Multiphoton absorption (MPA) has garnered significant attention in the fields of optical power limiting and ultrafast all-optical switching applications due to its effective control and modulation of optical signals^[1,2]. Nonlinear optical (NLO) materials play a crucial role in realizing such photonic devices, necessitating high optical nonlinearity, low linear optical losses, and ultrafast response time^[3]. In this regard, wide-gap semiconductors have emerged as promising candidates for these requirements owing to their weak absorption from one-photon transitions and large nonlinearity at communication wavelengths. Recent investigations have extensively explored the two-photon absorption (2PA) and three-photon absorption (3PA) properties of β-Ga2O3^[4], ZnO^[5], ZnS^[6], and GaN^[7] through experimental measurements and theoretical calculations. However, these bulk semiconductors exhibit relatively low nonlinear coefficients (2PA coefficient β2∼10cm/GW and 3PA coefficient β3∼10−3–10−4cm3/GW2), relatively slow relaxation time (tens to hundreds of picoseconds), as well as substantial lattice mismatch that hinder their application in integrated photonic platforms. The utilization of 2D materials offers new opportunities for the development of ultrafast and miniaturized photonic devices due to their significant third-order and high-order NLO absorption, ease of integration, and small lattice mismatch^[8].

A recent study has demonstrated that 2D GaP achieves ultrafast sub-30 fs all-optical switching based on the optical Kerr effect and 2PA^[1]. Similarly, hexagonal phase GaS also exhibits high stability, tunable bandgap (2.3–3.2 eV), and strong optical absorption^[9]. In the field of NLO, recent advancements in nanoscale GaS have shown excellent saturable absorption (SA) properties and successful applications in ultrafast mode-locking^[10]. Additionally, odd-layer GaS possesses a noncentrosymmetric nature with a strong second-order susceptibility of approximately 47.98 pm/V^[11]. In terms of MPA property research, previous experimental investigations on the 2PA of the GaS thin film at a single wavelength of 650 nm reveal that the 2PA coefficient is in the order of 103 scale, much higher than conventional wide-bandgap semiconductors^[12], making GaS a promising candidate for designing high-performance all-optical switches and optical limiters. Therefore, direct measurement and systematic analysis of the dispersion characteristics of both 2PA and 3PA spectra in the GaS thin film are crucial for promoting relevant application development.

In this Letter, we present a comprehensive investigation of the 2PA and 3PA spectra of GaS thin film in the spectral range of 540–1600 nm using a femtosecond Z-scan system. Both the 2PA and 3PA spectra exhibit excellent agreement with theoretical analysis based on the two-band approximation model for direct bandgap semiconductors. The measured maximum values are approximately 19.89 cm/GW at 620 nm for β2 and approximately 4.88cm3/GW2 at 1500 nm for β3, which significantly surpass those reported for previously studied wide-bandgap semiconductors. Based on a simplified two-energy level model, our findings reveal that the GaS film not only exhibits a large cross section of 2PA (4.82×10−52cm4·s·photon−1 at 620 nm) but also demonstrates a strong cross section of 3PA (8.17×10−80cm6·s2·photon−2 at 1500 nm). These results underscore the immense potential of wide-bandgap 2D materials possessing exceptional MPA for applications in NLO.

The GaS thin film was synthesized using a physical vapor deposition (PVD) technique. In the experimental setup, 20 mg of GaS powder was placed at the center of a tube furnace with a heating zone set at 940°C to facilitate source evaporation. Subsequently, the resulting GaS vapor was transported onto a sapphire substrate using high-purity argon gas with a flow rate of 40 sccm, leading to the formation of a continuous thin film that is approximately 650 nm thick. This morphology is confirmed by a scanning electron microscope (SEM), as shown in Fig. 1(a). Additionally, the micro-Raman spectrum shown in Fig. 1(b) exhibits three characteristic peaks at 182.8, 286.4, and 357.1cm−1 corresponding to the A1g1, E2g1, and A1g2 phonon modes respectively, thereby indicating successful preparation of the GaS film. Furthermore, X-ray diffraction (XRD) spectroscopy confirms the absence of byproducts such as Ga2S3 during synthesis. This is evident from Fig. 1(c), which exhibits three prominent peaks at 11.60°, 23.07°, and 34.81° corresponding to the (002), (004), and (006) planes for hexagonal GaS (PDF#08-0417). Moreover, the peaks at 30.72° and 35.53° originate from the substrate. The narrow half-widths observed indicate a high degree of crystallinity and purity in our samples. Further characterization through UV-Vis absorption measurements reveals significant optical absorption within the wavelength range of 400 to 500 nm [Fig. 1(d)]. Based on Tauc plot analysis, the intrinsic bandgap energy of GaS is estimated to be approximately 2.3 eV, consistent with previous reports^[11].

The dispersion of the nonlinear coefficient (β) in the GaS film was investigated using a broadband Z-scan system. In this study, we utilized an optical parametric amplifier (TOPAS-Prime) pumped by a Ti:sapphire femtosecond laser operating at a center wavelength of 800 nm with a repetition rate of 1 kHz as the laser source. The output laser from the optical parametric amplifier covered a wavelength range spanning from 540 to 1600 nm. Figure 2(a) depicts the open aperture Z-scan curves obtained under excitation wavelengths ranging from 540 to 1040 nm. The decrease in normalized transmittance as the sample approaches the focal point (Z=0) clearly indicates a reverse saturable absorption (RSA) response, which can be attributed to phenomena such as nonlinear scattering, excited state absorption (ESA), and MPA^[13]. Considering our GaS film’s uniform thickness without any scattering centers, we can neglect the contribution of nonlinear scattering, which is further confirmed by relocating the photodetector position and observing no scattering signal^[14]. According to the estimated bandgap above the excitation photon energy, the phenomenon of ESA is eliminated, and MPA becomes the dominant effect for RSA. The NLO response of the GaS film was further characterized by fitting ln(1−T) versus ln(I) plots using linear regression analysis. As shown in Fig. 2(b), the fitted slop values closely approximate 1, with values of 0.95, 1.07, 1.04, 0.97, 1.10, 1.09, and 0.98, indicating degenerate 2PA^[15]. Additionally, when excited within a photon energy range between Eg and 1/2Eg via wavelengths spanning from 540 to 1040 nm, it supports a 2PA transition process. Considering that the femtosecond pulse width is shorter than the excited state lifetime, we propose a two-energy-level model to elucidate the optical absorption process as illustrated in Fig. 3(a)^[16], where under laser excitation most electrons in the first excited state (S1) rapidly return back into the ground state (S0) while higher excited state absorptions can be ignored.

Based on the aforementioned analysis, we can quantitatively analyze the β2 variation by employing Eq. (1) as follows^[17,18]: TMPA(z)=1−1M3/2βMI0M−1Leff(M)(1+z2/z02)M−1,(1)where z0 represents the Rayleigh length of the laser beam, I0 denotes the incident intensity at focus, Leff is the effective thickness of the GaS film, and M corresponds to M photon absorption.

The fitting results, represented by the solid lines in Fig. 3(b), exhibit excellent agreement with the experimental data points. The β2 values of the GaS film are 11.21±1.76cm/GW (540 nm @ 2.29 eV), 10.79±2.59cm/GW (580 nm @ 2.14 eV), 19.89±2.92cm/GW (620 nm @ 2.00 eV), 19.71±3.48cm/GW (660 nm @ 1.88 eV), 19.19±2.95cm/GW (800 nm @ 1.55 eV), 8.82±1.82cm/GW (940 nm @ 1.32 eV), and 4.51±1.71cm/GW (1040 nm @ 1.19 eV). The results demonstrate that the β2 values reach their maximum at 0.7 Ephoton/Eg and subsequently exhibit a declining trend with increasing Ephoton/Eg. More importantly, the β2 values of the GaS film are considerably larger compared to wide bandgap semiconductors, such as β-Ga2O3^[4,19], GaN^[7], and ZnO^[5] listed in Table 1. This can be attributed to the quantum confinement in 2D materials and strong light–matter interaction^[20,21]. It is worth noting that the β2 value of GaS is lower compared to previously reported β-GaS film, which can be attributed to the much higher intensity of our excitation laser. The evidently remarkable potential of the GaS film for NLO devices within the visible spectral region is apparent. Furthermore, the dispersion behavior observed in the GaS film aligns well with theoretical predictions based on Wherrett’s two-band and perturbation theory^[22,23], in which the 2PA dispersion relation can be described by β2=292π(e2ℏc)2f2fℏPn2Eg3(2ℏω/Eg−1)3/2(2ℏω/Eg)5,(2)where c represents the speed of light, Eg denotes the material’s bandgap, n signifies its refractive index, P corresponds to the Kane momentum parameter^[7], f2 accounts for numerical factors, and ℏω is the photon energy.

According to the simplified two-level system, rate equations and nonlinear propagation equations can be expressed as ∂N0∂t=−σ2PAN0I22ℏω+N1τ10,(3)∂N1∂t=σ2PAN0I22ℏω−N1τ10,(4)N=N0+N1.(5)

The propagation equation in the sample is described as dIdz=−α(I)=−α0−αNL,(6)where αNL=σ2PAN0I2. By employing the Runge–Kutta method to fit the experimental data in Fig. 3(c) with Eqs. (3)–(6), it is possible to extract the 2PA cross section (σ2PA) and relaxation time (τ10). The τ10 of GaS has been measured approximately at (85 ± 19) ps^[12], which serves as the fitting range. Based on this, the fitted σ2PA values of GaS are determined as follows: 3.16×10−52cm4·s·photon−1 (540 nm), 2.72×10−52cm4·s·photon−1 (580 nm), 4.83×10−52cm4·s·photon−1 (620 nm), 4.40×10−52cm4·s·photon−1 (660 nm), 4.35×10−52cm4·s·photon−1 (800 nm), 2.18×10−52cm4·s·photon−1 (940 nm), and 1.23×10−52cm4·s·photon−1 (1040 nm). The results further demonstrate that the wavelength-dependent 2PA cross section is consistent with the 2PA coefficient. Moreover, owing to its strong 2PA coefficient and ultrafast carrier lifetime characteristics, the GaS film exhibits great potential for integrated optical limiter devices.

To further investigate the dispersion characteristics of 3PA, we conducted open-aperture Z-scan measurements by increasing the excitation wavelength from 1200 to 1600 nm. As shown in Fig. 4(a), a similar RSA phenomenon is observed. Initially, we assume that the RSA induced by 3PA occurs when the incident photon energy is above 1/3Eg but less than 1/2Eg. Furthermore, Fig. 4(b) demonstrates a well-fitted linear relationship between ln(1−T) and ln(I). The slopes obtained under different excitation wavelengths are close to 2, indicating degenerate 3PA for the GaS film. Under laser excitation, 3PA involves the simultaneous absorption of three photons, which leads to a transition from the ground state level (S0) to the excited state level (S1), as illustrated in Fig. 5(a).

Based on Eq. (1), the β3 values of the GaS film are calculated and summarized in Fig. 5(b). The values are as follows: (1.85±0.44)cm3/GW2 (1200 nm @ 1.03 eV), (3.35±0.58)cm3/GW2 (1300 nm @ 0.95 eV), (5.88±0.40)cm3/GW2 (1500 nm @ 0.83 eV), and (4.46±0.59)cm3/GW2 (1600 nm @ 0.78 eV). The β3 of the GaS film is larger than those of β-Ga2O3, GaS, GaN, ZnO, and ZnS and is comparable with monolayer MoS2, CsPbBr3, and (C4H9NH3)2PbBr4 perovskites in Table 1 due to its 2D quantum confinement. Furthermore, the dependence of β3 on Ephoton/Eg can be well fitted by Wherrett’s two-band perturbation theory equation as follows^[22]: β3=31028π2(e2ℏc)3ℏ2P3n3Eg7(3ℏω/Eg−1)1/2(3ℏω/Eg)9.(7)

To gain a deeper understanding of the 3PA mechanism, a two-energy-level model is also proposed. In this scenario, the rate equation model can be given as ∂N0∂t=−σ3PAN0I33ℏω+N1τ10,(8)∂N1∂t=σ3PAN0I33ℏω−N1τ10,(9)N=N0+N1,(10)dIdz=−σ3PAN0I3.(11)

Here, σ3PA represents the 3PA cross-section. By fitting the Z-scan experimental data in Fig. 5(c) with Eqs. (8)–(11), we obtain σ3PA values of 1.59×10−80cm6·s2·photon−2 (1200 nm), 2.93×10−80cm6·s2·photon−2 (1300 nm), 8.17×10−80cm6·s2·photon−2 (1500 nm), and 6.59×10−80cm6·s2·photon−2 (1600 nm). These results further demonstrate that the larger σ3PA values are responsible for the strong 3PA coefficient exhibited by the GaS film. Based on these findings, it can be concluded that the GaS film possesses significant 2PA and 3PA properties, making it a promising candidate for applications such as optical power limiting and ultrafast all-optical switching.

We conduct a systematic investigation of the dispersions of 2PA and 3PA coefficients in the GaS film using a Z-scan system equipped with a laser source ranging from 540 to 1600 nm. The observed decrease in both β2 and β3 values of the GaS film with increasing Ephoton/Eg is consistent with the predictions of the two-band perturbation theory. Based on a two-energy-level model, our results reveal that the 2PA and 3PA coefficients are strongly influenced by absorption cross sections, reaching maximum values of 4.47×10−52cm4·s·photon−1 at 620 nm and on the scale of 10−80cm6·s2·photon−2 at 1500 nm, respectively. These findings demonstrate that GaS films exhibit strong MPA properties and ultrafast carrier relaxation, making them promising candidates for optical power limiting and integrated all-optical switching devices.