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Abstract
Spatiotemporal mode-locked (STML) fiber lasers have become an excellent platform in nonlinear optics research due to the rich nonlinear evolution process. In order to overcome modal walk-off and realize phase locking of different transverse modes, it is usually crucial to reduce the modal group delay difference in the cavity. In this paper, we use long-period fiber grating (LPFG) to compensate the large modal dispersion and differential modal gain in the cavity, realizing the spatiotemporal mode-locking in step-index fibers cavity. The LPFG inscribed in few-mode fiber could induce strong mode coupling, which has wide operation bandwidth based on dual-resonance coupling mechanism. By using dispersive Fourier transform involved intermodal interference, we show that there is a stable phase difference between the transverse modes constituting the spatiotemporal soliton. These results would be beneficial for the study of spatiotemporal mode-locked fiber lasers.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As a typical nonlinear system, passively mode-locked fiber lasers have been attracting the attention of researchers. For a long time, the researches of passive mode-locked fiber lasers have been based on single-mode fibers (SMFs). Plenty of nonlinear evolution processes are also found in passively mode-locked fiber lasers based on SMFs, including buildup of soliton molecule [1], soliton explosion [2], soliton rain [3], soliton collision [4–6], rouge wave [7,8] and optical turbulence [9,10]. In recent years, the study of nonlinear phenomena in multi-mode fibers (MMFs) has become a research hotspot, including mode-locked fiber lasers based on MMFs. Compared with SMFs, the addition of the spatial dimension leads to more complex nonlinear dynamic process in the MMFs. Therefore, mode-locked fiber lasers based on MMFs are known as spatiotemporal mode-locked (STML) fiber lasers. STML results from a delicate balance between modal dispersion, chromatic dispersion and various nonlinear effects. Remarkably. the research of STML lasers was reported in 2017, Wright et al. reported STML in the normal dispersion region through spectral filtering and spatial filtering, which is also the first time to experimentally verify the feasibility of STML [11]. In recent years, STML lasers have gradually attracted the interest of researchers, and a large number of spatiotemporal solitons dynamics have been revealed, such as multimode soliton molecule generation [12], multiple soliton and its building process [13,14], spatiotemporal vortices [15], dispersion-managed soliton [16], self-similar soliton [17], wavelength-tunable lasers [18], transition between Q-switching and STML [19] and transition between noise-like pulses and Q-switching [20]. Therefore, the complicated nonlinear dynamics in STML lasers is gradually being revealed.
In order to achieve stable STML, the modal dispersion in the cavity needs to be effectively suppressed. Therefore, the graded-index (GRIN) MMF with small transverse mode dispersion were used in the cavity of STML laser [11–19,21–23], which also includes all-fiber STML lasers [23]. In 2021, Ding et al. experimentally demonstrated that MMF mode-locked laser based on step-index (STIN) MMF can also achieve STML [24]. Although STIN MMFs have larger modal dispersion, saturable absorber (SA) in the cavity can compensate the modal dispersion. Active STIN MMFs are more readily available commercially, and phase-matched passive fibers can be used in cavity to obtain higher energy pulses. STIN MMFs are also applied in all-fiber spatiotemporal mode-locked lasers [25]. Zhang et al. fused GRIN MMF with spectral filter to achieve equivalent virtual spatial filtering and spectral filtering. Thence, this research can better facilitate the fabrication of high-power lasers. However, as an important part of implementing STML, SA is not flexible and controllable enough for compensating modal dispersion.
Long-period fiber grating (LPFG) is a passive mode coupling device, which has the characteristics of low cost, small insertion loss, high mode coupling efficiency, and good stability. By introducing LPFG into few-mode fiber (FMF), all-fiber mode converter can be realized [26–29]. In addition, LPFG also plays an important role in compensating for group delay in mode-division multiplexed (MDM) transmission system to reduce the complexity of digital signal processing for compensating modal dispersion. Liu et al. propose and experimentally demonstrated that uniform LPFG can reduce group delay spread efficiently in 9-LP mode FMF [30], which is also meaningful to STML lasers with large modal dispersion. With the strong mode coupling formed by the grating, the gain equalization between modes can also be achieved [31]. In addition, through the mode conversion of Bragg grating, the Q-switched pulse of LP01 and LP11 can also be realized in the laser [32]. Under certain conditions, dual-resonance coupling phenomenon will occurred in gratings. Guo et al. experimentally demonstrated that broadband mode conversion in 1µm can be realized with the dual-resonance coupling in STIN fibers [33], which provides conditions for controlling modal dispersion and realizing spatiotemporal mode locking in a wide wavelength range.
In this paper, an STML laser based on home-made FMF-LPFG is built. When the pump power reaches 5.5W, the laser can achieve stable mode locking, and output a few-mode pulse sequence with a pulse interval of 35.1 ns and a repetition frequency of 28.49 MHz. The spatiotemporal soliton consists of LP01 and LP11 modes. By using dispersive Fourier transform (DFT) involved intermodal interference, we show that there is a stable phase difference between the transverse modes constituting the spatiotemporal soliton. Due to the dual-resonance coupling effect, the LPFG has a spectral conversion bandwidth about 139 nm for LP01 and LP11 modes. The wide operating bandwidth of the grating can avoid the phase distortion of the spatiotemporal soliton due to the phase mismatch effect of the grating. The results are beneficial for the study of STML lasers with large modal dispersion, and are helpful to the STML lasers with all-fiber structure.
2. Experimental setup
The schematic of experimental setup and measurement system are shown in Figure. 1. A 980 nm pump beam from laser diode is coupled into the 2.3 m ytterbium-doped double-clad fiber (Liekki YB1200-10/125DC) by combiner, and the output beam passes through the FMF-LPFG and then enters the spatial optical path through the lens L1. Among them, LP01 and LP11 realize mode coupling in FMF-LPFG. Here, we replace FMF with SMF-28e, which supports three transverse modes at 1030 nm, namely LP01, LP11 and LP21. Nonlinear polarization rotation mode-locking is adopted in STML laser, which consists of half-wave plate (HWP), isolator (ISO) and quarter-wave plates (QWPs). A 70:30 beam splitter is used to extract the beam out of the cavity for measurement, and filter (Thorlabs, bandpass filter with center wavelength of 1030 nm and 3 dB bandwidth of 10 nm) is placed in the Nonlinear Polarization Rotation (NPR) to achieve spectral filtering. The circular structure of the spatial filter can selectively filter out the radially symmetric mode, so it can also achieve spatial filtering. The output beam in the NPR is reflected by the mirror M1 and then coupled back to the gain fiber through the lens L2.
Fig. 1. The schematic of FMF mode-locked laser. (LD, laser diode; HWP, half-wave plate; QWP, quarter-wave plate; BS1-3, beam splitter; ISO, isolator; M1-2, mirror; L1-5, Lens; PD1-3, photodetector).
The measurement system of STML laser is divided into three parts, namely beam profile measurement, time domain measurement and spectrum measurement. 30% of the beams in the cavity pass through the 90:10 beam splitter BS2, 10% of the beams are detected by the beam analyzer (Duma BeamOn U3), and 90% of the beams are divided into two beams with equal power through the beam splitter BS3. Optical spectrum analyzer (OSA, YOKOGAWA AQ6370D) of 0.02 nm spectral resolution and oscilloscope 1 (Lecroy 620Zi, 2 GHz bandwidth and 20GS/s sampling rate) plus a photodetector PD1 (Thorlabs) are used to measure spectrum and pulse train. Therefore, the measurement system can achieve simultaneous measurement of mode field, spectrum and time domain. In order to realize the real-time observation of the dynamic process, we use the DFT technology to measure the real-time spectrum. The oscilloscope 2 (Tektronix DPO75902SX, 33 GHz bandwidth and a 100 GHz sampling rate) receives two electrical signals at the same time, one of which is detected by the photodetector PD2 (Thorlabs), and the other is detected by the photodetector PD3 (Keyang photonics, KY-PRM-I-FA, 18 GHz bandwidth). In order to stretch the pulse in time domain, we inserted a 1.5 km dispersion-compensating fiber (DCF, YOFC, DCF-G.652C/250) before PD3. Due to the limitation of experimental conditions, the measurement of real-time pulse train and real-time spectrum are not performed simultaneously with other measurements.
3. Results and discussion
3.1 Dual-resonance coupling LPFG
The FMF-LPFG inscribed in SMF is a significant part of STML laser. We fabricated the FMF-LPFG with a CO2 laser (CO2-H30 Han’s laser). The CO2 laser radiate vertically downward to one side of the fiber and move along the fiber axis, thus destroying the physical structure of the fiber, realizing the asymmetric distribution of the refractive index, and promoting the coupling of the fundamental mode to the higher-order mode. According to our previous results [33], the transmission spectrum and mode field are shown in Fig. 2. In order to realize the mode coupling between LP01 and LP11, the phase matching conditions need to be satisfied between the two transverse modes. The number of grids is 40 and the period of FMF-LPFG is 449µm which is calculated by Eq. (1):
${\beta _p}$ and ${\beta _q}$ represent propagation constants of mode p and q respectively.
The transmission spectrum of FMF-LPFG is shown in Fig. 2(a), which is overserved with a supercontinuum source and an OSA. We coil the FMF with appropriate curvature at the output to filter the higher-order mode through large bending loss, and ensure there is only LP01 mode component in the spectrum. Because each mode has a dispersion inflection point near 1µm, two resonance peaks at 934 nm and 1073 nm can be seen in the transmission spectrum, which are calculated by Eq. (2):
where ${n_{eff,p}}$ and ${n_{eff,q}}$ are the effective refractive index of mode p and q respectively. That is to say, FMF-LPFG has a spectral conversion bandwidth about 139 nm for LP01 and LP11 modes. There is one period corresponding to two resonance wavelengths, which is called the phenomenon of dual-resonance coupling of different modes [33]. Besides, many oscillations can be seen in Fig. 2(a), which is mainly caused by intermodal interference. Fig. 2(b) shows the output mode field of FMF-LPFG, which is closed to LP11. A 1030 nm continuous wave (CW) light source and a CCD are used in the measurement. When the fundamental mode is input, mode coupling is achieved through FMF-LPFG. Thus, the output mode field corresponds to LP11. In FMF-LPFG, mode coupling occurs between different transverse modes. When the phase matching conditions between the modes are satisfied, that is, near the resonance wavelength, the highest mode coupling efficiency is realized. The mode coupling within the STML laser based on FMF-LPFG satisfies the equation [34]:Mi corresponds to the transform matrix of modes in each part of the cavity, including gain fiber, few-mode fiber, FMF-LPFG, NPR, saturable absorber, and et al. Among them, FMF-LPFG has the greatest influence on the mode coupling, and its corresponding transformation matrix:
3.2 STML verification
When the pump power is 5.5W, stable mode locking can be achieved by rotating the HWP and QWPs in the cavity. The output characteristics of the STML laser are shown in Fig. 3. The pulse train measured by oscilloscope 1 is shown in Fig. 3(a), the pulse interval is 35.1 ns. Fig. 3 (b) shows the output spectrum from 1000 nm to 1100 nm, and the bandwidth is about 19 nm. Fig. 3 (c) and (d) shows the RF spectrum with 1kHz resolution and 500 MHz span, respectively, where the fundamental frequency is 28.49 MHz and exhibits a 69.74 dB signal-to-noise ratio. Fig. 3 (e) shows that the output beam profile is a mixture of several modes. Because the energy ratio of the different modes is different from Fig. 2(b) before the pulses enter the FMF-LPFG, according to Eq. (5) and (6), it can be seen that the mode field is a mixture of LP01 and LP11, and the mode field is different from Fig. 2(b). In addition, the evolution of pulse train and real-time spectrum under stable mode-locking state are obtained through DFT technology, and the relevant results are shown in Fig. 4. It can be seen from Fig. 4(a) and (b) that the pulse is in stable transmission state, and after the pulse is stretched in DCF, different modes are separated in time domain due to different group velocities (FMF and DCF can support the same transverse modes, so no new higher-order modes are generated), so two peaks can be seen in Fig. 4(b).
Fig. 3. The characteristics of stable mode-locking state. (a) Pulse train measured by oscilloscope 1, (b) Spectrum measured by OSA with 0.02 nm resolution, (c) RF spectrum with 1kHz resolution, (d) RF spectrum in 500 MHz span (e) beam profile.
The characteristic of STML is that the phase of longitudinal and transverse modes are locked simultaneously. The verification methods are mainly divided into spectral filtering and spatial sampling [17]. Their essence is to extract a single mode as much as possible, so as to prove that each mode is in mode-locking state. Here, we do not adopt the above two methods, but verify STML by the mode interference. It is well known that when the laser is in the STML state, the group velocity difference between different transverse modes will remain unchanged, so the phase difference between the transverse modes will become a fixed value, resulting in interference. In order to verify STML, we observe the stretched pulse by dislocation fusion of multi-segments of fiber behind lens L4, as shown in Fig. 5(b). Compared to the original DFT measurement system, as shown in Fig. 5(a), we put a section of HI1060 fiber in the middle of FMFs. Meanwhile, two sections of FMFs are loaded on the fiber fusion splicer. The HI1060 fiber is spliced with a 1.5 km FMF behind, and the output beam is measured by oscilloscope 2 after passing through PD3. Comparing two different measurement methods, there will be different interaction processes between the modes. Here we take LP01 and LP11 as examples. The different modes in Fig. 5(a) will occur in the FMF due to the different group velocities, so LP01 and LP11 are separated in time domain, as shown in Fig. 4(b). In Fig. 5(b), due to the addition of HI1060 fiber, after dislocation alignment, LP11 is coupled to LP01 and intermodal interference occurs. Therefore, LP01 is the dominate mode in output, and the intermodal interference is also retained.
Fig. 5. The schematic of (a) original DFT measurement system and (b) DFT measurement system with misalignment.
Since the dislocation alignment loss is too large, we increased the pump power to 7W. At this time, the laser is in a stable multi-pulse mode-locking state. The real-time pulse train and real-time spectrum are shown in Fig. 6(a) and (b). It can be seen that two peaks appear after each pulse is stretched, indicating that walk-off occurs in different transverse modes due to different group velocities. Fig. 6(c) shows the corresponding spectrum measured by DFT, and it can be seen that the energy of the slower transverse mode is higher. Because the laser mentioned is a dissipative mode-locked laser, the wavelength is in the normal dispersion region, so the slower transverse mode is a relatively low-order mode, indicating that the relatively low-order mode occupies a higher energy in the output pulse. The real-time spectrum with misaligned alignment in multi-pulse mode-locking state and corresponding spectrum measured by DFT are shown in Fig. 6(d) and (e). Because of the addition of HI1060 fiber, Fig. 6(d) has a certain time delay compared with Fig. 6(b). It is found from Fig. 6(e) that the higher-order transverse mode has a higher energy proportion, and this mode should be LP11 which is the result of mode mismatch between different modes during the transmission process. Meanwhile, it can be seen from Fig. 6(e) that the interference peak appears on the two peaks at the same time. For a clearer observation, we zoom in on Fig. 6(d) and (e). It can be seen from Fig. 6(f) and (g) that interference peaks in each main peak have equal time interval and remain stable with increasing transmission distance, indicating that the two modes form stable interference in the fiber with a fixed phase difference, which also proves that the STML is achieved in laser.
Fig. 6. The (a) pulse train and (b) real-time spectrum in multi-pulse mode-locking state, (c) corresponding spectrum measured by DFT. (d) The evolution of real-time spectrum with misaligned alignment in multi-pulse mode-locking state, (e) corresponding spectrum measured by DFT, (f) close-up of (d) and (g) close-up of (e).
The STML laser studied in this paper is STML laser based on FMF, but it is also applicable to STML lasers based on MMFs. The highly flexible fiber grating cans flexibly compensate for larger modal dispersion in the cavity by adjusting its grating period and other relevant parameters, which is not only applicable to LP01 and LP11, but also to more mode groups. Meanwhile, the introduction of fiber gratings is of great significance for the research of STML lasers with all-fiber structure.
4. Conclusion
In conclusion, we build an STML laser based on FMF-LPFG. By introducing FMF-LPFG into the STML laser, the energy of different transverse modes in the cavity is continuously redistributed. At the same time, the addition of FMF-LPFG effectively compensates the large intermodal dispersion in STIN MMF, and STML is realized. To verify the reliability of STML, compared to the conventional spectral filtering and spatial sampling, mode interference is adopted. Interference between LP01 and LP11 occurs in laser by dislocation alignment of different types of fibers. Finally, stable interference fringes are observed through DFT, which proves that phase-locking is achieved between different modes and STML is achieved. This is beneficial to actively control the dynamic process in the STML laser, which is of great significance for the further study of STML.
Funding
National Key Research and Development Program of China (2018YFB0504400); National Natural Science Foundation of China (12274238, 61835006, 62205159); Natural Science Foundation of Tianjin City (19JCZDJC31200); Special Project for Cooperation in Basic Research of Beijing, Tianjin and Hebei (21JCZXJC00010).
Disclosures
The authors declare no conflicts 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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