1 Introduction

Mid-infrared (mid-IR) laser sources, operating within the 3–5 μm band, are pivotal in various applications like environmental monitoring, medical diagnostics, and spectroscopy. However, the direct availability of solid-state lasers in this band is limited. Consequently, the optical parametric oscillator (OPO) has emerged as a promising technique for generating radiation within this range. Over the past decade, considerable attention has been directed towards mid-IR OPOs utilizing various crystals such as ZnGeP2 (ZGP) [1, 2], CdSiP2 (CSP) [3, 4], KTiOAsO4 (KTA) [5,6,7,8], and periodically poled LiNbO3 (PPLN) [9, 10]. Of these crystals, KTA exhibits mature growth techniques, offering good crystal quality in large sizes. With a broad transmission range spanning from 0.35 to 5.3 μm and high damage threshold, KTA presents itself as a viable candidate. At 3.5 μm waveband, KTA typically exhibits lower absorption in comparison to KTP, enhancing its effectiveness in mid-IR OPOs. This lower absorption minimizes energy loss and thermal effects, resulting in high conversion efficiency and stable operation. Moreover, KTA-based OPOs effectively utilize 1 μm lasers as pump sources, facilitating non-critical phase-matching conditions. In previous studies [5,6,7,8], mid-IR KTA OPOs were implemented in various configurations, including extracavity OPOs, diode side-pumped intracavity OPOs (IOPOs), and diode end-pumped IOPOs. Among these, the end-pumped intracavity scheme stands out due to its compactness, low threshold, and high efficiency. However, while idler waves with controllable pulse repetition rates around dozens of kHz have been achieved in different IOPOs [2, 5, 9, 10], the pulse widths remain relatively long (exceeding 3 ns), resulting in relatively low pulse peak powers (below 30 kW). Therefore, there’s a pressing need to develop high-quality pump laser sources with high peak powers and shorter pulse widths. Recent advances in dual-loss modulation, involving simultaneous use of an active Q-switch and a passive Q-switch in the resonator, offer promise in this regard [11,12,13]. In this technique, the active modulator manages the pulse repetition rates to efficiently store energy in the laser medium, while the saturable absorber (SA) acts as a passive modulator, refining the pulses. The combined effect of double modulation losses on the pulse edges leads to the generation of shorter pulse widths and higher peak powers. With such dual-loss modulation-based Q-switched lasers to pump OPOs, idler wave pulses with short pulse widths and stable pulse repetition rates as well as high peak powers can be achieved [14]. By integrating the dual-loss modulation technique with intracavity KTA-OPOs, the aim is to create efficient mid-IR laser sources with substantially enhanced peak powers and shorter pulse widths. Such an approach could pave the way for impactful advancements in mid-IR laser technology.

SAs play a crucial role in the functionality of dual-loss modulation Q-switched lasers. Recent advancements in two-dimensional (2D) materials, characterized by low symmetrical structures and in-plane anisotropic properties, have opened new avenues for the advancement of pulsed lasers [15,16,17]. Materials such as graphene [18], transition metal dichalcogenides [19], topological insulators [20], black phosphorus [21], and MXenes [22] have found widespread applications in Q-switching and mode-locking laser technologies, representing two primary approaches for generating laser pulses. Among these materials, ternary chalcogenides have garnered significant attention due to their potential for development and future prospects [23]. The incorporation of a third element in 2D ternary chalcogenides offers an additional degree of freedom in comparison to their binary counterparts [24], allowing control over physical properties, electronic structures, and functionalities through compositional adjustments [25]. As a promising fluorescent sensor material [26], Ta2NiS5, belonging to the orthorhombic crystal structure, has emerged as a promising material, boasting a bandgap of 0.30 eV [24] and temperature-dependent magnetic susceptibility [27]. Its exceptional properties have been harnessed in various laser systems. For instance, Ma et al. demonstrated passively Q-switched and mode-locked fiber lasers operating at 1.0 and 1.5 μm wavelengths using Ta2NiS5 SA [28]. Yan et al. achieved a passively Q-switched bulk laser at 1.9 μm, showcasing a shortest pulse duration of 313 ns [29]. Given the remarkable saturable absorption capabilities of Ta2NiS5 in solid-state lasers, there is an imperative to explore its potential in generating high-power, short-duration pulsed laser radiation. Leveraging Ta2NiS5’s saturable absorption properties in solid-state lasers, it is envisioned that an idler-resonant KTA IOPO with high peak power and short pulse duration can be realized using a dual-loss modulated Q-switched laser with an acousto-optic modulator (AOM) and a Ta2NiS5 SA as the pump laser source.

Theoretical models employing dynamic rate equations have proved effective in guiding experiments involving lasers or OPOs based on Ta2NiS5 [30, 31]. Notably, theoretical calculations incorporating the Gaussian spatial distribution of fundamental photons have shown better alignment with experimental results in comparison with plane-wave approximation outcomes [32, 33]. However, few studies have explored Gaussian rate equations in the context of IOPOs. The parameters related to the saturable absorption of Ta2NiS5, such as ground-state absorption cross-section, excited-state absorption cross-section, and excited-state lifetime, remain less reported due to ongoing research on the energy band structure of Ta2NiS5. Consequently, investigations into the dynamics of lasers modulated by Ta2NiS5, particularly an IOPO driven by it, have not been extensively documented in the literature.

In this study, 2D Ta2NiS5 nanoflakes were synthesized via the liquid-phase exfoliation method. Comprehensive characterizations of their morphologies and nonlinear optical responses were conducted. By utilizing a Ta2NiS5 SA and an AOM, an idler-resonant KTA IOPO driven by a dual-loss modulated Q-switched laser was demonstrated. Operating under an incident pump power of 18.9 W and an AOM modulation rate of 1 kHz, the generated signal and idler waves with the shortest pulse widths were 872 ps and 2.86 ns, corresponding to the maximum peak powers of 392 kW and 57.6 kW, respectively. In addition, our experiments revealed also that the IOPO, pumped by a doubly Q-switched laser employing an AOM and a Ta2NiS5 SA, produced signal and idler waves with shorter pulse widths and higher peak powers compared to the IOPO driven by a singly Q-switched laser with an AOM. Moreover, a set of dynamic coupling rate equations accounting for Gaussian spatial distribution in an idler-resonant DIOPO incorporating Ta2NiS5 SA was given. Notably, the numerical outcomes aligned closely with the experimental findings, underscoring the potential of Ta2NiS5 as a material for generating short-pulsed lasers.

2 Experiment

2.1 Fabrication and characterization of Ta2NiS5 SA

The liquid-phase exfoliation technique was employed to produce Ta2NiS5 SAs. Initially, a commercially available 99.999% pure Ta2NiS5 ternary chalcogenide powder was mixed with absolute ethyl alcohol. Subsequently, ultrasonic processing (80 W, 3 h) followed by centrifugation (6000 rpm, 20 min) was conducted to exfoliate the sample and obtain a dispersed solution. This dispersed solution, consisting of the exfoliated material, was then deposited onto a quartz glass substrate using the spin-coating method (500 rpm, 5 min). The resulting solution was dried under vacuum at a constant temperature of 330 K to achieve the formation of Ta2NiS5 SAs.

The layered structure of the ternary Ta2NiS5 compound is illustrated in Fig. 1a, characterized by a series of molecular bonds arranged in zigzag patterns. This arrangement leads to distorted NiS4 tetrahedra and TaS6 octahedra that are connected through S–S edges [29]. These layers are stacked along the b axis through weak van der Waals forces, with a single layer exhibiting a thickness of 0.63 nm in Ta2NiS5 [26]. Scanning electron microscopy (SEM) and Transmission electron microscopy (TEM) were utilized to observe the morphology of Ta2NiS5 nanoflakes. As depicted in Figs. 1b and c, stacked layers are evident on the smooth surface of Ta2NiS5 samples, indicating successful exfoliation. Additionally, Raman spectroscopy was conducted on the as-prepared Ta2NiS5, measuring from 50 to 300 cm−1 under a 532 nm laser. The obtained Raman spectra in Fig. 1d display three Raman active modes: B2g (twisting motion), 2Ag and 3Ag (stretching motions) for Ta2NiS5 [34]. Furthermore, atomic force microscopy (AFM) was employed to capture the morphology of the as-prepared Ta2NiS5 nanosheets. Figures 1e and f illustrate uniformly dispersed exfoliated Ta2NiS5 nanosheets on the substrate, with an average thickness of 20 nm, corresponding to approximately 31 layers. These findings indicate the 2D nature of the exfoliated Ta2NiS5 nanosheets, exhibiting a few layers in thickness. In summary, the observed properties confirm that the as-prepared Ta2NiS5 nanosheets qualify as 2D materials due to their few-layer structure and exhibit topological characteristics typical of such materials.

Fig. 1
figure 1

Characterization of the Ta2NiS5: a Structural model of layered simples, b SEM image, c TEM image, d Raman spectra, e AFM image, and f corresponding height distributions along the marked lines 1 and 2 in e

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The linear transmission spectral of Ta2NiS5 SAs were obtained using a UV–Vis-NIR spectrophotometer. As illustrated in Fig. 2a, the spectral transmission profile spans a wide range from 400 to 1800 nm, exhibiting noticeable fluctuations. The observed fluctuations around 1600 nm are attributed to lamp changes during the measurements. When the incident light possesses energy higher than the material’s optical band gap, electrons from the valence band are continuously excited to the conduction band. Consequently, as per the Pauli Exclusion Principle, the conduction and valence bands become fully occupied by electrons and holes, impeding interband transition. This behavior demonstrates the saturated absorption characteristics of the material. The nonlinear transmission properties of the Ta2NiS5 SAs were investigated by employing double-channel synchronous detection within the 1 μm waveband using a custom-built pulsed laser, as presented in Fig. 2b. The fitted data reveal a modulation depth (\(\Delta T\)) of 11.94%, corresponding to a saturation intensity (Isat) of 2.63 MW/cm2. Additionally, the non-saturable absorption (\(\alpha_{NS}\)) was determined to be 7.06%. The inset of Fig. 2b illustrates the linear relationship between transmittance and peak energy density at low-power density, with a linear fitting yielding a slope of 1.67. These properties underscore the substantial potential of Ta2NiS5 SAs for facilitating Q-switched laser performance.

Fig. 2
figure 2

a Linear transmission spectral and b nonlinear transmission properties and the linear relation for low-power density (inset) of the as-prepared Ta2NiS5 SAs

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2.2 Experimental configuration

Figure 3 demonstrates the schematic setup for an idler-resonant KTA IOPO pumped by a dual-loss-modulated Q-switched laser with an AOM and Ta2NiS5 SA. A 50 W 808 nm fiber-coupled commercial diode laser (FAP-I system, Coherent Inc.) with a core diameter of 400 µm was utilized as the pump source. To relay the pump beam into the laser medium, a coupling lens system with a focal length of 45 mm and a coupling efficiency of 90% was employed. The laser medium comprised an a-cut YVO4/Nd:YVO4 composite crystal with a Nd3+ concentration of 0.3 at% and dimensions of 3 × 3 ×  (2 + 10) mm3. The pump surface of YVO4 was antireflection (AR) coated at 808 nm, while both surfaces of YVO4/Nd: YVO4 composite crystal were AR coated at 1064 nm. The nonlinear crystal was a 5 × 5 × 20 mm3 X-cut (θ = 90°, ϕ = 0°) KTA crystal arranged in a type-II noncritical phase matching configuration to achieve the most effective nonlinear coefficient and acceptance angle. Both surfaces of the KTA crystal were AR coated at 1064 and 1535 nm (R < 0.2%) and high-transmission (HT) coated at 3470 nm (T > 95%). An AOM (GSQ27-3, central frequency: 27.12 MHz, RF power: 50 W, 26th Institute, CETC, China) measuring 47 mm in length and AR coated at 1064 nm (R < 0.2%) on both surfaces served as the active modulator. The modulation repetition rate of the AOM ranged from 1 to 50 kHz. The YVO4/Nd:YVO4 and KTA crystals were thermally managed by wrapping them with indium foil and placing them in copper holders connected to a water-cooling system to maintain a constant water temperature of 14 °C for stable laser output. The Ta2NiS5, prepared as an SA, was integrated into the fundamental laser system. M1 was a flat mirror with AR coated at 808 nm (R < 0.2%) on its entry face and high-reflectivity (HR) coated at 1064 nm (R > 99.8%), HT coated at 808 nm (T > 97%) on the opposite face. M2 was a flat mirror composed of infrared silica glass (JGS3). Its entry surface was coated for AR at 1064 nm (R < 0.2%), and the other surface was coated for HT at 1064 nm (T > 99.5%), and HR at both 1535 and 3470 nm (R > 99.8%). M3 was a flat mirror made of Al2O3, coated for HR at 1064 nm (R > 99.9%), HT at 1535 nm (T > 99.3%), and partial-reflectivity (PR) at 3470 nm (R = 91%). Thus, the idler-resonant OPO cavity oscillated between M2 and M3, whereas the fundamental wave oscillated between M1 and M3. Thus, the respective lengths of the OPO and fundamental wave cavities were set as 28 and 112 mm, respectively.

Fig. 3
figure 3

Experimental setup for the idler-resonant KTA IOPO pumped by a dual-loss-modulated Q-switched YVO4/Nd:YVO4 laser with AOM and Ta2NiS5 SA

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To spectrally isolate the idler, signal, and residual fundamental waves, distinct optical components were employed in the experimental setup. A CaF2-based dichroic mirror DM1 (HR coated at both 1535 and 1064 nm, and HT coated at 3470 nm) was utilized to segregate the idler wave. Additionally, a BK7 glass-based dichroic mirror DM2 (HR coated at 1535 nm and HT coated at 1064 nm) was employed to separate the signal wave from the residual fundamental wave. The measurements of pulse characteristics and signal wave spectrum were conducted using specific instruments. The DPO-7104C digital oscilloscope (with a rise time of 350 ps, 1 GHz bandwidth, 20 G samples/s, Tektronix Inc., USA), along with a fast InGaAs photodetector featuring a rising time of 0.4 ns (New Focus, 1611) were employed for analyzing the signal pulse. Furthermore, a Wavescan laser spectrometer (with a resolution of 0.4 nm, APE GmbH, Germany) was utilized for spectral analysis. For the characterization of the idler wave, the DPO-7104C digital oscilloscope, in conjunction with an HgCdZnTe photoconductive detector having a response time of 1 ns, was used to measure the pulse characteristics. Moreover, the idler wave spectra were recorded using another Wavescan MIR laser spectrometer (APE GmbH, Germany). To measure the output powers of the signal and idler waves accurately, a PM100D laser power meter (Thorlabs., USA) was employed in the experimental setup.

In the experimental setup, the configuration changed between idler-resonant DIOPO when incorporating the Ta2NiS5 SA into the laser cavity and idler-resonant SIOPO when the Ta2NiS5 SA was absent from the cavity.

3 Results and discussions

Figure 4 illustrates the average output powers of signal and idler waves concerning incident pump powers for various AOM modulation rates for both the SIOPO and DIOPO. The plot depicts a near-linear increment in the average output powers of signal and idler waves with increasing pump power and AOM modulation frequency for both setups. Notably, due to the additional insertion loss attributed to the Ta2NiS5 SA in the resonator, the thresholds of signal and idler waves for DIOPO were higher compared to those of SIOPO at identical AOM modulation rates. At a modulated frequency of 4 kHz, the recorded highest average output powers of signal and idler waves were 566 mW and 269 mW, respectively, for SIOPO. Conversely, for DIOPO, these values were measured at 467 mW and 228 mW, respectively, corresponding to an overall optical-to-optical conversion efficiency of 3.68%. In comparison to the singly Q-switched fundamental laser, the doubly Q-switched fundamental laser has lower conversion efficiency due to larger losses of the dual-loss modulation, resulting in lower conversion efficiency of the OPO system. In addition, due to low repetition (1–4 kHz) for obtaining shorter pulse width, a shorter upper-level lifetime of Nd:YVO4 crystal may lead to heavy spontaneous emission, resulting in low conversion efficiency of the fundamental laser. Moreover, the related parameters in the experiment may not be optimal, such as the cavity, the length of KTA, the transmission of the OPO mirror, and so on. If the related parameters are optimized, the conversion efficiency can be improved.

Fig. 4
figure 4

Dependence of the average output powers of the signal and idler waves for a SIOPO and b DIOPO on the pump powers at various modulation frequencies

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Figure 5 displays the characteristic output spectrum of the DIOPO when subjected to an incident pump power of 18.9 W and an AOM modulation rate of 1 kHz. In this spectrum, the fundamental wave and signal wave were identified at 1064 nm and 1535 nm, respectively. Simultaneously, the idler wave was observed at a wavelength of 3467 nm.

Fig. 5
figure 5

A typical output spectrum of DIOPO at an incident pump power of 18.9 W and an AOM modulation rate of 1 kHz

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Figure 6 shows the pulse widths of the signal and idler waves for two IOPOs versus the incident pump powers for different modulation frequencies (Symbol). It’s evident that, for both SIOPO and DIOPO, the pulse widths of signal and idler waves decrease as the incident pump power increases. Additionally, under identical incident pump power conditions, the pulse widths of signal and idler waves are notably shorter for DIOPO than those for SIOPO. Specifically, at an incident pump power of 18.9 W and a modulation frequency of 1 kHz, the shortest pulse widths achieved were 872 ps and 2.86 ns for DIOPO’s signal and idler waves, respectively. In contrast, for SIOPO, the corresponding values were 2.4 ns and 4.54 ns, respectively.

Fig. 6
figure 6

Pulse widths of the signal and idler waves for a SIOPO and b DIOPO versus the incident pump powers at various modulation frequencies. Symbols for experimental data, curves for theoretical result

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Figure 7 illustrates the temporal pulse shapes of undepleted and depleted fundamental, signal, and idler waves for both SIOPO and DIOPO, set at an incident pump power of 18.9 W and an AOM modulation rate of 1 kHz. The visual comparison in Fig. 7 indicates that due to the pulse-shortening effect induced by dual-loss modulation, the pulse duration of the fundamental wave in DIOPO was notably shorter than that in SIOPO under equivalent pump power and AOM repetition rate. To assess the pulse width discrepancies of signal and idler waves in the two IOPOs, two pulse width compression ratios, Ts and Ti, were introduced, where Ts = Tss/Tsd and Ti = Tis/Tid, in which Tss, Tsd and Tis, Tid are the pulse widths of signal and idler waves in SIOPO and DIOPO, respectively. At an incident pump power of 18.9 W and an AOM modulation rate of 1 kHz, the calculated pulse width compression ratios were 2.75 for the signal wave and 1.58 for the idler wave. In comparison to the singly Q-switched fundamental laser, the doubly Q-switched fundamental laser can generate a shorter pulse width due to larger losses of the dual-loss modulation. In addition, both the active and the passive losses can attenuate the leading and trailing edges of the pulse, obtaining a shorter pulse width. A shorter fundamental laser in the DIOPO can generate a shorter OPO pulse than those in the SIOPO. Additionally, both Figs. 6 and 7 highlight another observation: the pulse widths of signal waves were shorter than those of idler waves for both SIOPO and DIOPO. This difference is attributed to the larger cavity loss in signal waves, resulting in a shorter photon lifetime compared to idler waves. However, it’s essential to note that the measured pulse duration might be broader than the actual pulse due to the time resolution limitations of the experimental facilities [35].

Fig. 7
figure 7

Temporal pulse shapes of the undepleted and depleted fundamental, signal, and idler waves for two IOPOs at the incident pump power of 18.9 W and AOM modulation rate of 1 kHz

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From Fig. 7, it can be also seen that the pulse shapes of the depleted fundamental wave may not exhibit the expected sharp decline from the peak value when the OPO oscillation is built up. The reason may be that the low overall conversion efficiency (3.68%) of our OPO system in the nonlinear interactions impacts the expected pulse shape dynamics. In addition, the imperfections or suboptimal coatings of the used KTA crystal play a crucial role in the performance of the OPO, resulting in less pronounced pulse depletion.

In an effort to delve deeper into the operational mechanism of the dual-loss-modulated IOPO setup, our experiment also included the recording of pulse characteristics from a singly AOM modulated SIOPO laser, providing a comparison baseline. At an incident pump power of 18.9 W, the temporal sequences of pulse widths for signal waves from both IOPO configurations and the fundamental wave from the Ta2NiS5 SA singly Q-switched fundamental laser are compiled in Fig. 8. As evident from Figs. 7, 8a, and b, the DIOPO not only generates a signal wave with a shorter pulse width but also exhibits reduced amplitude-to-amplitude fluctuations and higher stability when compared to the SIOPO. Upon deactivating the AOM, the laser operated in the state of a singly Q-switched fundamental laser with Ta2NiS5 SA. However, the IOPO operation, pumped by the Ta2NiS5 SA singly Q-switched laser, couldn’t be achieved, potentially due to insufficiently high power density under this specific configuration. Nonetheless, the pulse widths and repetition rates of the Ta2NiS5 SA singly Q-switched fundamental waves’ envelopes were notably larger than those of the signal wave from the DIOPO laser, displaying a more considerable pulse-to-pulse amplitude fluctuation, as illustrated in Fig. 8c. From our observations, we infer that the stable operation of the OPO is generally contingent upon the stability of the fundamental laser. Within a dual-loss modulated Q-switched laser system, the incorporation of a passive SA can effectively suppress the number of longitudinal modes, consequently narrowing the line width. The stable temporal behavior of the DIOPO pulses can be attributed to the mode-suppressing effect of the Ta2NiS5 SA. To substantiate this, we measured the linewidths of the fundamental lasers in both singly and doubly Q-switched configurations. The linewidth of the singly Q-switched laser was measured to be 1.84 nm, whereas the linewidth of the doubly Q-switched laser incorporating the Ta2NiS5 SA was 0.26 nm. The significant narrowing of the spectral linewidth in the doubly Q-switched configuration confirms the mode-suppressing effect of the SA, leading to enhanced stability in the temporal behavior of the DIOPO pulses. This restriction imposed on the fundamental wave’s longitudinal modes contributes significantly to the stable operation of both the fundamental laser and the OPO. Therefore, the utilization of the Ta2NiS5 SA has the potential to substantially enhance the stability of Q-switched IOPO systems.

Fig. 8
figure 8

Typical temporal pulse trains of the signal wave from a SIOPO, b DIOPO, under an incident pump power of 18.9 W and a repetition rate of 1 kHz, and c A typical temporal pulse train of the fundamental wave from a singly Q-switched fundamental laser with Ta2NiS5 SA under an incident pump power of 18.9 W

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In accordance with the formulas E = PA/f and PP = E/t, where PA represents the average output power, f denotes the AOM modulation rate, t signifies the pulse width, the single pulse energy E and the peak power PP can be estimated. Figures 9 and 10 present the single pulse energies and peak powers of signal and idler waves in relation to incident pump powers at various modulation frequencies for both IOPO configurations (Symbol). Despite the average output powers and single pulse energies of signal and idler waves are lower in DIOPO than those in SIOPO under identical incident pump power and AOM modulation rate, the pulse peak powers in DIOPO were still notably higher in comparison to those in SIOPO, primarily due to the pulse shortening effect in DIOPO. At the maximum incident pump power of 18.9 W and AOM modulation rate of 1 kHz, the maximum pulse peak powers for SIOPO were calculated to be 182.8 and 46 kW for signal and idler waves, respectively, whereas for DIOPO, these values were 392 and 57.6 kW, respectively. Two pulse peak power enhancement factors, referred to as compression ratios (Ps and Pi), were introduced: Ps = Psd/Pss and Pi = Pid/Pis, where Pss, Psd, Pis, and Pid represent the pulse peak powers of signal and idler waves in SIOPO and DIOPO, respectively. At an incident pump power of 18.9 W and AOM modulation rate of 1 kHz, the pulse peak power enhancement factors for the signal and idler waves were determined to be 2.14 and 1.25, respectively.

Fig. 9
figure 9

Pulse energies of the signal and idler waves for a SIOPO and b DIOPO versus the incident pump powers at different modulation frequencies. Symbols for experimental data, curves for theoretical result

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Fig. 10
figure 10

Peak powers of the signal and idler waves for a SIOPO and b DIOPO versus the incident pump powers at different modulation frequencies. Symbols for experimental data, curves for theoretical result

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4 Theoretical analysis

Rate equations can be used to describe the dynamics of Q-switched lasers [36]. By considering IOPO, the coupled rate equations for an idler-resonant Q-switched YVO4/Nd:YVO4/KTA IOPO can be given the numerical simulation can be done.

The rate equations take into account both the Gaussian transversal and longitudinal distributions of photon density within the cavity, incorporating the thermal influence of the gain medium. The representation of the average intracavity photon density \(\phi (r,t)\) in the TEM00 mode is articulated as follows:

$$\phi (r,t) = \phi (0,t)\exp ( - \frac{{2r^{2} }}{{w_{l}^{2} }}),$$
(1)

Within this equation, \(w_{l}\) symbolizes the average beam radius pertaining to the fundamental wave.

$$\phi_{j} (r,t) = \frac{{w_{l}^{2} }}{{w_{j}^{2} }}\phi_{j} (0,t)\exp ( - \frac{{2r^{2} }}{{w_{j}^{2} }})\left( {j\, = \,g,a,y,k,f,s,i} \right),$$
(2)

where \(\phi_{j} (0,t)\) (j = s, i) denotes the intracavity photon densities related to the signal wave and idler wave; \(\phi_{j} (0,t)\) (j = g, a, y, k) pertains to the intracavity photon densities of the fundamental wave at specific locations: YVO4/Nd:YVO4 crystal, AOM, Ta2NiS5 SA, and KTA crystal [37]. \(w_{j}\) (j = g, a, y, k) represents the radius of the TEM00 mode at these respective positions. The connection between photon densities and the electric field is denoted by: \(\phi_{j} (r,t) = {{\varepsilon_{j} E_{j}^{2} (r,t)} \mathord{\left/ {\vphantom {{\varepsilon_{j} E_{j}^{2} (r,t)} {4\hbar \omega_{j} }}} \right. \kern-0pt} {4\hbar \omega_{j} }}\left( {j\, = \,f, s, i} \right)\). By considering the transverse Gaussian distribution of beam energy, \(E_{j} (r,t) = E_{j} (0,t)\exp ( - {{r^{2} } \mathord{\left/ {\vphantom {{r^{2} } {w_{j}^{2} }}} \right. \kern-0pt} {w_{j}^{2} }})\left( {j\, = \,f, s, i} \right)\) can be derived, where \(E_{j} (r,t)\) stands for the electric field. \(E_{j} (0,t)\left( {j\, = \,f,s,i} \right)\) represents the electric field of the fundamental wave, the signal wave, and the idler wave along the laser axis, respectively.

Under the Gaussian assumption, the excited-state population densities \(n_{y1} (r,t)\) of the saturable absorber can be described as:

$$n_{y1} (r,t) = n_{y1} (0,t)\exp ( - \frac{{2r^{2} }}{{w_{y}^{2} }}),$$
(3)

In the idler-resonant IOPO, it serves as a pure singly resonant oscillator (SRO) where the output coupler remains fully transparent to the signal wave. This implies that the signal wave does not experience the cavity. Similar to most second harmonic generation (SHG) processes, where the frequency-doubled laser also experiences no cavity, the behavior of the signal wave in our IOPO is comparable. Therefore, the dynamic process governing the generation of the signal wave in the idler-resonant IOPO can be applied when considering the SHG process of the frequency-doubling laser. In these analogous scenarios, the photon density of the signal wave can be expressed as [38]:

$$\phi_{s} (0,t) = \frac{{\hbar \mu_{0} d_{eff}^{2} \omega_{f} \omega_{s} \omega_{i} }}{{4n_{f}^{2} n_{i}^{2} }}(2l_{KTA} )^{2} \phi (0,t)\phi_{i} (0,t),$$
(4)

where \(\mu_{0}\) represents the permeability of vacuum, while \(\omega_{f,s,i}\) denote the angular frequencies corresponding to the fundamental, signal, and idler waves, respectively. The effective nonlinear coefficient is denoted as deff, and \(n_{f,s,i}\) represent the refractive indices of the fundamental, signal, and idler waves in the KTA crystal, respectively.

Considering the transmission exceeding 99.3% for the signal wave by the output coupler (M3), it effectively halts the evolution of the signal wave field within the idler-resonant IOPO. Accounting for this, the coupled rate equations governing the idler-resonant IOPO with the saturable absorption characteristic of Ta2NiS5 SA are derived based on prior research works [32, 33, 36, 37]:

$$\begin{gathered} \int_{0}^{\infty } {\frac{d\phi (r,t)}{{dt}}} \cdot 2\pi rdr = \int_{0}^{\infty } {\frac{1}{{t_{r} }}} \cdot \left[ {2\sigma l_{g} n(r,t)\phi_{g} (r,t) - (\sigma_{g} - \sigma_{e} )l_{y} n_{y1} (r,t)\phi_{y} (r,t) - \sigma_{e} l_{y} n_{y0} \phi_{y} (r,t) - L} \right] \hfill \\ \cdot 2\pi rdr - \int_{0}^{\infty } {\frac{1}{{t_{r} }}} \cdot \left[ {\delta_{T} (t)\phi_{g} (r,t) + \delta_{a} (t)\phi_{a} (r,t)} \right] \cdot 2\pi rdr \hfill \\ - \int_{0}^{\infty } {\frac{{\delta Iw_{k}^{2} }}{{8\hbar w_{l}^{2} }} \cdot \frac{{l_{opo} }}{l}} \cdot \frac{{l_{KTA} }}{{l_{opo} }} \cdot E_{f} (r,t)E_{i} (r,t) \cdot 2\pi rdr, \hfill \\ \end{gathered}$$
(5)
$$\int_{0}^{\infty } {\frac{{d\phi_{i} (r,t)}}{dt}} \cdot 2\pi rdr = \int_{0}^{\infty } {\left[ { - \frac{{\phi_{i} (r,t)}}{{\tau_{i} }} + \frac{\delta I}{{8\hbar }} \cdot \frac{{l_{KTA} }}{{l_{opo} }} \cdot E_{f} (r,t)E_{i} (r,t)} \right]} \cdot 2\pi rdr,$$
(6)
$$\int_{0}^{\infty } {\frac{dn(r,t)}{{dt}}} \cdot 2\pi rdr = \int_{0}^{\infty } {\left[ {R_{in} e^{{ - {{2r^{2} } \mathord{\left/ {\vphantom {{2r^{2} } {w_{p}^{2} }}} \right. \kern-0pt} {w_{p}^{2} }}}} - \sigma c\phi_{g} (r,t)n(r,t) - \frac{n(r,t)}{\tau }} \right]} \cdot 2\pi rdr,$$
(7)
$$\int_{0}^{{r_{y} }} {\frac{{dn_{y1} (r,t)}}{dt}} \cdot 2\pi rdr = \int_{0}^{{r_{y} }} {\left[ {\frac{{n_{y0} - n_{y1} (r,t)}}{{\tau_{y} }} - \frac{{\varepsilon_{p} }}{{4\hbar v_{p} }} \cdot \sigma_{g} cn_{y1} (r,t)E_{f}^{2} (r,t)} \right]} \cdot 2\pi rdr,$$
(8)

where \(n(r,t)\) represents the population inversion intensity of the fundamental laser medium; \(n_{y0}\) and \(n_{y1} (r,t)\) denote the ground-state and the excited-state population density for the Ta2NiS5 SA; \(\sigma\) and \(\tau\) stand for the stimulated emission cross-section and the stimulated radiation lifetime of the laser gain medium; \(L\) signifies the round-trip loss, incorporating the intrinsic loss of the laser cavity. Equation (5) delineates the loss of fundamental wave photons due to thermal effects, AOM modulation, and nonlinear transformations within the KTA crystal. Within the equation, \(- [(\sigma_{g} - \sigma_{e} )l_{y} n_{y1} (r,t)\phi_{y} (r,t){ + }\sigma_{e} l_{y} n_{y0} \phi_{y} (r,t)]\) signifies the loss of fundamental wave photons attributed to the Ta2NiS5 SA. The accrual of the idler wave photons with time is portrayed by Eq. (6). Equation (7) explicates the loss of population inversion within the laser medium. Within these equations, \(t_{r}\) denotes the round-trip time of the fundamental wave. Notably, l, lg, la, lKTA, ly, and lopo represent the physical lengths of the fundamental laser cavity, the gain medium, AOM crystal, KTA crystal, Ta2NiS5 SA, and OPO cavity. εj and \(v_{j}\) (j = f, s, i) depict the dielectric constants and the frequencies of the three waves. \(\tau_{j}\) (j = s, i) symbolizes the OPO cavity lifetimes of the signal and idler waves. \(\delta = \varepsilon_{0} d_{eff}\), where \(\varepsilon_{0}\) corresponds to the dielectric constant of the vacuum. For simplicity, perfect phase matching in the KTA crystal is assumed, i.e., \(I = 1\), \(w_{k} = w_{i}\). In Eq. (5), δT denotes the diffractive loss induced by the thermal effect of the gain medium. δa(t) can be expressed as:

$$\delta_{a} (t) = \delta_{a} \exp \left[ { - \left( {\frac{t}{{t_{AO} }}} \right)^{2} } \right],$$
(9)

where tAO and δa represent the turn-off time and the intrinsic diffraction loss of the AOM. δa(t) characterizes the loss function attributed to the gradual AOM induced by the exponential time delay.

Regarding the Ta2NiS5 SA, the change in the population inversion density is depicted in Eq. (8). In Eq. (5), \(\sigma_{g}\) and \(\sigma_{e}\) represent the ground-state and the excited-state absorption cross-sections, which are critical parameters defining the saturable absorption properties of the Ta2NiS5 SA. As indicated in Ref. [37], the relationship between them can be expressed as:

$$\frac{{\sigma_{g} }}{{\sigma_{e} }} = \frac{{\ln T_{0} }}{{\ln T_{\max } }},$$
(10)
$$\frac{{(\sigma_{g} - \sigma_{e} ) \times T_{0} }}{hv} = k,$$
(11)
$$T_{0} = \exp ( - n_{y0} \sigma_{g} l_{y} ),$$
(12)
$$T_{\max } = \exp ( - n_{y0} \sigma_{e} l_{y} ),$$
(13)

The determination of small-signal transmittance (T0) at low power density, maximum transmittance (Tmax) at high power density, and the slope k can be derived from Fig. 2b. Subsequently, utilizing Eqs. (10) and (11), the values of \(\sigma_{g}\) and \(\sigma_{e}\) for the 20 nm-thick Ta2NiS5 SA can be estimated as 5.91 × 10−19 cm2 and 2.053 × 10−19 cm2, respectively. Additionally, accounting for the inhomogeneous broadening mechanism, the excited-state lifetime (\(\tau_{y}\)) of Ta2NiS5 SA is calculated to be 1.152 ms [37].

Table 1 presents the key parameters illustrating the saturable absorption characteristics of the 2D Ta2NiS5 SA, while Table 2 outlines the additional parameters employed in the numerical simulation.

Table 1 The pivotal parameters defining the saturable absorption characteristics of 2D Ta2NiS5 SA
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Table 2 Additional parameters for theoretical solutions
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The calculated output pulse energies for both the idler and signal waves are given by:

$$E_{i} = \frac{{\pi w_{k}^{2} chv_{i} }}{4}\ln (\frac{1}{{R_{i} }})\int_{0}^{\infty } {\phi_{i} (0,t)dt} ,$$
(14)
$$E_{s} = \frac{{\pi w_{k}^{2} chv_{s} }}{4}\int_{0}^{\infty } {\phi_{s} (0,t)dt} ,$$
(15)

The coupled rate equations governing the idler-resonant SIOPO pumped by a singly Q-switched laser with an AOM were derived by excluding the terms related to Ta2NiS5 SA in the aforementioned equations. Utilizing the parameters outlined in Table 1 and Table 2, the numerical simulation of the coupled rate equations for both IOPOs was conducted based on the derived rate Eqs. 5, 6, 7, 8. The computed temporal profiles of signal and idler waves for both IOPOs, at an incident pump power of 18.9 W and an AOM modulation rate of 1 kHz, are illustrated in Fig. 11. The simulated values for the pulse widths, pulse energies, and peak powers of signal and idler waves for the two IOPOs, respectively, were given with curves in Figs. 6, 9 and 10. Notably, the simulations are in agreement with the experimental data.

Fig. 11
figure 11

Calculated temporal shapes of the signal and idler waves for a SIOPO and b DIOPO at an incident pump power of 18.9 W and an AOM modulation rate of 1 kHz

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5 Conclusions

Ta2NiS5 nanosheets were fabricated via the liquid-phase exfoliation method and their morphology and nonlinear optical response were studied. By using Ta2NiS5 nanoflakes as a saturable absorber (SA), an idler-resonant KTA IOPO driven by a dual-loss-modulated Q-switched YVO4/Nd:YVO4 laser with AOM is realized. Achieving maximum output powers of 467 mW for the signal wave and 228 mW for the idler wave, the system displayed an overall optical-to-optical conversion efficiency of 3.68%. Impressively short pulse widths of 872 ps and 2.86 ns for the signal and idler waves respectively were obtained, corresponding to peak powers of 392 and 57.6 kW, affirming Ta2NiS5 as a promising SA in pulse laser applications. Experimental results showcased the DIOPO’s capability to compress pulse widths compared to the SIOPO. Furthermore, the coupled rate equations for the idler-resonant DIOPO are given and the numerical simulations of the equations agree with the experimental results.