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 WS2 NPs are chemically synthesized using hard mesoporous silica. The nonlinear refractive index n2 and third-order susceptibility χ(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 WS2 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.

The IF-like WS2 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 WS2 NPs can then be obtained by removing the template in H2S gas. A typical scanning electron microscopy (SEM, JEOL, JSM-7000F) image of the synthesized WS2 NPs is shown in Fig. 1(a). The multilayer structure with an interlayer distance of ∼0.67nm is clearly characterized using a high-resolution transmission electron microscopy (HRTEM, JEOL, JEM-2100F) image [Fig. 1(b)]. The synthesized WS2 NPs hold IF-like features with an average diameter of ∼26.5nm, 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 WS2 structure (JCPDS card No: 08-0237) [Fig. 1(c)].

Two Raman peaks are observed at ∼353 and ∼420cm−1 under the excitations of 532 and 633 nm continuous-wave (CW) lasers [Fig. 1(d)]. The Raman active lattice vibrations at the Γ point of the hexagonal Brillion zone are modes 421cm−1 and 356cm−1 in the detected region [31,32]. Furthermore, second-order Raman transition, i.e., two longitudinal acoustic (2LA) phonons at ∼353cm−1, are also observed for excitation energies close to the band gap. More interesting is the activation of the B1u mode, which is silent in planar 2D TMDs. Its excitation arises from the curved layers and structural disorder of WS2 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 WS2 NPs to that without WS2 NPs. Figure 1(e) shows the transmittance spectrum of WS2 NP dispersion ranging from 400 to 900 nm (Andor SR500I), which is used to characterize the effective number of WS2 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 WS2 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.

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 Δn [34]. As the laser beam propagates along the z axis, the field E reorients the direction of WS2 NPs in the normal plane. According to the Kerr effect, the refractive index of the suspension can be described by n=n0+n2I, where n0 is the linear refractive index, n2 is the nonlinear refractive index of WS2 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 WS2 dispersions of a thickness L, the incident light will gain an intensity-dependent phase [34] Δφ(r)=(2πn0λ)∫0Leffn2I(r,z)dz,(1)where I(r,z) is the intensity distribution of the focused laser beam, r∈[0,+∞) is the transverse coordinate in the beam, and the host solvent is ethanol with a refractive index of n0=1.36. Leff represents the effective interaction length contributing to the SSPM process, which can be calculated by Leff=∫L1L2(1+z2/z02)−1dz=z0atan(z/z0)|L1L2, where z0=πω02/λ, is defined by the waist width ω0 and wavelength of the laser beam; L=L2−L1 is the thickness of the quartz cuvette. In the experiment, L=10mm and ω0=74.2μm at the front surface of the cuvette. For simplicity, the incident Gaussian laser with a cylindrical symmetry along the z axis will gain an additional phase shift Δφ(r)=Δφ0exp(−2r2/ω02) after passing through the WS2 dispersions. Here, Δφ0 is the phase shift at the diffraction ring center, i.e., r=0 [34]. By using Eq. (1), we obtain Δφ0=2πn0n2LeffI/λ with I(0,z)=2I [35], which indicates that a larger intensity results in more phase shift. Since the temporal slot between pulses is 12.5 ns, all of the interference arises from the SSPM within each single pulse. Radiation fields from the area around two different points have the same wave vector and can cause interference. Maximum constructive or destructive interference is determined by Δφ(r1)−Δφ(r2)=mπ, where m is an odd or even integer corresponding to dark or bright stripes, respectively. The total number of diffraction rings can be estimated as N=[Δφ(0)−Δφ(∞)]/2π=Δφ0/2π, which linearly increases as laser intensity increases [Fig. 3(a)]. In addition, at a given incident intensity, more rings pour out at longer wavelength irradiation.

The nonlinear refractive index can be expressed as [20,23] n2=λ2n0LeffdNdI.(2)The slope S=dN/dI 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, χtotal(3)=λcn02.4×104×π2LeffS [20,23,36].

As introduced previously, third-order nonlinear susceptibility χ(3) is of great significance for indicating nonlinear performance of the nonlinear materials. Here, the third-order nonlinear susceptibility of monolayer WS2 NPs can be estimated using the counterpart of multiple layer structures with χtotal(3)=Neff2χmonolayer(3) [20,23], where Neff represents the effective number of WS2 layers in the NPs, and χmonolayer(3) represents the contribution of one layer WS2 out of Neff layers to the third-order susceptibility of WS2 NPs. Therefore, χmonolayer(3) can be calculated with the following equation: χmonolayer(3)=n02n2(cm2/W)0.0395×Neff2.(3)The transmission of monolayer WS2 is 99.3%–99.7% at the selected wavelength [23,37]. Therefore, according to the transmission measurement in Fig. 1(e), the effective layer number Neff is estimated to be 48–140 at the wavelength ranging from 720 to 800 nm. Thus, the third-order susceptibility χmonolayer(3) for monolayer WS2 is estimated to be in order of ∼10−7esu, which is two orders higher than the counterparts of popular 2D materials with planar features. Similar results are obtained when the solvent is replaced by methylbenzene. In addition, the n2 of ethanol is ∼9 orders smaller than the counterpart of the WS2 nanosheet [23], so the influence of the solvent on the final third-order nonlinearity can thus be excluded. As shown in Fig. 3(b), the third-order susceptibility χmonolayer(3) for monolayer WS2 varies slightly around 720–800 nm. Both the nonlinear refractive index and third-order susceptibility χmonolayer(3) obtained by the SSPM experiment are listed for an explicit comparison (Table 1). Regarding the planar 2D WS2, the introduced additional freedom by curved features plays an encouraging role in boosting up the nonlinear characteristics. On the other hand, compared with other 2D materials, such as black phosphorus (BP), IF-like WS2 NPs with superior n2 and χmonolayer(3) highlight a better idea of improving nonlinear optical properties.

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 ∼0.5s (Fig. 4). Subsequently, both the horizontal and vertical diameters of the rings collapse and reach a steady state after ∼2.8s and ∼4.5s, 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], WS2 NPs in the upper part of the dispersions are precipitated into the lower part, resulting in a smaller density of WS2 NPs in the upper half of the dispersions, and then a reduced Neff, naturally, with a reduced n2. 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 WS2 NPs.

The maximum value of the vertical radius of the outermost ring and its half-cone angle are denoted by RH and θH, respectively. The half-cone angle can be written as θH=λ/2π(dΔφ/dr)max, which can be further simplified, for a Gaussian beam, to be θH≈n2IC, where C=[−(8IrLeff/ω02)×exp(2r2/ω02)]max with r∈[0,+∞) being a constant. The distortion angle can be expressed as θD≈Δn2IC, where Δn2 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 Δn2/n2 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 θD<θH. When the incident intensity reaches the wavelength-dependent threshold of approximately 30–40W/cm2, 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 WS2 NPs to continuously sink below the laser beam. After a period of thermal convection, when the density of WS2 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 Ti3C2Tx MXene exceeds the counterparts of the proposed WS2 NPs. However, the underlying mechanism here is different from those observed in Ti3C2Tx MXene with a narrow direct bandgap [41]. Because of the multiple layers in WS2 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 χ(3)≈Ne4/ε0m3ωe06d2, where e is the element charge, ε0 is the vacuum permittivity, N is the density of electrons of the material, ωe0 is the oscillation frequency of electrons, ωe0=me4/32π2ε02ℏ3, d is the lattice constant, and m is the effective mass of the conduction electron [1]. If d is identified with the Bohr radius a0=4πε0ℏ2/me4, we obtain that χ(3)∝m−7. Due to the distortion and curved features in WS2 NPs, the effective mass of electrons in IF-like WS2 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 n2 and χ(3).

The mechanism of the SSPM phenomenon in WS2 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 n2 and χ(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 WS2 NPs. Initially, an arbitrary angle related to the interaction energy exists between the direction of the WS2 NPs polarization and the laser-induced electric field. As interaction energy is minimized, WS2 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 WS2 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 MoS2 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 WS2 NPs.

In conclusion, a novel type of IF-like WS2 NPs is successfully synthesized using the hard template method with a diameter of 26.5 nm. By characterizing the nonlinear refractive index n2 and third-order susceptibility χ(3) using the SSPM method with a visible fs pulse laser, we obtain n2∼10−5cm2/W and χ(3)∼10−7esu, 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.