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Abstract
The Mamyshev oscillator (MO) can generate high-performance pulses. However, due to their non-resonant cavities, they usually are not self-starting, and there is almost no effort to reveal the pulse buildup dynamics of the MO. This paper investigates the dynamic of single pulse (SP) and multi-pulse formation in a self-starting MO. It indicated that both SP self-starting and multi-pulse self-starting can be obtained by adjusting the oscillator parameters. More importantly, increasing pump power could only result in bound state pulses (BSPs) if SP self-starting was formed. With the increase of the pump power, the pulse number in BSPs would increase. However, multiple pulses could not be formed only by increasing the pump power, and the BSPs obtained here underwent SP generated from noise, amplified, and then bounded, which is different from conventional passive mode-locked fiber lasers (CPMLFLs). On the other hand, if multiple pulses were self-initiated, BSPs, pulse bunch, and harmonic mode-locked pulses (HMLPs) could be obtained by adjusting the polarization state and pump power in the cavity. Furthermore, once any of the above states are formed, if the oscillator polarization state and filter interval are unchanged, only increasing the pump power from zero, the original state can still be obtained, which is consistent with the characteristics of the CPMLFLs. These findings will provide new insights into the pulse dynamics of self-starting MO, which will be significant for studying ultrafast laser technology and nonlinear optics.
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1. Introduction
Ultrafast lasers have received widespread attentions due to their extensive applications in industrial fields such as optical communication, material processing, metrology, and basic scientific [1–4]. Mode-locked fiber lasers are considered an ideal platform for generating ultrafast optical pulses due to their compactness and robustness [5]. Generally, to achieve mode-locked in fiber lasers, a saturable absorber (SA), as a key component, was often used since it can introduce power-related losses in the laser cavity and select the strongest pulse from the background noise to form mode-locked. There are two commonly used SAs, one is material type SAs, such as graphene [6], black phosphorus [7], carbon nanotubes [8,9], transition metal dichalcogenides [10], topological insulators [11], PbS [12], ferroferric oxide [13], etc., and the other is equivalent SAs based on nonlinear polarization rotation [14], nonlinear amplifier loop mirror [15], single mode-multimode-single mode fiber structure [16] or figure-9 structure [17]. Due to the decrease in material performance over time and the lower damage threshold, fiber lasers based on material SAs have lower pulse peak power. The equivalent SA is based on the nonlinear effect of optical fibers, so its transmittance curve is usually a sine function of the pulse peak power, which also limits the peak power and energy of the generated mode-locked pulse.
To generate higher energy pulses, a new type of fiber ultrashort pulse source, the Mamyshev oscillator (MO), has been proposed. The mode-locked mechanism of MO utilizes a combination of cascade spectral broadening induced by self-phase modulation (SPM) and offset spectral filtering effect to achieve a transfer function related to peak power, which serves as an effective SA for pulse generation [18]. Since Regelskis et al. proposed and demonstrated the first fiber MO, delivering pulse with 1.06 $\mathrm {\mu}$m central wavelength and 2.8 nJ pulse energy [19], researchers have mainly focused on how to increase the pulse energy and narrow the pulse duration of the MO [20–22]. For example, by designing an intracavity parabolic pulse evolution, Wise’s group obtained a pulse with a pulse energy of 50 nJ. Using a larger core fiber, Sidorenko et al. scaled the pulse energy of MO to 190 nJ. By introducing a single-polarization large-mode-area photonic crystal gain fiber, the pulse energy was further increased to the order of 1 $\mathrm {\mu}$J.
However, none of the above MOs are self-starting. In fact, due to the step-like saturable absorption effect, the continuous wave (CW) signal is significantly suppressed, making it difficult for MO to start [20]. In order to start MOs, researchers have proposed several methods, such as injecting external seed or using an acousto-optic modulator [19,20,22], adding an additional starting arm [23,24], and adjusting the pump power [25,26]. In addition to using the above methods to achieve mode-locked by introducing sufficiently strong fluctuations in the cavity, another method to achieve self-starting is to overlap bandwidths of the two offset spectral filters to generate mode-locked [27]. This operation regime of fiber pulse generators was investigated in 2018 [28], and then Bao’s group reported self-starting spatiotemporal mode-locked in a multimode MO [29]. Our group demonstrated a self-starting Yb-doped fiber MO and observed multiple dynamic patterns of pulses [30]. Very recently, we reported a self-starting Yb-doped fiber MO and studied its starting, extinction, and output dynamics by designing and using a special temperature-sensitive band-pass filter [31]. However, most researchers have focused on Yb-doped fiber lasers. Due to the relatively narrow gain bandwidth of Er-doped fibers, their self-starting is more difficult than that of Yb-doped fiber MOs. So far, only a few studies have reported using filter overlap to achieve self-starting in Er-doped fiber MOs [32]. However, compared to Yb-doped MOs, Er-doped MOs are useful in many applications, such as remote sensing, LiDAR, and medicine.
On the other hand, using overlapping bandwidths of the two offset spectral filters can successfully start MOs and form multiple pulses as the pump power increases due to the narrow filter interval. So, it can also be used as a research platform for multi-pulse dynamics. In fact, the study of bound state pulses (BSPs) helps to exploit larger telecommunication capacity in optical fiber transmission lines [33], and harmonic mode-locked pulses (HMLPs) with higher repetition rates can be widely used in many applications such as optical communication, frequency measurement and ranging, or data storage [34–36]. Therefore, the study of multi-pulse MO has also attracted many researchers’ attention [37]. In 2019, Luo’s group reported on the multi-pulse dynamics in an Er-doped fiber MO, which used an external seed to initiate the pulse operation of the oscillator [38]. Very recently, they further investigated the buildup dynamics of the oscillator pulse from the seed signal in an Er-doped fiber MO and demonstrated that the multi-pulse starting operation evolved from multiple seed pulses, rather than the pulse splitting effect in MO with large offset spectral filtering [39]. Their research findings are very interesting. So, in self-starting Er-doped fiber oscillators, whose parameters are related to the self-starting characteristics of single pulse (SP) and multi-pulse? What are the dynamic characteristics of SP and multi-pulse? All the questions above form the original intention of our research.
In this paper, we investigated the dynamic of SP and multi-pulse formation in the self-starting process of MO. It was demonstrated that both SP self-starting and multi-pulse self-starting can be obtained under different filter intervals and pump power. When an SP self-starting occurred in MO, increasing the pump power could obtain the BSPs. When multi-pulse self-starting occurred in MO, BSPs, pulse bunch, and HMLPs could be obtained by adjusting the pump power and intracavity polarization state. Moreover, we demonstrated that the filter interval and intracavity polarization state played a crucial role in generating SP or multi-pulse oscillations during the self-starting process.
2. Experimental setup
The schematic of the Er-doped MO is illustrated in Fig. 1. The laser cavity contains two arms, arm1 and arm2. In each arm, a section of Er-doped fiber (EDF, Er-80-4/125, Thorlabs) with a peak core absorption of 55 dB/m at 976 nm was used as a gain medium and was core-pumped via a wavelength division multiplexer (WDM) by a single-mode LD, which had a maximum pump power of 1.0 W. The length of EDF$_{1}$ and EDF$_{2}$ are 1.9 m and 3.0 m, respectively. Two isolators, ISO$_{1}$ and ISO$_{2}$, were placed on each arm to ensure unidirectional transmission. The change of the polarization state within the oscillator was achieved by adjusting the polarization controllers (PC$_{1}$ and PC$_{2}$) on each arm, and the laser outputs were obtained from the 10% output couplers (OC$_{1}$ and OC$_{2}$). Each OC was connected to a band-pass filter (filter $_{1}$ and filter$_{2}$), where filter$_{1}$ had a fixed central wavelength of 1549.4 nm and a 3-dB bandwidth of 3.0 nm, and filter$_{2}$ was a tunable filter, its 3-dB bandwidth was 1.0 nm, and its central wavelength can be tuned from 1530 nm to 1580 nm by adjusting a helical micrometer. The total fiber cavity length was 25.8 m, corresponding to the cavity dispersion of -0.245 ps$^2$. The inset of Fig. 1 shows the spectral transmission characteristics of the two filters. The central wavelength of filter$_{2}$ is 1546.0 nm at the helical micrometer of 5.18 mm. An optical spectrum analyzer (Yokogawa AQ6370D), a 1-GHz oscilloscope (Keysight DSO-X 3104 T) and 8-GHz oscilloscope (RTP084B) together with a 3-GHz photodetector, a commercial optical autocorrelator (FR-103XL), and a radio frequency (RF) spectrum analyzer (Agilent N9020A) were used to monitor the spectra, pulse train, the pulse shape, and the stability of operation. In addition, a 33-GHz oscilloscope (Keysight DSOZ504A) and a 45-GHz photodetector ( New Focus 1014) were also used to observe the bound pulses.
Fig. 1. Schematic of the Er-doped MO. WDM: wavelength division multiplexer; EDF: Er-doped fiber; ISO: isolator; OC: output coupler; PC: polarization controller (Inset: the spectral transmission characteristics of the two filters).
3. Experimental results and discussion
In our experiment, by adjusting the pump power, the central wavelength of the filter$_{2}$, and the intracavity polarization state, we obtained multi-pattern pulses based on SP self-starting and multi-pulse self-starting, respectively.
3.1 MO with SP self-starting
When the central wavelength of the filter$_{2}$ was set to 1545.8 nm, i.e., the filter interval was set to 3.5 nm, the power of pump$_{1}$ and pump$_{2}$ was 136 mW and 182 mW, respectively, then, by simply rotating the waveplates, the MO started to operate in an SP regime, that is, the MO formed mode-locked based on SP self-starting.
Figure 2(a) shows the output spectra obtained from two output ports. The uniformly spaced pulse sequence has a period of 129 ns, as shown in Fig. 2(b), which is consistent with the total cavity length of 25.8 m. The average output power was 0.24 mW and 0.78 mW, corresponding to output pulse energies of 0.031 nJ and 0.1 nJ, respectively. The autocorrelation trace in Fig. 2(c) shows that the full widths at half maximum (FWHM) are 1.53 ps and 1.25 ps, respectively, assuming Gaussian profiles (dotted green line). Figure 2(d) shows the RF spectrum at a range of 100 MHz and a resolution bandwidth (RBW) of 2 kHz, and it can be seen that the repetition rate is 7.8 MHz and the signal-to-noise ratio (SNR) is 53 dB, indicating that the output pulse is particularly stable.
Fig. 2. Characteristics of SP from the two output ports: (a) the output spectra, (b) the oscilloscope train, (c) the autocorrelation trace, and (d) the RF spectrum at a range of 100 MHz and a resolution bandwidth of 2 kHz.
From Fig. 2, we can see that the pulses from the two output ports have different spectral widths, pulse durations, and energies, although the output coupling ratios were the same for both arms. However, since the MO acted as a two-stage amplifier, the length of EDF$_{2}$ was longer than that of EDF$_{1}$, and the pump power at arm2 was bigger than that of arm1, the output power of port2 was much larger than that of port1. Therefore, we just recorded the data from output port2 for the following research.
In this case, we kept the polarization state of the oscillator and the filter interval unchanged and only alternately increased the power of two pumps in steps of 10 mW each time. When the power of pump$_{1}$ and pump$_{2}$ was increased to 152 mW and 206 mW, respectively, we obtained a two-pulse BSP, as shown in Fig. 3.
Fig. 3. Two-pulse BSP: (a) the output spectrum, (b) the trace of the 33-GHz oscilloscope, and (c) the autocorrelation trace.
The output spectrum exhibits significant periodic modulation, with a modulation period, $\Delta \lambda$, of 0.06 nm, as shown in Fig. 3(a). From formula [40] $c\cdot \Delta T\cdot \Delta \lambda =-{{\lambda }^{2}}$, where $\it {c}$ is the light speed, $\Delta T$ represents the pulse interval, and $\lambda$ corresponds to the center wavelength, it can be inferred that the interval between two pulses is 133 ps, which corresponds to the trace of the 33-GHz oscilloscope, shown in Fig. 3(b). The average output power of the pulse was 1.08 mW. Through curve fitting, we found the pulse duration is 1.30 ps, as shown in Fig. 3(c), so the pulse interval is 102.3 times the pulse duration, confirming that they were loose-bound pulses. Such a large pulse interval makes it impossible to observe BSPs on the autocorrelator. This is because the time measurement range of the optical autocorrelator we used is 185 ps.
In the experiment, further only alternately increasing the power of pump$_{1}$ and pump$_{2}$ to 211 mW and 271 mW, respectively, while keeping the polarization state of the oscillator and the filter interval unchanged, we obtained the three-pulse BSP, as seen in Fig. 4. Figure 4(a) shows the output spectrum, which has dense spectral periodic modulations of $\sim$ 0.04 nm, which corresponds to the interval between two pulses, $\Delta T$, of $\sim$ 233 ps, as shown in the trace of the 33-GHz oscilloscope of Fig. 4(b). Figure 4(c) shows the autocorrelation trace, and the pulse duration is 1.24 ps, so $\Delta T$ is 187.9 times the pulse duration, the large interval confirming that they were loose bound pulses. Furthermore, due to the large pulse interval, we are also unable to observe BSPs on the autocorrelator.
Fig. 4. Three-pulse BSP: (a) the output spectrum, (b) the trace of the 33-GHz oscilloscope, and (c) the autocorrelation trace.
After the bound states of three pulses appeared, we continued alternately to increase the pump power while keeping the polarization state of the oscillator and the filter interval unchanged. Nevertheless, when the pump power of pump$_{1}$ and pump$_{2}$ increased to 220 mW and 286 mW, respectively, there was no bound state of multiple pulses or other types of multiple pulse distributions, but the mode-locked disappeared.
Visualization 1 shows the formation of BSPs. As the pump power increased, small pulses were generated from noise and gradually amplified as the pump power increased until the pulse energy was the same as the original pulse. Then, two pulses with equal energy and unstable spacing oscillate in MO. The two pulses are bound together under the dispersion wave caused by CW (see Fig. 3(a) and Fig. 4(a)), forming a two-pulse BSP. With the increase of the pump power, the process of changing from a two-pulse BSP to a three-pulse BSP is the same as above.
It should be noted that unlike the BSPs obtained in conventional passive mode-locked fiber lasers (CPMLFLs), which are generated simultaneously [41], the BSPs obtained here underwent SP generated from noise, amplified, and then bounded.
3.2 MO with multiple pulses self-starting
When the central wavelength of the filter$_{2}$ was set to 1546.0 nm, i.e., the filter interval was set to 3.3 nm, by increasing the power of pump$_{1}$ and pump$_{2}$ to 170 mW and 220 mW, respectively, the MO started to operate in multiple pulses regime, that is, the MO formed mode-locked based on multiple pulses self-starting.
3.2.1 Bound-state pulses
When MO operated in a disordered multi-pulse state, by adjusting the power of pump$_{1}$ and pump$_{2}$ to 178 mW and 231 mW, respectively, and carefully adjusting the waveplates while keeping the filter interval unchanged, BSPs were obtained. Figure 5 shows the features of a two-pulse BSP. Figure 5(a) is the oscilloscope train of the pulse. The pulse-to-pulse separation is 129 ns. The output spectrum exhibits significant periodic modulation, with a modulation period, $\Delta \lambda$, of 0.18 nm, as shown in Fig. 5(b). It corresponds exactly to the pulse interval, $\Delta T$, of 47.27 ps, as shown in the autocorrelation trace of Fig. 5(c). The pulse duration is 1.44 ps, so the pulse interval is 32.83 times the pulse duration, indicating that this is a loosely BSP. The average output power of the pulse was 1.14 mW.
Further increasing the power of pump$_{1}$ and pump$_{2}$ to 190 mW and 238 mW, respectively, and carefully adjusting the waveplates while keeping the filter interval unchanged, another two-pulse BSP was obtained, as shown in Fig. 6. Figure 6(a) is the oscilloscope train of the BSP. The modulation period, $\Delta \lambda$, of the output spectrum, is 0.20 nm, as shown in Fig. 6(b), consistent with the pulse interval, $\Delta T$, of 39.36 ps, as shown in Fig. 6(c). The pulse duration is 1.29 ps, as shown in the illustration of Fig. 6(c), and the pulse interval is 30.5 times the pulse width, indicating that this is also a loosely BSP. The average output power was 1.21 mW.
Fig. 5. Features of a two-pulse BSP: (a) the oscilloscope train, (b) the output spectrum, and (c) the autocorrelation trace (Insets: corresponding autocorrelation trace with fitted Gauss curves).
Fig. 6. Two-pulse BSP in another polarization state: (a) the oscilloscope train, (b) the output spectrum, and (c) the autocorrelation trace (Insets: corresponding autocorrelation trace with fitted Gauss curves).
3.2.2 Pulse bunch
Unlike the case of SP self-starting, if we further increased the power of pump$_{1}$ and pump$_{2}$ while keeping the filter interval and intracavity polarization state unchanged, we did not obtain a more-pulse bound state but instead obtained an unevenly distributed pulse beam, named pulse bunch. Figure 7 shows the features of the pulse bunch obtained at the power of 304 mW for pump$_{1}$ and 366 mW for pump$_{2}$, respectively. From Fig. 7(a), we can see that its spectrum is no longer modulated. In a 1-GHz oscilloscope, we observe that the pulses of the pulse bunch are unevenly distributed in the oscillating cavity and that one pulse in the pulse bunch is almost twice as strong as the others, as shown in the inset of Fig. 7(b). Figure 7(b) shows the pulse train in a 33-GHz oscilloscope. Figure 7(c) shows the zoom of the pulse in the dashed box, which can be seen as two pulses bunched together. The interval between the two pulses of the bound pulses can be measured using a 33-GHz oscilloscope with a photodetector of 45-GHz to be 112 ps. The average output power of the pulse bunch was 2.44 mW. Figure 7(d) shows the RF spectrum up to 60 MHz with an RBW of 3.0 kHz, from which we can see that the repetition rate of the pulse bunch is $\sim$54.2 MHz, and the SNR is 48 dB, indicating that the laser operated in a relatively stable regime.
Further carefully adjusting the waveplates while keeping the filter interval and the pump power unchanged, another pulse bunch containing 7 pulses was obtained, as shown in Fig. 8. Figures 8(a) and (b) show the output spectrum and pulse train of the pulse bunch. In this case, there are two pairs of bound pulses in the pulse bunch, and the pulse intervals are 111 ps and 109 ps, respectively, as shown in Fig. 8(c). The average output power of the pulse bunch was 2.44 mW. The repetition rate of the pulse bunch is also $\sim$54.2 MHz, and the SNR is 47 dB, as shown in Fig. 8(d).
Fig. 7. Features of pulse bunch: (a) the output spectrum, (b) the pulse train in a 33-GHz oscilloscope (Insets: pulse train in a 1-GHz oscilloscope), (c) the zoom of the pulse in the dashed box, and (d) the RF spectrum up to 60 MHz with an RBW of 3.0 kHz.
Fig. 8. Features of pulse bunch in another polarization state: (a) the output spectrum, (b) the pulse train in a 33-GHz oscilloscope (Insets: pulse train in a 1-GHz oscilloscope), (c) the zoom of the pulse in the dashed box, and (d) the RF spectrum up to 60 MHz with an RBW of 3.0 kHz.
3.2.3 Harmonic mode-locked pulses
When the MO is operated in the above pulse bunch state, by increasing the power of pump$_{1}$ and pump$_{2}$ to 309 mW and 370 mW and carefully adjusting the waveplates, all pulses can move along the oscillator until the interval is equal, and HMLPs are formed. We obtained seventh-order HMLPs in the experiment, as shown in Fig. 9.
Figure 9(a) shows the spectrum and corresponding autocorrelation trace assuming the Gaussian profile. The FWHM of the pulse is 1.41 ps. From Fig. 9(b), we can see that there are 7 equally spaced pulses within one oscillator cycle. Figure 9(c) shows the wideband RF spectrum up to 180 MHz with an RBW of 2.0 kHz, corresponding to a repetition frequency of about 54 MHz. The SNR of the HMLP is greater than 45 dB. The average output power was 2.5 mW.
Fig. 9. Characteristics of 7$\rm {^{th}}$-order HMLPs: (a) the output spectrum (Inset: autocorrelation trace), (b) the pulse train, and (c) the RF spectrum of 7$\rm {^{th}}$-order HMLPs.
After the 7$\rm {^{th}}$-order HMLPs formed, we reduced the pump power while keeping the polarization state of the oscillator and the filter interval unchanged. It is found that when the pump power of pump$_{1}$ and pump$_{2}$ reduced to 255 mW and 334 mW, the 6$\rm {^{th}}$-order HMLPs formed, whose output spectrum, autocorrelation trace, pulse train, and RF spectrum are shown in Figs. 10(a-c). However, at this point, the energy of the optical pulse within the HMLP is not equal, and its stability is poor. If we continue to alternately reduce the pump power, we cannot obtain lower-order HMLPs, but rather, the mode-locked disappears. This is also different from the CPMLFLs.
Fig. 10. Characteristics of 6$\rm {^{th}}$-order HMLPs: (a) the output spectrum (Inset: autocorrelation trace), (b) the pulse train, and (c) the RF spectrum of 6$\rm {^{th}}$-order HMLPs.
It should be noted that once any of the above states are formed, including BSPs, pulse bunch, and HMLPs, if the oscillator polarization state and filter interval are unchanged, only increasing the pump power from zero, the original state can still be obtained. That is to say, under a certain polarization state and filter interval, all the above states can achieve self-starting, which is consistent with the characteristics of the CPMLFLs. At the same time, it is found that the larger the filter interval, the greater the pump power required to realize MO self-starting. In addition, although both SP auto-starting and multi-pulse auto-starting can be generated, the MO parameter range to achieve SP auto-starting is smaller. For example, by fixing the filtering interval to 3.5 nm and adjusting the polarization state of the oscillator to a suitable position, SP auto-starting could only be achieved by increasing alternately the power of pump$_{1}$ and pump$_{2}$ from 119 mW and 163 mW to 140 mW and 188 mW, respectively. When the power of pump$_{1}$ and pump$_{2}$ exceeded 140 mW and 188 mW respectively, the MO would achieve multi-pulse self-starting, but when the power of pump$_{1}$ and pump$_{2}$ was less than 119 mW and 163 mW, respectively, the MO could not achieve self-starting. On the other hand, when the power of pump$_{1}$ and pump$_{2}$ was fixed at 128 mW and 172 mW respectively, in a suitable polarization state of the oscillator, the filter interval was only in the range of 3.0 $\sim$ 3.5 nm, SP self-starting could be realized, when the filter interval was less than 3.0 nm, multi-pulse self-starting could be realized, and when the filter interval was greater than 3.5 nm, self-starting could not be realized.
4. Conclusion
In conclusion, we have investigated the dynamic of single pulse (SP) and multi-pulse formation in a self-starting Er-doped Mamyshev oscillator (MO), and it indicated that both SP self-starting and multi-pulse self-starting could be obtained by adjusting the oscillator parameter. Specifically under large filter interval and small pump power, SP self-starting is prone to occur. At this time, as the pump power increases, additional small pulses will form from noise and then be gradually amplified as the pump power increases until the pulse energy is the same as the original pulse. Then, two pulses with equal energy and unstable spacing oscillate in MO. Under the action of the dispersive wave caused by continuous wave, the two pulses are bound together, forming a two-pulse bound state pulse (BSP), which is unlike the BSPs obtained in conventional passive mode-locked fiber lasers, in which the bound pulses are generated simultaneously. At small filter intervals, it is easy for multiple pulses to self-starting. In this case, BSPs, pulse bunch, and harmonic mode-locked pulses (HMLPs) have be obtained by adjusting the polarization state and pump power in the cavity. Furthermore, once HMLPs are formed, the order of the HMLP will decrease when the pump power is reduced. At the same time, the energy of the optical pulse is no longer equal, and further reducing the pump power leads to the disappearance of mode-locked. In addition, once any of the above states, including BSPs, pulse bunch, and HMLPs, were formed, if the oscillator polarization state and filter interval are unchanged, and only by increasing the pump power from zero the original state can still be obtained, that is, under a certain polarization state and filter interval, all the above states can achieve self-starting. These findings provide some new insights into the self-starting pulse dynamics of MOs and help design MOs that avoid the formation of multiple pulses to achieve high-energy pulses in practical applications.
Funding
National Natural Science Foundation of China (12074098, 62305103); Key Program of the Natural Science Foundation of Hebei Province (F2022205006); S&T Program of Hebei (226Z1706G, 236Z1807G); Hebei Graduate Education and Teaching Reform Research Project (YJG2024043).
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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