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Abstract
We numerically study a hybrid mode-locked holmium-doped fiber laser based on the coupled nonlinear Schrödinger equations, by introducing nonlinear polarization rotation and saturable absorber in a same cavity. Maps of mode-locking states related to the intracavity wave-plates are created to evaluate the stability of mode-locking. Nonlinear polarization rotation imposes negative effect of linear absorption and even reverse saturable absorption to the pulses in the hybrid cavity. Saturable absorber compensates the negative effect from nonlinear polarization rotation, then consequently contributes to the stability of single-pulse mode-locking states. The extension of pulse time duration indicates the dynamic synergy between nonlinear polarization rotation and saturable absorber rather than the static combination mechanism of them.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Passively mode-locking is a common approach for generating the stable ultrashort pulses with high peak power. From 1990’s, passively mode-locked fiber lasers are extensively investigated [1–4], whose working wavelength has been extended from the communication band of 1.55 µm to the bands of 1 µm and 2 µm. At the band of 2 µm, both Ho- and Tm-doped fibers can be used as the gain media in fiber lasers, which can be applied in the fields of LIDAR, remote sensing, telecommunication, and micromachining [5–8]. Usually, the mode-locked Tm-doped fiber lasers generate ultrashort pulses at the wavelengths slightly below 2 µm, which are corresponding to the strong water absorption lines and with great importance in surgery [9]. On the other hand, the undisturbed propagation of laser beam is required in practice because the present of strong water absorption peaks can affect the beam quality [10]. The lasers operating above 2 µm are needed for fitting into the atmospheric window [11,12]. Therefore, the Ho-doped fiber lasers are suitable to generate the laser pulse at this wavelength range. Many experimental results are devoted to the field of mode-locked Ho-doped fiber laser. Especially, the all-fiber design has been reported by several groups, which is benefit for miniaturization and portability of Ho-doped fiber lasers [13–17].
Passively mode-locking in a fiber laser is usually realized with the help of saturable absorbers (SAs). As a typical SA, semiconductor saturable absorber (SESAM), which was demonstrated in 1990’s [18], has been widely used in research and industry nowadays. From the early years of this century, various novel carbon nanomaterials, such as carbon nanotubes (CNTs), graphene, graphene oxide, have been flourishing in the field of mode-locking lasers. CNT working as a SA to suppress amplitude noises at 1550 nm was first proposed in 2003 [19]. Since their unique electronic and optical properties, CNTs have been used as mode-lockers with fast response time (subpicosecond) and broad operation band [20,21]. Because they can be easily integrated into various fiber configuration and keep the alignment-free and all-fiber format, CNTs are rapidly adopted in passively mode-locking fiber lasers [22,23]. With the help of all polarization-maintaining cavity design [24], single-wall carbon nanotubes (SWNTs) based mode-locking fiber lasers can provide good stability, which are competitive candidates working as the seed sources for the applications of precision spectroscopy and metrology. Because the wavelength outside 1.55 µm band is not fit for the bandgap of conventional CNT, graphene that has no bandgap is the preferred SA in many mode-locking lasers working in the region of 2 µm for Ho-doped fiber laser [25] and even mid-infrared. Recently, the improved CNTs with new structure parameters can work at 2 µm with the modulation depth increased [26–28]. However, SWCNT based fiber SA devices is easily to damage so that the output power of mode-locking ultrashort fiber laser is limited.
Besides saturable absorption, other nonlinear effects in optical fibers working as the artificial saturable absorber (ASA) are vigorous for mode-locking. Nonlinear polarization rotation (NPR), nonlinear optical loop mirror (NOLM), and nonlinear amplifying loop mirror (NALM) have been proposed to mode-lock the fiber laser. Among these nonlinear effects, NPR has been widely investigated. Different from the SA based mode-locking schemes, NPR just uses common optical devices, such as polarizers, wave plates, and polarization beam splitter. It can cover a wide wavelength range from near-infrared to mid-infrared, and can suffer high pump power over 1 W [29]. However, as the polarization state of the pulse is sensitive to environmental variation, it is challenging to realize the long-term stability with good repeatable performance. The key problem of self-starting also limits the application of NPR mode-locking scheme.
One practical approach to overcome the disadvantages of the SA and ASA schemes is to introduce the hybrid scheme in a laser cavity. Based on SESAM and NPR, several kinds of hybrid mode-locking fiber lasers were reported. A. Ruehl et al. discussed the application of hybrid mode-locking scheme in wave-breaking free fiber lasers at 1 µm [30]. SESAM reduced the mode-locking threshold drastically, stabilized the laser against competitive cw-operation, and facilitated the suppression of chaotic pulsations. This hybrid scheme can even stabilize the Er-doped fiber lasers working at harmonic mode-locking states [31,32]. As another candidate, the CNT based hybrid scheme was applied in various mode-locking fiber lasers. The Tm-doped all-fiber laser mode-locked by SWCNT and NPR was demonstrated to generate higher-order solitons at 1860-1980nm with long-term stability [33]. An isolator-free Er-doped fiber laser was then proposed by the same group [34]. The research on hybrid mode-locking fiber lasers paid much attention to the experimental evaluation of the new schemes. Some numerical investigations have been devoted to the hybrid scheme [35–37], where NPR is easily described by the transmission of the cavity. However, more information from the full vector model is needed to illustrate how the hybrid mode-locking scheme promotes the performance of a mode-locking fiber laser.
In this paper, we focus on the numerically study of the three different mode-locking schemes of involving SWCNT-based SA, NPR, and the hybrid SA-NPR in a Ho-doped fiber laser. In section 2, the model of pulse generation and propagation inside the cavity is introduced. We simulate the mode-locking in the laser by using a full vector model based on the coupled nonlinear Schrödinger equations. In section 3, the simulation results and discussion are presented. We confirmed that in a same fiber cavity, the hybrid scheme can generate more stable single-pulse states than the NPR when the wave plate angles vary. The stability of hybrid mode-locking is analyzed from the saturable absorption properties in the three mode-locking schemes. Different mode-locking states are examined. Finally, we conclude the main points of this paper.
2. Simulation model
The diagram of the Ho-doped fiber laser is shown in Fig. 1. The ring cavity consists of a 1 m long Ho-doped fiber (HDF) with the second order dispersion of −107 ps2/km as the gain media, and a 1.7 m long passive single mode fiber (SMF) with the second order dispersion of −97 ps2/km. Two groups of λ/2 and λ/4 wave plates are set before and after a polarization dependent isolator (PD-ISO), which help to find the mode-locking working points in the schemes of the hybrid and NPR. The SA is inserted into the cavity after the coupler. We choose a 50/50 coupler here to extract the pulses from the cavity. The output coupling ratio is with great importance for guiding the signal from the cavity. For getting a maximum output power or other purposes, an optimized coupling ratio should be considered [38,39]. The parameters in simulation are listed in Table 1.
Table 1. Simulation Parameters in Simulation
In simulation, the full-vector model is built for the pulse generation and propagation in the laser cavity. The extended coupled nonlinear Schrödinger equations are used for calculation [40]
Here, the linear polarizer in x polarization axis is chosen, where px = 1, py = 0. The transmission matrix for the wave plates in our simulation is
3. Simulation results and discussion
In our simulation, we used the self-written code, which was confirmed by comparing with the results in Ref. [33] and [43]. White Gaussian noise was set up as the initial condition. The three different mode-locking schemes, SWCNT-based SA, NPR, and the hybrid, were adapted separately for comparison. Figure 2 shows the results of SA scheme. Figure 2(a) is the time domain evolution of the output pulses, which shows the stable single-pulse generated after around 200 roundtrips. The output pulse at the 2000th roundtrip is shown in Fig. 2(b), which has a Gaussian shape pulse duration of 0.90 ps and peak power of 936.4 W. From the results of pulse time duration, peak power, and repetition rate (approximated as 66 MHz corresponding to a cavity length of 3 m), we can estimate that the average output power is 55 mW. Considering the conversion efficiency from Ref. [16] and [28], the pump power should be in the range of 1.5 W to 2 W.
Fig. 2. SA mode-locking scheme. (a) Time domain evolution of the output pulses; (b) Output pulse at the 2000th roundtrip; (c) Output optical spectrum.
Different from the SA scheme that only generates one output state, the NPR and hybrid mode-locking schemes have various output states, of which the output states are related to the wave plates’ angles. In calculation, we fixed the polarizer at x-axis, and varied the angles of the two wave plates (A and B) from 0° to 90° to x-axis with a step of 2.5°. The angles of the other half wave plate and quarter wave plate were kept at 45° and at 5°, respectively. Figure 3 shows the three typical output states in NPR scheme. The stable single-pulse output state could be obtained after 200th roundtrip when the wave plates angles were set at A = 60° and B = 72.5° as shown in Fig. 3(a). When the angles of wave plates A and B were set at 55° and 85° respectively, the unstable single-pulse output state could be generated. We can observe the single pulse train with peak power variation in Fig. 3(b). The situation of unstable multi-pulse output can be seen from Fig. 3(c), at which the wave plates angles are A = 2.5° and B = 42.5°. Different from the single pulse output states, at the unstable multi-pulses states there are many small chaotic pulses output along with roundtrip number.
Fig. 3. Time domain evolution of the output pulses. (a) stable single-pulse in NPR scheme (the angle of wave plates A = 60°, B = 72.5°); (b) unstable single-pulse in NPR scheme (the angle of wave plates A = 55°, B = 85°); (c) unstable multi-pulses in NPR scheme (the angle of wave plates A = 2.5°, B = 42.5°).
Figure 4 presents the different output pulse states as maps for the NPR and hybrid schemes. The symbols correspond to different pulse states. The red circles and filled circles represent the unstable and stable single-pulse states, respectively. The blue hexagons stand for the unstable multi-pulse states. The blue rectangles correspond to the state of no-oscillation. At this state, the intracavity loss is so large that no output can be detected from the cavity. The points without any mark stand for the states with chaotic output which cannot be classified. We can see from Fig. 4 that the number of the red filled circle in the hybrid scheme is more than that in the NPR scheme. This suggests the hybrid mode-locking scheme can transform the unstable pulse states of the NPR scheme into stable single-pulse state at the same wave plate angles.
Fig. 4. Maps of the different pulse states of Ho-doped mode-locking laser. (a) NPR mode-locking scheme; (b) Hybrid mode-locking scheme.
The mode-locking pulses are affected by the factors of the net dispersion, accumulated nonlinear phase shift, properties of the SA, etc. For a SA, modulation depth is vital for generating and stabilizing the pulse. Usually, the modulation depth of a SA is defined as the maximum change in absorption induced by the incident light. However, in an NPR mode-locking fiber laser, the change of saturable absorption is not directly related to the incident light on the polarizer or polarization beam splitter, but dynamically responds to the laser pump power, the angles of the wave plates, the output coupling ratio, etc. Thus, we prefer to use the “power absorption change” (ΔL) to define the “modulation depth” in the discussion. The power absorption L(T) at the polarizer as a function of window time can be expressed by
where Pout(T) and Pin(T) are pulse power after and before the polarizer. By mapping the window time T to input pulse power under the instantaneous response assumption, we can obtain the information of L(Pin) as a function of input power. Consequently, ΔL(Pin) can be defined as the absorption difference between the peak and pedestal of the pulse.To illustrate a calculated ΔL(Pin), in Fig. 5 we choose a stable single pulse generated by the NPR mode-locking scheme at the angle of wave plates (A = 60°, B = 72.5°), as an example. Figure 5(a) illustrates the pulse time information before and after the polarizer. It can be found that the pulse power is attenuated at the polarizer. By using Eq. (4) and mapping the input power, we can observe the “saturable absorption” property in Fig. 5(b), where the absorption difference between the peak and pedestal of the pulse is 6.42%. There are absorbance changes at low power range in Fig. 5(b), which is due to interaction between the pedestal of the pulse and the dispersive wave. In the following, the ΔL(Pin) with the number of positive, zero, and negative will be discussed.
Fig. 5. The pulse generated by the hybrid mode-locking scheme at the angle of wave plates (A = 60°, B = 72.5°). (a) The pulse before and after the polarizer; (b) The pulse absorption curve in the polarizer.
In Fig. 6 the contour map of ΔL is plotted in function of the angle of wave plates for NPR and SWCNT mechanisms in the hybrid scheme. It is shown in Fig. 6(a) that ΔL can be negative, zero, and positive for NPR mechanism, which suggests the difference mode-locking process inside the cavity. Figure 6(b) indicates that for SWCNT mechanism, ΔL is positive for all the different mode-locking status. We choose three different cases to analyze in detail: 1) stable single-pulse output from the hybrid scheme but unstable multi-pulses output from the NPR scheme; 2) stable single-pulse output from the hybrid scheme but unstable single-pulse output from the NPR scheme; 3) stable single-pulse output from the hybrid and NPR schemes. In each case, the angles of the wave plates are the same.
Fig. 6. The contour map of ΔL in function of the angle of wave plates for the hybrid scheme. (a) The NPR mechanism, and (b) the SWCNT mechanism.
Firstly, we check the mode-locking states at the angle of wave plates (A = 2.5°, B = 42.5°) and (A = 7.5°, B = 22.5°), at which in hybrid scheme the laser works at stable single-pulse states, and in NPR the laser generates at unstable multi-pulses. Figure 7 shows the absorption curves of the pulse in the hybrid scheme and the NPR at two different angles of wave plates, where the insets are the time domain evolution of the output pulses. At the case that wave plates angle are set at A = 2.5° and B = 42.5° in the hybrid mode-locking scheme, as shown in Fig. 7(a), the polarizer and SA provide the ΔL of −4.2% and 7.94%, respectively. The total ΔL is 3.74%. On the other hand, in the NPR mode-locking scheme, as indicated in Fig. 7(b), the polarizer provides a small ΔL of 0.99%, only unstable multi-pulses are generated. When the wave plates angle A and B are set at 7.5°and 22.5° in the hybrid scheme, as illustrated in Fig. 7(c), the polarizer provides a ΔL of −2.89% to the pulse and the SA provides 7.95%. The total ΔL is 4.44%, and the stable single-pulses are generated. However, in the NPR scheme, as shown in Fig. 7(d), the polarizer provides a ΔL of −3.08%, and the unstable multi-pulses are generated as illustrated in the inset.
Fig. 7. The absorption curves of the pulse in the hybrid mode-locking scheme and NPR. (a) and (b) the angle of wave plates (A = 2.5°, B = 42.5°); (c) and (d) the angle of wave plates (A = 7.5°, B = 22.5°). The insets are the time domain evolution of the pulses at corresponds the mode-locking scheme and angle of wave plates.
At the two mode-locking states mentioned above, the saturable absorption is too small to support the stable single-pulse state in the NPR scheme. After introducing SA into the cavity, the pulse evolution of polarization state is changed inside the cavity. The absorbance of NPR mechanism in the hybrid scheme at low powers can have a lower value than that at high powers. This means that the NPR mechanism plays the role of reverse saturable absorber on the pulse, where ΔL is negative. However, although ΔL is negative provided by the NPR mechanism, with the help of SA, the total ΔL is positive. The cavity can provide enough saturable absorption for mode-locking.
Next, we examined the mode-locking states at the wave plates angle of (A = 40°, B = 52.5°) and (A = 55°, B = 85°), where for the hybrid scheme the fiber laser works at stable single-pulse state, and for the NPR scheme the laser generates unstable single-pulses. Figure 8 plots the absorption curves of the pulse in the hybrid scheme and the NPR in the case of the two different angles of wave plates, where the insets are the time domain evolution of the output pulses from the cavity. In the hybrid scheme, an interesting phenomenon is that ΔL = 0 at the polarizer. The NPR mechanism only introduces linear loss to the pulse, as shown in Figs. 8(a) and (c). Thus, the pulse shaping is only based on the SA. As mentioned in the Ref. [40], when the mode-locking pulse has evolved into a soliton, it has uniform polarization across the pulse profile. This is also verified by checking the phase information in the calculated data.
Fig. 8. The absorption curves of the pulse in the hybrid mode-locking scheme and NPR. (a) and (b) the angle of wave plates (A = 40°, B = 52.5°); (c) and (d) the angle of wave plates (A = 55°, B = 85°). The insets is the time domain evolution of the pulses at corresponds the mode-locking scheme and angle of wave plates.
In the cases discussed above, the NPR mechanism does not contribute to the saturable absorption in the cavity. The SA compensates the negative effect from NPR and provide the saturable absorption for mode-locking. Thus, the unstable states are converted into stable single pulse states, and the stability of the hybrid mode-locking fiber is increased. The full vector model reveals that the previous model considering the NPR mechanism as a SA simply is inadequate for the accurate design or analysis of a hybrid mode-locking fiber laser.
Finally, we examine the mode-locking states at the wave plates angle of A = 60° and B = 72.5°, at which both the hybrid scheme and NPR can generate stable single-pulse. Figure 9(a) and (b) illustrate the absorption curves of the pulses in the hybrid scheme and NPR, where the insets are the time domain evolution of the output pulses from the cavity. In the hybrid scheme, both SA and NPR mechanism contribute saturable absorption to the pulses, where the polarizer provides a ΔL of 1.34% and the SWCNT provides 6.87%. The NPR mechanism for the hybrid scheme makes the main contribution to the pulse shaping, as shown in Fig. 9(c). On the other hand, the SA stimulates the saturable absorption and starts the mode-locking. This suggests that the NPR mechanism performs the positive role of saturable absorption, and the enhanced mode-locking stability of the hybrid scheme benefits from the cooperation of SA. In Fig. 9(d), the temporal information of the output pulses is presented, where the three mode-locking schemes are compared. The hybrid scheme possesses two saturable absorption mechanisms in the same cavity; however, the output pulse has the broadest time duration of 2.22 ps. The two independent saturable absorption mechanisms in a same mode-locking fiber laser cavity do not necessarily generate an even shorter pulse. For confirming the difference in output pulse time duration, we checked another mode-locking state at the wave plates angle of A = 90° and B = 0°. From Fig. 10(a), it can be found that the polarizer provides a ΔL of 3.48% and the SWCNT provides 7.64%. For NPR scheme, ΔL is as high as 52.72% as shown in Fig. 10(b), and it also mainly contributes to the pulse shaping as illustrated in Fig. 10(c). From Fig. 10(d), it can be observed that a shortest time duration of 0.45 ps is generated by the NPR scheme. The hybrid scheme generates a broadest pulse with the time duration of 1.39 ps. These results are different from the description in Ref. [45], where the hybrid mode-locked erbium-doped fiber laser with SWCNTs and NPR generates the pulse with time duration significantly shorter than that in a SWCNTs-only mode-locked laser. This reveals that in pursuit of a shorter pulse, just introducing two saturable absorption schemes into a Ho-doped fiber laser cavity is not necessarily useful. The abnormal extension of pulse time duration indicates the dynamic synergy between NPR and SA rather than the static combination mechanism of them. If we change the pump power in a hybrid mode-locked fiber laser, at some specific wave plate angles, the laser will be even switched from the stable mode-locking state into the unstable state. Thus, the intracavity losses, pump power, and net dispersion (Ho-doped fiber laser cavity with larger anomalous dispersion), should be comprehensively considered. Another point is that the contribution of SA and NPR to pulse shaping related to the sequence of intracavity single-turn pulses through the SA and NPR structures during the simulation. In our simulation, if SA and NPR structures changed the position, SA will suffer double the power (the output coupling ratio is 50%), and the absorption will change. The pulse back to the cavity will be different, so after the propagation along the SMF, the NPR effect to the pulse will be different too.
Fig. 9. At the angle of wave plates (A = 60°, B = 72.5°). (a) The absorption curves of the pulse in the hybrid mode-locking scheme, and (b) is NPR; (c) The pulse before and after SA and polarizer in the hybrid scheme; (d) The comparison of the output pulses in SA, NPR and hybrid scheme.
Fig. 10. At the angle of wave plates (A = 90°, B = 0°). (a) The absorption curves of the pulse in the hybrid mode-locking scheme, and (b) is NPR; (c) The pulse before and after SA and polarizer in the hybrid scheme; (d) The comparison of the output pulses in SA, NPR and hybrid scheme.
4. Conclusion
In conclusion, the stability of a hybrid mode-locked Ho-doped fiber laser was numerically investigated. The single-wall carbon nanotube working as saturable absorber (SA) and nonlinear polarization rotation (NPR) were introduced into a same laser cavity in our calculation. The pulse generation and propagation inside the cavity were simulated by a full vector model based on the coupled nonlinear Schrödinger equations. Maps of the different pulse states were plotted versus the intracavity wave plate angles. By introducing SA into a Ho-doped fiber laser cavity, the hybrid mode-locking scheme can convert the unstable pulse state of the NPR scheme into a stable single-pulse state, which enhances the stability of mode-locking. Such phenomenon was discussed by analyzing the dynamic saturable absorption in the laser cavity. The NPR mechanism performs three different absorption states, including reverse saturable absorption, linear absorption, and saturable absorption. The saturable absorber can compensate the negative effect from nonlinear polarization rotation, then consequently contribute to the stability of single-pulse mode-locking states. The abnormal extension of pulse time duration indicates the dynamic synergy between nonlinear polarization rotation and saturable absorber rather than the static combination mechanism of them. We believe that the work contributes to the deep understanding of hybrid mode-locking dynamics and provides helpful guidance for the fiber laser design and optimization.
Funding
National Natural Science Foundation of China (61875043).
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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