Graphdiyne (GDY), a novel 2D carbon allotrope formed by the hybridization of sp and ${\mathrm{sp}}^2$ carbons, features a variety of outstanding characteristics such as high electron and hole mobility, direct natural band gap, and superior thermal stability [1], which benefit its application in numerous fields such as batteries [2], biomedicine [3], catalysis [4], and photodetectors [5,6]. Compared to zero band gap graphene, GDY has a natural band gap energy calculated to be between 0.46 and 1.22 eV [7–9], endowing itself as an excellent semiconductor material for electronic devices. Recently, GDY has attracted considerable attention in photonic and optoelectronic applications with the discovery of their broad band working wavelength and high optical nonlinear susceptibility [10–12]. The third-order susceptibility is a complex quantity, where the real part is responsible for the Kerr effect that leads to a number of nonlinear phenomena such as self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM); the imaginary part is responsible for nonlinear absorption. The saturable absorption effect of the 2D material family has been extensively investigated for mode-locking pulsation since 2009 [13–15], and great progress has been made in the field of high-order harmonic ultrashort pulse generation [16–20]. The first demonstration, to the best of our knowledge, of GDY used as a saturable absorber for pulse generation in femtosecond level was reported by Zhao et al., which opens up a path for its application in ultrafast photonics and optoelectronics [21]. Recently, the growing prospect of the exploitation of nonlinear Kerr effects in 2D materials for all-optical signal processing has triggered intense research interests in the application of the real part of third-order susceptibility. Among these nonlinear phenomena, FWM is particularly valued, as it offers application scopes in wavelength conversion, frequency comb generation, supercontinuum generation, and optical parametric amplification [22]. The nonlinear refractive index $n_2$ of GDY ($\sim {10}^{-9}{\mathrm{m}}^2/\mathrm{W}$) is large among various 2D materials such as graphene ($\sim {10}^{-11}{\mathrm{m}}^2/\mathrm{W}$), antimonene ($\sim {10}^{-19}{\mathrm{m}}^2/\mathrm{W}$), and ${\mathrm{MoS}}_2$ ($\sim {10}^{-10}{\mathrm{m}}^2/\mathrm{W}$) [10,23–25]. This offers opportunities towards the development of GDY-enabled FWM devices with compact size. The reported length of nanomaterial-based FWM devices that employ a lateral interaction structure (e.g.,  microfiber or side-polished fiber) varies from hundreds of micrometers (μm) to a few centimeters (cm) [24,26–29]. In addition, the GDY is constituted by the sp and ${\mathrm{sp}}^2$ hybridized alkyne bonds and benzene rings [30], where the high $\mathit{\pi }$ conjugation endows GDY with superior stability. Thus, GDY can effectively prevent its photo-oxidation and photodegradation under strong light exposure, which is pursued in fabricating stable optical devices. For comparison, although the $n_2$ of black phosphorus (BP) is at the same order as that of GDY [31], the application of BP generally needs additional steps such as encapsulation and surface coordination to guarantee its stability due to the intrinsic instability of phosphorene under ambient conditions [32–35], which increases the fabrication complexity of the device.

The typical all-fiber scheme for nanomaterial-based FWM employs two continuous-wave (CW) lasers as the pump, and the total pump power ranges from a few hundred milliwatts (mW) to as high as several watts (W) [36]. The heat cumulated from such high CW power can induce damage to the nanomaterials and their substrate, however, the conversion efficiency is still limited. An effective way to reduce the pump power and improve the conversion efficiency would suggest use of a pulse-pulse pump, replacing the CW sources altogether. However, compared to the CW-CW source, where interaction between the two beams is always satisfied, the pulse-pulse pump strictly requires that the two pulse trains arrive at the nonlinear device simultaneously. In other words, the two pulse trains should be synchronized in both time (i.e., possessing the same repetition frequency) and space (i.e., propagating without time delay). So far, most of the schemes for synchronous ultrashort pulses generation are based on the synchronization of two oscillators [37–39]. The mismatch length of these schemes is sensitive to the environmental perturbations. In addition, the pulses from these schemes are only synchronized in time, while the synchronization in space cannot be obtained due to the chromatic dispersion in the fiber. A simple and low-cost solution to produce a suitable pump source remains challenging.

Here, we propose an all-fiber FWM scheme using a dual-wavelength pulse-pump that is synchronized in both time and space. The dual-wavelength pulses are filtered from an erbium-doped (Er-doped) mode-locked fiber laser (MLFL) and then synchronized in space by an optical delay line. Thanks to the high peak power of the pulse-pump, FWM conversion efficiency of $-39.05\mathrm{dB}$ in GDY can be achieved under a low average pump power of 6.9 mW, overcoming the drawback of the CW-pump scheme. We also find an application for our FWM scheme in calculating the effective nonlinear coefficient $\gamma$ of the integrated GDY microfiber.

The GDY film on copper foil is prepared via Glayser–Hay coupling according to Li’s methods [1].

The morphology characterizations of GDY are shown in Fig. 1. Figure 1(a) exhibits the SEM image of GDY. The high-resolution transmission electron microscopy (HR-TEM) image of GDY is shown in Fig. 1(b), where the inset shows the corresponding selected area electron diffraction (SAED) pattern, which represents orderly layered 2D material. Figure 1(c) manifests the Raman spectra of the as-prepared GDY film, as the $1360{\mathrm{cm}}^{-1}$ and $1575{\mathrm{cm}}^{-1}$ peaks referred to the D and G bands, which were related to the stretching vibration of aromatic rings. The X-ray photoelectron spectroscopy (XPS) profiles of the narrow scan for element C included four subpeaks of C 1s at 284.4, 284.9, 286.2, and 288.5 eV, representing the orbitals in C-C (${\mathrm{sp}}^2$), C-C (sp), C-O, and $\mathrm{C}=\mathrm{O}$ bonds, respectively, in Fig. 1(d).

The proposed all-fiber FWM scheme is shown in Fig. 2. The experimental setup consists of two main parts. The first part is an Er-doped fiber (EDF) laser mode-locked by nonlinear polarization evolution (NPE). A 1.03 m long Liekki Er80-8/125 EDF is used as the gain medium that is pumped by a 980 nm laser diode. The ring cavity also consists of a three-paddle polarization controller (PC) that adjusts the polarization of the pulse in the cavity, an inline polarizer (ILP) that allows only one polarization state to pass through, an isolator that ensures unidirectional circulating of the light, and an optical coupler (OC) that delivers 50% power to the second part. The MLFL has a total length of 5.68 m and a net cavity dispersion of $-0.115{\mathrm{ps}}^2$. Figure 3(a) shows the mode-locked spectrum at 1565.3 nm with an output power of 6.5 mW from the MLFL.

The second part of the scheme transforms the pulses from the first part to be synchronized dual-wavelength pulses. The pulses from the MLFL are firstly filtered by a dense wavelength division multiplexer (DWDM) with a pass channel at 1566.3 nm and channel spacing of 100 GHz. The signals from the reflection channel of the first DWDM are then delivered to the second DWDM with pass channel at 1563.9 nm and channel spacing of 100 GHz. As the pulse trains selected by the two DWDMs are filtered from the same mode-locked laser, they possess the same repetition frequency (i.e., synchronized in time). However, the same repetition frequency cannot guarantee space synchronization due to the different fiber lengths of the two paths. An optical delay line with 330 ps tuning range is thus inserted after the first DWDM to compensate the length mismatch. The two pulse-trains are then amplified by the two EDF amplifiers (EDFAs), respectively. An attenuator is inserted to adjust the amplified power from EDFA1. PC2 and PC3 are used to adjust the phase matching between the two pulse trains when they are combined by the 3 dB OC.

The tunable band-pass filter with a bandwidth of 3.2 nm suppresses the FWM signal brought by the fiber pigtail of the OC and the massive amplified spontaneous emission (ASE) brought by the EDFAs. The optical spectrum of the dual-wavelength pulses after the tunable filter shows central wavelengths of 1563.9 nm (termed as pump1) and 1566.27 nm (termed as pump2), respectively [Fig. 3(b)], where full width at half-maximum of the bandwidth is 0.64 nm and 0.66 nm, respectively. The oscilloscope trace and radio frequency (RF) spectrum are shown in Figs. 3(c) and 3(d), respectively. The two pulse trains on the oscilloscope have a fixed distance [Fig. 3(c)], and the RF spectrum shows a single fundamental frequency of $\sim 35.22\mathrm{MHz}$ with a signal to noise ratio (SNR) $>50\mathrm{dB}$ [Fig. 3(d)]. Both the oscilloscope trace and RF spectrum confirm that the two pulse trains are synchronized with the same repetition frequency. We note that Fig. 3(c) is recorded by removing the 2.76 m attenuator to distinguish the two pulse trains on the oscilloscope. The pulse duration, measured by an autocorrelator, is 8.7 ps for pump1 and 8.5 ps for pump2 [Figs. 3(e) and 3(f)]. The time bandwidth products of the two pulses are calculated to be 0.7, which indicates that the pulses are chirped. With the measured average power of 2.26 mW and 4.67 mW, the peak power of the two pulses is calculated to be 7.37 W and 15.6 W, respectively. The GDY deposited on the microfiber (GDY-microfiber) is employed as the nonlinear device here for FWM generation. The microfiber is fabricated by stretching the standard single-mode fiber (SMF-28e) under an oxy-hydrogen flame. The GDY is deposited on the tapered fiber by an optical deposition method. The loss is 0.42 dB for the bare microfiber and increases to 1.59 dB after optical deposition with GDY. Figure 4 shows the optical microscopic image of the fabricated sample with a deposition length of 860 μm and a taper waist diameter of 5.6 μm.

With the pump scheme described in Section 3, we firstly perform the experiment with the bare microfiber without GDY. When the delay line is properly tuned so that the two pump-pulses are exactly synchronized in the space, signals with different wavelengths arise in the spectrum. The new frequency signals could be further optimized to the maximum by properly setting PC2 and PC3 for phase matching [Fig. 5(a)]. According to the energy conservation of the FWM theory [22], we confirm that the generated signals at the short wavelength of 1561.5 nm and the long wavelength of 1568.6 nm are the first-order anti-Stokes bands and the first-order Stokes bands, respectively. In this case, the FWM signal is completely contributed by fibers, and the conversion efficiency is $-34.99\mathrm{dB}$ (the 0.42 dB loss of the microfiber is included).

We then perform the experiment again with the GDY-microfiber while the other components in the experiment maintain unchanged. The pump passing through the GDY-microfiber is 1.17 dB lower due to the absorption of the GDY, while the intensity of first-order anti-Stokes bands and Stokes bands is improved by 0.634 dB [Fig. 5(a)]. The FWM conversion efficiency is $-34.36\mathrm{dB}$ (the 0.42 dB loss of the microfiber and the 1.17 dB absorption of the GDY are included). We note that the intensity of the FWM signals is actually improved by 1.804 dB ($1.17+0.634\mathrm{dB}$) when considering that the GDY can also cause 1.17 dB loss to the FWM signal contributed by fibers. Thus, we deduce that GDY contributes 34% to the whole FWM signal, and the true conversion efficiency of GDY alone is $-39.05\mathrm{dB}$. The spectral variations versus time delay are recorded in Fig. 5(b), where we define the time delay that results in maximum signals as “0 ps.” The new frequency signals would be faded out when the time delay is tuned away from 0 ps, indicating that the signals are indeed generated by FWM. We note that the first-order Stokes and anti-Stokes bands do not center at the exact same wavelength, as the delay is changed from $-5\mathrm{ps}$ to 0 ps and to 5 ps. We attribute this to the chirp of the pump pulses. When the delay is $<0\mathrm{ps}$, the leading edge of pump1 interacts with the trailing edge of pump2; and the situation is opposite when the delay is $>0\mathrm{ps}$. As pulses are chirped, frequencies are different at the leading edge and trailing edge, leading to the first-order Stokes and anti-Stokes bands centered at the different wavelengths. The first-order anti-Stokes signal is then filtered out by a DWDM with center wavelength of 1561.4 nm when delay $=0\mathrm{ps}$ [Fig. 5(a)]. The oscilloscope trace of the filtered signal is a set of pulse trains, which would be vanished by changing the time delay, indicating that the pulse signal indeed belongs to the first-order anti-Stokes band other than incomplete isolation from the pump [Fig. 5(c)]. The RF of the filtered signal centered at 35.22 MHz has an $\mathrm{SNR}>50\mathrm{dB}$ [Fig. 5(d)].

According to the FWM theory, the frequency of the newly generated signal should match $2{\omega }_i-{\omega }_j$ and $2{\omega }_j-{\omega }_i$, respectively, where ${\omega }_i$ and ${\omega }_j$ are the frequency of the pump. Thus, the wavelength of the newly generated signal should be ${\lambda }_i{\lambda }_j/(2{\lambda }_j-{\lambda }_i)$ and ${\lambda }_i{\lambda }_j/(2{\lambda }_i-{\lambda }_j)$, respectively, where ${\lambda }_i$ and ${\lambda }_j$ are the wavelength of the pump. By substituting the pump wavelength of 1563.9 nm and 1566.27 nm into the formula, the newly generated signal should be at 1561.5 nm and 1568.6 nm, respectively, which are consistent with our experimental results. The FWM conversion efficiency $R$ is determined by the following equation [40]: $$R=\eta P_1{P}_2{(\gamma L)}^2\exp (-\alpha L),$$(1)where $P_1$ and $P_2$ are the power of the pump, $L$ is the light propagation length, $\alpha$ is the absorption, $\gamma$ is the effective nonlinear coefficient, and $\eta$ is the transition effectivity given as $$\eta =\frac{1}{1+{(\mathrm{\Delta }k/\alpha )}^2}\{1+\frac{4\exp (-\alpha L){\sin }^2(\mathrm{\Delta }kL/2)}{{[1-\exp (-\alpha L)]}^2}\},$$(2)where $\mathrm{\Delta }k$ is the phase-matching factor expressed as $$\mathrm{\Delta }k=\frac{{\lambda }_k^2}{2\pi c}{({\omega }_i-{\omega }_j)}^2[D_c+\frac{{\lambda }_k^2}{2\pi c}(|{\omega }_i-{\omega }_j|)\frac{\mathrm{d}{D}_c}{\mathrm{d}\lambda }],$$(3)where $D_c$ is the chromatic dispersion, $\mathrm{d}{D}_c/\mathrm{d}\lambda$ is the dispersion slope, and ${\lambda }_k$ ($k=i,j$) is the wavelength corresponding to the wave at frequency. The chromatic dispersions of the bare microfiber and GDY-microfiber are calculated by the finite-difference eigenmode (FDE) solver, where the boundary conditions are set as a perfectly matched layer. The chromatic dispersion and the dispersion slope of the microfiber are calculated as 10.5 ps/(nm · km) and $0.02{\text{ps/}(\mathrm{nm}}^2·\mathrm{km})$, respectively. The chromatic dispersion and the dispersion slope of the GDY-microfiber are calculated as $-85\text{ps/(nm}·\mathrm{km})$ and $-0.4{\text{ps/}(\mathrm{nm}}^2·\mathrm{km})$, respectively. The absorptions of the microfiber and GDY-microfiber are $112.5{\mathrm{m}}^{-1}$ and $431.5{\mathrm{m}}^{-1}$, respectively. Substituting the above values into the Eq. (3), the phase-matching factors of the microfiber and GDY-microfiber are calculated to be $0.052{\mathrm{m}}^{-1}$ and $-0.4{\mathrm{m}}^{-1}$, respectively. However, the change in phase-matching factor has little effect on the transition effectivity $\eta$. The $\eta$ of the microfiber and GDY-microfiber are both calculated to be $\approx 1$ with a difference on the order of ${10}^{-8}$. This result can also be estimated as follows: the term ${(\mathrm{\Delta }k/\alpha )}^2$ in Eq. (2) is on the order of ${10}^{-7}$, and the second term in the bracket of the right side of the Eq. (2), whose value is limited by the term ${\sin }^2(\mathrm{\Delta }kL/2)$, is also on the order of ${10}^{-7}$. Thus, the $\eta$ could be considered as one despite that the device is coated with GDY or not. The gain in FWM conversion efficiency of the GDY-microfiber compared to the bare microfiber is mainly due to the improvement of nonlinear coefficient $\gamma$ other than the change in phase matching. This conclusion is consistent with previous reports [36,41].

$\mathit{\gamma }$ of the device is closely relevant to $n_2$ of the materials and the structure of the device. Table 1 shows typical results of FWM based on 2D materials, where the pump power, $n_2$ of the materials, and the geometric parameters of these devices are also presented. Although the $n_2$ of BP and GDY are at the same order, the maximum conversion efficiency of $-39.05\mathrm{dB}$ obtained in this work is still improved by two orders of magnitude with only 6.93 mW average pump power. Our synchronized dual-wavelength pulse scheme is thus of great advantage in improving FWM conversion efficiency with a low incident average pump power, reducing possible damages to the 2D material-based devices.Table 1.

Four-Wave Mixing Phenomenon Experimentally Demonstrated in 2D Materials

This all-fiber FWM scheme can be a simple tool to estimate the $\gamma$ of the nonlinear device. Compared with Z-scan that is usually employed to measure the nonlinear coefficient of the film type 2D material-based devices, the FWM scheme proposed here could be applied to microfiber type devices where nonlinear coefficients are not available by typical Z-scan measurement. As we set $\text{delay}=0\mathrm{ps}$ in our experiment, $P_1$ and $P_2$ represent the peak powers of the pump1 and pump2, respectively. By tuning the variable attenuator, the FWM spectra versus different $P_2$ are recorded [Fig. 6(a)]. We take both the anti-Stokes and Stokes signals into account and average the conversion efficiency to cancel the coupling power uncertainty. The conversion efficiency of the anti-Stokes and Stokes signals increases linearly with the increases of $P_2$, and the averaged result could be well fitted by the linear function with a slope of $6.9\times {10}^{-6}$ [Fig. 6(b)]. As the slope $k=P_1{(\gamma L)}^2\exp (-\alpha L)$, by substituting $P_1=7.37\mathrm{W}$, $\alpha =431.5{\mathrm{m}}^{-1}$, and $L=860\mathrm{\mu m}$ into the formula, we estimate the $\gamma$ of the GDY in our experiment to be $1354{\mathrm{W}}^{-1}/\mathrm{km}$.

Although the result of $-39.05\mathrm{dB}$ in the manuscript has been improved by 20 dB compared to previous work by employing the all-fiber synchronized dual-wavelength pump source, there still needs more research to further improve the FWM efficiency. In the future, we will optimize the integration strategy and device parameters such as the absorption, deposition length, and taper waist to improve the nonlinearity of the device.

In summary, we have proposed an FWM scheme based on synchronized dual-wavelength pump pulses. The GDY-microfiber integrated in our scheme achieves a maximum conversion efficiency of $-34.36\mathrm{dB}$ under the low average pump power of 6.93 mW. As the GDY contributes 34% to the whole FWM signal, the conversion efficiency of the GDY is $-39.05\mathrm{dB}$, which is improved by at least 20 dB compared to previously reported results. In addition, our proposed compact design provides an effective approach to measure the nonlinear coefficient of nanomaterials-based devices.