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Manipulating the polarization dynamics in a >10-GHz Er3+/Yb3+ fiber Fabry-Pérot laser

Wei Lin, Xuewen Chen, Xu Hu, Tao Luo, Yiheng Fan, Wenlong Wang, Zhaoheng Liang, Lin Ling, Molei Hao, Xiaoming Wei, and Zhongmin Yang

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Abstract

In this work, we report on the vector and scalar soliton dynamics that result from inevitable fiber birefringence in an 8-mm Er3+/Yb3+ fiber based Fabry-Férot (FP) laser that has a free spectral range of up to 12.5 GHz. The generation of polarization-evolving vector solitons can largely degrade the performance of application systems, and the underlying mechanisms and manipulation technologies are yet to be explored. To realize the transition from vector to scalar (linearly polarized) state, we here incorporate the polarization selection effect (PSE) in the simulation model and the numerical results verify that only a small amount of PSE is sufficient for manipulating the soliton dynamics. It also reveals that, prominent polarization-dependent intensity discrimination can be acquired via geometry-induced oblique incidence to the Bragg mirror of the semiconductor saturable absorber mirror (SESAM), and we obtain switchable operating states by tilting the SESAM in the experiments. These efforts create a feasible method to manipulate high-repetition-rate pulse and may shed light on understanding the dissipative soliton dynamics in ultrafast fiber FP lasers.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Soliton mode-locking provides a powerful mechanism [1,2] to generate ultra-low-noise ultrashort pulses [3,4], in which a large amount of longitudinal modes (e.g., >105) are perfectly phase-locked. Notably, high-repetition-rate (HRR) solitons, which are separated by a short time interval (e.g., <ns), stand out in the areas requiring high refresh rate or radiofrequency bandwidth. HRR soliton lasers, on one hand, exhibit a remarkable reduction of pulse energy in comparison with the low-repetition-rate counterparts; on the other hand, they enable high average power through using rare-earth-ion doped fiber amplification [5–7]. The research on high-power GHz-level-HRR pulsed laser, especially, is largely motivated by the recent progress on femtosecond laser ablation [8] that is anticipated to be promising for high-speed and precise micromachining [9], as well as other frontier applications, like astrophysical spectrometer calibration [10,11] and arbitrary optical waveform processing [12]. More importantly, HRR pulsed lasers operating at microwave repetition rate, particularly in the X- and K-band (i.e., 8∼27 GHz), can facilitate the generation of low-noise microwave signal [13,14] and bridge optical frequency and microwave [15,16]. Furthermore, mediated by the optical frequency synthesis/division [17,18], HRR pulsed lasers become crucial for cutting-edge microwave photonic applications, including optical atomic clock [19,20], photonic radar [21], photonic analog-to-digital converter [22], 5G mobile communication [23], and etc.

Different technologies can be applied to generate stable solitons with electronic repetition rate covering the X- and K-band, including solid-state mode-locked laser, high-Q microresonator, and mode-locked fiber laser (MLFL). (1) Solid-state mode-locked laser: it demonstrated the pioneering experiments of 10-GHz self-referenced optical frequency comb [24]. Although the solid-state mode-locked laser exhibits superior noise performance in comparison with other schemes [3], it usually requires delicate optical alignment [25]. As the repetition rate scales up to >10 GHz, it becomes more difficult to realize the transition from the Q-switched mode-locked (QSML) to continuous-wave mode-locked (CWML) state [26]. (2) High-Q microresonator: it opens new avenue to generate solitons with ultrahigh repetition rates (e.g., tens of GHz to THz) [27], since the power threshold of soliton formation decreases with increasing repetition rate [28]. Although both crystalline and SiN microresonators with X- and K-band repetition rates have been reported [13,14,29,30], it is still technically challenging to fulfill scalable fabrication and manage the severe thermal effect of the centimeter-scale microresonator. (3) MLFL: By adopting the heavily-doped gain fiber [31–33], Fabry-Pérot (FP) fiber laser has been verified to be promising for generating solitons with >10-GHz repetition rates [34–38], as it advances in reliability, packaging footprint, thermal management, beam quality, etc. However, both theoretical and experimental investigations are still required to tackle the problems existed in fiber FP laser with GHz repetition rates, including unexpectedly low pulse energy in CWML [37], Q-switching pulse dynamics [39], vectorial behavior of solitons, and soliton trapping mechanism [40].

Being substantially distinctive from dissipative Kerr solitons (DKSs) in coherently-driven passive Kerr microresonators, dissipative solitons in MLFLs are the stationary localized structure formed by the interplay of dynamic saturable gain and loss, dispersion, nonlinearity [41,42], which are underpinned by the complex Ginzburg-Landau equation, instead of the Lugiato-Lefever equation [43]. Compared to the parametric gain with femtosecond-scale response time for DKS, rare-earth doped gain fibers have fairly long lifetimes, i.e., µs∼ms. Thus, the gain dynamics, as the origin of QSML, should be considered in the rare-earth-doped fiber laser [44]. Furthermore, because of the intrinsic fiber birefringence, vectorial nature of HRR solitons, resulted from polarization-mode dispersion, fiber chromatic dispersion, nonlinear birefringence (i.e., cross-phase modulation), and dynamic saturable absorption, should be further explored [45,46]. It is pivotal to understand versatile polarization dynamics existed in the cases with moderate fiber birefringence [40], as well as to deal with the polarization rotating phenomenon that disrupts the stable operation. However, there is by far few studies on vectorial dynamics in centimeter-scale mode-locked fiber lasers operating with >10 GHz fundamental repetition rate).

In this paper, we theoretically and experimentally study the features of vector and scalar solitons in an Er3+/Yb3+ fiber FP laser with a fundamental repetition rate of >10 GHz. In order to manipulate the polarization dynamics, resonator geometry engineering is applied to obtain either normal or oblique incidence on the semiconductor saturable absorber mirror (SESAM), such that the intracavity polarization selective effect (PSE) is induced to switch between the vector and scalar solitons without destructing the all-fiber configuration. From a practical viewpoint, the proposed method can enable removing the polarization rotating dynamics in ultrashort fiber resonator. Moreover, the linearly polarized (scalar) soliton state is characterized in details, and good long-term stability is demonstrated. The phase noise of the generated Ku-band microwave signal at 12.5 GHz is found to be comparable with that produced by other schemes at high-frequency band (typically >100 kHz).

2. Theoretical analysis of vector solitons in ultrashort Er3+/Yb3+ fiber resonator with geometry engineering

Figure 1(a) shows the configuration of fiber FP resonator that consists of a dielectric film (DF), a short segment of highly-doped Er3+/Yb3+ gain fiber (GF), and a SESAM. Geometry engineering is exploited to produce an asymmetric fiber end-facet [inset of Fig. 1(a)], which permits either normal or oblique incidence on the SESAM, as schematically shown in Fig. 1(c). In the following, we first analyze the PSE in the case of oblique incidence. Then, we incorporate the polarization-dependent reflectance to model the fiber FP resonator, such that vectorial features of the solitons are numerically investigated.

 figure: Fig. 1.

Fig. 1. (a) Configuration of the SESAM mode-locked Er3+/Yb3+ fiber Fabry-Pérot (FP) resonator. DF, dielectric film; GF, gain fiber; SESAM, semiconductor saturable absorber mirror. Inset: side view of the fiber end-facet attached to the SESAM with interior angles ${\theta _1}$ and ${\theta _2}$. (b) Photo of the ultrashort fiber resonator with geometry engineering. (c) Normal and oblique incidences caused by geometry engineered ferrule. (d) Reflectance of the SESAM for p- and s-polarized lights. Solid and dashed curves are plotted according to Eq. (5) and Eqs. (1 –3), respectively. Inset: the SESAM with tilted input, assuming that orthogonally polarized components u and v are parallel to the orientations of TE (s-polarized) and TM (p-polarized) modes, respectively. (e) Quantitative description of polarization selection for different angles of incidence. Top panel: contrast ratio (CR) between reflectance ${\mathrm{{\cal R}}_s}$ and ${\mathrm{{\cal R}}_p}$ for different angles of incidence ${\theta _0}$. Bottom panel: energy ratio evolution of s-polarized soliton with varying ${\theta _0}$. Regimes of vector and scalar solitons, together with the critical CR for polarization switching (i.e., ${\theta _0}\sim {3^\circ }$), are shown for clarity.

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2.1 PSE induced by oblique incidence on SESAM

For thin films, the oblique incidence can break the symmetry of the reflectance with respect to lights that have different orientations of polarization, yielding the PSE [47]. The SESAM, as a nonlinear mirror, is a combination of saturable absorber and thin-film Bragg mirror with quarter-wavelength structure [48]. Here we interpret the PSE on the SESAM via calculating the polarization-dependent reflectance, i.e.,

$${\mathrm{{\cal R}}_{p,s}} = {\left|{\frac{{{\eta_{G({p,s} )}} - {\eta_{I({p,s} )}}}}{{{\eta_{G({p,s} )}} + {\eta_{I({p,s} )}}}}} \right|^2},$$
where $\mathrm{{\cal R}}$ represents the reflectance, and the subscripts $p,s$ designate the p- and s-polarized components, respectively. ${\eta _G}$ and ${\eta _I}$ denote the admittance of the incident medium (i.e., the GF here) and input optical admittance, respectively. They are given as
$${\eta _{I({p,s} )}} = \frac{{{M_{21}} + {M_{22}}{\eta _{H({p,s} )}}}}{{{M_{11}} + {M_{12}}{\eta _{H({p,s} )}}}},\; M = \left[ {\begin{array}{cc} {{M_{11}}}&{{M_{12}}}\\ {{M_{21}}}&{{M_{22}}} \end{array}} \right] = H{({LH} )^m},$$
$$H = \left[ {\begin{array}{cc} {cos{\delta_H}}&{isin{\delta_H}/{\eta_H}}\\ {i{\eta_H}sin{\delta_H}}&{cos{\delta_H}} \end{array}} \right],\; \; L = \left[ {\begin{array}{cc} {cos{\delta_L}}&{isin{\delta_L}/{\eta_L}}\\ {i{\eta_L}sin{\delta_L}}&{cos{\delta_L}} \end{array}} \right],$$
$${\eta _{G(p )}} = \frac{{{n_G}}}{{cos{\theta _0}}},\; \; {\eta _{G(s )}} = {n_G}cos{\theta _0},$$
$${\eta _{H/L(p )}} = \frac{{n_{H/L}^2}}{{\sqrt {n_{H/L}^2 - n_G^2si{n^2}{\theta _0}} }},\; \; \; {\eta _{H/L(p )}} = \sqrt {n_{H/L}^2 - n_G^2si{n^2}{\theta _0}} $$
where the subscripts G, H, and L designate the media of GF, high- and low-index films, respectively. n is the refractive index, ${\theta _0}$ is the angle of incidence, $\delta $ is the phase shift when light travels through a single-layer film. For 10 pairs of alternative high- and low-index films in quarter-wavelength structure here [48], we have $m = 10$, and
$${\delta _{H/L}} = \frac{{\pi \sqrt {n_{H/L}^2 - n_G^2si{n^2}{\theta _0}} }}{{2{n_{H/L}}}}$$

For a small angle of incidence, we simplify Eqs. (1-3) by utilizing the approximation

$${\delta _{H/L}}\sim \pi /2,$$
and we have
$${{\cal{R}}_p}({\theta _0}) = \frac{{\frac{{{n_G}}}{{cos{\theta _0}}} - \frac{{n_H^2}}{{\sqrt {n_H^2 - n_G^2si{n^2}{\theta _0}} }}{{\left( {\frac{{{n_H}}}{{{n_L}}}} \right)}^{4m}}{{\left( {\frac{{n_L^2 - n_G^2si{n^2}{\theta_0}}}{{n_H^2 - n_G^2si{n^2}{\theta_0}}}} \right)}^m}}}{{\frac{{{n_G}}}{{cos{\theta _0}}} + \frac{{n_H^2}}{{\sqrt {n_H^2 - n_G^2si{n^2}{\theta _0}} }}{{\left( {\frac{{{n_H}}}{{{n_L}}}} \right)}^{4m}}{{\left( {\frac{{n_L^2 - n_G^2si{n^2}{\theta_0}}}{{n_H^2 - n_G^2si{n^2}{\theta_0}}}} \right)}^m}}},$$
$${{\cal{R}}_s}({\theta _0}) = \frac{{{n_G}cos{\theta _0} - \sqrt {n_H^2 - n_G^2si{n^2}{\theta _0}} {{\left( {\frac{{n_H^2 - n_G^2si{n^2}{\theta_0}}}{{n_L^2 - n_G^2si{n^2}{\theta_0}}}} \right)}^m}}}{{{n_G}cos{\theta _0} + \sqrt {n_H^2 - n_G^2si{n^2}{\theta _0}} {{\left( {\frac{{n_H^2 - n_G^2si{n^2}{\theta_0}}}{{n_L^2 - n_G^2si{n^2}{\theta_0}}}} \right)}^m}}},$$

Both complete and reduced interpretations of the reflectance ${\mathrm{{\cal R}}_{p,s}}$, i.e., Eqs. (1–3) and Eq. (5), are plotted in Fig. 1(d), where the key parameters used in the computation include ${n_G} = 1.5$, ${n_L} = 3$, ${n_H} = 3.5$. The calculated results shown by solid and dashed curves validate the rationality of Eq. (4) for ${\theta _0} < {10^ \circ }$

$$CR({{\theta_0}} )= 10lg[{{\mathrm{{\cal R}}_s}({{\theta_0}} )/{\mathrm{{\cal R}}_p}({{\theta_0}} )} ],$$

It can be used to quantitatively evaluate the strength of PSE. Consequently, for ${\theta _0} < {10^ \circ }$, the CR is computed to be less than 0.025 dB [top panel of Fig. 1(e)], implying a nearly polarization-independent reflectance. However, the PSE, albeit weak, has not been explored for vector soliton studies in ultrashort fiber resonators.

2.2 Characteristics of vector solitons in ultrashort Er3+/Yb3+ fiber FP laser

In this section, we establish a numerical model of ultrashort Er3+/Yb3+ fiber FP resonator to examine the influence of the PSE on the formation of vector solitons. By unfolding the bidirectional linear configuration [40], the fiber FP resonator can be treated as unidirectional ring fiber cavity, incorporating DF, SESAM, two segments of GFs (c.f. [44]). The master equations with regard to discrete intracavity elements are provided as:

At $z \in ({0,L} )\cup ({L,2L} )$,

$$\begin{aligned}\frac{{\partial {u^{(n )}}({z,t} )}}{{\partial z}} &= ik{u^{(n )}} + g{u^{(n )}} - \delta \frac{{\partial {u^{(n )}}}}{{\partial t}} + \left( {\frac{g}{{{\mathrm{\Omega }^2}}} - \frac{{i\beta }}{2}} \right)\frac{{{\partial ^2}{u^{(n )}}}}{{\partial {t^2}}}\\&+ i\gamma \left( {{{|{{u^{(n )}}} |}^2} + \frac{2}{3}{{|{{v^{(n )}}} |}^2}} \right){u^{(n )}} + i\frac{\gamma }{3}{v^{(n )}}^2{u^{(n )}}^\ast ,\end{aligned}$$
$$\begin{aligned}\frac{{\partial {v^{(n )}}({z,t} )}}{{\partial z}} &={-} ik{v^{(n )}} + g{v^{(n )}} + \delta \frac{{\partial {v^{(n )}}}}{{\partial t}} + \left( {\frac{g}{{{\mathrm{\Omega }^2}}} - \frac{{i\beta }}{2}} \right)\frac{{{\partial ^2}{v^{(n )}}}}{{\partial {t^2}}}\\&+ i\gamma \left( {{{|{{v^{(n )}}} |}^2} + \frac{2}{3}{{|{{u^{(n )}}} |}^2}} \right){v^{(n )}} + i\frac{\gamma }{3}{u^{(n )}}^2{v^{(n )}}^\ast ,\end{aligned}$$
$$g = \frac{{{g_0}}}{{1 + \smallint ({{{|{{u^{(n )}}({z,t} )} |}^2} + {{|{{v^{(n )}}({z,t} )} |}^2}} )dt/{E_g}}}.$$

At $z = L,$

$$\left[ {\begin{array}{c} {{u^{(n )}}({{L^ + },t} )}\\ {{v^{(n )}}({{L^ + },t} )} \end{array}} \right] = {e^{\frac{1}{2}\alpha ({q - {q_0}} )}}\sqrt {{\mathrm{{\cal R}}_s}({{\theta_0}} )- q} \left[ {\begin{array}{cc} 1&0\\ 0&{{\mathrm{{\cal R}}_p}({{\theta_0}} )/{\mathrm{{\cal R}}_s}({{\theta_0}} )} \end{array}} \right]\left[ {\begin{array}{c} {{u^{(n )}}({L,t} )}\\ {{v^{(n )}}({L,t} )} \end{array}} \right],$$
$$\frac{{\partial q}}{{\partial t}} = \frac{{q - {q_0}}}{{{\tau _a}}} - q\frac{{{{|{{u^{(n )}}({L,t} )} |}^2} + {{|{{v^{(n )}}({L,t} )} |}^2}}}{{{E_a}}}.$$

At $z = 2L,$

$$\left[ {\begin{array}{c} {{u^{({n + 1} )}}({0,t} )}\\ {{v^{({n + 1} )}}({0,t} )} \end{array}} \right] = \sqrt {1 - {q_l}} \left[ {\begin{array}{c} {{u^{(n )}}({2L,t} )}\\ {{v^{(n )}}({2L,t} )} \end{array}} \right].$$
where ${u^{(n )}}$ and ${v^{(n )}}$ are the slowly-varying field envelopes on the slow and fast axes in the n-th roundtrip (RT). The fiber birefringence causes the differences in wavenumber and group velocity, i.e., $k = \pi /{L_B}$ and $\delta = k{\lambda _c}/2\pi c$. To include the polarization-dependent reflectance on the SESAM for oblique incidence, a diagonal matrix accounting for the PSE, i.e., $({{\mathrm{{\cal R}}_s}/{\mathrm{{\cal R}}_P}} )$ is introduced. Without loss of generality, we assume that the fields u and v are consistent for the s- and p-polarized components in Eq. (8a), respectively. In this way, the relationship between the feature of vector solitons and angle of incidence ${\theta _0}$ can be dissected. In the simulation, the values of the key parameters are listed in Table 1.

Tables Icon

Table 1. Key parameters used for simulating the fiber FP resonator with a repetition rate of 12.5 GHz

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Figure 2 showcases different polarization dynamics by changing the angle of incidence ${\theta _0}$. The calculated optical spectra, illustrated in Figs. 2(a) and (b), identify the formations of vector solitons (state 1) and scalar soliton (state 2) for normal and oblique incidence, respectively. The shot-to-shot evolution of polarization ellipse, in the case of normal incidence [inset of Fig. 2(a)], manifests the RT-evolving polarization dynamics and thereby identifies the formation of polarization-rotating vector solitons (PRVSs). In the case of PRVSs, the orthogonally-polarized components parallel to the slow and fast axes of the birefringent GF have almost identical spectral bandwidth (i.e., ∼4 nm), and undergo periodic variation of relative phase difference by retaining only a weak intensity difference accordingly. The weak intensity difference is evident by comparable energy ratios of 0.52 and 0.48 for components on the slow and fast axes [top panel of Fig. 2(c)]. The well-preserved symmetry in intensity throughout the polarization rotation is also quantitatively depicted by the Stokes parameter ${s_1}$ of <0.07 [bottom panel of Fig. 2(c) for state 1]. Such evolving behavior of PRVSs can be analogous to the counterparts existed in normal-dispersion GHz fiber laser [40]. After substituting ${\theta _0} = {6^\circ }$ for ${\theta _0} = {0^\circ }$, s-polarized field (parallel to the slow axis) attenuates less due to the action of PSE, and its energy ratio then gradually increases to ∼1 after ∼6000 RTs [top panel of Fig. 2(c) for state 2]. The deterministic switching to the linear polarization is simultaneously demonstrated by the value of ${s_1}$, which grows from ∼0.05 to 1. The relevant mapping of the evolutionary Stokes parameters on the Poincaré sphere is shown in Fig. 2(d).

 figure: Fig. 2.

Fig. 2. (a) Optical spectra of the orthogonally polarized states along slow and fast axes for normal incidence ${\theta _0} = {0^\circ }$ (labelled as state 1). Inset depicts the ensemble of polarization ellipses for 10-RT polarization rotating vector solitons (PRVSs). (b) Optical spectrum of scalar solitons, polarized on the slow axis for oblique incidence ${\theta _0} = {6^\circ }$ (labelled as state 2). (c) Dynamic evolution when switching from state 1 to 2. Top and bottom panels respectively show the evolutionary energy ratio between the two polarization states and normalized Stokes parameter ${s_1}$. (d) Corresponding polarization dynamics visualized on the Poincare sphere. ${s_1},{s_2},{s_3}$ are normalized Stokes parameters.

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3. Cavity assembly by using geometry engineered Er3+/Yb3+ fiber and mode-locking results

In this section, we elaborate the preparation of the GF sealed in a geometry engineered ferrule and the procedure of cavity assembly. By performing mode-locking experiments in such fiber resonator, the soliton dynamics is then characterized. In particular, we compare the noise property of the generated microwave signal with that of other radio-frequency (RF) sources and discuss its potential in the synthesis of low-noise microwave.

3.1 Assembly of the geometry engineered fiber resonator

We first use a ceramic ferrule (with inner/outer diameter of 125 µm/2.5 mm) to seal a segment of 8-mm GF (homemade heavily Er3+/Yb3+ co-doped phosphate fiber with ∼122-µm cladding diameter [31]). In this case, the entire phosphate fiber is glued inside of the ferrule [ferrule 1 in Fig. 3]. Subsequently, the ferrule loaded with the GF is angle-polished by using a FC/APC connector polishing disc (Thorlabs, D50-FC/APC), and is inserted into a non-angle disc (Thorlabs, D50-FC) to perpendicularly polish the tilted end-facet. In this way, we can fabricate a geometry engineered ferrule, of which the side view shows asymmetry geometry, as interpreted by angles ${\theta _1} = {96.3^\circ }$ and ${\theta _2} = {91.7^\circ }$ [photo in Fig. 3(a)]. The aforementioned fabrication procedure is illustratively described in Fig. 3(a).

 figure: Fig. 3.

Fig. 3. (a) A two-step fabrication procedure of the geometry-engineered ferrule loading the gain fiber. Angles ${\theta _1}$ and ${\theta _2}$ are measured to be ${96.3^\circ }$ and ${91.7^\circ }$. (b) Illustration of polarization selection by rotating the ferrule (Ferrule 2) attached to a SESAM. The geometry engineered (GE) end-facet is in yellow color. Detailed procedure is pictorially explained in the dashed box. Please note that, we enlarge the tilted angle of ferrule 1 for better visualization in the schematic diagram, and in practical situation the slightly tilted ferrule 2 will not affect the connection via the matching sleeve. (c) Experimental setup. TC, temperature controller; WDM, wavelength division multiplexer; LD, lasing diode; ISO, isolator; PC, polarization controller; PBS, polarization beam splitter. For better understanding of polarization decomposition and synthesis, the principal axes of the laser output and the PBS are illustrated on right bottom. When orientating the PC to align the two sets of principal axes, the polarization decomposition is fulfilled to separately characterize the vector solitons. On the other hand, if the orientation of the principal axis for the output pulse is $\pi /4$ relative to the PBS, it yields polarization synthesis.

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As sketched in Fig. 3(b), the FP mode-locked fiber resonator is made of a DF-coated ferrule (not shown), GF filled ferrule 1, and a SESAM (Batop, SAM-1550-10-5ps-1.0b). To connect the DF-coated ferrule and ferrule 1, a ceramic sleeve with a matched size (7 mm in length) is employed. Moreover, the SESAM is sandwiched by ferrules 1 and 2 inside another sleeve. To perform mode-locking, a 974-nm pump laser is launched into the resonator via a 980/1560 nm wavelength-division multiplexer (WDM), and the laser emission is extracted from the 1560-nm signal port and then pass through a fiber isolator (ISO). To allow further characterizing the vectorial trait of the laser output, a polarization resolvable measurement implemented by a combination of polarization controller (PC) and polarization beam splitter (PBS) is thus performed [40]. Due to the contact with geometry engineered end-facet of ferrule 1, the SESAM attached to ferrule 2 is rotated to adjust the angle of incidence on the SESAM [c.f. Figure 1(c)]. In this process, we check the states of CWML in different rotating conditions, and consequently recognize two distinctive mode-locking operations that produce vector and scalar solitons, respectively.

3.2 Different operating states of the mode-locking

In the case of generating vector solitons, the output powers from the two ports of PBS are close to each other, i.e., 1.5 and 1.35 mW, as summarized in Fig. 4. In Fig. 4(a), optical spectra of the slow and fast axes measured by an optical spectrum analyzer (Yokogawa AQ6370B, 0.02 nm resolution) have almost consistent intensities, the feature of which is well consistent with that of the numerically-predicted PRVSs [see Fig. 2(a)]. The good symmetry in intensity is intrinsically associated with the solution of vector solitons existed in coupled nonlinear Schrodinger equation [49,50]. The soliton trapping effect [51], as the predominant mechanism of group-velocity locking, usually gives rise to significant frequency shift between the orthogonally polarized components [52], such that the walk-off effect caused by the polarization mode dispersion can be well balanced by the chromatic dispersion effect. However, such opposite drift in wavelength for the vector soliton can be significantly weakened in HRR fiber lasers [40] and is only exhibited as minor short-wavelength discrepancy, as shown in Figs. 2(a) and 4(a) for simulation and experiment, respectively. Besides, sideband structure [53,54] is absent because of the ultra-small net intracavity dispersion [55]. To further identify the relative phase change between the two polarization components of PRVSs, we then rotate the PC to initiate coherent polarization mixing, and monitor the pulse output from either port of the PBS in both the RF and time domains in the process of polarization synthesis. The variation of the existing phase difference can be translated into intensity modulation, which can be easily captured by using photodetector (PD, Newport 818-BB-51F) and electronic devices, i.e., signal analyzer (Rohde & Schwarz FSWP26) and real-time oscilloscope (Keysight DSOV204A) in this work. Figure 4(b) shows the RF spectrum of the CWML experiment, in accordance with the simulated result. The RF spectrum displays significant side peaks and implies a polarization rotating frequency (PRF) of ∼1.18 GHz. It is intriguing to find that the ratio of repetition rate ${f_r}$ to PRF ${f_{PRF}}$ is half-integer, i.e., ${f_r}/{f_{PRF}} ={\sim} 10.5$, which suggests that the polarization rotation in the case of HRR mode-locking is no longer restricted to the integer multiple of cavity roundtrip time. This is distinctive from the case of low-repetition-rate mode-locking [56], and more details are provided in Appendix 1. Correspondingly, the oscilloscopic trace becomes intensity-modulated in a time period of ∼850 ps by polarization synthesis [Fig. 4(c)], which intuitively portrays the RT-evolving phase difference between the orthogonally polarized electric fields. Particularly, for vector solitons in HRR fiber laser, the dynamic phase evolution usually presents since the nonlinear birefringence induced by the cross-phase modulation is inadequate to compensate the linear fiber birefringence [46]. According to the theoretical analysis in Sec. 2, we anticipate that the SESAM that closely contacts with the flat part of the end-facet (ferrule 1) can render a normal incidence scenario, in which the resultant PSE is too weak to suppress one of the polarization components.

 figure: Fig. 4.

Fig. 4. (a) Optical spectra measured at ports 1 and 2 of the PBS when the laser delivers vectoral solitons. The light fields polarized along slow and fast axes of the polarization-maintaining pigtail fiber are in red and blue colors, respectively. Inset: closeup that manifests the longitudinal mode spacing. (b) Experimental (top) and simulated (bottom) radio frequency (RF) spectrum of a coupled lightwave output after polarization synthesis. (c) Oscilloscopic traces of the fast-axis output in the process of polarization synthesis via continuously adjusting the PC. The waveforms are vertically offset for better visualization. Syn. and Dec. are short for polarization synthesis and decomposition.

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By appropriately rotating the ferrule 2, we can alter the contacting condition between the SESAM and the end-facet of ferrule 1, such that stronger PSE can be produced for generating scalar solitons. By measuring the output powers at the two ports of the PBS, the polarization extinction ratio (PER) is estimated to be ∼26 dB after optimizing the setting of the PC. The corresponding polarization-resolved optical spectra are also measured and shown in Fig. 5(a), wherein the intensity contrast between the orthogonal polarization components is about 20 dB. The relevant RF spectrum with a signal-to-noise ratio of >80 dB, i.e., Fig. 5(b), is free from any side peaks. In the meantime, a fairly good linear polarization is confirmed as only pulse train along the slow axis is observed, as shown in Fig. 5(c). The pulse train at a 12.5-GHz repetition rate is manifested as nearly sinusoidal oscillation due to the limited bandwidth of the PD and oscilloscope.

 figure: Fig. 5.

Fig. 5. (a) Optical spectra of the slow-axis (red) and fast-axis (blue) polarized outputs when the laser delivers scalar solitons. A contrast of ∼20 dB between the two outputs is denoted. Inset: magnified modulated pattern of the spectrum. (b) Corresponding RF spectra measured at the slow-axis output. Top panel: a span of 200 MHz with a resolution bandwidth (RBW) of 1 kHz. Bottom panel: a span of 26.5 GHz with an RBW of 10 kHz. (c) Oscilloscopic traces of the two polarization states. The sinusoidal-like trace (in red) accounts for the slow-axis polarized component.

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Benefiting from the excellent performance of the homemade Er3+/Yb3+ co-doped phosphate fiber with high small-gain coefficient of >9 dB/cm at 1534 nm (Appendix 2) and low-loss fiber resonator with a Q factor of >5×105, the pump threshold for CWML, i.e., ∼69.1 mW in this work, reaches a record low level in comparison with other GHz fiber lasers at 1.5 µm [34–36,38]. As shown in Fig. 6(a), an average power of >5 mW is achieved at a moderate pump power of 85.2 mW. The pulse duration is measured to be ∼2.5 ps by using an autocorrelator (APE Pulsecheck USB 50), assuming a sech2 pulse shape [Fig. 6(b)]. Given that the 3-dB bandwidth is ∼3 nm, a transform-limited pulsewidth of 854 fs is expected after dechirping. To evaluate the long-term stability of the linearly polarized soliton (LPS), we fixed the pump power at 75 mW, and simultaneously record the optical spectrum and the output power for 30 minutes. As illustrated in Fig. 6(c), no visible wavelength drift is observed, and the distinguishable longitudinal mode spacing of ∼0.1 nm over the whole spectrum is clearly presented. Figure 6(d) shows the power stability, wherein the peak-to-peak instability is less than ±0.41%, and a relative standard deviation of only 0.13% is computed according to the histogram on the right side. In addition, we also carry out complementary experiment of 5-GHz mode-locking by contacting the geometry engineered end-facet with the DF-coated ferrule, relevant results are provided in Appendix 2.

 figure: Fig. 6.

Fig. 6. Performance of scalar solitons generated in the Er3+/Yb3+ fiber FP resonator. (a) Output power as a function of the launched pump power. QSML, Q-switched mode-locked; CWML, continuous wave mode-locked. (b) Autocorrelation trace (blue) and sech2-fitting curve (green). Here, the pulses signal is amplified to an average power of ∼35 mW for measuring the autocorrelation trace. (c) Evolution of the optical spectrum over 30 minutes. A representative optical spectrum plotted in linear scale is displayed on the right panel. (d) Evolution of the output power over 30 minutes (left panel) and its relevant histogram (right panel). A peak-to-peak instability of ±0.41% and relative standard deviation (RSD) of 0.13% are calculated.

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When considering both optical and electronic domains, the HRR fiber laser have two carrier frequencies: one corresponds to the central wavelength of ∼1563 nm in the optical domain; the other is determined by the fundamental repetition rate in the electronic domain, i.e., 12.5 GHz here. This feature enables generating stable microwave signal covering a frequency range of 8∼18 GHz. As a crucial performance metric, the repetition rate noise of the solitons is further evaluated by using a phase noise analyzer (Rohde & Schwarz FSWP26). The single-sideband (SSB) phase noise ranging from 10 Hz to 10 MHz is presented in Fig. 7, wherein it hits -89 dBc Hz-1 at 10 kHz, -112.8 dBc Hz-1 at 100 kHz, and reaches down to -134.4 dBc Hz-1 at 1 MHz. The phase noise power spectral density (PNPSD) well obeys the 1/f2.6 dependence, indicating a joint contribution from the white frequency noise and flicker frequency noise [60]. The corresponding timing jitter integrated from 10 Hz to 10 MHz is 20.7 ps (right y-axis of Fig. 7). Similar PNPSD lineshape was previously obtained in a 10 GHz graphene-fiber microresonator with active feedback control [38]. For a better vision of its potential of generating low-noise microwaves, we briefly list the phase noise performance of different approaches, i.e., the insert of Fig. 7. It is manifested that the fiber laser shows a comparable (or even better) phase noise performance at frequencies higher than 100 kHz, while the performance at lower frequencies needs be further optimized [61,62]

 figure: Fig. 7.

Fig. 7. Phase noise and integrated timing jitter of 12.5-GHz scalar solitons. The 1/f2.6 fitting curve is also provided. For comparison, phase noise performances of other sources are summarized in the table (left bottom), including X-band SiN microresonator [14], MgF2 microresonator [16], Er3+/Yb3+ co-doped fiber laser [38], integrated optoelectronic oscillator (OEO) [57], dielectric resonator oscillator [58], and Si OEO [59].

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4. Conclusion

In conclusion, we report on vector and scalar solitons in a SESAM mode-locked Er3+/Yb3+ fiber laser with a fundamental repetition rate up to 12.5 GHz. The versatile polarization dynamics are studied by exploring the resonator geometry engineering, which permits either normal or oblique incidence on SESAM. The influence of the potential PSE on the vectorial feature of the soliton is firstly investigated by numerical simulation. In the experiment, the resonator geometry engineering is realized through a two-step procedure, which enables manipulating the state of mode-locking, e.g., switching from PRVSs and LPS. It thus provides a promising method to suppress the dynamic polarization evolution in HRR all-fiber lasers for stable operation. For the linear polarization output, the long-term stability of the LPS is verified by continuously recording the optical spectrum and output power. Moreover, the relatively low-noise property of the 12.5 GHz microwave signal at high-frequency band is further examined, i.e., -134.4 dBc Hz-1 at 1 MHz.

Appendix 1 – Polarization rotating period with half-integer multiple of cavityroundtrip time

By setting the beat length ${L_B}$ integer multiple and half-integer multiple of the cavity length $2L$, i.e., ${L_B}/2L = 10,$ and ${L_B}/2L = 9.5$, the evolving behavior of the PRVSs is provided in Fig. 8. In the former case, as illustrated in Figs. 8(a) and (b), the RT-evolving pulse energy and Stokes parameter ${s_2}$ after polarization synthesis perfectly recover every 11 RTs. Accordingly, the RF spectrum exhibits side peaks at ${f_{PRF}} ={\pm} 1.134$ GHz, i.e., ${f_r}/{f_{PRF}} = 11$. Once ${L_B}/2L$ becomes half-integer, ${f_r}/{f_{PRF}}$ can no longer be integer as well, as disclosed in Figs. 8(c) and 8(d). The RT-evolving quantities undergo similar ‘period-doubling bifurcations’ as designated by blue and red arrows.

 figure: Fig. 8.

Fig. 8. Polarization rotation vector solitons with integer and half-integer multiple of roundtrip time. (a) Evolution of the pulse energy (top) and Stokes parameter ${s_2}$ (bottom) after polarization synthesis. (b) RF spectrum after polarization synthesis. Here, the ratio of repetition rate ${f_r}$ to polarization rotating frequency ${f_{PRF}}$ equals to 11. (c,d) Corresponding results when the ratio of repetition rate ${f_r}$ to polarization rotating frequency ${f_{PRF}}$ equals to 10.5.

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Appendix 2 – Gain characterization of Er3+/Yb3+ co-doped phosphate fiber and 5-GHz mode-locking

The small-signal gain coefficient of the homemade gain fiber is tested. The injected signal power is kept below -10 dBm to avoid probable gain saturation. The measured result is shown in Fig. 9(a), wherein the maximum gain coefficient reaches 9.13 dB/cm.

 figure: Fig. 9.

Fig. 9. (a) Gain measurement of the homemade Er3+/Yb3+ co-doped phosphate fiber. (b-d) 5-GHz-repetition-rate mode-locking by connecting angled-polished end-facet with DF ferrule: optical spectrum (b), oscilloscopic trace (c), and RF spectrum (d).

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Because of a similar multilayer-stack structure of the DF, we also expect the production of the PSE by contacting the geometry-engineered end-facet with the DF-coated ferrule. As a complementary experiment, a 5-GHz-mode-locking experiment is also conducted. We successfully eliminate the polarization rotation phenomenon with ${f_{PRF}}\sim 120$ MHz, and obtain scalar solitons, as depicted in Fig. 9(b-d).

Funding

Science and Technology Planning Project of Guangdong Province (2020B1212060002); Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X137); National Natural Science Foundation of China (U1609219); Guangdong Key Research and Development Program (2018B090904003); Double First Class Initiative (D6211170); Mobility Programme of the Sino-German (M-0296); Natural Science Foundation of Guangdong Province (2021B1515020074); NSFC Development of National Major Scientific Research Instrument (61927816)).

Disclosures

The authors declare no conflict of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)
Fig. 1.
Fig. 1. (a) Configuration of the SESAM mode-locked Er3+/Yb3+ fiber Fabry-Pérot (FP) resonator. DF, dielectric film; GF, gain fiber; SESAM, semiconductor saturable absorber mirror. Inset: side view of the fiber end-facet attached to the SESAM with interior angles ${\theta _1}$ and ${\theta _2}$. (b) Photo of the ultrashort fiber resonator with geometry engineering. (c) Normal and oblique incidences caused by geometry engineered ferrule. (d) Reflectance of the SESAM for p- and s-polarized lights. Solid and dashed curves are plotted according to Eq. (5) and Eqs. (1 –3), respectively. Inset: the SESAM with tilted input, assuming that orthogonally polarized components u and v are parallel to the orientations of TE (s-polarized) and TM (p-polarized) modes, respectively. (e) Quantitative description of polarization selection for different angles of incidence. Top panel: contrast ratio (CR) between reflectance ${\mathrm{{\cal R}}_s}$ and ${\mathrm{{\cal R}}_p}$ for different angles of incidence ${\theta _0}$. Bottom panel: energy ratio evolution of s-polarized soliton with varying ${\theta _0}$. Regimes of vector and scalar solitons, together with the critical CR for polarization switching (i.e., ${\theta _0}\sim {3^\circ }$), are shown for clarity.

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Fig. 2.
Fig. 2. (a) Optical spectra of the orthogonally polarized states along slow and fast axes for normal incidence ${\theta _0} = {0^\circ }$ (labelled as state 1). Inset depicts the ensemble of polarization ellipses for 10-RT polarization rotating vector solitons (PRVSs). (b) Optical spectrum of scalar solitons, polarized on the slow axis for oblique incidence ${\theta _0} = {6^\circ }$ (labelled as state 2). (c) Dynamic evolution when switching from state 1 to 2. Top and bottom panels respectively show the evolutionary energy ratio between the two polarization states and normalized Stokes parameter ${s_1}$. (d) Corresponding polarization dynamics visualized on the Poincare sphere. ${s_1},{s_2},{s_3}$ are normalized Stokes parameters.

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Fig. 3.
Fig. 3. (a) A two-step fabrication procedure of the geometry-engineered ferrule loading the gain fiber. Angles ${\theta _1}$ and ${\theta _2}$ are measured to be ${96.3^\circ }$ and ${91.7^\circ }$. (b) Illustration of polarization selection by rotating the ferrule (Ferrule 2) attached to a SESAM. The geometry engineered (GE) end-facet is in yellow color. Detailed procedure is pictorially explained in the dashed box. Please note that, we enlarge the tilted angle of ferrule 1 for better visualization in the schematic diagram, and in practical situation the slightly tilted ferrule 2 will not affect the connection via the matching sleeve. (c) Experimental setup. TC, temperature controller; WDM, wavelength division multiplexer; LD, lasing diode; ISO, isolator; PC, polarization controller; PBS, polarization beam splitter. For better understanding of polarization decomposition and synthesis, the principal axes of the laser output and the PBS are illustrated on right bottom. When orientating the PC to align the two sets of principal axes, the polarization decomposition is fulfilled to separately characterize the vector solitons. On the other hand, if the orientation of the principal axis for the output pulse is $\pi /4$ relative to the PBS, it yields polarization synthesis.

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Fig. 4.
Fig. 4. (a) Optical spectra measured at ports 1 and 2 of the PBS when the laser delivers vectoral solitons. The light fields polarized along slow and fast axes of the polarization-maintaining pigtail fiber are in red and blue colors, respectively. Inset: closeup that manifests the longitudinal mode spacing. (b) Experimental (top) and simulated (bottom) radio frequency (RF) spectrum of a coupled lightwave output after polarization synthesis. (c) Oscilloscopic traces of the fast-axis output in the process of polarization synthesis via continuously adjusting the PC. The waveforms are vertically offset for better visualization. Syn. and Dec. are short for polarization synthesis and decomposition.

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Fig. 5.
Fig. 5. (a) Optical spectra of the slow-axis (red) and fast-axis (blue) polarized outputs when the laser delivers scalar solitons. A contrast of ∼20 dB between the two outputs is denoted. Inset: magnified modulated pattern of the spectrum. (b) Corresponding RF spectra measured at the slow-axis output. Top panel: a span of 200 MHz with a resolution bandwidth (RBW) of 1 kHz. Bottom panel: a span of 26.5 GHz with an RBW of 10 kHz. (c) Oscilloscopic traces of the two polarization states. The sinusoidal-like trace (in red) accounts for the slow-axis polarized component.

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Fig. 6.
Fig. 6. Performance of scalar solitons generated in the Er3+/Yb3+ fiber FP resonator. (a) Output power as a function of the launched pump power. QSML, Q-switched mode-locked; CWML, continuous wave mode-locked. (b) Autocorrelation trace (blue) and sech2-fitting curve (green). Here, the pulses signal is amplified to an average power of ∼35 mW for measuring the autocorrelation trace. (c) Evolution of the optical spectrum over 30 minutes. A representative optical spectrum plotted in linear scale is displayed on the right panel. (d) Evolution of the output power over 30 minutes (left panel) and its relevant histogram (right panel). A peak-to-peak instability of ±0.41% and relative standard deviation (RSD) of 0.13% are calculated.

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Fig. 7.
Fig. 7. Phase noise and integrated timing jitter of 12.5-GHz scalar solitons. The 1/f2.6 fitting curve is also provided. For comparison, phase noise performances of other sources are summarized in the table (left bottom), including X-band SiN microresonator [14], MgF2 microresonator [16], Er3+/Yb3+ co-doped fiber laser [38], integrated optoelectronic oscillator (OEO) [57], dielectric resonator oscillator [58], and Si OEO [59].

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Fig. 8.
Fig. 8. Polarization rotation vector solitons with integer and half-integer multiple of roundtrip time. (a) Evolution of the pulse energy (top) and Stokes parameter ${s_2}$ (bottom) after polarization synthesis. (b) RF spectrum after polarization synthesis. Here, the ratio of repetition rate ${f_r}$ to polarization rotating frequency ${f_{PRF}}$ equals to 11. (c,d) Corresponding results when the ratio of repetition rate ${f_r}$ to polarization rotating frequency ${f_{PRF}}$ equals to 10.5.

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Fig. 9.
Fig. 9. (a) Gain measurement of the homemade Er3+/Yb3+ co-doped phosphate fiber. (b-d) 5-GHz-repetition-rate mode-locking by connecting angled-polished end-facet with DF ferrule: optical spectrum (b), oscilloscopic trace (c), and RF spectrum (d).

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Tables (1)

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Table 1. Key parameters used for simulating the fiber FP resonator with a repetition rate of 12.5 GHz

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Equations (16)

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R p , s = | η G ( p , s ) − η I ( p , s ) η G ( p , s ) + η I ( p , s ) | 2 ,
η I ( p , s ) = M 21 + M 22 η H ( p , s ) M 11 + M 12 η H ( p , s ) , M = [ M 11 M 12 M 21 M 22 ] = H ( L H ) m ,
H = [ c o s δ H i s i n δ H / η H i η H s i n δ H c o s δ H ] , L = [ c o s δ L i s i n δ L / η L i η L s i n δ L c o s δ L ] ,
η G ( p ) = n G c o s θ 0 , η G ( s ) = n G c o s θ 0 ,
η H / L ( p ) = n H / L 2 n H / L 2 − n G 2 s i n 2 θ 0 , η H / L ( p ) = n H / L 2 − n G 2 s i n 2 θ 0
δ H / L = π n H / L 2 − n G 2 s i n 2 θ 0 2 n H / L
δ H / L ∼ π / 2 ,
R p ( θ 0 ) = n G c o s θ 0 − n H 2 n H 2 − n G 2 s i n 2 θ 0 ( n H n L ) 4 m ( n L 2 − n G 2 s i n 2 θ 0 n H 2 − n G 2 s i n 2 θ 0 ) m n G c o s θ 0 + n H 2 n H 2 − n G 2 s i n 2 θ 0 ( n H n L ) 4 m ( n L 2 − n G 2 s i n 2 θ 0 n H 2 − n G 2 s i n 2 θ 0 ) m ,
R s ( θ 0 ) = n G c o s θ 0 − n H 2 − n G 2 s i n 2 θ 0 ( n H 2 − n G 2 s i n 2 θ 0 n L 2 − n G 2 s i n 2 θ 0 ) m n G c o s θ 0 + n H 2 − n G 2 s i n 2 θ 0 ( n H 2 − n G 2 s i n 2 θ 0 n L 2 − n G 2 s i n 2 θ 0 ) m ,
C R ( θ 0 ) = 10 l g [ R s ( θ 0 ) / R p ( θ 0 ) ] ,
∂ u ( n ) ( z , t ) ∂ z = i k u ( n ) + g u ( n ) − δ ∂ u ( n ) ∂ t + ( g Ω 2 − i β 2 ) ∂ 2 u ( n ) ∂ t 2 + i γ ( | u ( n ) | 2 + 2 3 | v ( n ) | 2 ) u ( n ) + i γ 3 v ( n ) 2 u ( n ) ∗ ,
∂ v ( n ) ( z , t ) ∂ z = − i k v ( n ) + g v ( n ) + δ ∂ v ( n ) ∂ t + ( g Ω 2 − i β 2 ) ∂ 2 v ( n ) ∂ t 2 + i γ ( | v ( n ) | 2 + 2 3 | u ( n ) | 2 ) v ( n ) + i γ 3 u ( n ) 2 v ( n ) ∗ ,
g = g 0 1 + ∫ ( | u ( n ) ( z , t ) | 2 + | v ( n ) ( z , t ) | 2 ) d t / E g .
[ u ( n ) ( L + , t ) v ( n ) ( L + , t ) ] = e 1 2 α ( q − q 0 ) R s ( θ 0 ) − q [ 1 0 0 R p ( θ 0 ) / R s ( θ 0 ) ] [ u ( n ) ( L , t ) v ( n ) ( L , t ) ] ,
∂ q ∂ t = q − q 0 τ a − q | u ( n ) ( L , t ) | 2 + | v ( n ) ( L , t ) | 2 E a .
[ u ( n + 1 ) ( 0 , t ) v ( n + 1 ) ( 0 , t ) ] = 1 − q l [ u ( n ) ( 2 L , t ) v ( n ) ( 2 L , t ) ] .
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