1. Introduction

First theoretical studies of graphene, a single sheet of carbon atoms arranged in a honeycomb lattice, date back over seventy years, and were realized in the context of graphite [1]. However, the first isolation of graphene was only achieved in 2004 [2]. Since then, graphene has attracted a huge amount of attention due to its unique properties and its wide potential for many applications [3]. The electronic properties of graphene are what inspired most researchers initially. Due to its infinite two-dimensional expansion, electrons possess a linear relation between energy and momentum and thus refer as massless Dirac fermions [4]. Moreover, the valence and conduction bands of graphene overlap slightly, making this material a zero-bandgap semiconductor. These properties make graphene a good candidate for applications in nano-electronics [5]. Graphene also shows remarkable nonlinear optical characteristics that are used more and more for photonics [6] and as a saturable absorber in modulators or in mode-locked lasers [7,8]. Theoretical and experimental studies have also predicted a high and broadband nonlinearity due to the zero-bandgap and in particular a large Kerr nonlinear index of refraction ($n_2$) [9–13]. Finally, graphene is compatible with CMOS fabrication processes, a property that is very promising for integrated all optical signal processing applications. As such, graphene has recently been successfully deposited on top of silicon nanophotonic waveguides, resulting in a modification of the nonlinearity of the overall structure [14].

Along with the studies on graphene films, carbon-based quantum dots have also attracted a significant interest. The quantum dots (QDs) are now widely used for their photoluminescence properties [15] and are very promising for applications in all-optical signal processing. Since the pioneering work made by Ashkin more than 35 years ago [16], it is known that most of the QDs possess nonlinear absorption. They exhibit large Kerr nonlinearities, especially close to the linear absorption band. In addition, their NL optical properties could be fully controlled by the size [17], the shape and the concentration of quantum dots, leading to quasi on-demand NL optical properties [18]. In [19] thermo-optical properties of carbon nanodots having 3 nm average size has been described at 800 nm in the femtosecond regime at high repetition rate. Moreover, nanodiamonds and carbonaceous quantum dots have been considered in [20] in the picosecond and nanosecond regimes. Due to their specific sp²/sp³ bonded carbon atoms ratio they exhibit different NL properties showing negative NL refraction in the both regimes at 532 nm. Nanodiamonds have exhibited NL absorption in the nanosecond regime at 532 nm and 1064 nm. This absorption has been attributed to reverse saturation absorption (RSA) [20,21]. Carbon quantum dots also offer the possibility to be doped or to anchor on them others chemical species in order to enhance their NL properties [22,23]. Beside these studies, graphene family quantum dots have also been explored. Aqueous suspensions of finely dispersed large sheets (500 nm diameter) of graphene oxide (GO) were found to exhibit important nonlinear optical absorption, both at 532 nm and at 1064 nm, under both nanosecond and picosecond laser pulses while the nonlinear refraction was difficult to be observed clearly [24]. GO quantum dots, composed of elongated strips of graphene with straight edges, have been considered as well as their corresponding reduced forms especially graphene quantum dots structures (GQDs) [25]. As most of carbon-based quantum dots they benefit from low toxicity, photoluminescence, chemical stability and high quantum confinement effect properties. Therefore, GQDs have been successfully used for biological [26], opto-electronics [27], and environmental [28] applications. Studies on their NL properties have however mostly focused on the nonlinear absorption properties of graphene families QDs structures in the visible and infrared domains [20,24]. All considered structures exhibited typical π-π* transition absorption band with a maximum in the range 230-270 nm. Excitation around 532 nm being resonant with this absorption band through multiphoton absorption process, large NL losses has been reported and attributed to two-photon absorption (TPA) and nonlinear scattering (NLS) process for high incident laser intensities. At 1064 nm and 532 nm in the ns regime, nonlinear scattering and two-photon-absorption were found to have strong effects on the NL responses of the graphene nanostructures [24,29]. NL absorption have also been reported theoretically in the infrared domain for wavelengths in the range 2-6 µm [30]. The NL refraction properties of these QDs structures has only being briefly reported experimentally in wavelength ranges far from the π-π* transition absorption band, revealing low NL refraction behavior [24], or anyway under the experimental resolution of the measurement system to detect it. Finally, it had to be mentioned a recent computational study, in which the linear and nonlinear optical properties have been investigated in terms of polarizability and first hyperpolarizability [31] on GQDs of various sizes, shapes, and edges.

In the present paper, we characterize experimentally in the picosecond regime, the NL refraction index and absorption of GQDs in the ultraviolet (355 nm), in the visible (532 nm) and in the infrared (1064 nm) domains. We consider suspension of GQDs in water, a liquid for which NL properties are known and very low [32,33]. The characterization is performed using a Z-scan [34] based technique called the D4σ method [35]. It allows accurate NL measurements, as previously reported [36,37]. Comparison with other reported values is made. It is expected that GQDs will inherit from both the rich NL optical properties of graphene as well as from the versatility and the control of QDs. Therefore, the quantification of the NL properties of GQDs, especially close to the linear absorption band, where the NL refraction is expected to be large, could pave the way towards the functionalization of these structures for applications in laser domain, in all-optical signal processing and for high contrast bio-imaging and bio-sensing applications [39].

2. Experimental procedure

Because the production of high‐quality GQDs do not result in well‐defined particles [25], we consider commercially available fluorescent blue GQDs in water at a concentration of 1 mg/ml, from STREM Chemical Incorporation. Quantum dots are composed of few-layer of graphene with a maximal thickness of 100 nm and a diameter lower than 5 nm. Strong quantum confinement of excitons in these quantum dots allows the opening of bandgap in the electronic structures. To confirm this behavior, we first characterize the linear optical properties of these GQDs. The absorbance spectrum of a 2 mm thick colloid of GQDs, as represented in Fig. 1(a), reveals two absorption bands approximately situated at 350 nm and 235 nm. The latter band represents the typical $\pi \to {\pi }^{*}$ transition of an aromatic pi system [40] while the former one represents the $n\to {\pi }^{*}$ transition revealing the presence of graphene oxide nanoparticle (NP) in the studied colloid or a partial oxidation of these GQDs [41] in water. GQDs has an electronic structure that can be tuned by the QD size which is related to changes in the absorption band at 350 nm. We then excite the solution of GQDs under picosecond (ps) laser pulses centered at 355 nm, i.e. very close to the absorption band where the absorption coefficient is measured: $\alpha =2525m^{-1}$, leading to an absorption cross section ${\sigma }^{(1)}=5\times {10}^{-23}{m}^{-2}$. The obtained photoluminescence spectrum at 355 nm excitation is represented in Fig. 1(b) and exhibits a maximum emission in the blue at 486 nm, as expected from the GQDs datasheets [42]. The wavelength width of the photoluminescence emission is about 100 nm at full width at half maximum (FWHM).

 

Fig. 1 (a) Spectral absorbance of 2 mm solution of GQDs at 1 mg/ml concentration in water. The absorption is maximal at 234 nm and 347 nm; (b) Photoluminescence of GQDs excited at 355 nm in the ps regime. The first vertical line is the incident laser wavelength. Maximum emission is found at 486 nm.

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We next characterize the NL optical properties of GQDs, using the fundamental, the second and the third harmonics of a 10 Hz-repetition rate Nd: YAG laser. This corresponds to wavelengths equal to 1064 nm, 532 nm and 355 nm, respectively. The pulse durations (FWHM) are 17 ps, 12 ps and 10 ps respectively. The D4σ-Z-scan technique was applied to characterize the intensity dependent refractive behavior of the GQDs and the standard open-aperture Z-scan technique was used to determine the NL absorption coefficient. All measurements were performed at room temperature with linearly polarized light incident perpendicularly to the sample surface.

The setup used to implement the D4σ−Z-scan is presented in Fig. 2. We recall here briefly the method; more experimental details can be found in [35–38].

 

Fig. 2 Experimental D4σ-Z-scan setup. The sample (NLM) is scanned along the beam direction around the focal plane (z = 0). The labels refer to: lenses (L₁, L₂ and L₃), beam splitters (BS₁ and BS₂), mirrors (M₁ and M₂), Camera (CCD).

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The experimental setup is composed of a 4f-imaging system ($f_1=f_2=20\text{cm}$). The nonlinear sample under investigations is moved along the beam direction in the focus region. An image of the output beam, transmitted by the sample, is then recorded by a CCD camera. As previously shown, the variations in the transmitted beam profile are directly related to the induced NL phase-shift and absorption. We then perform simulations of the propagation of the beam from the object to the image plane based on Fourier optics. We take into account the transmittance of the NL medium by means of the evolution equations of the NL phase-shift and intensity along the propagation using the thin medium approximation. The same numerical procedure is repeated for each sample position defined by the motor step used in the experiment (1 mm). Finally, fits of the data representing the beam waist relative variation (BWRV) spatial profile of the output beam by the corresponding simulations provide the NL refractive index $n_2$ of the considered sample. On the other hand, the NL absorption coefficient C₂ is obtained from the fits of the conventional open-aperture Z-scan profile. Two sets of acquisitions are performed for each measurement. The first set is in the NL regime and the second one in the linear regime, obtained by reducing the intensity of the incident laser; this is necessary to remove from the NL measurements the diffraction, diffusion and/or imperfection contributions due to the inhomogeneity of the sample. Note that we perform absolute measurements, thus avoiding an intensity calibration based on a reference material. The main source of uncertainty comes thus from the absolute measurement of the laser pulse energy. The accuracy of the joulemeter used is about 10% at all considered wavelengths. More experimental details can be found in [35–38].

3. Results and discussion

We assume that the nonlinear response of the material exhibits an effective cubic nonlinearity defined by a NL absorption coefficient, $C_2\left(m/W\right)$, a NL refractive index $n_{2eff}\left(m^2/W\right)$, and we take into account the linear absorption, $\alpha \left(m^{-1}\right)$. In these conditions, the amplitude transmittance of the sample is described by $T\left(z,u,v\right)={\left[1+q\left(z,u,v\right)\right]}^{-1/2}\exp \left[j\Delta {\phi }_{{}_{\text{NL}}}^{\text{eff}}\left(z,u,v\right)\right],$ where z represents the position of the sample in the focus region. $u=x/\lambda f_1$ and $v=y/\lambda f_1$ are the normalized spatial frequencies with λ denoting the incident laser wavelength. $q\left(z,u,v\right)=C_2{L}_{eff}I\left(z,u,v\right)$ where $I(z,u,v)$is the intensity of the laser beam inside the sample and $L_{\text{eff}}=\left(1-e^{-\alpha L}\right)/\alpha$. The NL phase-shift is $\Delta {\phi }_{{}_{\text{NL}}}^{\text{eff}}\left(z,u,v\right)=2\pi n_{2eff}{L}_{eff}{I}_{\text{eff}}\left(z,u,v\right)/\lambda$ where $I_{\text{eff}}$ is the effective intensity seen by the sample: $I_{\text{eff}}\left(z,u,v\right)=I\left(z,u,v\right)\log \left[1+q\left(z,u,v\right)\right]/q\left(z,u,v\right)$. The peak on-axis NL phase-shift at the focus point is $\Delta {\phi }_0=\Delta {\phi }_{{}_{\text{NL}}}^{\text{eff}}(0,0,0)$ which can be deduced from the BWRV signals.

We first characterize the NL response of GQDs in the UV domain at a wavelength of 355 nm, i.e. in the absorption band. The experimental results and the fits are represented in Fig. 3(a) for the NL refraction index and Fig. 3(b) for the NL absorption. The red circles represent the data obtained for a 1 mm thick solution of GQDs in water under laser intensity of 12 GW/cm². The blue squares are the corresponding data for 2 mm thick cell filled with pure deionized water in the same experimental conditions. The thickness is doubled to obtain a sufficient signal for n₂ measurement. A very good agreement between the experimental data and the fittings can be observed for the NL refraction signals (BWRV) and the NL absorption open aperture Z-scan profiles. As expected, we clearly notice the influence of the GQDs in the NL signal. Considering the NL absorption response (Fig. 3(b)), the transmittance of the GQDs increases when the sample is moving close to the focus point, i.e. when the incident light intensity increases. This is undoubtedly the signature of a saturable absorption behavior which is characterized by a negative value of $C_2=\left(-1.4\pm 0.4\right)\times {10}^{-11}m/W$. The NL transmission as a function of increasing light intensity can also be fitted using a parameter called the saturation intensity and defined as the optical intensity required to reduce the absorption to half of its unbleached value. This yields to a saturation intensity equal to $I_S=4.36GW/c{m}^2$, which is comparable to materials such as carbon nanotubes [43] and four orders of magnitude higher than values obtained for graphene films [44] where the saturation absorption strength could be modulated over a wide range by varying the number of graphene layers. As expected no signature of NL absorption is found for water (blue curves in Fig. 3(b)) under the same experimental parameters.

 

Fig. 3 NL responses at 355 nm of 1 mm thick GQDs-water solution (red circles) and 2 mm thick water (blue squares) at the same intensity: 12 GW/cm²; (a) NL refraction; (b) NL absorption. The solid and dashed lines are the numerical fittings.

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Taking into account the contributions of the cell’s wall and the NL medium, the total effective NL phase-shift deduced from the BWRV signal, is written, as: $\Delta {\phi }_0=\Delta {\phi }_c+\Delta {\phi }_{\text{l}}$, where the subscript c stands for the cell in fused silica and l for the NL medium. Therefore, in the case of a significant NL absorption, the effective NL refractive index of the NL medium alone is given by:

We next characterize the NL response of GQDs, far from the absorption band, in the visible and infrared domain at wavelengths of 532 nm and 1064 nm, respectively. At these both wavelengths the NL response of the GQDs is similar to the one of pure water. To illustrate this behavior, an example of the experimental results and numerical simulations is shown in Figs. 4(a) and 4(b) for the infrared wavelength. Therefore, these GQDs have no significant NL response in the visible and infrared domains, i.e. far from the absorption band $\left(C_2<0.01\times {10}^{-11}m/W\right)$.

 

Fig. 4 NL responses at 1064 nm of 2 mm thick GQDs-water solution (red circles) and 2 mm thick pure water (blue squares) at the same intensity: 69 GW/cm²; (a) NL refraction; (b) NL absorption. The solid and dashed lines are the fittings.

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In contrast with the large and broadband nonlinear response of graphene [45], the three-dimensional confinement of excitons in the quantum dots modifies largely the electronic configuration and therefore the NL optical properties of graphene which behave mostly as semiconducting carbon nanotubes [46]. The NL properties of GQDs are therefore found to be significant only in the region close to the absorption band. As the sharp resonances exhibited by the GQDs are directly related to the size of the quantum dots, it is therefore expected that the position of the maximal GQDs NL response can be spectrally shifted, in function of the desired applications.

The NL Kerr index, obtained from the BWRV signal with the empty cell, is $n_{2c}=\left(0.8\pm 0.2\right)\times {10}^{-20}{m}^2/W$ at 532 nm and $n_{2c}=\left(0.65\pm 0.2\right)\times {10}^{-20}{m}^2/W$ at 1064 nm. Considering Eq. (1) in the case of no NL absorption, we obtain the same NL index of refraction for water and for GQDs solution at 532 nm: $n_{2w/GQD}=\left(1.5\pm 0.4\right)\times {10}^{-20}{m}^2/W$. At 1064 nm the results for water and for GQDs in solution are also very close: $n_{2w}=\left(0.65\pm 0.15\right)\times {10}^{-20}{m}^2/W$ and $n_{2GQD}=\left(0.8\pm 0.2\right)\times {10}^{-20}{m}^2/W$, respectively. All the obtained experimental measurements are summarized in Table 1.

Table 1. GQDs in water at a concentration of 1 mg/ml. Average values of the measured NL coefficients at different wavelengths. The third column shows the range of incident intensity. α, n₂ and C₂ denote the linear absorption, the NL refractive index and the NL absorption coefficients respectively.

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These values obtained for the GQD in water consider both contributions of graphene and water. To determine the NL refraction and absorption coefficients of a single NP from which the quantum dots are made of, we next focus our discussion in the UV regime, where the NL response of GQDs is significant. The thin sample approximation allows us to neglect the diffraction of the laser beam inside the sample. The path of the laser light through the NL medium can therefore be considered roughly as a cylinder of volume: $V=S_{beam}L=\pi {\omega }_{0f}^{2}L$, where ω_0f is the beam waist of the laser in the focus plane and S_beam is the surface of the incident beam (Fig. 5). Using the results of the generalized Maxwell Garnett model [47] in the range of the very low nanoparticle filling factor ($f\ll 1$) and considering the value of the host dielectric constant of water ($n\approx 1.3$) lower than that of the GQD ($n\approx 2$), one can observe that the effective NL susceptibility of the colloid ${\chi }_{eff}^{\left(3\right)}$ varies linearly with f: Eq. (3) in [18] reduces to:

 

Fig. 5 The light beam in the cell with the different geometrical parameters considered for the calculation of the total phase shift. Inside the beam the GQDs are considered stacked, side by side, filling totally a cylinder having the same section than the beam with a thickness $L_{GQD}$. Above and below the beam, the same number of GQDs but randomly distributed.

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To determine the nonlinear refraction coefficient of a single graphene dot, neglecting the phase shift of the water, one could consider again that the total NL phase-shift $\Delta {\phi }_T$ induced in the sample over a length L by the incident light is equivalent to that obtained by the sum of the NL phase-shifts induced in every NP over $L_{GQD}$, where the GQDs are stacked and uniformly distributed side by side to recover the total surface of the beam (see Fig. 5). Note that $L_{GQD}=N_{layer}{L}_{1layer}$, with $N_{layer}$ being the number of layers that would constitute the cylinder shown in Fig. 5 taking into account the concentration of the GQDs in water and $L_{1layer}$ representing the thickness of the monolayer of graphene. Then, the measured n₂ at line 4 of Table 1 (in the UV) is related to the NL index coefficient of a monolayer $n_{2,1lay}$ by the following:

We finally open the discussion on the NL response of the GQDs considered in this study and the comparison of the results with those found in the literature. At 355 nm in the picosecond regime, we are the first to our knowledge to make these measurements. But at other wavelengths one must be very careful when synthetizing the results. Observing the absorption spectrum of the measured solutions we notice a notable difference between the products used even if the denomination, pulse duration and the concentration are sometimes equivalent. For example, in our case we do not find two-photon absorption (2PA) when the material is irradiated at 532 nm whereas in the reference [20] the authors found a strong NL response at the same wavelength and in the same picosecond regime. The absorption spectrum shown in Fig. 1(a) shows that in our case the 2PA just falls between the two absorbing peaks at 234 nm and 347 nm (in a transmission window). Whereas in the reference [20] considering the continuous and highly absorbent spectrum in the UV range, the presence of 2PA is justified. Other measurements were made by the same group on graphene oxide (GO) preparations always in the same picosecond regime and at the same wavelengths (532 nm and 1064 nm) [24]. In all cases the colloids showed high NL absorption and low NL refraction. Large GO sheets of micrometer range were present in the characterized colloids. But if one look again to the absorption spectra of the elaborated products, we always notice a high absorption at 266 nm where the 2PA phenomenon appears. Finally, in reference [29] where the optical limiting properties of graphene families, including graphene oxide nanoribbons, were investigated at 532 and 1064 nm, not only the difference in the results could be explained by a longer impulse regime (nanosecond) with higher thermal effects, but also the absorption spectrum of the compositions still presents a flat continuous decreasing behavior in the UV band with a relatively small peak around 230 nm. In summary, it is necessary to be very careful in the NL characterization domain when the materials are elaborated in different ways and in different labs which arises the difficulty of comparison to the complexity of the NL phenomena depending on too many parameters.

Furthermore, we consider that the electronic band can be schematically represented by a three-level model (see Fig. 6.3.1 in [50]). When the GQDs are under ultraviolet laser light illumination, the electronic excitation pathway is the following: due to ultraviolet absorption, electrons pass from the ground state to the upper exited state, where the lifetime is long. Therefore, at sufficiently high incident light intensity, the rate of excitation is such that the ground state becomes depleted, and the absorption subsequently saturates. Then, fast phonons non-radiative relaxation occurs, and electrons pass at a lower level from where they decay, in a time interval of the nanosecond, i.e. larger than the laser pulse duration [19,39,51], giving rise to the phenomenon of blue fluorescence. Finally, note that the signs of the NL coefficients measured in this paper agree with Eqs. (2.14 a)-(2.14 b) given in [52] where these equations are analytically formulated for the nonlinearity near and on the one and two-photon resonances, assuming a two-level system.

4. Conclusions

We have characterized, for the first time to our knowledge, the NL response of GQDs suspension in water in the UV domain. We have observed that the Kerr refractive index and the nonlinear absorption are significant only at 355 nm, very close to the linear absorption band. No NL response is present in the visible and infrared regions. At 355 nm, we have brought out a clear saturable absorption effect, $C_2=\left(-1.4\pm 0.4\right)\times {10}^{-11}m/W$, and the nonlinear refractive index is $n_{2GQD}=\left(5.7\pm 1.2\right)\times {10}^{-19}{m}^2/W$ ($C=1mg/ml$). Thanks to a simple model, we have estimated the nonlinear refractive index and nonlinear absorption values of a single layer graphene NP constituting the GQDs,$n_{2,1lay}=5.3\times {10}^{-16}{m}^2/W$ and $C_{2,1lay}=-1.3\times {10}^{-8}m/W$, respectively. The obtained Kerr index value is in the order of magnitude of bulk materials. More interestingly, it corresponds to a focusing effect and it is three orders of magnitude lower than that found in the literature for monolayer graphene in the infrared domain. We expect this first study can foster several others considering an applicative point of view for the functionalization of the GQDs structures in optoelectronic devices.