1.

INTRODUCTION

The performance of a fiber laser can be largely contingent on its active gain medium, * which is an optical fiber doped with rare-earth elements such as erbium or ytterbium [1], [2]. By definition, a few-mode fiber (FMF) allows the propagation of a finite number of higher-order spatial modes in conjunction with the fundamental one within the same medium, which has attracted considerable research consideration for the past decade [3], [4]. The development of the FMFs originated from mode-division multiplexing (MDM), which is compatible with the passive optical network (PON), as an appealing tactic of the space division multiplexing (SDM) technologies to overcome the interminable capacity crunch [5], [6]. With each data channel modulated into an individual spatial/polarization modes in an FMF to increase the overall number of parallel channels, ultra-high capacity long-haul SDM transmission can be realized [7]-[9]. A variety of different mode types propagating inside an FMF may include the transverse modes, the supermodes, principle modes, and twisted modes like orbital angular momentum of light (OAM); yet the most commonly used one is still the linearly polarized (LP) modes [10], [11]. In principle, the amount of modes is restricted by the Abbe number or V number, refractive index profiles and their relative phases contributing to the propagating field [12]. The crosstalk and modal delay can be dynamically compensated by the use of the adaptive multiple-input multiple-output (MIMO) equalizer by disentangling the spatial modes via the equalization matrix inverse to the original FMF channel matrix in either time domain (TD) or frequency domain (FD) [13]-[15]. Beyond telecommunication, the FMF can find a wide range of other applications, to quote just a few, optical amplifiers, mode converters, few-mode sensors or lasers, as summarized in Fig. 1 [16], [17]. Technically, doped fiber amplifiers provide light amplification without lasing. The few-mode amplifiers enable sharing of a single pump between multiple signal modes and amplifying many channels with different modes simultaneously regarding the modal intensity overlap, thus greatly reducing the overall cost and power consumption of the MDM systems [18], [19]. As another critical component to enhance the compactness and flexibility of the SDM systems, mode converters can alter the mode shapes by passive optical elements like filters, lenses or the FMF itself as the medium [20], [21]. Furthermore, the variation of modal phase difference in an FMF can be applied to gauge temperature and strain in the interferometric sensors [22], [23]. Additionally, in recent years, there is an emergent interest in applying FMF in the fiber laser applications [24], [25].

Figure 1.

Schematic diagram of a few-mode fiber (FMF) with multiple linearly polarized (LP) spatial modes along with its potential applications.

As mentioned above, the unique FMF characterization, including modal content, birefringence, dispersion as well as bending losses, is advantageous for numerous applications involving optical transport networks and high laser power generation [26], [27]. The FMF exhibits a higher power threshold than that of a conventional single-mode fiber (SMF), for the reason that the phase noise is intensely impacted by stimulated Brillouin scattering (SBS) and FMF is more tolerant to SBS due to larger core diameter [28]. The few-mode optical amplifiers can be categorized into few-mode Erbium-doped fiber amplifiers (EDFA) as well as few-mode distributed Raman amplifiers, whose efficiency is directly associated with the integral of the transverse field overlap of the spatial modes in an FMF [29], [30]. The characterization of FMF is different from that of a SMF mainly due to the differential modal group delay (DMGD) and absolute chromatic dispersion values of each mode within the medium [31], [32]. Modal dispersion or DMGD is defined as a distortion mechanism occurring in an FMF, for the reason that the propagation velocities for all modes are not the same, which causes the optical signals to spread in time [33]-[35]. Alternatively, the mode-dependent loss (MDL) arises from the inline component imperfections and breakdowns the modal orthogonality, degrading the capacity of MIMO channels consequently [36], [37]. The mode coupling effects and subsequent impairments like mode dispersion and MDL in-between the spatial channels can be compensated by MIMO digital signal processing at the receivers [38], [39], such as least mean square (LMS) algorithm and recursive least square (RLS) algorithm [40], [41].

Above and beyond, the fiber laser gain can be provided through the fiber nonlinearities, for instance the four-wave mixing (FWM) or stimulated Raman scattering (SRS) within the gain medium [42]. The intermodal nonlinear characteristics attracted lots of attention for long-haul SDM telecommunications data links in the past decade, because FMF has a relatively larger effective mode area than that of SMF, even at the fundamental LP₀₁ mode, which allows the suppression of the conventional intra-modal nonlinearities at high optical power such as self-phase modulation (SPM) [43], [44]. For one case in point, in order to produce FWM efficiently in SMF, it is indispensible that the pump wavelength shall coincide with the zero-dispersion wavelengths (ZDW) [45]. On the other hand, it has been experimentally demonstrated that intermodal FWM might still be effusively phase matched in the presence of large chromatic dispersion and polarization mode dispersion (PMD) in each spatial mode [46], [47]. Hereafter, the pump wavelength no longer has to be matching with ZDW in MDM systems, and there is no need to minimize CD variation in FMFs [48], [49]. As an alternative, we would be able to control the intermodal fiber nonlinearity by carefully balancing the group velocity dispersion (GVD) in FMFs, thus achieving the anticipated FWM conversion efficiency [50]. In another example, since Brillouin scattering depends on the strong correlation in-between the longitudinal acoustic and optical modes, every spatial mode within the FMF can have a slightly different Brillouin property, and thus can be used to enhance the performance monitoring functionalities [51], [52]. On the whole, such unique nonlinear characteristics can be applied in the design and development of compact light sources for generating ultrashort pulses with stable peak power [53].

To cut a long story short, the inclusive performance of a mode-locked soliton fiber laser is dependent on the balance between optical Kerr effect and cavity dispersion [2], [54]. And to realize high output energy in the present ultrafast fiber lasers, the scaling to higher peak powers with a larger mode area has become progressively more imperative [55], because the key challenges include low repetition rate and possible pulse breaking due to excessive nonlinear phase accumulation [1], [56]. To resolve the aforementioned issues, multiple-pulsing or bound solitons with discrete fixed pulse separations have been purposed in passively mode-locked fiber lasers, in which case the random phase and inter-pulse interference would impair the performance [57], [58]. In the FMF cases, anomalous GVD has been replaced by a combination of chromatic dispersion, DMGD, mode dependent gain or loss; while the optical Kerr nonlinearity portion has been substituted with a new intermodal nonlinear dynamics, which is worth further studying to manage the spatial temporal stability of the fiber laser performance [59], [60]. In this paper, we systematically investigated the formation and dynamics of synchronized soliton pulses in multiple spatial modes operating in the existence of a large modal dispersion. For a mode-locked laser cavity design at 1560 nm with maximum output power of 150 mW utilizing FMF, whose large mode area can reduce nonlinearities caused by high-peak power intensities, the loop length of FMF laser cavity is 5-meter-long with 2000-ppm erbium-doped concentration. The outline of this paper is listed in the following. Section 2 describes the principle of few-mode soliton fiber lasers, and provides the modeling by solving the complex cubic–quintic Ginzburg–Landau equation. Section 3 presents the results and discussions evaluating high power lasers with an enlarged mode area and mode coupling. Our results indicate that, power would be transferred from higher order modes to the fundamental one, due to soliton fission by dint of intermodal nonlinear coupling, to further stabilize mode-locking and realize high-repetition-rate fiber laser. Section 4 draws the final conclusion.

2.

FEW-MODE SOLITON FIBER LASER MODELING

To specifically model the pulse propagation within the oscillator, by treating the entire laser cavity as one single FMF, the complex cubic–quintic Ginzburg–Landau equation can be expressed in the following general form as [61] :

where E(z, t) denotes the electric field envelope, z and t represent the propagation coordinate and time from the center of the pulse, g symbolizes the gain parameter, Ω and β₁ are associated with the filter bandwidth and GVD, α and δ signifies the cubic and quintic saturable absorber terms, γ_eff stands for the effective nonlinear coefficient [62]. The effective nonlinear coefficient of FMF depends on the nonlinear indices of the bulk materials as well as its waveguide properties, such as the mode shape as well as the degree of confinement. The effective nonlinear coefficient can be well-defined in the following form:

where ω represents the wavelength, Pol. denotes the state of polarization (SOP), n₂ (x, y) symbolizes the spatially dependent nonlinear refractive index of various fiber layers, while F(x,y) embodies the normalized mode field pattern [43]. The block configuration of the few-mode soliton fiber laser cavity is presented in Fig. 2. To simulate the mode locking of dissipative solitons in a 1.55 µm system, the saturable absorber precedes an Erbium-doped gain fiber lumped into independent segments that separates the saturating gain, GVD and nonlinearity in the laser cavity modeling.

Figure 2.

Block configuration of a few-mode Erbium-doped soliton fiber laser cavity.

The effective refractive index of the fiber depends upon the frequency, polarizations, modes, as well as the power magnitude of the signal. The linear dispersion effects such as chromatic dispersion, PMD and DMGD come from the first three terms respectively, while the FMF nonlinearities mainly originate from the change of power as well as the nonlinear phase shift within the medium [63]. The linear dispersion depends on the normalized group delay, which can be expressed as:

Meanwhile, the resultant normalized propagation constant b_eff can be obtained in the following equation:

where β symbolizes the propagation constant determining how fast electric vectors are oscillating during the propagation through the optical fiber, k denotes the free-space wave number, Δ represents the relative refractive index of the fiber, n_Core and n_Clad stand for the refractive indices of the core and the cladding, which regulates the material dispersion along the fiber [64]. On the other hand, the intensity-dependent nonlinear phase shift Φ_NL (t) can be written in the form of:

where I(t) stands for the optical intensity, L denotes the cavity length, and the nonlinear phase shift varies with time according to the pulse chirping [65].

In addition, the triple cladding index profile of the FMF is depicted in Fig. 3 for large effective modal area and low loss designing, which is composed of a high delta core bounded by a depressed trench followed by a raised ring. The loop length of FMF laser cavity is 5-meter-long with 2000-ppm erbium-doped concentration. Supposedly, for the 1560 nm wavelength, when we increase the effective mode area in FMF, the core radius needs to be enlarged, while the core delta has to be reduced to retain the FMF cutoff wavelength below the minimum wavelength of the laser application window.

Figure 3.

The triple cladding index profile designing of an FMF supporting multiple LP modes.

3.

RESULTS AND DISCUSSION

This section provides the numerical results of the soliton pulse broadening and the corresponding spatiotemporal oscillation in a dissipative-soliton few-mode fiber laser, which are in line with the theoretical prediction in the previous content. From Fig. 4(a) to Fig. 7(a), we can clearly notice that, the spectrum stretches and grows sharp peaks around its edges as the nonlinear phase shift increases. In the meantime, from Fig. 4(b) to Fig. 7(b), the soliton evolution in space and temporal domain are explored. As the stable performance of a soliton fiber laser relies on the quasi-balance between optical Kerr effect and cavity dispersion, soliton would be perturbed when we increase the value of nonlinear phase shift, and it will try to fine-tune itself to its original formation, while radiating some of its energy in the narrow temporal pulses as dispersive waves. Furthermore, as nonlinear phase shift continues to grow, the spectrum broadens further and eventually develops structure or fringes due to the spatiotemporal modulation instability. And when these sidebands occur, power would be transferred from higher order modes to the fundamental one, due to soliton fission by dint of intermodal nonlinear coupling, which would further stabilize the mode-locking process and realize high-repetition-rate fiber laser.

Figure 4.

(a) Soliton pulse spectrum with nonlinear phase shift of π; (b) The corresponding soliton evolution in space and temporal domain in FMF.

Figure 5.

(a) Soliton pulse spectrum with nonlinear phase shift of 2π; (b) The corresponding soliton evolution in space and temporal domain in FMF.

Figure 6.

(a) Soliton pulse spectrum with nonlinear phase shift of 4π; (b) The corresponding soliton evolution in space and temporal domain in FMF.

Figure 7.

(a) Soliton pulse spectrum with nonlinear phase shift of 8π; (b) The corresponding soliton evolution in space and temporal domain in FMF.

4.

CONCLUSION

In this paper, we have systematically investigated the formation and dynamics of synchronized soliton pulses in a few-mode fiber (FMF) operating by solving cubic-quintic Ginzburg-Landau equation. For a mode-locked laser cavity design, the loop length of FMF laser cavity is 5-meter-long with 2000-ppm erbium-doped concentration around the 1560 nm window. Due to the soliton fission by intermodal nonlinear coupling, energy is transferred from higher-order modes to fundamental mode, to stabilize mode-locking and to accomplish high-repetition-rate fiber laser. Henceforth our results expressively facilitate the analytical and experimental study of soliton fiber laser design utilizing FMF.