Abstract
Graphene’s giant nonlinear optical response along with its integrability has made it a vaunted material for onchip photonics. Despite a multitude of studies confirming its strong nonlinearity, there is a lack of reports examining the fundamental processes that govern the response. Addressing this gap in knowledge we analyse the role of experimental parameters by systematically measuring the nearinfrared spectral dependence, the subpicosecond temporal evolution and pulsewidth dependence of the effective Kerr coefficient (n_{2},_{eff}) of graphene in hundreds of femtosecond regime. The spectral dependence measured using the Zscan technique is corroborated by a density matrix quantum theory formulation to extract a n_{2},_{eff} ∝ λ^{2} dependence. The temporal evolution obtained using the timeresolved Zscan measurement shows the nonlinearity peaking at zero delay time and relaxing on a timescale of carrier relaxation. The dependence of the n_{2},_{eff} on pulse duration is obtained by expanding the input pulse using a prismpair setup. Our results provide an avenue for controllable tunability of the nonlinear response in graphene, which is limited in silicon photonics.
Introduction
The Dirac band structure of graphene^{1,2} has endowed this atomically thin twodimensional material with exceptional optical and transport properties. As a scaleinvariant host of chiral carriers^{3}, graphene exhibits a universal optical response in absorption, with a single sheet capable of absorbing ~2.3% of normally incident light. Its reduced dimensionality together with the peculiar Dirac type dynamics of the quasiparticles enable graphene to have a relatively strong nonlinear optical response in the presence of intense electromagnetic illuminations^{4,5}. Intrigued by its potential applications in photonics and optoelectronics several theoretical and experimental studies of the nonlinear optical properties of graphene have been carried out^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22}. Due to the centrosymmetric crystalline structure of graphene, evenorder nonlinearities are forbidden, and the first nonlinear contribution is a third order effect. In particular, the Kerr type nonlinearity is dominantly attributable to the third order nonlinear interactions in graphene. The refractive index, n, considering the first nonlinear term is intensity dependent and is given by the expression n = n_{0} + n_{2}I, where n_{0} is the linear refractive index, n_{2} is the Kerr coefficient, and I is incident intensity. This Kerrtype nonlinearity in graphene has been reported to be large, confirmed by studies characterising frequency mixing^{4,6,7}, harmonic generation^{8}, selfphase modulation^{9,10}, and selfrefraction^{5,11,12,13,14,15}.
Several theoretical works have attempted to provide a cohesive theory for the third order nonlinearity in graphene, addressing the magnitude^{3,16,17,18} of the nonlinear refractive index (\({n}_{2}\)), the spectral^{3,4,16} and temporal^{19,20} dependence, and effects of Fermi energy modulation^{3,17,18,21,22}. Although topological anomalies in graphene hinder the adaption of perturbation theory in the treatment of the optical response of graphene^{3,23}, fortunately, over the optical wavelength range, nonlinear response coefficients can still describe interactions of photons. Perturbative treatment of the nonlinear optical response of graphene yields explicit expressions for the higher orders conductivity tensors. Throughout this work we refer to the third order frequency mixing conductivity as \({\sigma }^{(3)}({\omega }_{p},{\omega }_{q},{\omega }_{r})\) where \({\omega }_{p}\), \({\omega }_{q}\) and \({\omega }_{r}\) are the frequencies of the photons undergoing nonlinear interaction through the graphene monolayer. The conductivity is proportional to the Kerr coefficient through the third order susceptibility. The studies show large tunability in \({n}_{2}\) through Fermi level modulation. Other factors such as the spectral and temporal dependence have seen relatively less explicit theoretical discussion.
The experimental studies probing the Kerrnonlinearity in graphene have shown variation in n_{2} spanning six order of magnitudes (10^{−12} to 10^{−7} cm^{2}/W)^{4,5,11,14}. A point of contention while comparing these results are the varying spectral and temporal properties of the excitation source, sample preparation techniques, and substrate material. The measurements are performed with the excitation wavelength ranging from the visible to the midinfrared, while the pulse duration ranges from hundred femtoseconds to tens of picoseconds, under different sample preparation methods^{24}. It has been recognised that the n_{2} depends on the wavelength, pulse duration and power in semiconductor systems^{25}. Particularly in graphene, the n_{2} expected to become larger with longer wavelengths^{4}. This trend has been observed and reported thereafter in the near^{15} and midinfrared^{12} range, while spectral dependence in other regions remains experimentally unexplored. It has also been shown in graphene oxide, that the nonlinear parameters of the system change when excited with nanosecond and picosecond pulses^{26}. The decrease of n_{2} with increasing incident intensity due to saturation effects has been observed and reported previously in several studies^{5,12,15,27}. Raman studies have shown ptype doping occurs when chemical vapour deposition (CVD) graphene is transferred onto quartz^{24}. Considering these factors, it is more apt to consider the measured nonlinear parameters in conjunction with the experimental conditions. This makes it imperative to understand the dependence of these factors to gain a fundamental understanding of the governing processes.
In this study we report a systematic analysis of the nearinfrared spectral dependence, the subpicosecond temporal evolution and pulsewidth dependence of the effective Kerr coefficient (n_{2,eff}) of graphene in hundreds of femtosecond regime using our pumpprobe integrated Zscan (PPZS) setup. We refer to the nonlinear refractive index (n_{2}) as n_{2,eff} because we find that the nonlinearity in graphene comprises of the conventional Kerrtype nonlinearity and a component stemming from graphene’s unique structure. In addition to that, our setup allowed for temporal measurements, unravelling the evolution of the nonlinearity in a pumpprobe style measurement to reveal an ultrafast nonlinearity. Thus, this study provides a foundation for the development of predictable models imperative for device design, allowing for large controllable tunability of the nonlinear response of graphene.
Originally introduced by SheikBahae et al.^{28}, the Zscan technique proved to be an experimentally facile yet sensitive method to extract the phase and magnitude of the Kerr coefficient. The optically induced selfrefraction is quantified by relating the phase modulation of the traversing beam to the transmittance in the farfield in Closed Aperture (CA) configuration, while the Open Aperture (OA) configuration captures the effect of absorption. This technique was modified by Wang et al.^{29} to extract the temporal evolution through the integration of a secondary time delayed beam to obtain the timeresolved Zscan measurement. The PPZS setup is schematically illustrated and further elucidated upon in the Methods and Materials section. The setup can be operated in multiple modes of measurement. In the singlebeam mode, the setup is a standard Zscan measurement. In dualbeam mode, with the addition of a lockin amplifier, chopper and crosspolarisation filtering, the setup can be used to perform a standard pumpprobe measurement for the extraction of relaxation time constants. When used in conjunction with the Zscan components, the dualbeam mode is used for the timeresolved Zscan measurements.
Results and Discussion
Spectraldependence
In order to verify the quality and monolayer nature of the samples, the CVD fabricated graphene on quartz (GoQ) samples (1 × 1 cm^{2}) were characterised by Raman spectroscopy, shown in Fig. 1a. The Gband, 2Dband and Dband peaks appear at 1576 cm^{−1}, 2661 cm^{−1} and 1328 cm^{−1}, respectively. The relative intensities and widths of the G and 2D peaks confirm that the sample is single layer. We also perform a temporal correlation measurement on the sample, shown in Fig. 1b, to obtain a τ_{1} relaxation time constant of about ~113 fs. The methodologies and results are further discussed in the Methods and Materials section.
The spectral dependence of n_{2,eff} is obtained in the singlebeam Zscan mode and is measured by scanning the 100 fs excitation source from 800 to 1050 nm with 50 nm increments. The range of the excitation is limited by tunability of the laser (690–1050 nm). Since the measurement is sensitive to the beam quality, the beam is characterised using a knife edge beam profiler for quality and stability over the period of the measurement. To ensure that the Zscan modulation observed in the farfield is originating from graphene, a bare quartz substrate is measured at 900 nm at high onaxis irradiance to show no effect (see Supporting Information Fig. S5). The onaxis irradiance ranges from 1.5 to 5 GW/cm^{2}, limited by the available power of the laser at each wavelength. A standard acquired Zscan trace showing the OA, CA and Fit is shown in Fig. 2a. Our study focuses on the nonlinear refraction in graphene and not the nonlinear or saturable absorption. The OA trace is simply acquired for the normalisation of the CA trace for absorption effects, and presented for illustrative purposes to explain the quantum theory as presented shortly. The effects of saturation resulting in decrease of n_{2,eff} with increasing irradiance is observed and shown at 950 nm in Fig. S6 in supporting information.
The cumulative results of the spectral dependence are presented in Fig. 2b along with the quantum theory based fit. A full set of Zscan fits from which these results are compiled is also provided for reference (Fig. S7 in supporting information). The values for n_{2,eff} in this regime are measured to be positive and range from 9.07 × 10^{−9} to 1.76 × 10^{−8} cm^{2}/W in this excitation range which fits well with our quantum model with relaxation coefficients of γ_{1} ≈ γ_{2} ≈ 10 meV. The fitting parameters, γ_{1/2}, are obtained from the time constants derived from the temporal crosscorrelation measurement, shown in Fig. 1b. However, the pulse width of the pump and probes signals pose some uncertainties on the obtained decay constant of τ_{1} ~ 113 fs (see Methods and Materials) as the correlated pulses are of similar pulse durations. This uncertainty is accounted for in our theoretical model and is illustrated by the shaded (yellow) region in Fig. 2b. Shorter wavelengths are observed to have a n_{2,eff} value that is lower as compared to longer wavelengths, with approximately the same onaxis irradiance. Our calculation suggests a quadratic dependence of n_{2,eff} on λ which agrees with the experimental results. It is noted that n_{2,eff} exhibits negligible dependence on the Fermi level for a lowdoped graphene monolayer. In our case n_{2,eff} is dominated by contributions from interband transitions and since the λ^{2} dependence is a direct consequence of the linear band diagram around the Dirac point, the most relevant transition occur at the zero detuning region where ω = 2kv_{F}. Detuning is defined as \({{\rm{\Delta }}}_{k}=\hslash \omega { {\mathcal E} }_{cv}\), where ω is the frequency of the excitation photon and \({ {\mathcal E} }_{cv}\) is the energy of the transition, see Fig. 3a. The theoretical model employed in our theory uses semiconductor Bloch equations (SBEs) to describe the cooperative intrainterband dynamics of the population difference \({\mathscr{N}}(k,t)\) (between the valence and conduction bands) and the polarisation (coherence) \({\mathscr{P}}(k,t)\) for the Bloch state k. The phenomenological relaxation coefficients, γ_{1/2}, account for the collective broadening effects for the population and coherence decay, respectively. The theoretical derivation is succinctly provided in the Methods and Materials section. Under our theory, the electromagnetic coupling for normal illumination is defined by \({{\rm{\Phi }}}_{k}=\frac{{\bf{E}}\cdot {\hat{\phi }}_{k}}{\hslash k}\), where E is the electric field and the unit vector \({\hat{\phi }}_{k}\) is defined as \({\hat{\phi }}_{k}=\hat{z}\times {\bf{k}}/k\), shown in Fig. 3a.
The nonlinear optical absorption has multiple origins namely bleaching effects culminating in saturation, and two photon absorption. Since the graphene sample used in our experiment is lowdoped (i.e μ ≪ ℏω), absorption bleaching due to optically induced Pauliblocking plays the leading role. This assumption is supported by observation of the absorption drops upon high optical irradiance, refer to the OA trace in Fig. 2a. Furthermore, since the Fermi energy level is small compared to the energy of the photons, it naturally follows that Pauli blocking around zero detuning region is widely demolished. Under high intensity illumination in a Kerrtype material, the charged carriers undergo ultrafast Rabi oscillations, much faster than their relaxation rate. As a consequence, the steady state population, \({{\mathscr{N}}}_{{\bf{k}}}^{st}\), is significantly modified by light, disturbing the distribution at equilibrium, \({{\mathscr{N}}}_{{\bf{k}}}^{eq}\). The origin of the Kerrtype nonlinearity is the optically induced change to the steadystate population difference. The relaxation dynamics as the population difference evolves is schematically shown in Fig. 3b. Before saturation takes place, the nonlinear contribution of the field to the population difference around the zero detuning region is a quadratic function of the field magnitude, \({{\mathscr{N}}}_{{\bf{k}}}^{st}{{\mathscr{N}}}_{{\bf{k}}}^{eq}\approx \frac{1}{{\gamma }_{1}{\gamma }_{2}}{{\mathscr{N}}}_{{\bf{k}}}^{eq}{{\rm{\Phi }}}_{{\bf{k}}}{}^{2}\). The induced nonlinear current oscillating at the frequency ω is then given by \({{\bf{J}}}_{NL}=\sum _{k}{\hat{\phi }}_{k}({{\mathscr{N}}}_{{\bf{k}}}^{st}{{\mathscr{N}}}_{{\bf{k}}}^{eq}){ {\mathcal L} }_{{\bf{k}}}(\omega ){{\rm{\Phi }}}_{{\bf{k}}}\), where the Lorenztian \({ {\mathcal L} }_{{\bf{k}}}(\omega )\triangleq 1/({\gamma }_{2}+i{{\rm{\Delta }}}_{{\bf{k}}})\) accounts for the interband transitions. The nonlinear current J_{NL} is the microscopic origin of the intensity dependence of the refractive index so that n_{2} ∝ J_{NL}/E^{3}. For a small enough γ_{2}, the Kerrtype nonlinear induced current is given by:
where g_{s} and g_{v} are the spin and valley degeneracy factors, respectively, D = 1/4π^{2} is the density of states, and β~π is a dimensionless quantity that arises from angular integration around the Dirac cone. Due to low doping, \({{\mathscr{N}}}_{{\bf{k}}}^{eq}\approx 1\) over the zero detuning circle. Therefore, the frequency dependence of nonlinear current is dominated by the 1/k^{2} term appearing in Eq. 1. Over the Δ_{k} = 0 circle the Bloch wavenumber k is linearly proportional to the frequency which in turn yields λ^{2} dependence of the Kerr nonlinear coefficient.
It is worth pointing out that the quadratic wavelength dependence of the Kerr coefficient is a direct consequence of the linear energymomentum dispersion of the Dirac quasiparticles. The calculations outlined above can be effectively applied to all twolevel condensed matter systems by appropriately replacing interband coupling into Eq. 1. However, exclusive to graphene is the 1/k dependence of the interband coupling Φ_{k} which results in the square wavelength dependence of the Kerr coefficient.
Temporal evolution
The temporal evolution of the nonlinearity is obtained using the dualmode PPZS setup. A singlebeam (pump) Zscan measurement is performed to locate the peak and valley positions of the sample. In order to locate the zerodelay position of the probe, the sample (GoQ) is placed at the peak position and the probe is scanned until a crosscorrelation signal of the pump and probe pulses is obtained, shown in the inset of Fig. 4. The FWHM of the signal is 150 fs, which gives a pulse duration of 110 fs for the probe, considering a pulse duration of 102 fs for the pump. A similar scan is run at the valley position and both timeresolved data sets are used to extract the differential peakvalley transmittance, \({\rm{\Delta }}{T}_{pv}({t}_{d})=\pm [\frac{T({t}_{d},{Z}_{p})}{{T}_{OA}({t}_{d},{Z}_{p})}\frac{T({t}_{d},{Z}_{v})}{{T}_{OA}({t}_{d},{Z}_{p})}]\), where Z_{p/v} are the positions of the peak and valley, respectively, t_{d} is the probe delay, and T_{OA} is the OA transmittance for normalisation. The sign of ΔT_{pv} is given by the sign of Z_{p} − Z_{v}. The pump power used for this measurement was 200 mW, equating to an intensity of ~3.5 GW/cm^{2}. The pump to probe power ratio was kept at 20:1. In general, the farfield aperture should only allow about 1% of light transmittance to isolate the effect of wave front distortion due to phase modulation, however, due to the weak signal of the probe the aperture is opened to allow 10% of transmittance with additional averaging at each acquisition point.
The temporal evolution of the nonlinearity is shown in Fig. 4b. The effect peaks at zero probe delay and relaxes on the timescale of τ_{1}, while at longer time scales, τ_{2}, ΔT_{pv} shows no discernible variation. The symmetric shape of the figure leads us to believe that in this measurement we are simply observing the probe following the pump signals rather than unravelling the nonlinear phenomena hidden at shorter time scales. This is also supported by the fact that the pulse duration of the pump and probe is similar to the measured temporal crosscorrelation extracted in Fig. 1b. Therefore, we can conclude that the relaxation dynamics contributing to the observation of the nonlinear refraction are simply too fast to be measured by our laser pulses in this manner. The variation of ΔT_{P−V} or the induced phase shift ΔΦ seen in Fig. 4 relates to the Gaussian power distribution within the pump pulse which reaches its maximum value (i.e. ~3.5 GW/cm^{2}) at the centre peak of the pulse. Therefore, according to \({n}_{2,eff}=\frac{\sqrt{2}{{\rm{\Delta }}{\rm{\Phi }}}_{0}}{{k}_{0}{I}_{0}{L}_{eff}}\) (see Methods and Materials), the n_{2,eff} for the peak power is extracted to be 1.12 × 10^{−8} cm^{2}/W. As the intensity decreases moving away from the peak, the n_{2,eff} increasing in accordance with saturation. This measurement clearly shows that on the time scale greater than the heating and cooling times, the probe follows the pump and Kerr nonlinearity is tunable and controllable over this time scale. In this context, the heating and cooling times refers to the timescale of carrier excitation to higher states and subsequent carrier relaxation to the ground state, respectively^{30}. The modulation is understood through the relation n = n_{0} + Δn, where Δn = n_{2,eff}I, and the Δn parameter is modified as the n_{2,eff} varies nonlinearly on the intensity prior to complete saturation^{5}.
For the purposes of alloptical switching the on/off time of the nonlinearity is controlled by the pulseduration and power of pulse. If the pulseduration is longer than the effective relaxation times of the nonlinearity, then the evolution of the observed nonlinearity simply follows the Gaussian power distribution of the pulse. In this way through modulation of the Δn term, by changing the pulse properties, the switch can be controlled. However, there is saturation that takes place at relatively high intensities, leading to a deviation between the pump power within the pulse and the observed Δn.
Pulsewidth dependence
It is well understood that the relative timescales of the excitation pulse and the system response times determine the induced dynamics in the system. To investigate the effects of pulse duration on nonlinear refraction, we stretch the ~100 fs pulse upto ~475 fs at 900 nm to observe the effects in the long pulse regime; longer than the duration of the nonlinearity or relaxation time, τ_{1}. Pulse stretching is achieved using a dispersionbased prismpair apparatus, where the pulse can be temporally expanded by varying the separation distance of prisms^{31}. A schematic of the setup is shown in Fig. S8 in supplementary information. In our setup we use NSF11 flint glass prisms. The maximum allowed pulse expansion is limited by the space on the optical table. The autocorrelation curves for the expanded pulses at which the Zscan is performed are given in Fig. 5a and the results of the pulsewidth dependence are provided in Fig. 5b. A set of Zscan fits for the presented data is also provided in Fig. S9 in supporting information.
The results show a clear dependence of n_{2,eff} on the excitation pulse duration, with n_{2,eff} becoming larger with increasing pulsewidth. This trend was theoretically speculated by Vermeulen et al.^{10} and shown here experimentally. The n_{2,eff} ranges from 1.02 × 10^{−8} to 1.7 × 10^{−8} cm^{2}/W in this pulse duration regime with a minimum between 200–300 fs. The decay constant τ_{1}~113 fs measured in Fig. 1b falls near the beginning of the data set. However, doping can modify the decay constant values with p and ndoping making the time constants long and shorter, respectively^{32}. This data trend coincides with previously reported Zscan studies on graphene performed with picosecond excitation where the n_{2,eff} is larger than what is reported in the femtosecond regime. In addition to this, a similar comparative analysis performed on carbon disulphide (CS_{2}), reference material used for calibration of Zscan measurements, revealed a similar dependence of the n_{2,eff} on pulse duration^{33}. In a recent publication^{10}, this pulsewidth dependence of n_{2,eff} is theoretically derived for a regime when the effective decay constant is larger than the pulse duration in Zscan measurements. The interplay of relative carrier heating and cooling times is said to induce a nonlinear response that may not originate only from the conventional electronic Kerrtype nonlinearity but also from what they refer to as saturable photoexcitedcarrier refraction (SPCR). As mentioned previously, the nonlinearity originates from the carrier population difference. Given a pulse of an arbitrary pulse duration that impinges on the system, the maximum excited carrier density is reached at the peak of the pulse. If the pulse duration is shorter than the system’s relaxation constant i.e., τ_{1}~113 fs, then the maximum carrier density is reached on the timescale of the pulse duration, the carriers then remain in the excited state even after the pulse passes, subsequently relaxing on a timescale of τ_{1}. However, if the pulse duration is longer than τ_{1} then carriers continuously undergo excitation and relaxation as the pulse passes with the carrier population difference continuously evolving, and therefore the optical nonlinearity becomes dependent on the duration of the pulse. The saturability in graphene deviates from saturability in other 2D materials due to the presence of its unique gapless band structure which facilitates spontaneous saturation near the Dirac point even when there is no field^{23}. Therefore, it is apt that we refer to the Kerrtype nonlinearity characterised using the Zscan method as n_{2,eff} due to the presence of this contributing mechanism.
Conclusions
The systematic measurement and analysis of the dependence of n_{2,eff} on the spectral and temporal properties of the pulse, has allowed us to gain a more fundamental understanding of the parameters governing the observed nonlinear optical effect in graphene. This sheds significant light on the widespread debate in the field regarding the large variation in the n_{2,eff} value for graphene due to varying experimental conditions and sample preparation techniques. The dependence of n_{2,eff} on the exciting wavelength revealed a quadratic (n_{2,eff} ∝ λ^{2}) relationship observed experimentally and confirmed theoretically. The quantum theory developed can be used to calculate the nonlinear response under varying the relaxation coefficients and Fermi energy, and the results can be extended to other spectral regions. The timeresolved Zscan measurement revealed that the heating and cooling dynamics within our graphene sample are simply too fast to be probed with ~100 fs pulses using this method. However, this method reveals a practical application in alloptical ultrafast switching of the nonlinearity, completely controlled by the pulseduration and power of the impinging laser pulse.
Through the pulseduration dependent measurement, we were able to confirm the theoretically predicted relationship between n_{2,eff} and the laser pulseduration in the hundreds of femtosecond regime, with n_{2,eff} growing larger with longer pulseduration. Our results also coincide with the trend that is observed over several papers^{4,5,11,14} performing the measurement with different pulse durations, further reconciling our measurements, theory and experiments. In our measurements we used a prismpair setup to expand the pulses which is highly sensitive to beam alignment, posing a challenge if the pulse duration needs to be changed dynamically. Alternatively, transmission diffraction gratings can provide large pulse expansion ratios with high efficiencies and easier alignment^{34}. Using the diffraction gratings a larger range of pulse durations can be scanned and a decisive relationship between n_{2,eff} and pulse duration can be established.
Throughout all our experiments, the value for n_{2,eff} remains positive. The sign of n_{2,eff} has been a constant point of disagreement in literature, with the sign alternating from positive to negative depending on the method of characterisation employed or the conditions of the experiment. Theoretical studies have revealed that n_{2,eff} is very sensitive to the Fermi energy of the sample^{3,17,18,21,22}. Fermi energy modulation can be practically achieved with ease by electrically gating graphene samples in contrast to changing the spectral and temporal properties of the pulse to achieve the same tuning. Therefore, it becomes absolutely critical to perform an experimental characterisation of the dependence of n_{2,eff} on Fermi energy. By scanning this parameter and observing the nonlinear response, it is possible to observe if or when the change in sign takes place, which would resolve the supposedly contradictory reports. With this study we have gained critical fundamental understanding of the underlying processes governing the nonlinear optical phenomena and the effect of experimental parameters which can be used to largely tune the effect.
Methods and Materials
Pumpprobe integrated Zscan measurement
The experimental setup for this measurement is illustrated in Fig. 6. When operating in single beam mode, the setup is a classic Zscan measurement. The OA is used to normalise for absorption and laser fluctuations in the CA trace. In dualbeam mode, we use degenerate pump and probe at 900 nm, in a collinear configuration with a 20:1 pump/probe ratio. The pump and probe are orthogonally polarised using a λ/2 waveplate (λ/2), polarising beamsplitter (PBS) and polariser (Pol1) before the sample. An analyser polariser (Pol2) placed after the sample is rotated to achieve extinction of the pump beam.
The normalised CA transmittance is fitted to the equation: \(T(x,{{\rm{\Delta }}{\rm{\Phi }}}_{0})\simeq 1\frac{4{{\rm{\Delta }}{\rm{\Phi }}}_{0}x}{({x}^{2}+9)({x}^{2}+1)}\), where x is the Rayleigh length (z_{R}) normalised position (z/z_{R}) and ΔΦ_{0} is the nonlinearity induced phase shift. When a pulsed source is used with pulsewidth comparable to the duration of the nonlineairty, the nonlinear refractive index is extracted via \({n}_{2}=\frac{\sqrt{2}{{\rm{\Delta }}{\rm{\Phi }}}_{0}}{{k}_{0}{I}_{0}{L}_{eff}}\), where k_{0} is the wave vector, I_{0} is the onaxis irradiance and L_{eff} is the effective length of the sample^{35}. In our analysis the value for ΔΦ_{0} is extracted using a nonlinear leastsquares curve fitting routine in MATLAB. The routine takes in the aforementioned function along with a starting value for ΔΦ_{0}, and a vector constraining the upper and lower bound for the value. The routine then works to find a minimum for the sum of squares of the input function in an iterative manner, with the allowed number of iterations being easily modified. In our measurements the value for ΔΦ_{0} falls in the range of [−0.04, −0.15], which was the upper and lower bound input, while a starting point of 0 with 5000 iterations generally allowed for convergence.
Sample characterisation
In order to verify the quality of the samples, graphene on quartz (GoQ) samples (1 × 1 cm^{2}) were characterised by Raman spectroscopy, shown in Fig. 1a. The samples were purchased from an external supplier and were fabricated using chemical vapour deposition (CVD) onto a large copper sheet. The copper sheet was then cut into the respective sample size and graphene was transferred from the copper onto the quartz substrate. The Raman spectroscopy was performed using a homebuilt setup, combining a 561 nm continuous wave (CW) laser and a highresolution spectrometer. Prior to every measurement the laser input was read to normalise the spectral data to output the Raman shifts. The D band, G band and 2D band peaks appear at 1328 cm^{−1}, 1576 cm^{−1} and 2661 cm^{−1} respectively. The D band signifies disorder in the graphene lattice and is typically weak in high quality graphene, as seen in our sample measurement. A point to note as it pertains to the D band is that it is a resonant band and has very weak modes underlying it. The shape and position of this band can vary significantly depending on the excitation laser wavelength. The G band is a signature of an inplane vibrational mode stemming from the stretching of the CC bond, characteristic to all sp^{2} carbon systems. The 2D band is the second overtone of the D band and is the result of a twophonon lattice vibrational process. It is always strong in graphene and does not represent defects regardless of the presence of the D band. The peak at ~2460 cm^{−1} has been referred to as the G* band and is attributed to a double resonance^{36}. The peak appearing at ~2325 cm^{−1} may arise from additional defects or resonances. The relative intensities of the 2D and G peaks confirm that the sample is single layer.
We also perform a temporal correlation measurement on the sample, shown in Fig. 1b. The measurement was performed in the dualbeam mode of the PPZS setup using the translation stage and cross polarisers to obtain extinction of the pump. An extinction of ~10^{−4} was achieved with this configuration. The pump/probe power ratio was kept at 20:1, with the pump power being 200 mW. Due to the short timescale of the temporal correlation of the two pulses, a stage capable of achieving a repeatable resolution of ~5 femtosecond was employed. Lower resolution could be reached but was seen to be unreliable for a mechanical stage. The data is obtained over a timescale of 1.5 picoseconds, limited by speed of the stage and the oscilloscope sampling rate. The data is fit to a single exponential function of the form: exp(t/τ), to obtain a τ_{1} relaxation time constant of about ~113 fs.
Theoretical modelling of the nonlinear optical response in graphene
The nonlinear response coefficients have been calculated via Semiconductor Bloch Equations (SBEs). In the continuous excitation regime, the optical response can be estimated based on the dynamics of (i) the microscopic population difference \({\mathscr{N}}({\bf{k}},t)\) and (ii) the microscopic polarization \({\mathscr{P}}({\bf{k}},t)\). In a moving frame {τ, k′} = {t, k − δk(t)}, where δk obeys \(\frac{\partial \delta {\bf{k}}}{\partial t}+{\rm{\Gamma }}\delta {\bf{k}}=\frac{e}{\hslash }{\bf{E}}(t)\) and Γ is an phenomenological intraband relaxation coefficient, the dynamics of the charge carriers is governed by:
where \({\rm{\Phi }}({\bf{k}},t)=\frac{e{\bf{E}}\cdot {\hat{\phi }}_{{\bf{k}}}}{\hslash k}\) is the matrix element of the external potential of the direct optical transition, and the unit vector \({\hat{\phi }}_{{\bf{k}}}\) is defined as \({\hat{\phi }}_{{\bf{k}}}=\hat{z}\times {\bf{k}}/k\). The frequency \(\hslash {\varpi }_{{\bf{k}}}=2{ {\mathcal E} }_{{\bf{k}}}\) is the energy difference between the energy levels of the conduction and valence bands. At \(t=0\) before the electromagnetic field being applied, the population difference \({\mathscr{N}}\) is relaxed at its equilibrium \({{\mathscr{N}}}_{eq}=f(\hslash {v}_{F}k)f(\hslash {v}_{F}k)\) where \(f\) is FermiDirac distribution function. The moving frame accounts for intraband dynamics which has significant contribution in high harmonic generation via coherent Bloch oscillations.
Since the crystalline structure of graphene is centrosymmetric, even order nonlinearities are prohibited and therefore the first nonlinear term is the third order. We describe the effect of this third order nonlinearity by considering three complex fields with the time dependence of \({e}^{i{\omega }_{p}t}\), \({e}^{i{\omega }_{q}t}\) and \({e}^{i{\omega }_{r}t}\) and their mixing through the third order conductivity of graphene. The third order optical response can be interpreted as a threephoton process and different terms contribute to the third order conductivity tensor namely pure intraband term, pure interband term and combination of the both. There are six distinct photon processes contributing to the third order optics of graphene. However, since intraband processes mostly concern harmonic generation, in optical range the interband process plays the leading role in Kerr type linearity. SBEs can be combined and recast in the form of a dampdriven harmonic oscillator equation. The distinctive nonlinear process can be then obtained using an iterative technique^{23}.
The conductivity tensors for the six distinct photon processes contributing to the third order optical effect in graphene are presented in the supplemental document. The leading term (purely interband term) is:
where \({{\mathscr{P}}}_{I}\) denoted the intrinsic permutation operator and the operators \({{\mathscr{V}}}_{{\bf{k}}}\) and \({{\mathscr{W}}}_{{\bf{k}}}\) are defined as:
The equivalent third order bulk susceptibility of graphene is related to the third order surface dynamic conductivity via:
where d_{gr} is the equivalent thickness of graphene which is typically around d ≈ 0.3 nm^{3} and ε_{0} is the free space permittivity.In the case of graphene, the definition of the nonlinear bulk susceptibility is ambiguous due to the arbitrariness in the definition of the thickness of the twodimensional structure. The Kerr coefficient is defined through the intensity dependent refractive index i.e. n = n_{0} + (n_{2} − ik_{2})I where I = 2ε_{0}Re{n_{0}}cE^{2} (c is the speed of light). The complex nonlinear coefficient n_{2}−ik_{2} is related to the bulk susceptibility χ^{(3)}(ω, ω, −ω) as^{3}:
Since bulk susceptibility and the linear refractive index n_{0} are both normalized to the thickness d_{gr}, the Kerr coefficient is less sensitive to a particular selection for the thickness of the 2D material. The estimated values for the Kerr coefficient of graphene and the expected spectral dependence are presented in supporting information document.
Data Availability
Experimental data presented in this article is available from the corresponding author upon request.
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Acknowledgements
S.T., B.S. and A.H.M. acknowledge the financial support received from the Natural Science and Engineering Research Council of Canada (NSERC) and internal funding from U.W. B.S. and S.S.N. also acknowledge the financial support provided by the Canada First Research Excellence Fund.
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S.T., B.S., and A.H.M. defined the project and conceived the experiment(s). S.T. and B.S. conducted the experiment(s), B.S. developed the theoretical model and S.T. analyzed the results, under the guidance of A.H.M. and S.S.N. S.T. and B.S. wrote the manuscript and all authors reviewed the manuscript.
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Thakur, S., Semnani, B., SafaviNaeini, S. et al. Experimental Characterization of the Ultrafast, Tunable and Broadband Optical Kerr Nonlinearity in Graphene. Sci Rep 9, 10540 (2019). https://doi.org/10.1038/s4159801946710x
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