through nonlinear amplification of pulses with varying ellipticity. A pulse of 1-nJ energy, 1-ps duration, and 15-MHz repetition rate at 1030 nm is launched into a 2-m-long gain fiber with polarization ${\hat e_1}$. Noise is introduced in the orthogonal polarization (${\hat e_2}$). The fiber is co-pumped by a 3-W multimode diode at 976 nm. $e = 0\;{\rm and}\;1$ represent linearly and circularly polarized seed pulses, respectively. The insets are the output spectra corresponding to $e = 0,0.2,0.6$, and 1. The blue and red lines indicate the output spectra of the same (${\hat e_1}$) and orthogonal (${\hat e_2}$) polarizations as the input pulse, respectively. Results of 10 simulations are shown to demonstrate the mean values (center line) and standard deviations (shaded area).

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Fig. 3.
Fig. 3. Pulse-energy evolutions of single-mode (SM; top dashed line) and multimode (MM; bottom solid lines) scenarios aiming for a GMN state. A pulse of 5-nJ energy, 0.9-ps duration, and 5-MHz repetition rate is launched into the fundamental mode of each fiber. In the MM scenario, each higher-order mode of the seed has 0.01% of the total energy. The SM and MM fibers are co-pumped with 15 W and 10 W, respectively. Output spectra are shown on the right, where the black lines represent the fundamental mode, while other colors represent higher-order modes.

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Fig. 4.
Fig. 4. Dissipative beam-cleaning process with 10-ns pulses. Simulations include 21 modes, with equal seed energy in the first six modes with random phases, and noise in the other modes. The pulse repetition rate is 10 kHz and the co-pumping power is 50 W. Evolutions of (a) mode occupations (energy ratios of each mode), (b) loss at 1030 nm of a cut-line through the fiber core, and (c) peak power during the process are shown. Deeply and lightly colored lines represent mean values and results from 20 simulations, respectively. White dashed line in (b) represents the transition from amplification to absorption of the signal. Intensity (left; $I$) and inversion (right; ${N_2}$) profiles are displayed with normalized color maps at several propagation distances in (c), accompanied by doped cores (black circles).

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Fig. 5.
Fig. 5. Dissipative beam-cleaning process with 1-ps pulses. The co-pumping power is 20 W. Evolutions of (a) mode occupations (energy ratios of each mode), (b) loss at 1030 nm of a cut-line through the fiber core, and (c) total pulse energy and peak power with propagation.

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Fig. 6.
Fig. 6. Spectral evolutions of the (top) fundamental ${{\rm LP}_{{01}}}$ mode and (bottom) summation of all higher-order modes (HOMs) in picosecond beam cleaning (Fig. 5). All spectra are normalized to one in the linear scale. White dashed line represents the transition from the amplification to the absorption regime.

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Equations (24)

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∂ z A p ( z , Ω ) = ( D ^ + G ^ + R ^ + N ^ ) A p ( z , Ω ) = i [ β p ( ω ) − ( β ( 0 ) + β ( 1 ) Ω ) ] A p ( z , Ω ) + g p ( z , Ω ) A p ( z , Ω ) + i ∑ ℓ Q p ℓ ( z ) A ℓ ( z , Ω ) + i ω n 2 c ∑ ℓ m n { f b S p ℓ m n R b F [ A ℓ [ h b ∗ ( A m A n ∗ ) ] ] } ( 1 − f R ) S p ℓ m n K F [ A ℓ A m A n ∗ ] + f R { f a S p ℓ m n R a F [ A ℓ [ h a ∗ ( A m A n ∗ ) ] ] + f b S p ℓ m n R b F [ A ℓ [ h b ∗ ( A m A n ∗ ) ] ] } } ,
S p ℓ m n K = 2 3 S p ℓ m n R a + 1 3 S p ℓ m n k ,
S p ℓ m n k = ∫ ( F → p ∗ ⋅ F → n ∗ ) ( F → ℓ ⋅ F → m ) d 2 x ,
S p ℓ m n R a = ∫ ( F → p ∗ ⋅ F → ℓ ) ( F → m ⋅ F → n ∗ ) d 2 x ,
S p ℓ m n R b = 1 2 [ ∫ ( F → p ∗ ⋅ F → m ) ( F → ℓ ⋅ F → n ∗ ) d 2 x + S p ℓ m n k ] ,
∂ z A → ( r → , Ω ) = g 0 / 2 1 + ∫ I ( r → , t ) d t I s a t f ( Ω ) P d ( r → ⊥ ) A → ( r → , Ω ) ,
∂ z A p ( z , Ω ) + ∑ ℓ m n S p ℓ m n R a B m n ( z ) [ ∂ z A ℓ ( z , Ω ) ] ≈ g 0 2 f ( Ω ) Γ p A p ( z , Ω ) ,
∑ ℓ T p ℓ ( z ) [ ∂ z A ℓ ( z , Ω ) ] = g 0 2 f ( Ω ) A p ( z , Ω ) ,
T p ℓ ( z ) = [ δ p ℓ + ∑ m n S p ℓ m n R a B m n ( z ) ] Γ p − 1 .
g p A p = [ T − 1 ( z ) g 0 2 f ( Ω ) A ( z , Ω ) ] p .
d I ± , μ ^ d z ( r → , ν ) = g ( r → , ν ) I ± , μ ^ ( r → , ν ) + σ e ( ν ) N 2 ( r → ) h ν ∑ p = 1 N m | F p μ ^ ( r → ⊥ ) | 2 ,
I s ± , μ ^ ( r → , ν = Ω + ω 0 2 π ) = | F [ A ± , μ ^ ( r → , t ) ] | 2 R r ( N Δ t ) 2 ,
N 2 ( r → ) = R a ( r → ) R a ( r → ) + R e ( r → ) + 1 τ N T ( r → ) ,
R i ( r → ) = ∑ μ ^ = x ^ , y ^ ∑ ± ∫ 1 h ν σ i ( ν ) I ± , μ ^ ( r → , ν ) d ν ,
I ± , μ ^ ( r → , ν ) = P p ± , μ ^ ( z , ν p ) A c l a d d i n g + ∫ [ I s ± , μ ^ ( r → , ν ) + I A S E ± , μ ^ ( r → , ν ) ] d ν ,
d P p ± d z ( z , ν p ) = P p ± ( z , ν p ) A c l a d d i n g ∫ A c o r e g ( r → , ν p ) d 2 x ,
d I s ± , μ ^ d z ( r → , ν ) = g ( r → , ν ) I s ± , μ ^ ( r → , ν ) ,
d I A S E ± , μ ^ d z ( r → , ν ) = g ( r → , ν ) I A S E ± , μ ^ ( r → , ν ) + σ e ( ν ) N 2 ( r → ) sh ν ∑ p = 1 N m | F p μ ^ ( r → ⊥ ) | 2 ,
R i ( r → ) = ∑ ± 1 h ν p σ i ( ν p ) P p ± ( z , ν p ) A c l a d d i n g + ∑ μ ^ = x ^ , y ^ ∑ ± ∫ 1 h ν σ i ( ν ) [ I A S E ± , μ ^ ( r → , ν ) + I s ± , μ ^ ( r → , ν ) ] d ν .
d P A S E ( p ) ± , μ ^ d z ( z , ν ) = P A S E ( p ) ± , μ ^ ( z , ν ) ∫ A c o r e g ( r → , ν ) | F p μ ^ ( r → ⊥ ) | 2 d 2 x + σ e ( ν ) sh ν ∫ A c o r e | F p μ ^ ( r → ⊥ ) | 2 N 2 ( r → ) d 2 x .
G p ( z , Ω ) = A p ± ( z ± Δ z , Ω ) A p ± ( z , Ω ) = ∑ n = 1 N m A n ± ( z , Ω ) A p ± ( z , Ω ) ∫ e ± g ( r → , ν ) Δ z F → p ∗ ( r → ⊥ ) ⋅ F → n ( r → ⊥ ) d 2 x ≈ 1 ± Δ z ∑ n = 1 N m A n ± ( z , Ω ) A p ± ( z , Ω ) ∫ g ( r → , ν ) F → p ∗ ( r → ⊥ ) ⋅ F → n ( r → ⊥ ) d 2 x ,
g p A p ± ≈ 1 2 ∑ n = 1 N m A n ± ( z , Ω ) ∫ A c o r e g ( r → , ν ) F → p ∗ ( r → ⊥ ) ⋅ F → n ( r → ⊥ ) d 2 x .
N ^ l i n e a r A x ∝ i F [ ( | A x | 2 + 2 3 | A y | 2 ) A x + O ( A x ∗ A y 2 ) ] ,
N ^ c i r c u l a r A + ∝ i F [ O ( | A + | 2 A + ) + O ( | A − | 2 A + ) ] .
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