Due to their outstanding optical properties, atomically thin, layered crystals of transition metal dichalcogenides (TMDs) provide a fertile playground to investigate intriguing quantum optical phenomena and develop high-performance photonic devices ^[1–4]. As one important member of TMDs, two-dimensional (2D) molybdenum diselenide (MoSe2) has been experimentally reported for its promising nonlinear optical (NLO) properties in a wide spectral range from visible to near-infrared ^[5–7], such as giant two-photon absorption, strong nonlinear self-focusing, and nonlinear intensity dependent absorption or scattering, leading to optical limiting ^[1,8]. In addition, 2D MoSe2 also exhibits strong saturable absorption with a saturated intensity of 590±225GW/cm2 and large third-order nonlinear optical susceptibility up to ∼1.45×10−15esu for 800 nm femtosecond laser pulses ^[1]. This strong NLO response makes MoSe2 a promising material for a passive mode locker to produce ultrafast pulsed lasers. This application has been demonstrated in a fiber laser and reported femtosecond pulses with a high repetition rate of 3.27 GHz ^[9]. A saturable absorber can act either as a fast-saturable absorber, i.e., the recovery time is much shorter than the laser pulse duration, or a slow-saturable absorber, i.e., the recovery time is well above the laser pulse duration ^[10]. However, these papers have not focused on whether few-layer MoSe2 is suitable as a slow-saturable absorber or fast-saturable absorber. The NLO performance of few-layer MoSe2 as a slow-saturable absorber is still largely unexplored. Ultrafast time-resolved measurements are also needed, as the excitation carrier dynamics plays an important role in the NLO performance, e.g., repetition rate and pulse duration in an ultrafast pulse laser ^[10]. The dependence of the exciton population on laser intensity and exciton–exciton annihilation was observed in a MoSe2 monolayer by femtosecond transient absorption ^[11]. Ultrafast spectroscopic data show that the formation of trions in a MoSe2 monolayer and the annihilation of trions followed a non-exponential decay ^[12,13]. However, the above reports do not answer the question of what role the relaxation of excited carriers plays in passive mode locking. Further research on the NLO properties of few-layer MoSe2 as a slow-saturable absorber and the concurrent excited carrier relaxation processes is therefore required.

For a fiber laser with higher power and shorter pulse duration, it is necessary to have a deeper understanding of the NLO behavior of its mode-locking materials. In this work, we simultaneously studied the NLO responses and corresponding excitation carrier dynamics in liquid-phase-exfoliated MoSe2 by Z-scan and degenerate pump-probe techniques. Our results show that few-layer MoSe2 can be used as a slow-saturable absorber for passive mode locking. We employed a slow-absorber model modified from the Frantz–Nodvik equation to analyze this behavior and extract absorptive cross sections of the ground and excited states, NLO modulation depth, and saturated intensity of the few-layer MoSe2. Our pump-probe results show that the decay of excited carriers in few-layer MoSe2 follows two relaxations. A bi-exponential decay model was utilized to extract these two exact lifetimes to be ∼2ps and 218 ps, where the slow relaxation can be used to start the pulsed laser oscillation from continuous wave (CW) noise in the passive mode locking while the fast one stabilizes the mode locking for short pulses.

We employed liquid-phase exfoliation techniques to prepare the 2D MoSe2 dispersions in distilled water with sodium cholate (SC) as the surfactant ^[14]. Bulk MoSe2 was dispersed into SC aqueous solution with 5 mg/mL and 10 mg/mL concentrations, respectively. A high-power sonic tip (Sonics VX-750) was employed to sonicate this dispersion for 90 min using 40% amplitude. To counter the temperature increase from sonication, we placed the vial holding the MoSe2 dispersion into an ice water bath and set the sonic tip to pulse 2 s on and 4 s off. During sonication, the bulk MoSe2 was exfoliated into a monolayer, bilayer, and few-layer flakes. To remove the large flakes, the sonicated dispersions were centrifuged at 2000 r/min for 90 min. The top half of the centrifuged dispersion was collected for characterizations. To study the dependence of NLO performance on the linear absorption, the obtained dispersion was diluted into two with different concentrations. Before the optical experiments, the geometry of the 2D MoSe2 flakes was verified via transmission electron microscopy (TEM) and atomic force microscopy (AFM). No bulk MoSe2 was found in the TEM images of the centrifuged dispersions. Figure 1(a) shows a typical few-layer nanoflake of MoSe2 in TEM, where the stacking of monolayers can be clearly seen. We also found MoSe2 monolayers in our dispersions, as shown in Fig. 1(b) in a high-resolution TEM image. Both TEM images imply the high quality of our MoSe2 dispersion. These flakes are mainly composed of few-layer, bilayer, and monolayer MoSe2. This is confirmed by Raman technique, which is a powerful tool to study the thickness of 2D materials. As indicated in Fig. 1(c), a strong peak located at ∼241cm−1 is originated from out-of-plane vibration (A1g) ^[15]. We also observed two peaks at ∼285cm−1 and 353cm−1, which are assigned to the in-plane vibrations E2g1 and B2g1, respectively. These peaks can be captured only in few-layer MoSe2 instead of bulk ^[2]. To study the distribution of the number of the flake layers, AFM was employed to measure the sample. The AFM sample was prepared using the liquid-phase-exfoliated MoSe2 dispersion before dilution. A drop of the dispersion was cast on a precleaned silicon substrate, followed by a ∼50°C vacuum treatment to evaporate the water. In a ∼5μm×5μm area shown in Fig. 1(d), a large number of MoSe2 nano-flakes were seen. The thickness profiles were plotted using the AFM data to obtain the number of flake layers. Due to the ∼0.67nm thickness of a MoSe2 monolayer ^[16], the number of layers in a flake with a thickness of L can be approximately calculated through dividing L by 0.67 nm. A statistical distribution of the number of flake layers was obtained, as shown in Fig. 1(e). It can be seen that most of the flakes in the centrifuged dispersion were less than 15 layers. From statistics, the layer number between 25 and 40 was also calculated, as seen in Fig. 1(e). It is likely to result from the stack of flakes, e.g., see the green circles in Fig. 1(e). Excluding these stacks in the estimates, flakes with between 3 and 15 layers account for more than 90% of the nanosheets in our dispersions.

Open-aperture Z-scan technique was utilized to study the NLO response of two MoSe2 dispersions with linear absorption coefficients of 5.22cm−1 and 6.51cm−1. To measure the linear absorption (A0), we obtained the linear transmission (T0) by measuring the transmitted intensity after the sample in a 1 mm thickness quartz cuvette and that after the same cuvette filled with deionized water. According to A0=α0l, we can obtain the linear absorption coefficient, as indicated in the insets of Figs. 2(a) and 2(b). The Z-scan technique measures the transmission of the sample as a function of incident intensity. A varied intensity was obtained by moving the sample along the propagation direction, i.e., z direction, of a focused laser beam. The schematic of this optical setup is shown in Fig. 2(c). In order to eliminate the effect of laser fluctuations on our results, a small portion of the original laser beam was used as the reference signal. The laser pulses from the Ti: sapphire mode-locked laser (Coherent, RegA 9000) were at a wavelength of 800 nm, with a duration of ∼100fs and a repetition rate of 100 kHz. The solid scatter data points in Figs. 2(a) and 2(b) exhibit the experimental open-aperture Z-scan results of the MoSe2 dispersions in 1 mm thickness quartz cuvettes. As the sample moves toward the focal point (z=0mm), a clear increase in the normalized transmission was observed. That is to say, the normalized transmission increases with the increase of the incident intensity, i.e., saturable absorption. The saturable absorption can be explained by Pauli-blocking ^[17]. The electrons are pumped to the conduction band from the valence band by absorbing incident photons. As the incident photon fluence increases, more electrons are occupying the conduction band until fully filled. The conduction band can then no longer accept more incoming electrons, which results in the transmission increase at higher incident light intensities.

The propagation equation describing how light attenuates in an NLO medium can be written as ^[18,19] Here, I is the laser intensity, z′ is the propagating distance in the material, and α(I)=α0+αNLI represents the total absorption coefficients, including linear absorptive (α0) and nonlinear absorptive (αNL) parts. By employing Eq. (1) to fit the experimental Z-scan results (see the solid lines in Fig. 2), we obtained the nonlinear absorption coefficients of MoSe2, which were calculated to be −0.017cm/GW and −0.044cm/GW for the samples with linear absorption coefficients of 5.22cm−1 and 6.21cm−1, respectively. The more absorber atoms (larger linear absorption coefficient) that contribute to the nonlinear response, the larger the nonlinear coefficient is. Furthermore, the imaginary part of the third-order nonlinear optical susceptibility (Imχ(3)) can be calculated using Imχ(3)=(10−7cλn296π2)αNL, where λ is the laser wavelength, c is the light speed, and n is the refractive index ^[20]. As shown in Table 1, Imχ(3) of the few-layer MoSe2 was estimated to be up to the order of 10−14esu.

As the excited carrier dynamics is strongly relevant to the saturable absorption, we studied this relaxation process in MoSe2 using a degenerate pump-probe technique. The schematic of this optical setup is plotted in the inset of Fig. 3(d). A train of intense laser pulses was employed as a pump to inject the excited electrons and holes into the valence and conduction bands of MoSe2, respectively. Another train of pulses with relatively low intensity was delayed by a motorized linear translation stage as a probe. Both laser beams were at a wavelength of 800 nm, with a pulse duration of ∼100fs and a repetition rate of 100 kHz utilizing the same laser as before. An optical chopper was employed to modulate the probe and pump beams at 422 Hz and 733 Hz, respectively. The polarization of the probe beam was rotated by 90° with the help of a half-wave plate. This polarization orthogonality between pump and probe eliminated coherent spikes in all the pump-probe traces ^[21]. After the sample, the pump pulses were blocked by an aperture and a polarizer, while the probe pulses were corrected by a silicon photodiode and analyzed by an SR7270 DSP lock-in amplifier. The obtained differential transmission of MoSe2, ΔT/T, is plotted as scatter points in Fig. 3. It can be seen from Figs. 3(a) and 3(b) that the maximum differential transmission increases with the increase in the pump energies from 10 nJ to 200 nJ, which also implies saturable absorption. This is true for both few-layer MoSe2 dispersions with different linear absorptions. Figures 3(c) and 3(d) plot the corresponding measurements at a long delay time, obviously showing a two-lifetime relaxation process. The bi-exponential decay processes of excited carrier were also observed in monolayer MoSe2 ^[11].

To obtain a deeper understanding of these relaxation processes, a bi-exponential equation with autocorrelations was utilized to model the experimental results ^[10,19]: where D1 and D2 represent relative amplitudes, t is the pump-probe delay time, σ is the laser pulse width and “erfc” represents the standard error function. All the fittings are favorable to the experimental results, as the solid lines in Fig. 3. The fitting parameters are listed in Table 2. The corresponding relation processes are plotted in the inset of Fig. 3(c). At time zero, the carriers were excited by intense photons from the ground state to the excited state into a non-thermal equilibrium distribution. Within ∼100fs, these carriers were subsequently thermalized to a quasi-thermal equilibrium state. Since the time of this process is shorter than our laser pulse duration, it did not show in our results. Then the quasi-thermal carriers relaxed to a lower energy state via carrier–carrier scattering and/or phonon emission. This relaxation time was measured to be ∼2.19ps. Finally, the excited carrier annihilated via indirect recombination with a ∼218.2ps lifetime. This relaxation time is slightly smaller than that of the MoSe2 monolayer (∼300ps), in which the optical transition is via direct band ^[22,23].

Since both relaxations are much longer (∼2ps and ∼218ps) than the exciting laser pulse duration (∼100fs), the Z-scan results in Fig. 2 can be further analyzed using a slow-absorber model (SAM) modified from the Frantz–Nodvik equation in the following ^[24]: Here, T0=e−σgNL is the linear transmission, Tmax=e−σeNL is the saturable transmission, and TFN=ln[1+T0(eσgFℏω−1)]/(σgF/ℏω). σg and σe represent the absorptive cross sections of ground and excited states, respectively. N represents the density of the absorptive centers (density of atoms for approximation), and L=1mm is the sample thickness. ω is the angular frequency of the incident photons. The results are shown in Fig. 4(a). The corresponding NLO parameters are listed in Table 1. The non-saturable absorption loss can be calculated via Ans=1−Tmax to be 44.8% and 45.0% with corresponding coefficients for the dispersions with linear absorption coefficients of 5.22cm−1 and 6.51cm−1, respectively. The modulation depth, ΔT, defined as the difference between the maximum and minimum transmissions, i.e., ΔT=Tmax−T0, is shown in Fig. 4(a) as well. For the few-layer MoSe2 dispersions with linear transmission of 47.8% and 34.9%, ΔT is measured to be 7.4% and 20.0%, respectively. It implies that the NLO modulation depth can be controlled by changing the linear absorption of the dispersion, which can be tuned by diluting or concentrating a sample. This difference is more obvious when the transmission is converted to the differential absorption, as shown in Fig. 4(b). Another important NLO parameter is the saturated intensity IS, which refers to the laser intensity at the half NLO modulation depth of a saturable absorber. The IS was fitted to be 39.37MW/cm2 and 234.75MW/cm2 for the two samples with linear absorptions of 5.22cm−1 and 6.51cm−1, respectively. Combining the result in Ref. ^[1] of IS=∼590MW/cm2 for a sample with a linear absorption of 7.93cm−1, we can see that the saturable intensity increases with the linear absorption. This is because more photons are needed to saturate an absorber with higher concentration of carriers. The ratios of absorption cross sections of excited states to ground states, σe/σg, were estimated to be 0.81 and 0.57 for the 47.8% and 34.9% transmission few-layer MoSe2, respectively. This is in agreement with other saturable absorption materials: the absorptive cross section of the excited state is smaller than that of the ground state ^{[10,19,20,25]}. The small σe/σg also indicates low non-saturable absorption of few-layer MoSe2 when it is working for passive mode locking.

In this work, we studied the NLO properties of few-layer MoSe2 using open-aperture Z-scan and degenerate pump-probe techniques based on an 800 nm femtosecond laser. The results reported in this work indicate that few-layer MoSe2 dispersions possess saturable absorption with an imaginary part of the third-order NLO susceptibility of ∼10−14esu. The analysis based on the slow-absorber model shows that the saturated intensities of few-layer MoSe2 are 39.37MW/cm2 and 234.75MW/cm2 for the linear absorption coefficients of 5.22cm−1 and 6.51cm−1, respectively, with a corresponding NLO modulation depth of 7.4% and 15.1%. The pump-probe results indicate that the excited-carrier relaxation in MoSe2 is composed of a bi-exponential process with lifetimes ∼2ps and ∼218ps, respectively. These two lifetimes may be attributed to carrier–carrier scattering and indirect recombination. As in graphene, this two-lifetime relaxation indicates MoSe2 as a promising material for passive mode locking to produce ultrashort laser pulses, where the slow relaxation facilitates the starting of pulsed laser oscillation from CW noise, while the fast one stabilizes the mode locking for short pulses.

Acknowledgment. G. Wang, K. Wang, A. A. Baker-Murray, and W. J. Blau thank the Science Foundation Ireland and European Commission under the Seventh Framework Programme. J. Wang acknowledges the NSFC grant, the Strategic Priority Research Program of CAS, the Key Research Program of Frontier Science of CAS, and the Program of Shanghai Academic Research Leader. P. Fan, J. T. Luo, and G. Liang thank the Shenzhen Key Lab Fund. We are grateful for the support of materials from Dr. Chuanfang Zhang and the discussions with Brian Jennings.