Vector soliton dynamics in a high-repetition-rate fiber laser

Wei Lin; Wei Lin; Wei Lin; Wenlong Wang; Wenlong Wang; Wenlong Wang; Bin He; Bin He; Bin He; Bin He; Xuewen Chen; Xuewen Chen; Xu Hu; Yuankai Guo; Yuankai Guo; Yue Xu; Xiaoming Wei; Xiaoming Wei; Xiaoming Wei; Xiaoming Wei; Xiaoming Wei; Zhongmin Yang; Zhongmin Yang; Zhongmin Yang; Zhongmin Yang; Zhongmin Yang

doi:10.1364/OE.423811

1. Introduction

The mode-locked fiber laser has been rapidly developed and widely adopted in various fields for its outstanding performance, in terms of the long-term reliability [1,2], power scalability [3], frequency stability [4], to name a few. Recently, the high-repetition-rate (HRR) mode-locked fiber laser has attracted considerable attention [5–12] due to the rise in demand in the fields of ablation-cooled laser processing [13], astronomical spectroscopy [14], coherent optical communications [15], dual-comb spectroscopy [16,17], etc. To obtain continuous-wave mode-locking (CWML) in HRR mode-locked fiber lasers, enough pulse energy is required to overcome the Q-switched instability (QSI) [18–20], which is largely dominated by the modulation depth of saturable absorbers [21–24]. In this regard, in contrast to those artificial saturable absorbers, such as nonlinear polarization evolution (NPE) [25,26] and figure-9 scheme [27,28], saturable absorbers with shallower modulation depths, e.g., down to ∼1% [18,19], are beneficial for suppressing the HRR pulse instability as well as decreasing the mode-locking threshold. Among various saturable absorbers [5,10,29], the semiconductor saturable absorber mirror (SESAM) has been widely adopted in mode-locked fiber lasers for its reliability, flexibility, commercial quality [30]. Despite its promising advantages, the SESAM-based mode-locked fiber laser can easily suffer from time-varying polarization dynamics, as it allows oscillations in both polarization axes — a pressing issue in GHz-repetition-rate mode-locked fiber lasers [31].

In SESAM-based mode-locked lasers, solitons along both principal axes of birefringent fiber resonator can interact with each other through nonlinear effects, which are dominated by the coupled nonlinear Schrödinger equation (C-NLSE) [32,33]. The resultant polarization dynamics was firstly recognized by Cundiff et al. [34], and then experimentally [35,36] and theoretically [37,38] investigated in detail. The generation of vector solitons in mode-locked fiber lasers using SESAMs, as well as other kinds of saturable absorbers [39,40], substantially results from the formation of vectorial solitary waves in birefringent fibers, which can be categorized as group velocity locked vector soliton (GVLVS), polarization rotation vector soliton (PRVS) and polarization locked vector soliton (PLVS). To interpret the formation of vector solitons, two mechanisms have been proposed: firstly, group velocity locking based on soliton trapping effect [41–43] can impart frequency shift between the orthogonal polarization components and suppress the walk-off phenomenon induced by fiber birefringence [44]; secondly, the combined effect of self-phase modulation, cross-phase modulation and four-wave mixing in low-birefringence fiber cavities can give rise to the polarization (phase velocity) locking and result in elliptically polarized PLVS [45–47].

In contrast to the single-pass lightwave, the circulating lightwave in laser cavity can deliver versatile dynamics, such as period doubling [48], passive synchronization with multiple roundtrip times [49], coexistence of PLVS and PRVS [50], scalar-vector soliton [51], coexisting fundamental and high-order solitons [52], multi-scale polarization phenomena [53,54], vector-resonance-multimode instability [55], parametric-instability-assisted turbulent-like behavior [56,57]. Notably, new characterization technologies, such as fast polarization analyzer [58,59], time stretch [60], time-lens [61], as well as advanced data processing algorithms, e.g., Gerchberg-Saxton algorithm and its counterparts [62,63], nonlinear Fourier transform [64,65], have enabled the real-time measurement that can be applied to observe non-repeatable, polarization-resolved intracavity phenomena, e.g., vector self-pulsing [66], precessing state of polarization [67], optical (polarization) rogue wave [68–70], dynamic soliton trapping [71], frequency-oscillating vector soliton [72], vectorial spectral breathing [73,74], etc. The ability of observing polarization evolution dynamics, in return, can inspire further investigation of intracavity effects upon the vector soliton in mode-locked fiber laser and provide guideline of vector soliton manipulation [75–78].

Up to now, there are few works focusing on polarization dynamics in HRR mode-locked fiber laser [79]. In this work, we theoretically and experimentally investigate the dynamics of vector solitons in a GHz-repetition-rate mode-locked mini-cavity that has a compact architecture with excellent thermal control but limited polarization control [80]. Our numerical model can predict vector soliton dynamics that evolve from PRVS to linearly polarized soliton (LPS) by enhancing the intracavity fiber birefringence. It also gives rise to a cavity induced soliton trapping mechanism responsible for group velocity locking in HRR fiber laser. Subsequently, analytical approach is utilized to gain a deeper understanding of the roundtrip-to-roundtrip polarization dynamics of PRVS, and thereby illustrates the criterion for the formation of PLVS. Finally, experiments are carried out to validate these theoretical predictions, and a steady-state operation of LPS is obtained.

2. Theoretical analysis of vector solitons in a HRR fiber laser

The theoretical analysis includes two parts: 1) we will briefly outline the cavity modeling, based on which we numerically investigate the vector soliton dynamics by changing the intracavity fiber birefringence; 2) then we perform analytical calculation to gain a deeper insight into the time-varying polarization dynamics discovered in the numerical studies.

2.1 Numerical studies

In this study, we employ a Fabry-Pérot (FP) fiber laser with ultrashort cavity length, which can support an ultrahigh fundamental repetition rate [81]. By neglecting the standing wave effect, the forward- and backward-propagating light fields of the FP cavity with a cavity length of L can be treated as an unidirectionally-propagating light field in a ring cavity with a cavity length of $2L$ [20]. From distance $z = 0$ to $z = L$, the two orthogonal polarization components co-propagating in the gain fiber (i.e., Yb-doped fiber in this case) can be characterized by C-NLSE,

(1a)$$\begin{aligned} \frac{{\partial {u^{(n )}}({z,t} )}}{{\partial z}} &= ik{u^{(n )}} + g{u^{(n )}} - \delta \frac{{\partial {u^{(n )}}}}{{\partial t}} + \left( {\frac{g}{{{\mathrm{\Omega }^2}}} - \frac{{i\beta }}{2}} \right)\frac{{{\partial ^2}{u^{(n )}}}}{{\partial {t^2}}}\\ &+ i\gamma \left( {{{|{{u^{(n )}}} |}^2} + \frac{2}{3}{{|{{v^{(n )}}} |}^2}} \right){u^{(n )}} + i\frac{\gamma }{3}{v^{(n )}}^2{u^{(n )}}^\ast \end{aligned}$$

(1b)$$\begin{aligned} \frac{{\partial {v^{(n )}}({z,t} )}}{{\partial z}} &={-} ik{v^{(n )}} + g{v^{(n )}} + \delta \frac{{\partial {v^{(n )}}}}{{\partial t}} + \left( {\frac{g}{{{\mathrm{\Omega }^2}}} - \frac{{i\beta }}{2}} \right)\frac{{{\partial ^2}{v^{(n )}}}}{{\partial {t^2}}}\\ &+ i\gamma \left( {{{|{{v^{(n )}}} |}^2} + \frac{2}{3}{{|{{u^{(n )}}} |}^2}} \right){v^{(n )}} + i\frac{\gamma }{3}{u^{(n )}}^2{v^{(n )}}^\ast \end{aligned}$$

where the vector ${[{{u^{(n )}},{v^{(n )}}} ]^T}$ represents the field envelopes of the two polarization components in the n-th roundtrip. The wavenumber difference $k = \pi /{L_B}$ and group velocity difference $\delta = k{\lambda _c}/2\pi c$ are utilized to govern the fiber birefringence. The saturable gain is managed by

(2)$$g = \frac{{{g_0}}}{{1 + ({{{||{{u^{(n )}}({z,t} )} ||}^2} + {{||{{v^{(n )}}({z,t} )} ||}^2}} )\mathrm{\Delta }t/{E_g}}},$$

where $||\cdot ||$ is the L²-norm, $\mathrm{\Delta }t$ is the resolution in the t-axis. At the end of the gain fiber, the saturable absorber (i.e., SESAM) imparts a non-instantaneous intensity discrimination, which is given by

(3)$$\frac{{\partial q}}{{\partial t}} = \frac{{q - {q_0}}}{{{\tau _a}}} - q\frac{{{{|{{u^{(n )}}({L,t} )} |}^2} + {{|{{v^{(n )}}({L,t} )} |}^2}}}{{{E_a}}},$$

where the parameter q is related to the saturable loss of SESAM. The total loss of SESAM also includes unsaturable loss ${q_a}$. The light fields after the SESAM can be written as

(4)$$\left[ {\begin{array}{c} {{u^{(n )}}({{L^ + },t} )}\\ {{v^{(n )}}({{L^ + },t} )} \end{array}} \right] = \sqrt {1 - {q_a} - q} \left[ {\begin{array}{c} {{u^{(n )}}({L,t} )}\\ {{v^{(n )}}({L,t} )} \end{array}} \right]$$

where $z = {L^ + }$ is the location after the SESAM. After the reflection by the SESAM, the light fields propagate through the gain fiber, i.e., from $z = {L^ + }$ to $z = 2L.$ It leads to the vector ${[{{u^{(n )}}({2L,t} ),{v^{(n )}}({2L,t} )} ]^T}$ by solving Eq. (1), and the output ${[{{u^{(n )}}_o,{v^{(n )}}_o} ]^T}$ in the n-th round-trip becomes

(5)$$\left[ {\begin{array}{c} {{u^{(n )}}_o(t )}\\ {{v^{(n )}}_o(t )} \end{array}} \right] = \sqrt {{q_l}} \left[ {\begin{array}{c} {{u^{(n )}}({2L,t} )}\\ {{v^{(n )}}({2L,t} )} \end{array}} \right].$$

In the next roundtrip $({n + 1} )$, we then have

(6)$$\left[ {\begin{array}{c} {{u^{({n + 1} )}}({0,t} )}\\ {{v^{({n + 1} )}}({0,t} )} \end{array}} \right] = \sqrt {1 - {q_l}} \left[ {\begin{array}{c} {{u^{(n )}}({2L,t} )}\\ {{v^{(n )}}({2L,t} )} \end{array}} \right],$$

The computation is iteratively continued until a (quasi-) stationary solution is obtained. The key parameters used in the numerical studies are shown in Table 1. Please note that here the fiber laser is in the normal dispersion regime that supports gain-guided [82] and dissipative vector soliton [43,83,84].

Table 1. Key parameters used in the numerical studies

View Table

In an ultrashort FP fiber cavity, it is challenging to add polarization controllers, and thus the primary birefringence can mainly be dominated by the polarization-mode dispersion (PMD) of the Yb-doped gain fiber. The inherent beat length is usually in an order of several meter [85]. For the case with a beat length of 2 m (${L_B}/2L = 10$), the spectral and temporal characteristics of the vector solitons are illustrated in Fig. 1. As can be observed, the orthogonal polarization components have identical intensity profiles except the intensity discrepancy, resulting in a pulsewidth of ∼3.4 ps and a bandwidth of ∼1.6 nm, i.e., Figs. 1(a) and 1(b), respectively. To mimic the oscilloscopic waveforms in the experiments that involve polarization synthesis, linear transform is applied, i.e.,

(7)$$\left( {\begin{array}{c} {u^{\prime}}\\ {v^{\prime}} \end{array}} \right) = \left[ {\begin{array}{cc} {cos\theta }&{sin\theta }\\ { - sin\theta }&{cos\theta } \end{array}} \right]\left( {\begin{array}{c} u\\ v \end{array}} \right) = M\left( {\begin{array}{c} u\\ v \end{array}} \right).$$

The pulse train is expressed as $\sum\nolimits_i {{{|{{u^{^{\prime}(i )}}({t - i{T_R}} )} |}^2}}$ (${T_R}$ is the roundtrip time). Figure 1(c) shows typical numerical pulse trains that cover a time span of 45 ns, in which three different rotation angles $\theta $ of π/2, 1.1, and π/4 are utilized. In the case of $\theta = \pi /2$, a uniform pulse train is obtained. For the other cases, the pulse trains are intensity-modulated (i.e., $\theta = 1.1$ and $\pi /4$). These results imply that the time-varying polarization dynamics of the vector solitons can be mainly dominated by the phase shift between the two polarization components [i.e., $\sphericalangle(uv^\ast)$] instead of their intensity variation.

Fig. 1. Simulated output results of polarization rotation (PR) vector solitons for $L = 0.1,\; {L_B} = 2$. Temporal (a) and spectral (b) waveforms at slow and fast axes. The pulses at slow and fast axes have an identical pulsewidth of ∼3.4 ps. (c) Typical pulse trains of field $u^{\prime}$ represented by Eq. (7) with different rotation angles $\theta $. Here, the pulse trains are vertically offset for a better visualization.

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According to the numerical results, the condition of $2k \gg \gamma ({{P_u} - {P_v}} )/3$ indicates a non-negligible group velocity difference $\delta $, and thus a mechanism is required for interpreting the group velocity locking. As shown in Fig. 1(b), there is no obvious spectral translation between the orthogonally polarized solitons, which is unexpected as for the frequency shift soliton trapping (FS-ST) [41]. To understand the mechanism behind, we quantify the frequency shift by taking advantage of the first moment that is given by

(8)$$\Delta \omega = \left|{\frac{{\mathop \int \nolimits_{ - \infty }^\infty \omega {{|{\tilde{u}(\omega )} |}^2}d\omega }}{{\mathop \int \nolimits_{ - \infty }^\infty {{|{\tilde{u}(\omega )} |}^2}d\omega }} - \frac{{\mathop \int \nolimits_{ - \infty }^\infty \omega {{|{\tilde{v}(\omega )} |}^2}d\omega }}{{\mathop \int \nolimits_{ - \infty }^\infty {{|{\tilde{v}(\omega )} |}^2}d\omega }}} \right|,$$

where $\tilde{u}$ and $\tilde{v}$ are the Fourier transforms of u and v, respectively. The frequency shift $\Delta \omega $ with different cavity lengths is shown in Fig. 2(a). As can be observed, the frequency shift gradually increases with the cavity length and approaches an asymptotic value that is defined by a phenomenological formula

(9)$$\Delta \omega = a - b/2L,$$

The fitting parameters used in Eq. (9) are: $a = 0.28087\; THz$ and $b = 0.01165\; m/ps$. For comparison, the typical optical spectra at the slow and fast polarization axes are shown in Fig. 2(b), wherein two different fiber lengths are considered, i.e., 0.1 and 1 m. It is worth noting that, Eq. (9) can also be interpreted by

(10)$$2\delta = \beta \Delta \omega + \mathrm{\Delta }t/2L,$$

where, the first term on the right hand side is the group velocity difference that is compensated by the FS-ST effect; the second term is related to the cavity induced soliton trapping (CI-ST) effect. In terms of Eq. (10), the frequency shift at the limit of $L \to \infty $ is calculated to be $\Delta {\omega _{cal}} = 0.293\; THz$, close to the phenomenological result of $a\sim 0.281\; THz$, as shown in Fig. 2(a). Equations (9) and (10) imply a cavity induced time shift of $\mathrm{\Delta }t\sim 3.5 \times {10^{ - 4}}\; ps$, which compensates the PMD-induced walk-off. In this regard, by setting $\Delta \omega = 0$, the CI-ST effect is dominating and exclusively responsible for group velocity locking in the case of

(11)$${L_B}/2L > 10.$$

The soliton trapping effects that manage the group velocity locking are illustrated in Fig. 2(c), also the inset of Fig. 2(a). The CI-ST can be attributed to the non-instantaneous saturable absorption (SA) effect [86]. As the ‘averaged time shift’ $\mathrm{\Delta }t/2L$ between the two polarization components is comparable with the group velocity difference $2\delta $, such mechanism of binding the orthogonally polarized pulses can become predominant in ultrashort fiber cavities.

Fig. 2. Numerical studies of the soliton trapping effect and polarization dynamics. (a) Frequency shift $\varDelta \omega $ as a function of the cavity length. Here, the beat length is fixed at 0.4 m. Black dots are the numerical results calculated by using Eq. (8), while the error bars denote the fluctuations of $\varDelta \omega $; black dashed curve is the phenomenological fitting that approaches an asymptotic line (blue, indicated as Pheno.). The red dashed line (i.e., Cal.) indicates the frequency shift $\varDelta {\omega _{cal}} = 2\varDelta /\beta $ required for the exclusive FS-ST. The inset schematically shows the competitive mechanisms of FS-ST and CI-ST, which are responsible for group velocity locking. (b) Optical spectra in the cases of $L = 1\; m$ and $0.1\; m$, respectively. (c) Conceptual diagram of the HRR fiber laser with coexisting FS-ST and CI-ST effects. (d) Energy evolutions of PRVS and LPS. Here, the polarization dynamics of PRVS and LPS are also investigated by the Stokes vector on the Poincaré sphere.

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In the simulations, when the fiber birefringence is further enhanced, the energy difference between the two polarization components is increased, such that LPS is ultimately generated, as shown in Fig. 2(d), where the transition of LPS is observed by shortening the beat length from 2 to 0.2 m for a fiber length of $L = 0.1\; m$. Similar tendency to enlarge energy difference between orthogonally-polarized pulses via fiber birefringence enhancement has also been applied to MHz-repetition-rate fiber lasers [42,87]. For an intuitive understanding, the normalized Stokes parameters ${s_{n1}}$, ${s _{n2}}$, and ${s _{n3}}$ of the Poincaré sphere are provided as,

(12)$$\begin{array}{c} {{s_0} = {\boldsymbol{C}^\dagger }\boldsymbol{C},\; \; \; \; {s_1} = {\boldsymbol{C}^\dagger }{\boldsymbol{\sigma }_1}\boldsymbol{C},\; \; \; {s_2} = {\boldsymbol{C}^\dagger }{\boldsymbol{\sigma}_2}\boldsymbol{C},\; \; \; {s_3} = {\boldsymbol{C}^\dagger }{\boldsymbol{\sigma }_3}\boldsymbol{C},\; } \\ {s _{ni}} = {s_i}/{s_0}\; \; ({i = 1,\; 2,\; 3} ), \end{array}$$

where, the column vector $\boldsymbol{C} = {[{u,v} ]^T}$ represents the state of vectorial light fields, notation $\dagger $ is the complex conjugate transpose, ${\boldsymbol{\sigma }_{1 - 3}}$ are Pauli spin matrices, which are defined as

(13)$${\sigma _1} = \left[ {\begin{array}{cc} 1 &0\\ 0 &{ - 1} \end{array}} \right]\; \; \; \; \; {\sigma _2} = \left[ {\begin{array}{cc} 0 &1\\ 1 &0 \end{array}} \right]\; \; \; \; \; {\sigma _3} = \left[ {\begin{array}{cc} 0 &{ - i}\\ i &0 \end{array}} \right].$$

For PRVS, the polarization rotation dynamics exhibit a ring-like trajectory on the Poincaré sphere by approximately retaining a constant ${s _{n1}} = 0.11$, as shown in the top panel of Fig. 2(d). This highlights the polarization rotation resulted from the periodic variation of phase shift between the orthogonal polarization components, while the fluctuation of the intensity difference is weak. Besides, there is no visible modulation on the total pulse energy (i.e., $\int_{ - \infty }^\infty {({{{|{u(t )} |}^2} + {{|{v(t )} |}^2}} )dt}$) and Stokes parameter ${s_0}$, otherwise strong radiofrequency (RF) sidelobes could be observed (left panel of Fig. 7). This is largely different from the case after the polarization synthesis (right panel of Fig. 7). On the other side, in the case of LPS, the cyclical variation of the Stokes vector converges to a stationary point of $s_n = [{ - 1,0,0} ]$, i.e., the bottom panel of Fig. 2(d).

2.2 Analytical studies

The Stokes parameters introduced in Eq. (12) can also facilitate the analytical study of vector soliton by exploring the finite-dimensional ordinary differential equations (ODEs) [88,89]. Intuitively, the non-negligible higher-order fiber birefringence $\delta $ of our fiber cavity might violate the prerequisite of analytical model utilized in prior works [45,47]. However, in HRR mode-locked fiber laser, the complementary CI-ST effect can balance the group velocity difference, thereby creating an analogous ‘weak birefringence condition’. Hence, it is interesting to investigate the C-NLSE by neglecting the derivative terms that include the coefficient $\delta $, i.e.,

(14a)$$\frac{{\partial u({z,t} )}}{{\partial z}} = iku - \frac{{i\beta }}{2}\frac{{{\partial ^2}u}}{{\partial {t^2}}} + i\gamma \left( {{{|u |}^2} + \frac{2}{3}{{|v |}^2}} \right)u + i\frac{\gamma }{3}{v^2}{u^\ast },$$

(14b)$$\frac{{\partial v({z,t} )}}{{\partial z}} ={-} ikv - \frac{{i\beta }}{2}\frac{{{\partial ^2}v}}{{\partial {t^2}}} + i\gamma \left( {{{|v |}^2} + \frac{2}{3}{{|u |}^2}} \right)v + i\frac{\gamma }{3}{u^2}{v^\ast }.$$

Although Eq. (14) has neglected the dissipation terms that are stringently required for a stable soliton-like solution in the normal dispersion regime [90], it can still extract certain useful information of the vector soliton dynamics. By combining Eqs. (12) and (14), we have

(15a)$$\frac{d}{{dz}}\mathop \int \nolimits_{ - \infty }^\infty {s_1}dt ={-} \frac{{2\gamma }}{3}\mathop \int \nolimits_{ - \infty }^\infty {s_2}{s_3}dt,$$

(15b)$$\frac{d}{{dz}}\mathop \int \nolimits_{ - \infty }^\infty {s_2}dt = 2k\mathop \int \nolimits_{ - \infty }^\infty {s_3}dt + \frac{{2\gamma }}{3}\mathop \int \nolimits_{ - \infty }^\infty {s_1}{s_3}dt,$$

(15c)$$\frac{d}{{dz}}\mathop \int \nolimits_{ - \infty }^\infty {s_3}dt ={-} 2k\mathop \int \nolimits_{ - \infty }^\infty {s_2}dt.$$

Rigorously, it is required to isolate the $z$-dependent complex exponents to derive Eq. (15), detailed derivation can be found in prior works [45–47]. To further simplify the calculation of the integrals, an identical normalized complex function $f(t )$ is assumed, i.e.,

(16)$$\begin{array}{c} u({z,t} )= U(z )f(t )\; \; \; \; \; \; \; \; v({z,t} )= V(z )f(t )\\ with\; \; \; \; \; max[{|{f(t )} |} ]= 1. \end{array}$$

By substituting Eq. (16) into Eq. (15), it yields

(17)$$\frac{{d{S_1}}}{{dz}} ={-} 2\xi {S_2}{S_3},\; \; \; \; \; \; \frac{{d{S_2}}}{{dz}} = 2k{S_3} + 2\xi {S_1}{S_3},\; \; \; \; \; \; \frac{{d{S_3}}}{{dz}} ={-} 2k{S_2},$$

where, $\xi $ and ${S_i}\; ({i = 1,2,3} )$ are defined as

\xi = \frac{\gamma }{3}\frac{{\mathop \int \nolimits_{ - \infty }^\infty {{|{f(t )} |}^4}dt}}{{\mathop \int \nolimits_{ - \infty }^\infty {{|{f(t )} |}^2}dt}},

{S_0} = {|U |^2} + {|V |^2},\; \; {S_1} = {|U |^2} - {|V |^2},\; \; \; \; \; \; \; {S_2} = {U^\ast }V + U{V^\ast },\; \; \; \; \; \; {S_3} ={-} i({{U^\ast }V - U{V^\ast }} ).

Equation (17) has a set of solutions with rotation phases that comprise elliptic Jacobi functions [47]. Here, we focus on the case of $k/\xi > {S_0}$. In the limit of $\xi \to 0$, we leverage multi-scale analysis to expand the parameter ${S_1}$, i.e.,

(18)$${S_1} = {S_{10}} + \varepsilon {S_{11}},\; \; \; \; \; \xi = \varepsilon \xi ^{\prime},$$

the value of the parameter $\varepsilon $ is small to have ${S_{11}}$ and $\xi ^{\prime}$ at $\cal{O}$(1). Then, by collecting the terms at $\cal{O}$(1) and $\cal{O}(\varepsilon )$, we can rewrite Eq. (17) as,

(19a)$$\frac{{d{S_{10}}}}{{dz}} = 0,\; \; \; \; \; \; \frac{{d{S_2}}}{{dz}} = 2k{S_3},\; \; \; \; \; \; \frac{{d{S_3}}}{{dz}} ={-} 2k{S_2},$$

(19b)$$\frac{{d{S_{11}}}}{{dz}} ={-} 2\xi ^{\prime}{S_2}{S_3}.$$

In a concise formalism, the relevant solution becomes

(20a)$${S_1}/{S_0} = cos\phi + \frac{{\xi {S_0}si{n^2}\phi }}{{4k}}cos({4kz + 2\theta } ),$$

(20b)$${S_2}/{S_0} = sin\phi sin({2kz + \theta } ),$$

(20c)$${S_3}/{S_0} = sin\phi cos({2kz + \theta } ).$$

To validate Eq. (20), $z$-varying Stokes vector of the representative PRVS depicted in Fig. 1 is taken as an example, wherein numerical computation indicates pivotal parameters therein: $\xi = 6.134 \times {10^{ - 4}},\; {S_0} = 32.721$. $\phi $ and $\theta $ are free parameters determined by fitting to the numerical calculation ${S_3}/{S_0}$, i.e., $\phi = 1.46\; $ and $\theta = 1.4532$. As shown in Fig. 3(a), acceptable agreement between numerical simulation and analytical calculation is obtained by using Eq. (20), and the discrepancy can be attributed to the inevitable violation of the identity ${S_1}^2 + {S_2}^2 + {S_3}^2 = {S_0}^2$. Notably, the expression of ${S_1}/{S_0}$ can facilitate the understanding of the intensity difference $({{{|U |}^2} - {{|V |}^2}} )$ that oscillates at a frequency of twice the phase rotation frequency ${f_{PRF}} = 2{f_r}L/{L_B}$. Its modulation depth $\mathrm{\varDelta }{S_1}/{S_0}$, which is proportional to the beat length ${L_B}$, is also verified, as shown in Fig. 3(b). As a result, enhancing the linear fiber birefringence cannot only enlarge the intensity difference between orthogonal fields u and v, but also effectively suppress the $z$-dependent modulation $\mathrm{\varDelta }{S_1}$.

Fig. 3. Polarization dynamics dissected by the Stokes vector $[{{S_1}/{S_0},\; {S_2}/{S_0},{S_3}/{S_0}} ]$. (a) Evolutions of the Stokes parameters as a function of the propagation distance, i.e., numerical simulation (dots) and analytical calculation (solid curves) using Eq. (20). (b) Modulation amplitude $\mathrm{\varDelta }{S_1}/{S_0}$ as a function of ${L_B}/L$. The slope is determined to be $5.08 \times {10^{ - 4}}$ by a linear fit (dashed line) to the computed data (dots).

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In addition, the criterion of the polarization locking, i.e.,

(21)$$k/\xi < {S_0},$$

indicates a minimum beat length

(22)$$\frac{{min{{\{{{L_B}} \}}_{PL}}}}{L} = \frac{\pi }{{\xi {S_0}L}}\sim 794.$$

For ${L_B}/L = 800$, the generation of PLVS is numerically identified, as shown in Figs. 8(e)-(h). Despite the polarization locking verified in the numerical simulation, it is practically challenging to minimize the intracavity fiber birefringence and realize a beat length ${L_B} > 100\; \textrm{m}$ that permits PLVS formation. In another situation with a weak fiber birefringence (${L_B}/L = 500$), the polarization rotation renders a more evident intensity modulation with respect to each polarization component, as shown in Fig. 8(c). The corresponding evolutionary trajectory visualized on the Poincaré sphere (Fig. 8(d)) exhibits significant discrepancy from that of Fig. 2(d).

3. Experimental studies of vector solitons in a HRR mode-locked fiber laser

So far, we have theoretically investigated the polarization dynamics of the HRR mode-locked fiber laser, and now we perform experimental studies of vector solitons by using an experimental configuration shown in Fig. 4(a). The laser cavity has a linear all-fiber configuration similar to that of prior works [11,91]. A short cavity length, i.e., about 8.3 cm, enables a fundamental repetition rate of up to 1.3 GHz. The gain medium is a piece of highly Yb³⁺-doped fiber, both ends of which are sealed in ceramic ferrules. A dichroic film is coated on a fiber ferrule that connects with one end of the gain fiber. The dichroic film has a transmission of 98% at the pump wavelength and a reflectivity of ∼74% at the signal wavelength (i.e., 1.0 µm). The other end of the gain fiber is attached to a SESAM (Batop SAM-1040-8-1ps). To adjust the fiber birefringence of the laser cavity, we carefully twist the gain fiber at the end that is close to the SESAM. The laser is pumped by a 976-nm laser diode (LD, LUMENTUM S27-7402-460) through a wavelength-division multiplexer (WDM). A fiber isolator (ISO) is utilized to prevent backward-propagating light. Polarization controller (PC), placed before a polarization beam splitter (PBS), is employed to perform polarization synthesis. The PBS separates the two polarization components along the slow and fast axes of the polarization maintaining (PM) fiber, and thus enables polarization-resolved measurement.

Fig. 4. Characteristics of vector soliton dynamics in experiment. (a) Configuration of the GHz-repetition-rate mode-locked fiber laser. Inset: details of birefringence adjustment, the fiber is twisted by manually rotating the ferrule near SESAM and subsequently being fixed. SESAM, semiconductor saturable absorber mirror; GF, gain fiber; DF: dichroic film; WDM, wavelength-division multiplexer; ISO, isolator; PC, polarization controller; PBS, polarization beam splitter. (b) Radiofrequency spectra measured at port 1 of PBS by gradually twisting the gain fiber, curves 1-4 indicate the output states evolving from PRVS (traces 1-3) to LPS (trace 4). The 12-roundtrip (12-RT) PRVS corresponding to trace 3 is further characterized in (c) and (d). (c) Oscilloscopic traces of PRVS with different PC orientations. (d) Typical optical spectra measured at port 1 and 2 of PBS.

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By gradually twisting the gain fiber, we can manipulate the sidelobes of the RF spectrum, as shown in Fig. 4(b), e.g., the frequencies of the sidelobes are gradually detuned from the fundamental frequency, i.e., 7.5 MHz, 48.5 MHz and 109.5 MHz for curves 1, 2, and 3, respectively. The details of birefringence adjustment are illustrated in the inset of Fig. 4(a). These detuning frequencies correspond to polarization rotation periods of 173, 27, and 12 roundtrips (RTs), respectively. Further enhancement of the gain fiber birefringence can suppress the frequency sidelobes and thus allow LPS operation, i.e., curve 4 of Fig. 4(b). In general, the evolution of the phase shift between the two polarization components can be detected by transferring to intensity modulation through polarization synthesis, i.e., linear coupling between the two polarization components. In this regard, Fig. 4(c) illustrates oscilloscopic traces that evolve from a uniform pulse train to a modulated one through polarization synthesis for the case of 12-RT PRVS. The experimental measurements are well matched with the numerical results as shown in Fig. 1(c). The characteristics of optical spectra of the two polarization components, i.e., Fig. 4(d), exhibits similar features as that of numerical simulations, as shown in Fig. 1(b). For the case of LPS, i.e., Fig. 5, the pulse train is always uniform, regardless of polarization synthesis, as shown in Fig. 5(c). This can also be intuitively verified by back-to-back compared with the simulated results, i.e., Figs. 5(a) and (b).

Fig. 5. Characteristics of LPS. Typical pulse train of LPS, including experimental measurement (a) and numerical simulation (b). (c) Oscilloscope traces with different PC orientations.

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So far, we have accessed diverse vector soliton dynamics from PRVS to LPS in the HRR mode-locked fiber laser by twisting the gain fiber. However, by this means, it is hard to retain the manipulated fiber birefringence unchanged for a long operating time. To tackle with this problem, we here propose another experimental approach to ensure the fiber birefringence enhancement during fiber assembly. The approach outlined in Fig. 6(a) contains three steps: 1) inserting one end of the gain fiber into a zirconia ferrule and gluing them only at the exit end of the ferrule; 2) twisting the gain fiber to generate the twist-induced birefringence; 3) gluing the twisted gain fiber at the entrance end of the ferrule. In this way, the established HRR mode-locked fiber laser can deliver reliable linearly polarized output in a non-polarization-maintaining configuration (in contrast to PM architectures in [8,92]), as shown in Figs. 6(b)–(d). Typically, the CWML threshold of such a twisted fiber laser is about ∼100 mW, see Fig. 6(b). By further increasing the pump power to about 155 mW, it operates in the mode-locking regime with an output power of ∼8 mW. After mode-locking, typical optical spectra from port 1 and 2 of PBS are shown in Fig. 6(c) and imply a good polarization extinction ratio. Simultaneously, the state of polarization mapping on the equator of the Poincaré sphere confirms the linearly polarized state, as visualized in the inset of Fig. 6(c). The autocorrelation trace of the LPS pulse train is also measured and shown in Fig. 6(d), and a 3.5-ps pulse duration is obtained, assuming a sech² pulse shape.

Fig. 6. Method of birefringence enhancement during fiber assembly and the achieved results of LPS operation. (a) Left: closeup of one end of the laser cavity, where the induced birefringence enhancement is applied by gluing the twisted fiber in the ferrule. Right: steps for fiber assembly. (b) Output power as a function of the pump power. QSML, Q-switched mode-locking; CWML, continuous-wave mode-locking. (c) Optical spectra measured at port 1 and 2 of PBS. Inset: the degree of polarization measured by a polarimeter, where the red dot located close to (1,0,0) of the Poincaré sphere suggests a linear polarization state. (d) Autocorrelation trace.

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4. Conclusion

In conclusion, we have investigated vector soliton dynamics in HRR mode-locked fiber laser. Firstly, the numerical simulation unveils the CI-ST effect on the group velocity locking in the ultrashort cavity laser, complementary to the universal FS-ST mechanism. Then, the transition from PRVS to LPS is numerically obtained by enhancing the fiber birefringence of the HRR mode-locked fiber laser. Analytical calculation is simultaneously utilized to gain a deeper insight into the PRVS evolution as well as the criterion of PLVS formation. Finally, switching from PRVS to LPS is experimentally demonstrated in a HRR mode-locked fiber laser, which largely validates the theoretical findings. Remarkably, a general and promising protocol for fabricating a reliable LPS HRR mode-locked fiber laser is also proposed.

Appendix 1 – Fourier transform of pulse energies before and after the polarization synthesis

The bottom row of Fig. 7 illustrates the Fourier transforms of the pulse energies in the simulation, while the corresponding pulse trains are plotted on the top row. A weak intensity oscillation, which is not obvious on the pulse train without polarization synthesis, i.e., the top panel of Fig. 7(a), gives rise to RF sidelobes located at $2{f_{PRF}}$ and with a contrast ratio of ∼65 dB. Once the polarization synthesis is applied, the pulse train exhibits strong intensity modulation that is synchronized with the period of polarization rotation, i.e., the top panel of Fig. 7(b), resulting in RF sidelobes at ${\pm} {f_{PRF}}$, as shown on the bottom panel of Fig. 7(b).

Fig. 7. Simulated pulse trains and the relevant Fourier transforms of pulse energies (a) before and (b) after polarization synthesis.

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Appendix 2 – More details about the shot-to-shot polarization dynamics for weak birefringence

Figure 8 shows the vector solitons in the cases with weak birefringence. Notably, in contrast to the time-varying polarization dynamics dominated by the phase rotation between the orthogonally polarized components, the intensity modulation of the pulse train becomes more dramatic for ${L_B}/L = 500$, as illustrated in Fig. 8(c). Correspondingly, the polarization evolution of the PRVS on the Poincaré sphere is distinguishable from the meridian, see Fig. 8(d). By further reducing the fiber birefringence, stationary elliptically polarized soliton is obtained for ${L_B}/L = 800$. The stability of the vector solitons is numerically analyzed, as shown in Fig. 8(g). In this case, a fixed point on the Poincaré sphere with approximately zero ${s _{n2}}$ manifests the formation of PLVS, i.e., Fig. 8(h).

Fig. 8. Characteristics of vector solitons with weak birefringence. (a) Temporal profiles and (b) optical spectra of the PRVS for the case of ${L_B}/L = 500$. (c) Corresponding temporal and (d) polarization evolutions at the slow axis. (e-h) Characteristics for the case of ${L_B}/L = 800$.

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Funding

National Natural Science Foundation of China (U1609219); Special Project for Research and Development in Key areas of Guangdong Province (2018B090904001, 2018B090904003); Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X137); Science and Technology Planning Project of Guangdong Province (2017B030314005); NSFC Development of National Major Scientific Research Instrument (61927816).

Disclosures

The authors declare no conflict of interest.

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1. General parameters	Value
Fiber length (L, cm)	Variable; 10 in Figs. 1, 3
Central wavelength ( $λ_{c}$ , nm)	1054
Output ratio at the dichroic film ( $q_{l}$ , %)	25
2. Gain fiber’s parameters	Value
Small-signal gain ( $g_{0}$ , m⁻¹)	20
Gain saturation energy ( $E_{g}$ , pJ)	6
Gain bandwidth in angular frequency ( $Ω$ , THz)	7π ^a
Beat length ( $L_{B}$ , m)	Variable; 2, 0.4 in Figs. 1, 2
Second-order dispersion ( $β$ , ps²/m)	0.03
Nonlinearity ( $γ$ , W⁻¹m⁻¹)	0.0025
3. SESAM’s parameters	Value
Unsaturable loss ( $q_{a}$ )	0.03
Modulation depth ( $q_{0}$ )	0.05
Absorption saturation energy ( $E_{a}$ , pJ)	5.65
Relaxation time ( $τ_{a}$ , ps)	1

Vector soliton dynamics in a high-repetition-rate fiber laser

Abstract

Corrections

1. Introduction

2. Theoretical analysis of vector solitons in a HRR fiber laser

2.1 Numerical studies

2.2 Analytical studies

3. Experimental studies of vector solitons in a HRR mode-locked fiber laser

4. Conclusion

Appendix 1 – Fourier transform of pulse energies before and after the polarization synthesis

Appendix 2 – More details about the shot-to-shot polarization dynamics for weak birefringence

Funding

Disclosures

References

Cited By

Figures (8)

Tables (1)

Equations (31)

Optics Express