1. Introduction

Graphene, a new two-dimensional carbon nanomaterial with sp²-hybridized carbon atoms, has received tremendous interest in recent years owing to its various remarkable properties and potential applications in electronics and photonics [1–3]. Graphene oxide (GO) can be synthesized by the oxidation of graphite, the presence of oxygen-containing groups in GO sheet makes it strongly hydrophilic and water soluble. GO sheets contain a mixture of electronically conducting sp²-hybridized carbon domains and insulating sp³-hybridized carbon matrix, with the heterogeneous atomic and electronic structure. Since materials with large π-conjugated structures usually show strong nonlinear optical properties, graphene has exhibited saturable absorption (SA) due to the large sp²-hybridized carbon π-conjugated and zero energy gap structure [4–6], while GO has showed SA at low intensity, and photoinduced absorption (PA) including two photon absorption (TPA) at high intensity due to the coexistence of sp² carbon domains and sp³ matrix in GO [7–9]. Nonlinear scattering (NLS) was also observed in graphene and GO dispersions in nanosecond time regime due to the heating of carbon atoms [10–12]. The excellent nonlinear absorption (NLA) and NLS properties enable their potential applications in saturable absorbers, optical switching and optical limiter.

Besides NLA and NLS processes, nonlinear refraction (NLR) can also plays an important role in the applications of optical switching and optical limiter, and NLR of graphene has attracted great interest because of its large sp²-hybridized carbon π-conjugated structure. Recently, Wu et al reported the spatial self-phase modulation in graphene dispersions arising from the NLR of graphene [13]. Bourlinos et al reported the significant third-order nonlinear optical and NLR response of graphene fluoride in aqueous dispersions [14]. Zhang et al reported Z-scan measurements of the nonlinear refractive index of loosely stacked graphene, and the results showed that graphene possesses a giant nonlinear refractive index [15]. However, the reports about NLR properties of GO with sp² carbon domains and sp³ matrix and the relations between NLR and NLA of GO in the regime from nanosecond to femtosecond are few [16].

In this paper, the NLR properties of GO sheets in N, N-Dimethylformamide (DMF) were studied in nanosecond, picosecond and femtosecond regimes, Thick quartz cell and small beam waist radius at focus were used to induce and enhance the NLR signal for picosecond and nanosecond pulses. Results show that the dispersion of GO in DMF exhibits negative NLR properties, mainly arising from transient thermal effect in nanosecond regime. In picosecond and femtosecond regime, the dispersion exhibits negative NLR properties arising from sp² carbon domains and sp³ matrix of GO. To illustrate the relations between NLR and NLA, we also studied the NLA properties of the dispersion, results show that the transition from SA behaviors to reverse saturable absorption (RSA) of GO dispersion (also GO sheets) occurred as the energy (intensity) of input pulse increases, while the dispersion keeps the negative NLR behaviors in nanosecond, picosecond and femtosecond time regimes.

2. Experiments

GO was prepared by the modified Hummers method [17, 18] and was dispersed in DMF, the presence of oxygen-containing groups in GO makes it dispersed in DMF very well. Nonlinear optical (NLO) properties of GO were measured by using Z-scan technique [19]. In Z-scan measurements, a Q-switched Nd:YAG laser (Continuum Surelite-II), a mode-locked Nd:YAG laser (Continuum model PY61) and a mode-locked Ti: sapphire laser (Spitfire pro, Spectra Physics) laser were used to generate 4.8~10.7 ns (FWHM) nanosecond pulses at 532 nm, 35 ps (FWHM) pulses at 532 nm and 120 fs (FWHM) pulses at 800 nm, respectively. The repetition rates of nanosecond and picosecond pulses are 10 Hz, and that of femtosecond pulses is 1 kHz. For nanosecond pulses Z-scan experiments, laser beam was focused by a 15-cm and 25-cm focal length lens and the beam waist radius ω₀ at focus were 12 µm and 22 µm, respectively. For picosecond pulses Z-scan experiments, the beam waist radius is 16 µm. For femtosecond pulses Z-scan experiments, the beam waist radius is 31µm. The sample was filled in a 5-mm thick quartz cell for nanosecond and picosecond Z-scan experiments, while the sample was filled in a 2-mm thick quartz cell for femtosecond pulses experiments. The concentration of GO used in nanosecond and picosecond pulses Z-scan experiments is 0.5 mg/mL corresponding to a linear absorption coefficient α₀ of 4.41 cm⁻¹ at 532 nm, the concentration is 1 mg/mL corresponding to a linear absorption coefficient α₀ of 2.95 cm⁻¹ at 800 nm for femtosecond pulses experiments. The NLA properties of the samples was performed by open-aperture Z-scan experiments, all NLR Z-scan curves of the samples were obtained from closed-aperture Z-scan curves divided by corresponding open-aperture Z-scan curves.

It should be pointed out that the NLR signals of GO sheets dispersion filled in thin quartz cell were too weak and the signal to noise is quite low for nanosecond and picosecond pulses, even with the maximum concentration of GO of 1 mg/mL, so we have to use thick quartz cell with long pathlength to enhance NLR response [20], use small beam waist radius at focus to increase the input pulse intensity, and then a 5-mm thick quartz cell and a moderate concentration of GO of 0.5 mg/mL were selected. For femtosecond pulses, nonlinear scattering and damage of sample can take place easily for sample in thick quartz cell due to the long pathlength and high pulse intensity [21–23]. To enhance the NLR response and avoid the nonlinear scattering and damage of sample, a moderate quartz cell of 2 mm-thick and the maximum concentration of GO of 1 mg/mL were selected. All the pulse intensity was controlled to be lower than the nonlinear scattering and damage threshold in the three regimes. Although the linear absorption coefficient α₀ of the dispersion is 4.41cm⁻¹ for nanosecond and picosecond pulses experiments at 532 nm, while α₀ is 2.95 cm⁻¹ for femtosecond pulse experiments at 800 nm, the differences in cell thickness, the concentration of GO and the value of α₀ in the three regimes experiments did not cause confusions for our analysis of the nonlinear optical properties and nonlinear optical mechanisms of GO sheets.

3. Results and discussion

Figure 1 gives the UV-visible absorption spectrum of GO in DMF and the absorption of DMF has been subtracted. Unlike common semiconductor with an optical absorption edge, the GO sheets show a maximum absorption at around 265 nm, and tailing to 1000 nm. The broadband absorption shown in Fig. 1 and the reported broadband fluorescence suggest a dispersion of energy gaps of GO [24, 25]. Since sp² carbon domains and sp³ matrix are mixed and coexist in GO sheet as shown in the inset of Fig. 1, so the absorption spectrum of GO sheets include the absorption of sp² carbon domains and sp³ matrix.

 

Fig. 1 Absorption spectra of GO in DMF. Inset shows the structure of GO.

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Figures 2(a) -2(d) show the open-aperture Z-scan curves and NLR Z-scan curves of GO dispersion with nanosecond pulses. As shown in Fig. 2(a), GO dispersion shows SA at lower input pulse energy (intensity), with larger pulse energy, RSA behavior occurred near focus and became dominant as pulse energy increases. Meanwhile, the NLR Z-scan curves of the dispersions are characterized by a prefocal peak followed by a valley as shown in Fig. 2(b), which indicates the negative NLR or self defocusing properties for different input pulse energy.

 

Fig. 2 Open-aperture Z-scan curves (a) and NLR Z-scan curves (b) of the dispersion of GO in DMF for different input pulse energy with the same pulsewidth τ_p of 4.8 ns. Open-aperture Z-scan curves (c) and NLR Z-scan curves (d) of the dispersion of GO in DMF for different τ_p with the same pulse energy of 5.75µJ (2.54 J/cm²). △T_p-v (e) and n_2eff (f) of dispersions for different τ_p and input pulse energy; Solid lines are theoretical fits. ω₀ = 12µm for (a)- (f).

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From Figs. 2(b) and 2(e) we can also see that the difference between normalized transmittance peak and valley ($\Delta T_{P-V}=T_{Peak}-T_{Valley}$) increases as the pulse energy increases. For example, the value of $\Delta T_{P-V}$increases from 0.22 to 0.58 as pulse energy increases from 1.90 µJ (corresponds the input fluence at focus of F₀ = 0.84 J/cm²) to 8.15 µJ (F₀ = 3.60 J/cm²) under the same pulsewidth τ_p of 4.8 ns. As shown in Figs. 2(c) and 2(d) the RSA behavior of the dispersion keeps unchanged as τ_p increases with the same input pulse energy (or F₀) at focus. However, the NLR was enhanced for longer pulsewidth. For example, the value of $\Delta T_{P-V}$ increases from 0.44 to 1.10 as τ_p increases from 4.8 to 10.7 ns with the same pulse energy of 5.75 µJ (F₀ = 2.54 J/cm²).

Different mechanisms exist for NLR, such as electronic polarization, molecular reorientation effects, population redistribution (or excited states refraction), free carrier refraction, and thermal effect [26]. In the case of nanosecond pulses, considering the strong linear absorption of the pulse energy for the dispersion, the thermal effect should be generally taken into account for the measurements of NLR. The thermal effect arises from acoustic wave propagation caused by medium density change after local heating, and its buildup time τ_ac is determined by the time required for a sound wave to propagate across beam size and τ_ac = ω₀/υ_s, where ω₀ is the beam waist radius and υ_s is the velocity of sound in the medium. Velocity of sound in DMF is υ_s = 1439 m/s [27]. For ω₀ = 12µm, the buildup time of the thermally induced optical nonlinearities τ_ac is about 8.3 ns. Since τ_ac is comparable to τ_p and the factor τ_p/τ_ac is smaller than 1.6, so thermally induced optical nonlinearities is in the transient regime, and acoustic and electromagnetic wave equations must be solved simultaneously in the transient regime while the nonlinear process is simulated numerically [28].

To evaluate the NLR coefficient, we fit the experimental data only by solving the propagation equation of electric field envelope E as Ref. 29, The NLA coefficient and change of refraction index were written as $\alpha (I)={\alpha }_0/{(1+I/I_S)}^{0.5}+{\beta }_{eff}I$ and $\Delta n(I)=n_2{}_{eff}I$, respectively [12]. Where $I$is the laser radiation intensity,$I_S$ is saturable intensity,${\beta }_{eff}$ is the effective TPA coefficient. Here an effective NLR coefficient n_2eff was used to simplify the multiple nonlinear refraction processes and n_2eff values can be obtained by theoretical fitting. As shown in Fig. 2(f), we can see that n_2eff is negative and the value of |n_2eff| increases from $2.50\times {10}^{-13}$to$1.70\times {10}^{-12}$cm²/W as τ_p increases from 4.8 to 10.7 ns with the same pulse energy of 5.75 µJ (F₀ = 2.54 J/cm²). As the pulse energy increases, the value of |n_2eff| decreases. For example, the |n_2eff| value decreases from $3.52\times {10}^{-13}$ to $2.32\times {10}^{-13}$ cm²/W as input pulse energy from 1.90 µJ (F₀ = 0.84 J/cm²) to 8.15 µJ (F₀ = 3.60 J/cm²) with the same pulsewidth τ_p of 4.8 ns. Decreasing |n_2eff| values with increasing input pulse energy indicates that the NLR responses are not arising from intrinsic third-order nonlinear optical response. This phenomenon was also studied in Ref. 30.

The NLR of nigrosine in DMF was also measured, we found that the NLR response and the value of |n_2eff| for the dispersion of GO in DMF were comparable with that for the solution of nigrosine in DMF under the same condition. Since the NLR of nigrosine solution mainly originate from thermal effect [31] and no nonlinear optical signal for DMF was observed in our nanosecond experiments, so the strong pulsewidth influence on NLR indicates the thermal effect exists in the dispersion in nanosecond regime [32]. To further confirm the thermal origin of the NLR, we also measured the NLR of GO under the conditions of two different beam waist radius ω₀ of 12 µm and 22 µm with the same input fluence F₀ at focus. As shown in Figs. 3(a) and 3(b), when ω₀ changes from 12 µm to 22 µm, τ_ac increases from 8.3 to 15.3 ns. For τ_p = 10.7 ns, the value of |n_2eff| decreases from $2.26\times {10}^{-12}$cm²/W ($\Delta T_{P-V}=0.76$) to $8.80\times {10}^{-13}$ cm²/W ($\Delta T_{P-V}=0.54$), with a decreased factor of 2.57 (the ratio of $\Delta T_{P-V}$is 1.41), while for τ_p = 4.8 ns, the value decreases from 2.5 × 10⁻¹⁴ cm²/W ($\Delta T_{P-V}=0.42$) to $1.3\times {10}^{-14}$ cm²/W ($\Delta T_{P-V}=0.4$), with a decreased factor of 1.92 (1.05). So |n_2eff| (or $\Delta T_{P-V}$) decreases greatly for larger beam waist, which also indicates the existence of thermal effect [31].

 

Fig. 3 NLR Z-scan curves of the dispersion of GO in DMF for two different beam waist radius ω₀ for τ_p = 4.8 ns, F₀ = 2.54 J/cm². (b) τ_p = 10.7 ns, F₀ = 1.22 J/cm². Solid lines are theoretical fits.

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SA of the dispersion is attributed to band filling effect of sp² domains and RSA can be mainly arising from TPA of sp³ matrix of GO sheets [7, 8, 12]. By fitting, $I_S$was obtained to be ~10⁷ W/cm², ${\beta }_{eff}$to be 3~$11.5\times {10}^{-8}$ cm/W, as τ_p increases from 4.8 to 10.7 ns with the same pulse energy of 5.75 µJ (F₀ = 2.54 J/cm²), while ${\beta }_{eff}$is near a constant as pulse energy increases with the same τ_p.

Since GO exhibited SA arising from the sp² domains, and RSA from the sp³ matrix in nanosecond, picosecond and femtosecond regime, respectively [7–9, 12], NLR from π electrons and free carriers of sp² carbon domains, bound electrons and free carriers of sp³ matrix may be involved, similar to the case in some semiconductors [33], so NLR response from GO sheets should not be excluded. However, it is difficult to observe the NLR of GO sheets in nanosecond regime due to strong thermal effect.

To measure the intrinsic NLR of GO sheets, shorter pulses (35 ps) were used. For ω₀ = 16 µm, τ_ac = 11.1ns, τ_p/τ_ac = 0.003, thus thermal effect can be ignored for the 35 ps pulse. From Figs. 4(a) and 4(b) we can see that GO dispersion shows SA at lower intensity (9.0 GW/cm²), with further increasing in the intensity, RSA takes place near focus. Since no NLA response of DMF was observed during the open-aperture Z-scan experiments, the SA and RSA behavior should be arising from GO sheets in the dispersion. The fits were obtained by using $I_S=$3.2GW/cm² and ${\beta }_{eff}~3.7\times {10}^{-10}$cm/W.

 

Fig. 4 For 35 ps pulse at 532nm: (a) Open-aperture Z-scan curves of the dispersion of GO in DMF (GO + DMF). (c) NLR Z-scan curves of the dispersion of GO in DMF (GO + DMF) and DMF, solid icons stand for the dispersion (GO + DMF), hollow icons stand for the solvent of DMF. Solid lines are theoretical fits. (b) Effective TPA coefficient β_eff of GO as a function of incident intensity. (d) Effective NLR coefficient n_2eff of DMF, the dispersion and GO sheets as functions of incident intensity.

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For NLR experiments, the NLR properties of solvent DMF cannot be ignored due to the strong peak on-axis intensity $I_0$ in the sample under picosecond pulse, so the refraction index change Δn (dispersion) for the dispersion can be expressed as $\Delta n$(dispersion)$=\Delta n$(GO)$+\Delta n$(DMF), where Δn (GO) is the refraction index change of GO sheets and Δn (DMF) is that of the solvent DMF. Results show that DMF has an intensity-independent positive NLR coefficient n₂ of $1.8\times {10}^{-15}$ cm²/W, while GO dispersion (GO + DMF) show negative NLR behaviors arising from GO sheets. As shown in Figs. 4(c) and 4(d), we found that n_2eff is negative and the value of |n_2eff| of dispersion decreases from $3.5\times {10}^{-15}$ cm²/W to $1.4\times {10}^{-15}$ cm²/W, as the intensity increases from 9.0 GW/cm² to 36.7 GW/cm². The value of |n_2eff| of GO sheets from calculation decreases from $5.3\times {10}^{-15}$ cm²/W to $3.2\times {10}^{-15}$cm²/W. Comparing the open-aperture Z-scan curves in Fig. 4(a) with NLR curves in Fig. 4(c), we note that GO dispersion (and then GO sheets) keeps the negative NLR behaviors though the SA behavior to RSA transition.

Figures 5(a) and 5(c) show the open-aperture Z-scan and NLR curves of the dispersion of GO in DMF under femtosecond pulse laser at 800 nm with different intensity. The results are similar to the case of picosecond experiments although the laser wavelength and pulse width are different. GO dispersion exhibits SA at lower intensity (82.1 GW/cm²), with further increase in the intensity, RSA occurs near focus. $I_S=$17.5 GW/cm² and β_eff was about $2.5\times {10}^{-11}$ cm/W from fitting, as shown in Fig. 5(b). In Fig. 5(d) the results of fits show that DMF has an intensity-independent positive NLR coefficient n₂ of $6.3\times {10}^{-16}$cm²/W. There is a basic agreement with n₂ of DMF in Ref. 21 taking into account the NLR of the quartz cell. We also can see that GO dispersion (GO + DMF) keeps the negative NLR behaviors and the value of |n_2eff| decreases from $5.3\times {10}^{-16}$ cm²/W to $4.3\times {10}^{-16}$ cm²/W, the calculated value of n_2eff of GO sheets nearly keeps a constant of $-1.1\times {10}^{-15}$cm²/W, as intensity increases from 82.1 GW/cm² to 224 GW/cm². We also should note that GO dispersion (and then GO sheets) keeps the negative NLR behaviors though the SA behavior to RSA transition, similar to the case in picosecond time regime.

 

Fig. 5 For 120 fs pulse at 800 nm: (a) Open-aperture Z-scan curves of the dispersion of GO in DMF (GO + DMF). (c) NLR Z-scan curves of the dispersion of GO in DMF (GO + DMF) and DMF, solid icons stand for the dispersion (GO + DMF), hollow icons stand for the solvent of DMF. Solid lines are theoretical fits. (b) Effective TPA coefficient β_eff of GO as a function of incident intensity. (d) Effective NLR coefficient n_2eff of DMF, the dispersion and GO sheets as functions of incident intensity.

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In femtosecond time regime, electronic polarization, population redistribution (or excited states refraction), free carrier refraction would contribute to the NLR of GO sheets. However, there may be additional molecular reorientation contribution in picosecond and nanosecond time regime. For GO sheets, since the negative NLR were accompanied by SA and RSA in picosecond and femtosecond time regime, the mechanism of NLR from GO sheets can be explained by the schematic drawing shown in Fig. 6 . As shown in Fig. 6, the sp² domain with a diameter of ~3 nm, has an narrow energy gaps of ~0.5eV [24,34–36], the optical absorption of electrons in sp² domains can be saturable easily and SA occur due to valence depletion and conduction band filling. Meanwhile, the π electrons and free carrier refraction in the conduction band of sp² domains may contribute to NLR properties.

 

Fig. 6 A schematic drawing of NLA and NLR arising from sp² domains and sp³ matrix of GO sheets.

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For sp³ matrix, the energy gaps are large (typically 2.7~3.1 eV) [37], when the intensity of laser pulse increases, the bound electrons in the valence band in sp³ matrix will transit to conduction band and become free carriers through TPA mechanism for 532nm or 800 nm pulse laser. Since the relaxation time of free carriers are in the order of hundreds of femtosecond, and a few picoseconds to scores of picoseconds [8], the bound electron NLR in the valence band and free carrier refraction in the conduction band of sp³ matrix will also contribute to the NLR response of GO sheets. In GO sheets, sp² domains and sp³ matrix are coexistence, so SA, TPA and possible free carrier absorption (FCA) are coexistence in a GO sheet. In femtosecond time regime, the NLR from π electrons, bound electrons in the valence band and free carriers in the conduction band will contribute to the NLR of the whole GO sheets. What’ more, in the time regime of picosecond or longer, molecular reorientation effects of GO sheets may be also involved. While in nanosecond time regime, the contribution of these processes can be neglected because of the dominant transient thermal effect.

It should be pointed out that there are many differences in NLR mechanisms between the graphene layer on a substrate, graphene sheets in solution dispersions and GO sheets in dispersions. For single layer or few-layer graphene on a substrate, the NLR from π electrons is dominant and NLR are not affected by cumulative thermal effect due to the very high thermal diffusion coefficient of graphene [15]. For graphene sheets in solution dispersions, both π electrons and the reorientation and alignment of the graphene sheets can contribute to the NLR [13, 14], thermal effect from solution dispersions can also contribute to NLR under continuous wave or long pulse laser. For GO sheets in dispersions, free carriers, π electrons and bound electrons of sp² domains and sp³ matrix can contribute to the NLR. Reorientation effects of the GO sheets and thermal effect should be considered according to the duration of pulse.

Moreover, unlike the case of single layer or few-layer graphene on a substrate, the NLR response and n_2eff value of graphene sheet or GO sheets dispersions is the sum of NLR signals from a large amount of sheets interacting with laser pulse. The NLR response or n_2eff value of single graphene sheet or GO sheet cannot be obtained accurately, only can be obtained by roughly qualitative estimation [13], because the sheets are dispersed and the sheets sizes are not uniform. So the values of n_2eff (or n₂) in the three cases, cannot be compared directly, because the duration of laser pulse and wavelength, the samples and the mechanisms contributing to NLR are different.

4. Conclusions

In summary, the NLR properties of GO in DMF were studied in nanosecond, picosecond and femtosecond time regimes by Z-scan technique, NLA properties related to NLR properties were also studied. Results show that the dispersion of GO in DMF exhibit negative NLR properties, which is mainly attributed to transient thermal effect in nanosecond time regime. GO dispersions also exhibits negative NLR in picosecond and femtosecond time regime, which may be mainly arising from electronic polarization and possible free carrier refraction of sp² domains and sp³ matrix of GO sheets. GO dispersions and GO sheets keep negative NLR behaviors although the SA behaviors to RSA transition as pulse energy (intensity) increases in nanosecond, picosecond and femtosecond regimes, this makes GO materials promising candidates for practical applications as broadband and broad domain saturable absorbers, optical switching and optical limiter.