## Abstract

Two-dimensional (2D) transition metal dichalcogenides (TMDs) attain increasing attention due to their exceptional nonlinear optical efficiencies, which hold promising potential for on-chip photonics and advanced optoelectronic applications. Planar TMDs have been proven to support orders-higher third-order nonlinear coefficients in comparison with common nonlinear materials. Interestingly, stronger light–matter interaction could be motivated when curved features are introduced to 2D TMDs. Here, a type of inorganic fullerene-like ${\mathrm{WS}}_{2}$ nanoparticles is chemically synthesized using hard mesoporous silica. By using the spatial self-phase modulation (SSPM) method, the nonlinear refractive index ${n}_{2}$ and third-order susceptibility ${\chi}^{(3)}$ are investigated in the visible range. It is found that ${n}_{2}\sim {10}^{-5}\text{\hspace{0.17em}}{\mathrm{cm}}^{2}/\mathrm{W}$ and ${\chi}^{(3)}\sim {10}^{-7}\text{\hspace{0.17em}}\mathrm{esu}$, two orders higher than the counterparts of planar ${\mathrm{WS}}_{2}$ structures. Our experimental findings provide a fresh thinking in designing nonlinear optical materials and endow TMDs with new potentials in photonic integration applications.

© 2020 Chinese Laser Press

## 1. INTRODUCTION

Nonlinear optical effects ignite exciting light–matter interactions and greatly enlarge optical applications such as frequency conversion, optical imaging, and information processing [1,2]. Remarkable achievements have been made based on novel working principles [3] and materials [4,5] and their marriage [6–8]. However, one of the main challenges hindering the full exploration of nonlinear effects is the low intrinsic nonlinear susceptibilities of conventional materials. Over the past decade, two-dimensional (2D) materials have attracted increasing attention due to their outstanding optical, electronic, and mechanical properties [9–12]. As a typical example, transition metal dichalcogenides (TMDs) possess layer-dependent electronic bandstructure and thus tunable linear and nonlinear optical properties [13,14]. In particular, third-order nonlinearity holds unique importance for applications in mode-locked lasers, sensors [15], and all-optical switching and modulation [16]. To characterize the third-order susceptibility, various methods have been proposed, such as Z-scan [17,18], four-wave mixing [19], and spatial self-phase modulation (SSPM) [20].

So far, the exploration of optical properties of TMD materials has been mainly focused on various flat 2D structures, including nanosheets or nanoflakes [21–23]. However, in addition to the size, shape, thickness, and material quality of TMDs, the geometric characteristics are also supposed to greatly affect their optical properties [24]. In contrast, inorganic fullerene-like (IF-like) 2D nanoparticles (NPs) with curved geometric features introduce an additional freedom to control and enhance the light–matter interaction strength [25,26]. Initially, they are widely investigated as an efficient lubrication material [27,28]. Recently, it was found that the curved features are prone to symmetry breaking to 2D materials and then making the silent phonon mode Raman active [26,29,30]. It is thus naturally speculated that other nonlinear effects may be enhanced with the curved features.

Here, IF-like ${\mathrm{WS}}_{2}$ NPs are chemically synthesized using hard mesoporous silica. The nonlinear refractive index ${n}_{2}$ and third-order susceptibility ${\chi}^{(3)}$ are characterized using the SSPM method in the visible range. It is found that the nonlinear optical responses of the proposed structures are orders stronger than the counterparts of planar 2D ${\mathrm{WS}}_{2}$ films. Therefore, we believe that curved 2D materials could play a growing role in designing optical materials with superior efficiencies at each order of nonlinearity and are endowed with new potentials in high-speed optical signal processes and photonic integration applications.

## 2. EXPERIMENT

The IF-like ${\mathrm{WS}}_{2}$ NPs are chemically synthesized using ordered three-dimensional (3D) mesoporous silica (EP-FDU-12) as hard templates. The average diameter of the pores and thickness of the wall are 27 nm and 5 nm, respectively. The precursor, i.e., phosphotungstic acid (PTA), is incorporated into the template via a solvent evaporation process. The ${\mathrm{WS}}_{2}$ NPs can then be obtained by removing the template in ${\mathrm{H}}_{2}\mathrm{S}$ gas. A typical scanning electron microscopy (SEM, JEOL, JSM-7000F) image of the synthesized ${\mathrm{WS}}_{2}$ NPs is shown in Fig. 1(a). The multilayer structure with an interlayer distance of $\sim 0.67\text{\hspace{0.17em}}\mathrm{nm}$ is clearly characterized using a high-resolution transmission electron microscopy (HRTEM, JEOL, JEM-2100F) image [Fig. 1(b)]. The synthesized ${\mathrm{WS}}_{2}$ NPs hold IF-like features with an average diameter of $\sim 26.5\text{\hspace{0.17em}}\mathrm{nm}$, which do not exhibit a quantum size effect [25]. It is obvious that the NPs show curved multilayered features with a layer number $>5$ [Fig. 1(b)]. The peaks in the X-ray diffraction (XRD, Bruker, D8 Advance) pattern match well with the standard ${\mathrm{WS}}_{2}$ structure (JCPDS card No: 08-0237) [Fig. 1(c)].

Two Raman peaks are observed at $\sim 353$ and $\sim 420\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$ under the excitations of 532 and 633 nm continuous-wave (CW) lasers [Fig. 1(d)]. The Raman active lattice vibrations at the $\mathrm{\Gamma}$ point of the hexagonal Brillion zone are modes $421\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$ and $356\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$ in the detected region [31,32]. Furthermore, second-order Raman transition, i.e., two longitudinal acoustic (2LA) phonons at $\sim 353\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$, are also observed for excitation energies close to the band gap. More interesting is the activation of the ${\mathrm{B}}_{1\mathrm{u}}$ mode, which is silent in planar 2D TMDs. Its excitation arises from the curved layers and structural disorder of ${\mathrm{WS}}_{2}$ NPs [26,29,30].

Under the illumination of an incoherent white light source, the transmittance was obtained by normalizing the transmitted power of the ethanol solutions with ${\mathrm{WS}}_{2}$ NPs to that without ${\mathrm{WS}}_{2}$ NPs. Figure 1(e) shows the transmittance spectrum of ${\mathrm{WS}}_{2}$ NP dispersion ranging from 400 to 900 nm (Andor SR500I), which is used to characterize the effective number of ${\mathrm{WS}}_{2}$ layers in the SSPM experiment. There is no evident excitonic resonance feature in the transmission spectrum, which may be attributed to the decrease in the exciton binding energy due to the increase in the number of ${\mathrm{WS}}_{2}$ layers [33].

The experimental setup for SSPM is schematically shown in Fig. 2(a). A femtosecond (fs) pulse laser (Coherent, Chameleon Ultra II, repetition frequency 80 MHz, pulse width 100 fs at 800 nm) propagates along the $z$ axis and is loosely focused on the cuvette by a lens with a focal length of 200 mm. In the experiment, the incident power can be controlled using a set of neutral density (ND) filters. Then, diffraction patterns are recorded using a digital camera with a slow-motion function. Due to the SSPM effect, the transmitted light appeared as a set of conical shells, which form concentric rings on a 2D screen (Fig. 2). The outermost ring stripe is always brighter and wider than the inner ones. Interestingly, the initial concentric diffraction rings deform quickly [Fig. 2(b)]. The upper half of the ring pattern continuously collapses towards the center of the initial concentric rings and then enters a stable state. In contrast, the lower part distorts slightly. The evolution time from the generation of ring-shaped patterns to saturation of distortion phenomenon usually lasts from less than one second to several seconds, which relies on the impinging power.

## 3. RESULTS AND DISCUSSION

Generally, the SSPM phenomenon exhibits as a series of concentric diffraction rings on a projection screen when a high-intensity laser beam interacts with the nonlinear medium. The SSPM ring pattern is attributed to the laser-induced refractive index change $\mathrm{\Delta}n$ [34]. As the laser beam propagates along the $z$ axis, the field $E$ reorients the direction of ${\mathrm{WS}}_{2}$ NPs in the normal plane. According to the Kerr effect, the refractive index of the suspension can be described by $n={n}_{0}+{n}_{2}I$, where ${n}_{0}$ is the linear refractive index, ${n}_{2}$ is the nonlinear refractive index of ${\mathrm{WS}}_{2}$ NPs, and $I$ stands for the incident intensity of laser beam [1]. It should be noted that the self-focusing effect occurs when the beam enters into the Kerr media. The beam size rapidly converges into a minimum after a propagation length of less than one millimeter. Then, the beam propagates like a plane wave with a slightly increased diameter due to weak absorption and light scattering. Therefore, the self-focusing effect usually is not taken into account when measuring the nonlinear refractive index using SSPM [20].

After traversing the ${\mathrm{WS}}_{2}$ dispersions of a thickness $L$, the incident light will gain an intensity-dependent phase [34]

The nonlinear refractive index can be expressed as [20,23]

The slope $S=\mathrm{d}N/\mathrm{d}I$ can be readily obtained by fitting intensity-dependent ring numbers, which increases as wavelength increases at a given intensity [Fig. 2(a)]. Moreover, the total third-order susceptibility can be obtained, ${\chi}_{\text{total}}^{(3)}=\frac{\lambda c{n}_{0}}{2.4\times {10}^{4}\times {\pi}^{2}{L}_{\mathrm{eff}}}S$ [20,23,36].As introduced previously, third-order nonlinear susceptibility ${\chi}^{(3)}$ is of great significance for indicating nonlinear performance of the nonlinear materials. Here, the third-order nonlinear susceptibility of monolayer ${\mathrm{WS}}_{2}$ NPs can be estimated using the counterpart of multiple layer structures with ${\chi}_{\text{total}}^{(3)}={N}_{\mathrm{eff}}^{2}{\chi}_{\text{monolayer}}^{(3)}$ [20,23], where ${N}_{\mathrm{eff}}$ represents the effective number of ${\mathrm{WS}}_{2}$ layers in the NPs, and ${\chi}_{\text{monolayer}}^{(3)}$ represents the contribution of one layer ${\mathrm{WS}}_{2}$ out of ${N}_{\mathrm{eff}}$ layers to the third-order susceptibility of ${\mathrm{WS}}_{2}$ NPs. Therefore, ${\chi}_{\text{monolayer}}^{(3)}$ can be calculated with the following equation:

Figure 2(b) briefly demonstrates the evolution of the diffraction pattern. The concentric rings pour out from the center. The diffraction pattern approaches the maximum geometric size within $\sim 0.5\text{\hspace{0.17em}}\mathrm{s}$ (Fig. 4). Subsequently, both the horizontal and vertical diameters of the rings collapse and reach a steady state after $\sim 2.8\text{\hspace{0.17em}}\mathrm{s}$ and $\sim 4.5\text{\hspace{0.17em}}\mathrm{s}$, respectively (Fig. 4). In contrast, the vertical diameter shrinks to half of the maximum one, while the horizontal diameter only compresses to 82% of the maximum one. The third-order nonlinearity is estimated when the number of rings becomes stable.

The distortion of diffraction rings is mainly attributed to the change of local material concentration induced by the non-axis-symmetrical thermal convection [43,44]. When the laser is incident upon the dispersions, the temperature surrounding the laser beam becomes asymmetrical, as the temperature gradient above the laser beam rises while it remains nearly stationary below the laser beam. As the non-axis-symmetrical thermal conduction increases [45], ${\mathrm{WS}}_{2}$ NPs in the upper part of the dispersions are precipitated into the lower part, resulting in a smaller density of ${\mathrm{WS}}_{2}$ NPs in the upper half of the dispersions, and then a reduced ${N}_{\mathrm{eff}}$, naturally, with a reduced ${n}_{2}$. Therefore, the lower-half dispersions have a relatively stronger nonlinear optical response, leading to the vertical collapse of the SSPM diffraction rings. Notably, the vertical deformation of SSPM rings is of great significance for the study of the photorefractive index change of IF-like ${\mathrm{WS}}_{2}$ NPs.

The maximum value of the vertical radius of the outermost ring and its half-cone angle are denoted by ${R}_{H}$ and ${\theta}_{H}$, respectively. The half-cone angle can be written as ${\theta}_{H}=\lambda /2\pi {(\mathrm{d}\mathrm{\Delta}\phi /\mathrm{d}r)}_{\mathrm{max}}$, which can be further simplified, for a Gaussian beam, to be ${\theta}_{H}\approx {n}_{2}IC$, where $C={[-(8Ir{L}_{\mathrm{eff}}/{\omega}_{0}^{2})\times \mathrm{exp}(2{r}^{2}/{\omega}_{0}^{2})]}_{\mathrm{max}}$ with $r\in [0,+\infty )$ being a constant. The distortion angle can be expressed as ${\theta}_{D}\approx \mathrm{\Delta}{n}_{2}IC$, where $\mathrm{\Delta}{n}_{2}$ is the nonlinear refractive index change caused by intensity variation. Eventually, the change ratio of the nonlinear refractive index can be calculated [39,43].

An increased incident intensity induces a more obvious distortion. Figure 5 exhibits the relationship between incident intensity and $\mathrm{\Delta}{n}_{2}/{n}_{2}$ at different wavelengths. To a certain extent, the linear regulation of the refractive index change of the material can be achieved by adjusting the intensity of the applied optical field. Nevertheless, the distortion ratiocannot be infinitely large due to the limitation ${\theta}_{D}<{\theta}_{H}$. When the incident intensity reaches the wavelength-dependent threshold of approximately $30\u201340\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$, the distortion ratio is prone to saturation (Fig. 5). Even so, the Kerr effect itself is not saturated. Since the saturation of the distortion phenomenon is mainly influenced by the non-axis-symmetrical thermal convection, a vertically rising temperature gradient causes ${\mathrm{WS}}_{2}$ NPs to continuously sink below the laser beam. After a period of thermal convection, when the density of ${\mathrm{WS}}_{2}$ NPs above the laser beam is infinitely close to zero, the upper part of the diffraction rings gradually approaches complete collapse.

As shown in Table 1, only the third-order nonlinear performance of $\mathrm{T}{\mathrm{i}}_{3}{\mathrm{C}}_{2}{\mathrm{T}}_{x}$ MXene exceeds the counterparts of the proposed ${\mathrm{WS}}_{2}$ NPs. However, the underlying mechanism here is different from those observed in $\mathrm{T}{\mathrm{i}}_{3}{\mathrm{C}}_{2}{\mathrm{T}}_{x}$ MXene with a narrow direct bandgap [41]. Because of the multiple layers in ${\mathrm{WS}}_{2}$ NPs, no photoluminescence (PL) emission is observed in our experiment [46,47]. Therefore, no interband transition occurs. The electrons are delocalized by the polarized incident field. The nonlinear refractive index can be estimated by ${\chi}^{(3)}\approx N{e}^{4}/{\epsilon}_{0}{m}^{3}{\omega}_{e0}^{6}{d}^{2}$, where $e$ is the element charge, ${\epsilon}_{0}$ is the vacuum permittivity, $N$ is the density of electrons of the material, ${\omega}_{e0}$ is the oscillation frequency of electrons, ${\omega}_{e0}=m{e}^{4}/32{\pi}^{2}{\epsilon}_{0}^{2}{\hslash}^{3}$, $d$ is the lattice constant, and $m$ is the effective mass of the conduction electron [1]. If $d$ is identified with the Bohr radius ${a}_{0}=4\pi {\epsilon}_{0}{\hslash}^{2}/m{e}^{4}$, we obtain that ${\chi}^{(3)}\propto {m}^{-7}$. Due to the distortion and curved features in ${\mathrm{WS}}_{2}$ NPs, the effective mass of electrons in IF-like ${\mathrm{WS}}_{2}$ NPs is speculated to reduce in comparison with the counterparts in planar 2D materials [25]. Therefore, the reduced effective mass of electrons will contribute to a portion of the enhancement in ${n}_{2}$ and ${\chi}^{(3)}$.

The mechanism of the SSPM phenomenon in ${\mathrm{WS}}_{2}$ NPs dispersion is essentially an appearance of intensity-dependent change in the refractive index. In principle, the thermal effect can only play a crucial role when the pulse duration is longer than tens of picoseconds. Therefore, the thermal contribution plays a non-dominated role in ${n}_{2}$ and ${\chi}^{(3)}$ enhancement under the illumination of the fs pulse source. Nevertheless, its contribution may be comparable to the contribution of the reduced effective mass of electrons. The electrons and holes generated by photoexcitation will drift in directions that are antiparallel and parallel to the electric field, respectively, resulting in polarized ${\mathrm{WS}}_{2}$ NPs. Initially, an arbitrary angle related to the interaction energy exists between the direction of the ${\mathrm{WS}}_{2}$ NPs polarization and the laser-induced electric field. As interaction energy is minimized, ${\mathrm{WS}}_{2}$ NPs are reoriented and aligned. The isotropy of the carriers in each particle appears as a kind of coherence that contributes to the macroscopic SSPM phenomenon. While it has another explanation, the gap-dependent SSPM can be regarded as a purely coherent third-order nonlinear optical process, which is generated from the nonlocal ac electron coherence within the sample [35]. Since each ${\mathrm{WS}}_{2}$ NP is mimicked as a separated domain containing multiform carriers, anisotropic domains are reoriented to alignment attributed to the torque produced by interior electron coherence influenced by an external electromagnetic field and finally polarized. The dielectric polarization caused by the electron coherence effect can be regarded as the collective behavior of a large number of electrons within the sample. Similarly, the polarization induced by the drift of photoexcited carriers (holes) can also be considered as a collective behavior of carriers.

Recently, it was demonstrated that second-harmonic generation can be actively controlled via the generation of photocarriers in monolayer ${\mathrm{MoS}}_{2}$ using ultrashort pulses, which enables a promising time-resolved approach to characterize the second-order nonlinear response [48]. A similar approach is also promising for extension into unveiling the detailed physical mechanism of the enhanced third-order nonlinear properties of ${\mathrm{WS}}_{2}$ NPs.

## 4. CONCLUSION

In conclusion, a novel type of IF-like ${\mathrm{WS}}_{2}$ NPs is successfully synthesized using the hard template method with a diameter of 26.5 nm. By characterizing the nonlinear refractive index ${n}_{2}$ and third-order susceptibility ${\chi}^{(3)}$ using the SSPM method with a visible fs pulse laser, we obtain ${n}_{2}\sim {10}^{-5}\text{\hspace{0.17em}}{\mathrm{cm}}^{2}/\mathrm{W}$ and ${\chi}^{(3)}\sim {10}^{-7}\text{\hspace{0.17em}}\mathrm{esu}$, which are orders stronger than the counterparts of planar 2D materials. In addition, the enhanced third-order nonlinear response can be controlled flexibly by varying the excitation wavelength and incident intensity, which is beneficial for all-optical devices. Therefore, IF-like 2D materials will enrich the optical materials with superior efficiencies, and are endowed with promising potentials in photonic integration applications.

## Funding

Fundamental Research Funds for the Central Universities (Z201805196); Natural Science Foundation of Shaanxi Province (2018JM6001); Young Talent Recruiting Plans of Xi’an Jiaotong University.

## Disclosures

The authors declare no conflicts of interest.

## REFERENCES

**1. **R. Boyd, *Nonlinear Optics*, 3rd ed. (Elsevier, 2008).

**2. **S. Sakabe, C. Lienau, and R. Grunwald, *Progress in Nonlinear Nano-Optics* (Springer, 2015).

**3. **M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Colloquium: nonlinear metamaterials,” Rev. Mod. Phys. **86**, 1093–1123 (2014). [CrossRef]

**4. **A. Autere, H. Jussila, Y. Dai, Y. Wang, H. Lipsanen, and Z. Sun, “Nonlinear optics with 2D layered materials,” Adv. Mater. **30**, 1705963 (2018). [CrossRef]

**5. **T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, and O. Zilberberg, “Topological photonics,” Rev. Mod. Phys. **91**, 015006 (2019). [CrossRef]

**6. **A. Vakil and N. Engheta, “Transformation optics using graphene,” Science **332**, 1291–1294 (2011). [CrossRef]

**7. **H. Lin, B. C. P. Sturmberg, K.-T. Lin, Y. Yang, X. Zheng, T. K. Chong, C. M. de Sterke, and B. Jia, “A 90-nm-thick graphene metamaterial for strong and extremely broadband absorption of unpolarized light,” Nat. Photonics **13**, 270–276 (2019). [CrossRef]

**8. **G. Hu, X. Hong, K. Wang, J. Wu, H.-X. Xu, W. Zhao, W. Liu, S. Zhang, F. Garcia-Vidal, B. Wang, P. Lu, and C.-W. Qiu, “Coherent steering of nonlinear chiral valley photons with a synthetic Au-WS_{2} metasurface,” Nat. Photonics **13**, 467–472 (2019). [CrossRef]

**9. **D. J. K. S. Novoselov, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, “Two-dimensional atomic crystals,” Proc. Natl. Acad. Sci. USA **102**, 10451–10453 (2005). [CrossRef]

**10. **A. K. Geim and I. V. Grigorieva, “Van der Waals heterostructures,” Nature **499**, 419–425 (2013). [CrossRef]

**11. **Y. S. Ang, S. Sultan, and C. Zhang, “Nonlinear optical spectrum of bilayer graphene in the terahertz regime,” Appl. Phys. Lett. **97**, 243110 (2010). [CrossRef]

**12. **A. Wright, X. Xu, J. Cao, and C. Zhang, “Strong nonlinear optical response of graphene in the terahertz regime,” Appl. Phys. Lett. **95**, 072101 (2009). [CrossRef]

**13. **F. Xia, H. Wang, D. Xiao, M. Dubey, and A. Ramasubramaniam, “Two-dimensional material nanophotonics,” Nat. Photonics **8**, 899–904 (2014). [CrossRef]

**14. **D. N. Basov, M. M. Fogler, and F. J. Garcia de Abajo, “Polaritons in van der Waals materials,” Science **354**, aag1992 (2016). [CrossRef]

**15. **Y. Xu, Y. S. Ang, L. Wu, and L. K. Ang, “High sensitivity surface plasmon resonance sensor based on two-dimensional MXene and transition metal dichalcogenide: a theoretical study,” Nanomaterials **9**, 165 (2019). [CrossRef]

**16. **M. Taghinejad and W. Cai, “All-optical control of light in micro- and nanophotonics,” ACS Photon. **6**, 1082–1093 (2019). [CrossRef]

**17. **M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam *n*_{2} measurements,” Opt. Lett. **14**, 955–957 (1989). [CrossRef]

**18. **X. Jiang, S. Liu, W. Liang, S. Luo, Z. He, Y. Ge, H. Wang, R. Cao, F. Zhang, Q. Wen, J. Li, Q. Bao, D. Fan, and H. Zhang, “Broadband nonlinear photonics in few-layer MXene Ti_{3}C_{2}T_{x} (T = F, O, or OH),” Laser Photon. Rev. **12**, 1700229 (2018). [CrossRef]

**19. **Z. Zhang and P. L. Voss, “Full-band quantum-dynamical theory of saturation and four-wave mixing in graphene,” Opt. Lett. **36**, 4569–4571 (2011). [CrossRef]

**20. **R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. **11**, 5159–5164 (2011). [CrossRef]

**21. **J. Zhang, M. Ye, S. Bhandari, A. K. M. Muqri, F. Long, S. Bigham, Y. K. Yap, and J. Y. Suh, “Enhanced second and third harmonic generations of vertical and planar spiral MoS_{2} nanosheets,” Nanotechnology **28**, 295301 (2017). [CrossRef]

**22. **J. Sun, Y.-J. Gu, D. Y. Lei, S. P. Lau, W.-T. Wong, K.-Y. Wong, and H. L.-W. Chan, “Mechanistic understanding of excitation-correlated nonlinear optical properties in MoS_{2} nanosheets and nanodots: the role of exciton resonance,” ACS Photon. **3**, 2434–2444 (2016). [CrossRef]

**23. **G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS_{2}, WS_{2}, and MoSe_{2} nanosheet dispersions [Invited],” Photon. Res. **3**, A51–A55 (2015). [CrossRef]

**24. **J. Song, L. Zhang, Y. Xue, Q. Y. S. Wu, F. Xia, C. Zhang, Y.-L. Zhong, Y. Zhang, J. Teng, and M. Premaratne, “Efficient excitation of multiple plasmonic modes on three-dimensional graphene: an unexplored dimension,” ACS Photon. **3**, 1986–1992 (2016). [CrossRef]

**25. **G. L. Frey, S. Elani, M. Homyonfer, Y. Feldman, and R. Tenne, “Optical-absorption spectra of inorganic fullerene-like MS_{2} (M=Mo, W),” Phys. Rev. B **57**, 6666–6671 (1998). [CrossRef]

**26. **M. Krause, M. Virsek, M. Remskar, N. Salacan, N. Fleischer, L. Chen, P. Hatto, A. Kolitsch, and W. Moller, “Diameter and morphology dependent Raman signatures of WS_{2} nanostructures,” Chemphyschem **10**, 2221–2225 (2009). [CrossRef]

**27. **Y. B. L. Rapoport, Y. Feldman, M. Homyonfer, S. R. Cohen, and R. Tenne, “Hollow nanoparticles of WS_{2} as potential solid-state lubricants,” Nature **387**, 791–793 (1997). [CrossRef]

**28. **L. Rapoport, N. Fleischer, and R. Tenne, “Applications of WS_{2} (MoS_{2}) inorganic nanotubes and fullerene-like nanoparticles for solid lubrication and for structural nanocomposites,” J. Mater. Chem. **15**, 1782–1788 (2005). [CrossRef]

**29. **R. Hao, L. Zhang, L. Zhang, H. You, J. Fan, and J. Fang, “Curved 2D WS_{2} nanostructures: nanocasting and silent phonon mode,” Nanoscale **12**, 9038–9047 (2020). [CrossRef]

**30. **M. Staiger, P. Rafailov, K. Gartsman, H. Telg, M. Krause, G. Radovsky, A. Zak, and C. Thomsen, “Excitonic resonances in WS_{2} nanotubes,” Phys. Rev. B **86**, 165423 (2012). [CrossRef]

**31. **J. Verble and T. Wieting, “Lattice mode degeneracy in MoS_{2} and other layer compounds,” Phys. Rev. Lett. **25**, 362–365 (1970). [CrossRef]

**32. **X. Zhang, X.-F. Qiao, W. Shi, J.-B. Wu, D.-S. Jiang, and P.-H. Tan, “Phonon and Raman scattering of two-dimensional transition metal dichalcogenides from monolayer, multilayer to bulk material,” Chem. Soc. Rev. **44**, 2757–2785 (2015). [CrossRef]

**33. **G. D. Scholes and G. Rumbles, “Excitons in nanoscale system,” Nat. Mater. **5**, 683–696 (2006). [CrossRef]

**34. **S. Durbin, S. Arakelian, and Y. Shen, “Laser-induced diffraction rings from a nematic-liquid-crystal film,” Opt. Lett. **6**, 411–413 (1981). [CrossRef]

**35. **Y. Wu, Q. Wu, F. Sun, C. Cheng, S. Meng, and J. Zhao, “Emergence of electron coherence and two-color all-optical switching in MoS_{2} based on spatial self-phase modulation,” Proc. Natl. Acad. Sci. USA **112**, 11800–11805 (2015). [CrossRef]

**36. **B. Shi, L. Miao, Q. Wang, J. Du, P. Tang, J. Liu, C. Zhao, and S. Wen, “Broadband ultrafast spatial self-phase modulation for topological insulator Bi_{2}Te_{3} dispersions,” Appl. Phys. Lett. **107**, 151101 (2015). [CrossRef]

**37. **M. Bernardi, M. Palummo, and J. C. Grossman, “Extraordinary sunlight absorption and one nanometer thick photovoltaics using two-dimensional monolayer materials,” Nano Lett. **13**, 3664–3670 (2013). [CrossRef]

**38. **J. Zhang, X. Yu, W. Han, B. Lv, X. Li, S. Xiao, Y. Gao, and J. He, “Broadband spatial self-phase modulation of black phosphorous,” Opt. Lett. **41**, 1704–1707 (2016). [CrossRef]

**39. **L. Wu, Z. Xie, L. Lu, J. Zhao, Y. Wang, X. Jiang, Y. Ge, F. Zhang, S. Lu, Z. Guo, J. Liu, Y. Xiang, S. Xu, J. Li, D. Fan, and H. Zhang, “Few-layer tin sulfide: a promising black-phosphorus-analogue 2D material with exceptionally large nonlinear optical response, high stability, and applications in all-optical switching and wavelength conversion,” Adv. Opt. Mater. **6**, 1700985 (2018). [CrossRef]

**40. **G. Wang, S. Higgins, K. Wang, D. Bennett, N. Milosavljevic, J. J. Magan, S. Zhang, X. Zhang, J. Wang, and W. J. Blau, “Intensity-dependent nonlinear refraction of antimonene dispersions in the visible and near-infrared region,” Appl. Opt. **57**, E147–E153 (2018). [CrossRef]

**41. **L. Wu, X. Jiang, J. Zhao, W. Liang, Z. Li, W. Huang, Z. Lin, Y. Wang, F. Zhang, S. Lu, Y. Xiang, S. Xu, J. Li, and H. Zhang, “MXene-based nonlinear optical information converter for all-optical modulator and switcher,” Laser Photon. Rev. **12**, 1800215 (2018). [CrossRef]

**42. **L. Wu, W. Huang, Y. Wang, J. Zhao, D. Ma, Y. Xiang, J. Li, J. S. Ponraj, S. C. Dhanabalan, and H. Zhang, “2D tellurium based high-performance all-optical nonlinear photonic devices,” Adv. Funct. Mater. **29**, 1806346 (2019). [CrossRef]

**43. **G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. **104**, 141909 (2014). [CrossRef]

**44. **C. M. Vest and M. Lawson, “Onset of convection near a suddenly heated horizontal wire,” Int. J. Heat Mass Transfer **15**, 1281–1283 (1972). [CrossRef]

**45. **W. C. W. Ji, S. Lim, J. Lin, and Z. Guo, “Gravitation-dependent, thermally-induced self-diffraction in carbon nanotube solutions,” Opt. Express **14**, 8958–8966 (2006). [CrossRef]

**46. **A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C. Y. Chim, G. Galli, and F. Wang, “Emerging photoluminescence in monolayer MoS_{2},” Nano Lett. **10**, 1271–1275 (2010). [CrossRef]

**47. **K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, “Atomically thin MoS_{2}: a new direct-gap semiconductor,” Phys. Rev. Lett. **105**, 136805 (2010). [CrossRef]

**48. **M. Taghinejad, Z. Xu, H. Wang, H. Taghinejad, K.-T. Lee, S. P. Rodrigues, A. Adibi, X. Qian, T. Lian, and W. Cai, “Photocarrier-induced active control of second-order optical nonlinearity in monolayer MoS_{2},” Small **16**, 1906347 (2020). [CrossRef]