Tunable soliton molecules enabled by a nanotubes-based mode-locked fiber laser

Abstract

Soliton molecules are fascinating phenomena in ultrafast lasers which have potential for increasing the capacity of fiber optic communication. The investigation of reliable materials will be of great benefit to the generation of soliton molecules. Herein, an all-fiber laser cavity was built incorporating carbon nanotubes-based saturable absorber. Mode-locked pulses were obtained at 1565.0 nm with a 60 dB SNR and a 4.5 W peak power. Soliton molecules were subsequently observed after increasing the pump power and tuning polarization state in the same cavity, showing variable separation of pulses between 4.87 and 25.76 ps. Furthermore, these tunable soliton molecules were verified and investigated through numerical simulation, where the tuning of pump power and polarization state were simulated. These results demonstrate that soliton molecules are promising to be applied in optical communication, where carbon nanotube-based mode-locked fiber lasers serve as a reliable platform for the generation of these soliton molecules.

Graphical abstract

Keywords

ultrafast lasers / carbon nanotubes / mode-locking / soliton molecules

1 Introduction

Ultrafast lasers are widely employed in optical imaging, spectrum analysis, and advanced manufacturing due to high peak power, narrow pulse width, and high repetition frequency [1–4]. One of the ultrafast pulse generation techniques is mode-locking, which refers to phase locking of laser modes [5, 6]. In many cases, a soliton which travels at the group velocity of the medium will form in a properly designed laser cavity. Soliton molecules, also called bound states, refer to the situations where two or more solitons travel with a constant separation comparable to the pulse widths. The separation is maintained by the complicated interaction of mechanisms including cross-phase modulation, photoacoustic effect, and dispersive waves [7]. The separation and phase difference between solitons can provide extra dimensions of information encoding, extending the capacity of fiber optic communication.

In order to achieve mode-locked pulses, nonlinear optical modulation devices are generally adopted in laser cavities. Saturable absorbers (SAs) as passive modulators maintain the high-intensity light pulses and suppress the low-intensity fluctuations, so that one or several noise peaks are continually amplified by gain media and shaped by SAs to form a soliton pulse. Originally, artificial SAs including nonlinear polarization rotation, nonlinear optical loop mirror, nonlinear amplifying loop mirror, and multimode fibers-based SAs were applied to obtain ultrafast pulses [8]. Artificial SAs exhibit interesting properties such as wavelength independence and long operating time [9], while these SAs are sensitive to changes from environment and require further exploration in performance-tunable lasers [10]. Subsequently, real SAs such as semiconductor saturable absorber mirrors and nanomaterials came into view of researchers [11–13]. Nanomaterials have now received enormous attention in ultrafast photonics owing to their small size, low cost, and broadband absorption [14–17].

As one-dimensional nanomaterials, carbon nanotubes (CNTs) own excellent optical and photoelectric properties due to its unique structure [18]. Since the discovery of CNTs in 1991, it has shown a wide range of application prospects in the fields of biology, optoelectronics, sensing, and energy science [19–21]. Single-walled CNTs have a simpler energy level structure than double-walled and multi-walled CNTs, and thus have arouse great interest in nonlinear optical applications [22, 23]. In this work, the whole process of bound states generation is analyzed including the fabrication and characterization of CNTs-SA, as well as the theoretical and numerical explanation of the experimental results. This work aims to show the excellent optical properties of CNTs, and contribute to a further understanding of the relation between CNTs and soliton molecules.

2 Experimental section

The CNTs were fabricated as a polymer film with drying method, which provides a convenient approach to the preparation of reusable SA. In this section, drying method is introduced in detail, followed by characterizations of the raw CNTs powder and the film.

2.1 Fabrication of CNTs-based SA

The single-walled CNTs powder was produced with CVD method, which has diameters of less than 2 nm marked by the manufacturer. Firstly, CNTs and sodium dodecyl benzene sulfonate as the surfactant were both mixed into deionized water to reach a weight percentage of 0.04 wt%. The mixture then underwent 4 hours of ultrasonication at 120 W and 30 minutes of centrifugation at 5000 rpm in order to evenly disperse the CNTs and filter out impurities. Then, 1 wt% polyvinyl alcohol (PVA) solution in deionized water was made by heating to 70° and fiercely stirring for several hours so that the PVA dissolved completely. Finally, CNTs and PVA solutions were mixed in 1:4. The mixture was then ultrasonicated at 120 W for 30 minutes, poured onto a petri dish, and dried at room temperature for around one day until the film was formed. The CNTs film has a translucent, dark brown, and glossy appearance. A slice of the film was cut and sandwiched between two fiber patch cords to form a SA device for later characterization and experiment. The slice should fully cover the area of the fiber cladding, which was 125 μm in this case.

2.2 Characterization of the CNTs

The morphology, diameter distribution, linear transmittance and Raman spectrum of CNTs were examined. Fig.1(a) displays a transmission electron microscope (TEM) image of the sample dispersed in ethanol, showing the morphology of the CNTs. Inset shows the diameter distribution of CNTs by measurement and counting in several more magnified TEM images. Most of the CNTs had a diameter within 0.8 to 1.4 nm, which contains the diameters required for stimulated absorption at 1.5 and 2 μm [24]. The linear transmittance was measured with a spectrophotometer as shown in Fig.1(b), which exhibits potential for modulation at around 1.56 μm. Raman spectrum provides vibrational modes of substances, from which the information of atom-level structure can be analyzed. The Raman spectrum of the CNTs powder is shown in Fig.1(c), which provides molecule-level information of the sample. The large peak-intensity difference of D- and G-band indicates that the sample contained few double- or multi-walled CNTs, proving the purity of the sample. The radial breathing mode band of the Raman spectroscopy is enlarged as the inset of Fig.1(c), where the Raman frequency \omega of radial breathing mode (RBM) peak can approximately reflect the tube diameter d [24]. This relationship is commonly described as \omega =A/d+B, in which A and B are parameters that differs in multiple works. The coverage of 100 to 200 cm⁻¹ of the RBM band indicates a diameter range of 1.2 to 2.6 nm, with the 150 cm⁻¹ peak corresponding to 1.7 nm. The difference between the calculated and measured diameters may resulted from the fact that A and B are sensitive to the preprocessing methods and diameter range of the sample. The TEM image and Raman spectrum both show a diameter span of approximately 1.4 nm.

The nonlinear optical property of the SA was measured by a two-detector method, where the measuring setup consisted of a laser source, a variable optical attenuator (VOA), an optical coupler (OC), a SA, two power meters, and single-mode fibers. The laser source was a mode-locked fiber laser in the corresponding wavelength, and a VOA was used to adjust the laser input. An OC was used to split the laser pulse into two beams. One beam passed through the SA, while the other beam was directly determined as a reference. The two outputs were measured by a power meter, and the corresponding transmittance curve in Fig.1(d) was obtained through calculation. The CNTs-based SA possesses nonlinear absorption in 1.5 μm, indicating the remarkable properties of nanotubes in ultrafast photonics.

2.3 Setup of 1.5-μm laser cavity

The setup of Er-doped laser is shown in Fig.2. As for the pump in the fiber laser, a 980 nm laser diode was used for 1.5 μm mode-locked laser through a wavelength division multiplexer (WDM). The polarization controller (PC) was employed to facilitate the adjustment of polarization in fiber laser. The sandwich-structure CNT-based SA mentioned in the fabrication section was integrated into fiber laser. The 20% of the inner power was output by optical couplers (OCs) for measurement and observation, and an polarization-independent isolator (PI-ISO) assured the unidirectional propagation of mode-locked pulses. Furthermore, the gain fiber was a 0.5 m highly-doped LIEKKI ER110 Er-doped fiber, which provided the required gain in the cavity. The lengths of single-mode fiber pigtails of the rest components and patch cords added up to 5.6 m, contributing to a cavity length of 6.1 m and a total dispersion of −0.1 ps².

3 Results and discussion

In the Er-doped fiber laser, continuous wave laser was observed after increasing the pump power to 40 mW. After raising the pump power to 45 mW and slightly adjusting PC, the output showed a typical spectrum of mode-locking with a 1.68 nm full width at half-maximum (FWHM), as shown in Fig.3(a). The oscilloscope trace in Fig.3(b) exhibits a pulse time interval of 33.7 ns. Radio frequency (RF) analysis showed a 60 dB SNR as shown in Fig.3(c), indicating the short-time stability of the output. The RF analyzer also showed a repetition rate of 29.66 MHz, which is in accordance with the measured time interval. The output pulse shape was reflected in the autocorrelation trace in Fig.3(d). The pulse width is measured as 1.90 ps after sech² fitting and division by a factor of 1.54. The time-bandwidth product, single-pulse energy and peak power can be accordingly calculated as 0.391, 9.7 pJ and 4.5 W, respectively.

In the laser cavity, continuous wave laser, fundamental mode-locking, and multi-pulse mode-locking would occur successively when raising the pump power. Soliton molecules appeared at a pump power slightly higher than the threshold of fundamental mode-locking but lower than the threshold of multi-pulse mode-locking. Several representative fiber pressures of the PC should be selected, and during a relatively slow and steady rotation of the PC squeezer from one side to the other for each pressure, the spectrum analyzer would show characteristic spectrum of soliton molecules once or several times, proving the existence of soliton molecule solutions. The birefringence state should then be finely tuned around one of the solutions until steady state was achieved. In the case of our cavity, various soliton molecules appeared after increasing the pump power to 50 mW. Two or more pulses that propagates simultaneously with a stable distance comprise soliton molecules. Two-pulse bound states are characterized by a periodic modulation in the spectrum and three peaks with peak power of approximately 1:2:1 in the autocorrelation trace [25]. The pulse distances of the bound states could be tuned by slightly changing the pump power and PC. More specifically, after the formation of one steady state of soliton molecule by rotating the squeezer, the pulse separation could be continuously tuned within a certain range of around 0.5 to 2 ps by adjusting the PC pressure or adjustment of the pump power. Generally, higher pressure and lower pump power corresponded to larger separation, while a large variation of these factors will transfer the current soliton molecule state to another, so that the separation will undergo an inconsecutive change.

Fig.4(a)–(c) display the spectra of three soliton molecules, and Fig.4(d)–(f) show their corresponding autocorrelation traces of the pulses. In the spectral domain, the peak interval of the spectra \mathrm{\Delta } \lambda, also called a modulation period, can also be described in frequency domain as \mathrm{\Delta } \nu =\nu \cdot \mathrm{\Delta }\lambda /\lambda =c\cdot \mathrm{\Delta }\lambda /{\lambda }^2, where c is the light speed, and \lambda is 1566 nm in this case. The modulation periods \mathrm{\Delta }\nu and separations of pulses \mathrm{\Delta }\tau should satisfy \mathrm{\Delta }\nu \cdot \mathrm{\Delta }\tau = 1, which is obtained by applying Fourier transform to two adjacent same pulses in the time domain [7]. In our experiment, the modulation periods of the three soliton molecules \mathrm{\Delta }\lambda were 1.08, 0.46, and 0.35 nm, and the separations of the pulses \mathrm{\Delta }\tau were 7.75, 18.12, and 22.88 ps, respectively. It could be observed that \mathrm{\Delta } \nu \cdot \mathrm{\Delta } \tau =1 was approximately satisfied in both cases, proving that the outputs show typical results of soliton molecules. These results show that stable solitons of different wavelengths and soliton pairs can be generated in this CNTs-based fiber laser. In order to show the influence of pump power on a soliton molecule, the changes of the soliton molecule in Fig.4(a) with increasing power and unchanged polarization state are plotted in Fig.5. As shown in Fig.5(a), the output power of the soliton molecule increases as the pump power increases, while the pulse separation exhibits a tendency to decrease due to the enhancement of cross-phase modulation. Meanwhile, the pulse width gradually narrows and the spectral width gradually widens in Fig.5(b). These results represent the tunability of soliton molecules obtained with different pump powers.

Furthermore, numerical simulations of the bound states were conducted in order to investigate the properties of the bound states. In the simulations, the pulses exist in the form of complex intensity envelope in the time domain, where the square of its magnitude is pulse power, and its argument stands for temporal phase. Phase increasing or decreasing with time denotes up- or down-chirp, respectively. A lumped model is adopted where the numerical models of every part in the laser cavity are provided as a function, and the pulses in the observation window go through these functions in order. The pulse envelope satisfies nonlinear Schrödinger equation when travelling in the fibers:

(1)\mathrm{i}{\displaystyle \frac{\mathrm{\partial }A}{\mathrm{\partial }z}}={\displaystyle \frac{{\beta }_2}{2}}{\displaystyle \frac{{\mathrm{\partial }}^{2}A}{\mathrm{\partial }{T}^2}}-\gamma |A{|}^2 A+\mathrm{i}{\displaystyle \frac{g_0}{2}} \left(1+T_2^2{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{T}^2}}\right) A,

where A is the envelope, z is the distance which the pulses travel along the fiber, T is the time in time delay coordinate so that the coordinate moves in the group velocity of the fiber, {\beta }_2 is the second-order dispersion coefficient, \gamma is the nonlinear coefficient, g_0 is the small gain coefficient, and T_2 is the relaxation time of the doped ions. Here, attenuation is deemed as insignificant, and the rightmost part of the equation is set as zero for passive fibers. The small gain coefficient g_0 reflects the pump power so that it can be moderately changed during simulation. The CNTs SA provides nonlinear absorption, which is considered as a fast model [26]:

(2)T_{\mathrm{S}\mathrm{A}}= 1-{\alpha }_{\mathrm{n}\mathrm{s}}-{\displaystyle \frac{{\alpha }_0}{1+I/I_{\mathrm{s}\mathrm{a}\mathrm{t}}}},

where {\alpha }_{\mathrm{n}\mathrm{s}} is the non-saturable absorption coefficient, {\alpha }_0 is the modulation depth, I and I_{\mathrm{s}\mathrm{a}\mathrm{t}} are the intensity of incident light and the saturation intensity, respectively.

PC involves bending and pressure of the fiber. Bending rotates the electric field, and pressure provides birefringence to the fiber, where the fast and slow axes were perpendicular to and along the direction of the pressure respectively. For the bending, the electric field rotates in an angle {\theta }_{\mathrm{e}} that is smaller than the rotation of fiber {\theta }_{\mathrm{r}}:

(3){\theta }_{\mathrm{e}}={\displaystyle \frac{1}{2}}{k}_{\mathrm{b}}{\theta }_{\mathrm{r}},

where k_{\mathrm{b}} is 0.16 according to earlier work [27]. For the pressure, it induces a wavelength-dependent phase change \phi:

(4)\phi ={\displaystyle \frac{k_{\mathrm{p}}F}{\lambda d}},

where F is the force exerted on the fiber, \lambda is the wavelength, d is the fiber diameter, and k_{\mathrm{p}}= 6\times {10}^{-11}\mathrm{r}\mathrm{a}\mathrm{d}\cdot {\mathrm{m}}^2\cdot {\mathrm{N}}^{-1} [28]. The influence of pressure on the pulse envelop can then be expressed by the following operator:

(5){\hat{P}}_{\phi } =\left(\begin{array}{cc}{\mathcal{F}}^{-1}\cdot \mathrm{e}\mathrm{x}\mathrm{p}(\mathrm{i}\phi /2)\cdot \mathcal{F}& 0\\ 0& {\mathcal{F}}^{-1}\cdot \mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{i}\phi /2)\cdot \mathcal{F}\end{array}\right),

where \mathcal{F} is the Fourier operator. The total influence of PC on the pulse envelop can be simulated by the following sequence: rotation of {\theta }_r-{\theta }_e, phase change of \phi /2 for the slow axis and -\phi /2 for the fast axis, and then a rotation of {\theta }_e- {\theta }_r. This is thus expressed as

(6)\left(\begin{array}{c}{A}_y^{\prime }\\ A_x^{\prime }\end{array}\right)={\hat{R}}_{-{\theta }_r}{\hat{P}}_{\phi }{\hat{R}}_{{\theta }_r}\left(\begin{array}{c}{A}_y\\ A_x\end{array}\right),

where A_y and A_x stands for the slow and fast axes, and the rotation operator is expressed as

(7){\hat{R}}_{{\theta }_r}=\left(\begin{array}{cc}\cos ({\theta }_r-{\theta }_e)& -\sin ({\theta }_r-{\theta }_e)\\ \sin ({\theta }_r-{\theta }_e)& \cos ({\theta }_r-{\theta }_e)\end{array}\right) .

Since the rest part of the laser was a scalar model, the input is allocated to the slow axis since soliton is more stable in the slow axis of a birefringent fiber for simplification [29]. This model is also readily adaptable to vector simulation, where coupled mode equations will replace the scalar nonlinear Schrödinger equation.

Fig.6 depicts three simulation results of the bound states, where the normalized pulse profiles and phases are shown in Fig.6(a)−(c), and the spectra are given in Fig.6(d)−(f). The pulse profiles have also been centralized for clarity. In all three simulations, a cavity structure same as shown in Fig.2 was adopted, and the initial was a single pulse with a pulse width of 1 ps and a peak power of 0.1 W. The parameters of passive fiber, gain fiber and SA were all set according to the experiment. The passive fiber was configured as {\beta }_2=-20.7 {\mathrm{p}\mathrm{s}}^2\cdot {\mathrm{k}\mathrm{m}}^{-1} and \gamma =1.3{\mathrm{W}}^{-1}\cdot {\mathrm{k}\mathrm{m}}^{-1}, the gain fiber was configured as {\beta }_2=17.8{\mathrm{p}\mathrm{s}}^2\cdot {\mathrm{k}\mathrm{m}}^{-1}, \gamma = 2.7{\mathrm{W}}^{-1}\cdot {\mathrm{k}\mathrm{m}}^{-1}, and SA were set as {\alpha }_{\mathrm{n}\mathrm{s}}=0.28, {\alpha }_0=0.14, I_{\mathrm{s}\mathrm{a}\mathrm{t}}= 4\mathrm{W}. Initial Gaussian signals without chirp were added in these cavities. g_0 was initially set as 7.1 m⁻¹ and tuned between 6 and 8 m⁻¹ during the formation of the pulse to simulate the change of pump power. PC angle was tuned around 0° for the first two cases and around 90° for the last case. The time separation and spectral modulation periods of these results all agree with the experiment. Kelly sidebands can be observed in the spectra, which are typical of solitons but cannot be observed for small intensity in the real experiment. The phase differences of the pulses have also been marked, indicating that bound states may carry different phase differences, which provides an extra dimension of information modulation.

Generally, higher initial g_0 led to larger separation when the initial signal was a small single pulse and PC was kept unchanged. In our case, an initial g_0 larger than 7.1 m⁻¹ will split the initial pulse far away and immediately when the PC angle was kept unchanged as zero, while a smaller g_0 will result in two close solitons. However, when a steady state was reached, an increase of g_0 tended to pull the solitons closer. A relatively large initial angle of PC could increase the separation threshold of g_0, and the pulse was slowed down as the center moved to the negative direction of time. Tunable bound states can be achieved by properly tuning these parameters, while the subtle changes of the separation and phase difference induced by tuning PC were not intuitive.

In our experiments, CNTs exhibit a simple fabrication process and a broad saturable absorption, rendering them viable candidates as SAs for the realization of soliton molecules. Notably, their appropriate recovery time enables tunable spacing configurations for soliton molecules within a range spanning from proximity to the pulse width to values substantially exceeding it [30, 31]. Moreover, the adjustment of PC has no obvious effect on the modulation ability of CNTs in a laser cavity, which corresponds better to the static SA model used in the simulation. Comparatively, the NPR techniques, characterized by faster recovery dynamics, are typically advantageous for achieving narrow-spacing soliton molecules [32]. The SESAMs, which generally possess slower recovery times, demonstrate superior suitability for implementing wide-spacing soliton molecules [33].

According to the binding energy theory proposed by Komarov et al. [34], a laser cavity may support a series of soliton molecule separations with the minimum separation d_1 decided by the ground steady state, and the other possible separations are a multiple of this minimum, i.e., d_k=k d_1. This minimum can also be influenced by the conditions of the cavity, including dispersion, nonlinearity, and polarization states, making the possible solutions continuous within certain ranges. Thus, it is possible to achieve a steady state with a separation being exactly or close to a target value by finely tuning the PC in experiments as well as in the numerical simulation. The polarization state may affect the phase-sensitive interactions between solitons, and the pump power can exert its influence by effects such as gain depletion and recovery [35]. As for the shape of a single soliton in the molecule, it results from the balance between dispersion and nonlinearity inside the cavity. Larger unbalanced dispersion renders the pulse shape farther away from the transform limit, and higher pump power may decrease the pulse width. Based on the binding energy theory and our empirical experimental operations, we have done extensive research to find the achievable separations against pump power by tuning PC, as shown in Fig.7. These results can be divided into four regions where most results fall in the centers of the regions, which stand for the multiples of a possible minimum separation d_1. The d_1 has a decreasing tendency against the pump power, which agrees with existing theories that larger gain increase the cross-phase modulation effects and thus increase the attraction [7]. We achieved a maximum pulse spacing of 4.87 ps and a minimum of 25.76 ps. Soliton molecules with a separation outside this range turned to be highly unstable in our experiments. This is also in line with the binding energy theory because larger separation is more possible to appear at a lower pump power in the d_3 region of Fig.7, where the small pump power may not be able to provide enough attraction against disturbance for the solitons.

As Eq. (4) has suggested, PC can adjust the birefringence state inside the cavity, and exert wavelength-dependent phase shift to the pulse. This phase shift provides equivalent complex dispersive effect [36], which has influences on the basic binding energy, and thus the properties of the soliton molecules. On the one hand, the initial bending and twisting states of the laser cavity as well as the surrounding temperature will affect the birefringence of PC and other components [37]. In our work, experiments with different polarization states were performed immediately after adjusting the PC to minimize polarization drift caused by temperature changes in the laboratory. On the other hand, the effect of PC deserves further quantitative investigation in a vector model and experiments in order to gain more control over the bound states.

In order to achieve soliton molecules with larger separation, dispersion management strategies can be employed to suppress soliton interaction, thereby facilitating weakly-bound states. Besides, enhancing intracavity continuous-wave energy through optimized output coupler ratios promotes phase locking of solitons with larger separations. For soliton molecules with smaller separations, tightly-bound states may require higher pump power or SAs exhibiting high modulation depths to enhance the attraction between solitons [7, 38]. Larger pump power within the stable multi-soliton regime and highly-doped gain fibers with enhanced energy conversion efficiency can be used to increase output power and pulse energy. Furthermore, three- or more-soliton molecules require higher pump power to trigger further soliton splitting due to the soliton area theorem, while harmonic and multi-pulse mode-locking are more likely to realize in this case. Simultaneously, the generation and stabilization of multi-soliton molecules also need gain media with higher saturation thresholds and precise design of the intra-cavity dispersion and polarization to maintain balanced nonlinear interactions [39, 40].

4 Conclusion

Mode-locked pulsed lasers at different wavelengths were realized with single-walled CNTs-based SA. The fabrication and integration process of CNTs film were introduced, showing the convenience and controllability of drying method and sandwich structure. Continuous-wave mode-locking exhibited a 1.90 ps pulse width as well as a 60-dB SNR. Bound-state mode-locking was obtained in the same cavity, showing a variable pulse separation between 4.87 and 25.76 ps. These results demonstrate the versatility of bound states in optical communication and optical properties of CNTs as nonlinear modulators to generate bound state solitons.

References

[1] N. Picqué and T. W. Hänsch, Frequency comb spectroscopy, Nat. Photonics 13(3), 146 (2019)

[2] Z. Li, L. Xiao, Z. Feng, Z. Liu, D. Wang, and C. Lei, Sequentially timed all-optical mapping photography with quantitative phase imaging capability, Opt. Lett. 49(18), 5059 (2024)

[3] K. Sugioka and Y. Cheng, Ultrafast lasers-reliable tools for advanced materials processing, Light Sci. Appl. 3(4), e149 (2014)

[4] C. Zhang, L. Zhang, H. Zhang, B. Fu, J. Wang, and M. Qiu, Pulsed polarized vortex beam enabled by metafiber lasers, PhotoniX 5(1), 36 (2024)

[5] P. Grelu and N. Akhmediev, Dissipative solitons for mode-locked lasers, Nat. Photonics 6(2), 84 (2012)

[6] N. M. Kondratiev, V. E. Lobanov, A. E. Shitikov, R. R. Galiev, D. A. Chermoshentsev, N. Y. Dmitriev, A. N. Danilin, E. A. Lonshakov, K. N. Min’kov, D. M. Sokol, S. J. Cordette, Y. H. Luo, W. Liang, J. Liu, and I. A. Bilenko, Recent advances in laser self-injection locking to high-q microresonators, Front. Phys. (Beijing) 18(2), 21305 (2023)

[7] L. Gui, P. Wang, Y. Ding, K. Zhao, C. Bao, X. Xiao, and C. Yang, Soliton molecules and multisoliton states in ultrafast fibre lasers: Intrinsic complexes in dissipative systems, Appl. Sci. (Basel) 8(2), 201 (2018)

[8] H. Haus, E. Ippen, and K. Tamura, Additive-pulse modelocking in fiber lasers, IEEE J. Quantum Electron. 30(1), 200 (1994)

[9] S. M. Kobtsev, Artificial saturable absorbers for ultrafast fibre lasers, Opt. Fiber Technol. 68, 102764 (2022)

[10] A. I. Siahlo, L. Oxenlwe, K. S. Berg, A. T. Clausen, P. A. Andersen, C. Peucheret, A. Tersigni, P. Jeppesen, K. P. Hansen, and J. R. Folkenberg, A high-speed demultiplexer based on a nonlinear optical loop mirror with a photonic crystal fiber, IEEE Photonics Technol. Lett. 15(8), 1147 (2003)

[11] J. Sun, H. Cheng, L. Xu, B. Fu, X. Liu, and H. Zhang, Ag/MXene composite as a broadband nonlinear modulator for ultrafast photonics, ACS Photonics 10(9), 3133 (2023)

[12] U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. Aus der Au, Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers, IEEE J. Sel. Top. Quantum Electron. 2(3), 435 (1996)

[13] U. Keller and A. C. Tropper, Passively modelocked surface-emitting semiconductor lasers, Phys. Rep. 429(2), 67 (2006)

[14] F. Ceballos and H. Zhao, Ultrafast laser spectroscopy of two-dimensional materials beyond graphene, Adv. Funct. Mater. 27(19), 1604509 (2017)

[15] B. Fu, J. Sun, C. Wang, C. Shang, L. Xu, J. Li, H. Zhang, MXenes: Synthesis, and optical properties, and applications in ultrafast photonics, Small 17(11), 2006054 (2021)

[16] X. Liu, Q. Guo, and J. Qiu, Emerging low-dimensional materials for nonlinear optics and ultrafast photonics, Adv. Mater. 29(14), 1605886 (2017)

[17] W. Lyu, J. An, Y. Lin, P. Qiu, G. Wang, J. Chao, and B. Fu, Fabrication and applications of heterostructure materials for broadband ultrafast photonics, Adv. Opt. Mater. 11(12), 2300124 (2023)

[18] P. Avouris, M. Freitag, and V. Perebeinos, Carbon-nanotube photonics and optoelectronics, Nat. Photonics 2(6), 341 (2008)

[19] M. Terrones, H. Terrones, N. de Jonge, and J. Bonard, Carbon nanotube electron sources and applications, Philos. Trans. Royal Soc. A 362(1823), 2239 (2004)

[20] V. Schroeder, S. Savagatrup, M. He, S. Lin, and T. M. Swager, Carbon nanotube chemical sensors, Chem. Rev. 119(1), 599 (2019)

[21] J. Sun, Y. Wang, C. Zhang, L. Xu, and B. Fu, Spatiotemporal nonlinear dynamics in multimode fiber laser based on carbon nanotubes, Front. Phys. (Beijing) 19(5), 52201 (2024)

[22] K. Y. Lau, X. Liu, and J. Qiu, A comparison for saturable absorbers: carbon nanotube versus graphene, Adv. Photon. Res. 3(10), 2200023 (2022)

[23] X. Zhao, H. Jin, J. Liu, J. Chao, T. Liu, H. Zhang, G. Wang, W. Lyu, S. Wageh, O. A. Al-Hartomy, A. G. Al-Sehemi, B. Fu, and H. Zhang, Integration and applications of nanomaterials for ultrafast photonics, Laser Photonics Rev. 16(11), 2200386 (2022)

[24] L. Dai, Z. Huang, Q. Huang, C. Zhao, A. Rozhin, S. Sergeyev, M. Al Araimi, and C. Mou, Carbon nanotube mode-locked fiber lasers: Recent progress and perspectives, Nanophotonics 10(2), 749 (2020)

[25] L. Li, H. Huang, L. Su, D. Shen, D. Tang, M. Klimczak, and L. Zhao, Various soliton molecules in fiber systems, Appl. Opt. 58(10), 2745 (2019)

[26] E. Garmire, Resonant optical nonlinearities in semiconductors, IEEE J. Sel. Top. Quantum Electron. 6(6), 1094 (2000)

[27] R. Ulrich and A. Simon, Polarization optics of twisted single-mode fibers, Appl. Opt. 18(13), 2241 (1979)

[28] A. Smith, Single-mode fibre pressure sensitivity, Electron. Lett. 16(20), 773 (1980)

[29] K. J. Blow, N. J. Doran, and D. Wood, Polarization instabilities for solitons in birefringent fibers, Opt. Lett. 12(3), 202 (1987)

[30] L. Huang, H. N. Pedrosa, and T. D. Krauss, Ultrafast ground-state recovery of single-walled carbon nanotubes, Phys. Rev. Lett. 93(1), 017403 (2004)

[31] X. Xu, S. Ruan, J. Zhai, L. Li, J. Pei, and Z. Tang, Facile active control of a pulsed erbium-doped fiber laser using modulation depth tunable carbon nanotubes, Photon. Res. 6(11), 996 (2018)

[32] H. Qiang, Q. Qiao, H. Fu, J. Peng, K. Huang, and H. Zeng, Observation of soliton molecules in NPR mode-locked Er-fiber laser via birefringence management, IEEE Photonics Technol. Lett. 31(8), 639 (2019)

[33] B. Lu, Y. Wang, X. Qi, H. Chen, M. Jiang, L. Hou, K. Huang, J. Kang, and J. Bai, Observation of bound state solitons in tunable all-polarization-maintaining Yb-doped fiber laser, Laser Phys. 27(7), 075102 (2017)

[34] A. Komarov, K. Komarov, and F. Sanchez, Quantization of binding energy of structural solitons in passive mode-locked fiber lasers, Phys. Rev. A 79(3), 033807 (2009)

[35] D. Mao,Z. Yuan,K. Dai,Y. Chen,H. Ma, Q. Ling,J. Zheng,Y. Zhang,D. Chen,Y. Cui, Z. Sun,B. A. Malomed, Temporal and spatiotemporal soliton molecules in ultrafast fibre lasers, Nanophotonics, doi: 10.1515/nanoph-2024-0590 (2025)

[36] H. Zhang, D. Mao, Y. Du, C. Zeng, Z. Sun, and J. Zhao, Heteronuclear multicolor soliton compounds induced by convex-concave phase in fiber lasers, Commun. Phys. 6(1), 191 (2023)

[37] H. Shimizu, S. Yamazaki, T. Ono, and K. Emura, Highly practical fiber squeezer polarization controller, J. Lightwave Technol. 9(10), 1217 (1991)

[38] L. Gui, X. Li, X. Xiao, H. Zhu, and C. Yang, Widely spaced bound states in a soliton fiber laser with graphene saturable absorber, IEEE Photonics Technol. Lett. 25(12), 1184 (2013)

[39] P. Rohrmann, A. Hause, and F. Mitschke, Two-soliton and three-soliton molecules in optical fibers, Phys. Rev. A 87(4), 043834 (2013)

[40] J. H. Zhang, H. Q. Qin, Z. Z. Si, Y. H. Jia, N. A. Kudryashov, Y. Y. Wang, and C. Q. Dai, Pure-quartic soliton attracted state in mode-locked fiber lasers, Chaos Solitons Fractals 187, 115380 (2024)

RIGHTS & PERMISSIONS

Higher Education Press