Nonlinear optical properties of multilayer graphene in the infrared

: A negative value for the nonlinear refraction in graphene is experimentally observed and unambiguously verified by performing a theoretical analysis arising from the conductivity of the graphene monolayer. The nonlinear optical properties of multi-layer graphene are experimentally studied by employing the Z-scan technique. The measurements are carried out at 1150, 1550, 1900 and 2400 nm with a 100-femtosecond laser source. Under laser illumination the multi-layer graphene exhibits a transmittance increase due to saturable absorption, followed by optical limiting due to two-photon absorption. The saturation irradiance I sat and the two-photon absorption coefficient β are measured in the operating wavelength range. Furthermore, an irradiance-dependent nonlinear refraction is observed and discriminated from the conventional nonlinear refraction coefficient n 2 , which is not irradiance dependent. The values obtained for the irradiance-dependent nonlinear refraction are in the order of ∼ 10 − 9 cm 2 W − 1 , approximately 8 orders of magnitude larger than any bulk dielectrics.


Introduction
Graphene has a number of remarkable optoelectronic properties resulting from its unique band structure [1,2].The linear dispersion of the massless Dirac fermions implies that graphene can absorb photons with the same efficiency over a wide range of wavelengths.These properties, combined with the high carrier mobility as well as mechanical and thermal properties make graphene an ideal candidate for a wide range of optoelectronic applications [1][2][3].
Here we fabricated a high-quality, large-area multi-layer (5-7 layers) graphene sample to investigate its nonlinear optical response in the femtosecond regime at 1150, 1550, 1900 and 2400 nm by employing the Z-scan technique [24].Z-scan, a technique which is based on the principles of spatial distortion of a focused Gaussian beam by a sample translated along its propagation path, offers both great simplicity and very high sensitivity upon measuring the optical nonlinearities of the sample in question.Under strong laser illumination the multilayer graphene exhibits a transmittance increase due to saturable absorption, followed by a decrease in transmittance leading to an optical limiting effect due to 2PA.Particularly in the case of graphene which presents both saturable and two-photon absorption, Z-scan is a fast and elegant way to discriminate the SA and 2PA processes and investigate the threshold at which SA turns to 2PA.Furthermore, it enables to unambiguously determine the 2PA coefficients, β, and the saturation irradiances, I sat , in the operating wavelength range.
Furthermore, a nonlinear refraction effect is observed.This effect is irradiance-dependent due to the absorption saturation mechanism in graphene.Therefore we define this effect as irradiance-dependent nonlinear refraction and use the symbol, ( ) n I  discriminating it from the conventional n 2 , which would indicate an irradiance-independent nonlinear refraction coefficient.More specifically, as the irradiance is increased the nonlinear refraction saturates.
In this high irradiance regime, aside from (3) gr χ higher order odd terms of the nonlinear susceptibility, such as (5) gr χ , (7)   gr χ and so on become significant leading to the irradiance dependence of the nonlinear refraction.Thus making the conventional n 2 coefficient #261726 unsuitable to describe the nonlinear refraction since it is only related to the (3) gr χ term of the nonlinear susceptibility.
The nonlinear optical Kerr effect of few-layer graphene has previously been reported by Zhang et al. [25].In that work they performed Z-scan measurements by using 3.8 ps pulses at 1550 nm operating at a high repetition rate of 10 MHz.They reported a giant nonlinear refractive index, almost 9 orders of magnitude larger than bulk dielectrics, also pointing out its irradiance-dependence.It is precisely this exceptionally large value that leads to the contribution of higher order odd terms of the nonlinear susceptibility and subsequently the irradiance dependence of the nonlinear optical refraction in graphene as discussed in the previous paragraph.Furthermore, broadband four-wave mixing in few-layer graphene has been reported [26,27].This allowed the determination of the absolute value of the third order susceptibility for a single graphene layer, ≅ × esu reported in [26] corresponds to an equivalent n 2 of 1.5 × 10 −9 cm 2 W −1 [25].However, the high repetition rate and long pulse duration employed in [25] could lead to an uncertainty in excluding nonlinear mechanisms of thermal nature.
In this work we investigate the nonlinear optical Kerr effect of few-layer graphene in the femtosecond temporal regime to a broad range of wavelengths, namely from 1150 nm to 2400 nm.Most importantly we show that the nonlinear refraction of graphene is negative.We operate in a low repetition rate regime of 1 kHz with 100 fs pulse durations to avoid any contribution from thermal effects.The negative value of the nonlinear refraction is supported with a theoretical analysis arising from the conductivity of the graphene monolayer, proving that graphene presents a negative nonlinear refraction over a broad range of wavelengths, as also pointed out by Ooi et al. [28].Moreover, a thermal analysis confirms that there are no thermal contributions to the optical nonlinearity.The thermal analysis considers the temperature dynamics of a single layer.Since each layer absorbs the same fraction of the incident irradiance (2.3%), the temperature profiles for all layers will be identical.Furthermore, over the 100 fs pulse width used here, lateral diffusion can be neglected and therefore heat diffusion will occur at the same time in all layers.Moreover, there is no transverse temperature gradient, so diffusion occurs independently within each layer.The diffusion coefficient of graphene is given by ( ) where g k is the in-plane thermal conductivity of graphene, g c is the specific heat and g ρ is the density of graphene.
By using a modest value for the in-plane thermal conductivity of graphene in the order of 3 1 0 [29] and taking the density of graphene as -3 2.25gcm g ρ = [30] and the specific heat as -1 -1 0.7 JK g g c = [30], we obtained a diffusion coefficient, 6.35 Therefore, the 1 kHz repetition rate used in this work yields a diffusion length of 827 μm between successive pulses.The beam waists were measured at each of the operating wavelengths using the knife-edge technique and were found to be between 25 and 31 μm.This means that the diffusion length is ∼31 times bigger than the beam waists, leading to a drop of the peak temperature by about three orders of magnitude before the next pulse arrives on the sample, therefore not enabling the manifestation of any thermal accumulation effects.Note that if a higher value for the thermal conductivity is used, in the orders of the superior thermal conductivity reported for single-layer graphene [31], it will yield a longer diffusion length and consequentially will lead to an even bigger drop of the peak temperature between successive pulses.A negative refraction in graphite thin films using degenerate four-wave mixing has also been reported by Harutyunyan et al. [32].Although due to the complexity of the technique used in that work, it could not be unambiguously verified whether the negative refraction arose from the graphite films or the two counter-propagating beams forming a phase conjugating surface on the graphite films.This negative refraction property, which was experimentally studied in a broad wavelength range and explained theoretically in this work, defines graphene as a promising medium for super-resolution imaging [33][34][35].

Sample preparation
The sample is prepared as follows.Large-area (0.8 × 1.0 cm 2 ), multilayer graphene films were synthesized by a chemical vapor deposition method in a quartz tube using thermally-annealed copper foil (25-μm thick, 99.999% purity) as substrate [36,37].The graphene film on copper foil was examined by scanning electron microscopy (SEM; Nova nanoSEM 230) to ensure the highly crystalline and continuous features.Then, graphene films were transferred to quartz substrates using normal wet-transfer technology based on poly(methyl methacrylate) (PMMA) after etching away the copper foil with an aqueous solution of ammonium persulfate (0.01 g/L).The as-synthesized graphene films were characterized by optical microscopy, Raman and absorption spectroscopy.More detailed results and analyses can be found elsewhere [23].The graphene film thickness (5~7 layers) were determined by a combination of spectroscopic techniques, namely, (i) reflection contrast spectroscopy (Nikon Eclipse Ti with a 50 × objective lens, tungsten halogen lamp), (ii) micro-Raman spectroscopy (WITec Alpha300R with an excitation wavelength of 532 nm and a 50 × objective lens), and (iii) NIR one-photon absorption spectroscopy (a Shimadzu UV-3600 UV−vis−NIR spectrophotometer).In the wavelength range of 1150 -2400 nm, the absorbance of the film was in the range from 11% to 15%, in agreement with the monolayer absorbance (2.3%) multiplied by 5-7.

Nonlinear optical measurements
The nonlinear optical properties of our graphene sample were investigated using a Z-scan setup with a regeneratively amplified Ti:Sapphire source delivering ~100 fs pulses, pumping an optical parametric amplifier at a repetition rate of 1 kHz.The measurements are carried out in a broad range, ~1250 nm, at 4 different wavelengths: 1150, 1550, 1900 and 2400 nm, typical of Yb, Er, Tm and Cr:ZnSe lasers respectively.The beam was focused using a 200 mm focal length calcium fluoride lens after passing through a spatial filter in order to ensure its Gaussian shape.The sample was then translated through the focus along the beam path whilst recording the far-field transmittance through both open and closed aperture detectors.The waist of the beam was measured at each of the operating wavelengths using the knifeedge technique.It was found to be 25 μm at 1150 nm, 26 μm at 1550 nm, 28 μm at 2000 nm and 31 μm at 2400 nm.
For any given wavelength Z-scan measurements are undertaken at a range of pulse energies from 2 to 300 nJ which corresponded to irradiances from ~1 GWcm −2 to ~100 GWcm −2 .No laser induced damage was observed up to 100 GWcm −2 .Open-aperture measurements showed that the transmittance had a power dependence characteristic of saturable absorption in the relatively low-irradiance regime.As the input irradiance was increased, a rise in transmittance occurred due to saturable absorption, which was followed by a decrease in transmittance due to 2PA.
The competing SA and 2PA processes can be seen in Fig. 1 in which five consecutive Zscans with increasing pulse energies at 1550 nm are presented.At the low pulse energy of 50 nJ, as the sample was translated through the focus of the beam, only a rise in transmittance was observed due to SA.However, as the pulse energy was increased further, a small valley started to form on top of the transmittance peak indicating the onset of 2PA.At longer wavelengths the decrease in transmittance due to 2PA is more prominent; this suggests that there is a more significant contribution from 2PA due to the lower photon energy.At the highest irradiances investigated in this work, ∼100 GWcm −2 , the transmittance decreased to about 0.99 for the case of 2400 nm while for the case of 1150 nm the decrease only reached 1.05.This behaviour can be seen in Fig. 2, where the irradiance versus normalized transmittance (at z = 0) is plotted for all 4 wavelengths.
The analytical expression used to fit the open aperture Z-scan measurements presented in Fig. 2 is obtained by numerically solving the propagation equation of a beam through a thin medium which presents both SA and 2PA following the method reported by Wang et al. [38].The total nonlinear absorption coefficient α(I) at a given incident irradiance I is given by [21,[39][40][41]: In the above equation, α 0 is the linear absorption coefficient, I sat is the saturation irradiance and β is the 2PA coefficient.Inspecting Eq. ( 1) one can deduce that the first and second terms describe SA and 2PA processes respectively.The 2PA coefficient β and the saturation irradiance I sat are extracted from the transmission numerical fits of Fig. 2. In the numerical simulation for the best fit, the α 0 parameter used is extracted from the linear transmission spectra of the sample at each of the operating wavelengths.The results are shown in Table 1.By inspecting this Table, a decrease in the saturation irradiance with longer wavelengths is observed.This is in very good qualitative agreement with the theoretical analysis performed in this paper.Furthermore, Yang et al. [21] observed the same wavelength dependence in bilayer graphene.Their reported values for the saturation irradiance are I sat = 6 and 1.5 GWcm −2 at wavelengths of 780 and 1100 nm respectively, in very good agreement with our reported values in this work at longer wavelengths.
The 2PA coefficient β shows a small dependence on the wavelength, increasing with longer wavelengths.Yang et al. [21] reported giant 2PA in bilayer graphene with the same wavelength dependence as reported here.In addition, they derived a theoretical model showing the increase of β with longer wavelengths.Chen et al. [23] found the 2PA coefficient to be 16 cm/MW at 1100 nm in CVD-made 5-layer film of graphene, in good agreement with our reported values.The observed increase of β with wavelength, is also consistent with the theoretical prediction [21,23] that the 2PA coefficient is proportional to 4 ( ) hv − , where hv is the photon energy.References [21,23] treat graphene at visible and near-IR wavelengths, up to 1100 nm, in this work we have investigated the 2PA in graphene from 1150 nm up to 2400 nm.The simultaneous investigation of both SA and 2PA at longer wavelengths, and over a broad range, revealed the trend of the associated coefficients and the threshold at which SA turns to 2PA at this important wavelength region.This information will give insight in the optimum design of saturable absorbers for laser mode-locking in the wavelength region of ~1-2.5 μm.In addition, closed-aperture Z-scans were performed with a typical 50% aperture transmission.By introducing an iris in the beam path at the closed-aperture Z-scan regime, only an on-axis portion of the beam was collected.In this way the effect of the nonlinear phase shift can be measured; however, the effect of the absorptive nonlinearities is present in the closed-aperture trace too.In order to isolate the former from the latter, the closed aperture trace was divided by the open-aperture trace as explained by Sheik-Bahae et al. [24].This yelded a typical closed-aperture Z-scan trace, one that would be obtained in the absence of any absorptive nonlinearities.The closed-aperture Z-scan traces at an irradiance of ∼22 GWcm −2 are shown in Fig. 3 for all four wavelengths.The traces were obtained after implementing the above described method of dividing the closed-by the open-aperture Z-scan traces, with no averaging or normalization performed after that.This explains why the data of Fig. 3 are noisier than the data of Fig. 1.As can be seen from Fig. 3, the observed nonlinear refraction has a negative sign.The theoretical fits in Fig. 3 use the simplified closed-aperture fitting formula from Sheik-Bahae et al. [24].
The values of the irradiance-dependent nonlinear refraction obtained from the fit for each wavelength are presented in Table 1.As can be seen graphene presents giant irradiancedependent nonlinear refraction in the order of ∼10 −9 cm 2 W −1 .The order of magnitude for the n 2 coefficient reported in [25] was ∼10 −7 cm 2 W −1 ; they obtained the n 2 value for irradiances lower than 0.1 GWcm −2 before the nonlinear phase started saturating.In this work we operated in the nonlinear phase over-saturation regime, therefore measuring an irradiancedependent nonlinear refraction ( ) n I  .More specifically the Z-scans performed here were at irradiances of 22 GWcm −2 about two orders of magnitude higher than the irradiances used in [25].We obtained the irradiance-dependent nonlinear refraction coefficients for all of the operating wavelengths.The broad range at which the work was undertaken revealed for the first time the trend of the nonlinear refraction in graphene versus wavelength at such a wide range spanning ~1250 nm.

Theoretical analysis
In this section we present theoretical results that lend support to the above experimental findings.We concentrate in particular on the conductivity, which can give important indications on the validity of the measurements of the nonlinear refraction effect for the multilayer graphene.The conductivity is usually written for monochromatic waves; however, one can perform useful estimates with it, even in the case of short pulses.The analytical expression of the complex linear conductivity ( ) σ ω of a graphene monolayer is known from Falkovsky's work [42]: where T is the temperature, B k is the Boltzmann constant, F E is the Fermi energy (chemical potential), 1 g and 2 g are the inverse relaxation times [43].At zero temperature and in the case of photoexcited undoped graphene where 0 . 4 is purely real, and equal to the value for the universal conductivity of monolayer graphene.We now generalize Eq. ( 2) to the nonlinear case and for a multilayer graphene made of N layers.We basically model our multilayer as a single layer with an effective thickness such that the linear absorption coefficient α 0 , which is due to the N graphene layers plus the substrate, matches the experimentally observed one over the range of wavelengths relevant to our experiments.
Here we take the values , in order to fit the measured values of the linear absorption coefficient α 0 in Table 1.These values are effective and strongly depend on the number of layers and the type of substrate employed.In the case of photoexcited undoped graphene the Fermi energy vanishes, 0 F E = .The relation between the complex linear susceptibility and the conductivity in the multilayer graphene sample is taken to be simply as  where eff d is the effective thickness of the multilayer, which represents the combined thickness of each graphene layer in the multilayer and the contribution from the substrate.In our estimates we assume N = 7 layers, and we find that 8 , where d = 0.33 nm is the monolayer graphene thickness, fits well with the absorption data of our experiments.The theoretical value for the saturation irradiance is found by using the semiconductor Bloch equations [44] adapted to the graphene dispersion, and generalized for N layers.In particular, the rate equation (which is identical for the case of semiconductors and graphene, since it does not depend on the electronic dispersion [44]) gives the following formula: where F v is the Fermi velocity and ω 0 is central pulse frequency in the CW approximation.The wavelength dependence of sat I is plotted in Fig. 4(a) and qualitatively agrees well with the values in Table 1, also presented in 4 (a).The quantitative discrepancies observed with the experimental values are attributed to the fact that in the theoretical prediction the 2PA is neglected and the CW approximation for the conductivity is used.
Using the full nonlinear susceptibility (that can be derived by solving the rate equation of the semiconductor Bloch equations [44]) given by: ( ) ( ) in which the denominator is common to all rate equations based on two-level systems, we obtain ( ) ( ) ( ) where ( ) n I is the irradiance-dependent refractive index and ( ) k I is the irradiance- dependent extinction coefficient responsible for absorption.One can define the nonlinear refraction coefficient ( ) . This coefficient has the same physical dimension of the in the Taylor expansion of the nonlinear refractive index calculated at I = 0, which is independent of the irradiance in accordance with our earlier statement.However, ( ) , where the nonlinearity is non-perturbative.As a consequence, any measurement of the nonlinear refraction coefficient in graphene will strongly depend on the particular irradiance used.Thus, under conditions of oversaturation, the nonlinear refractive index coefficients do not decrease, the conventional (truncated) Taylor series diverges, and the interpretation of n 2 must be modified according to the above formulae and discussion.The calculated irradiancedependent nonlinear refraction coefficient ( ) , both real and imaginary part, is plotted in Fig. 4(b) against wavelength at 22 GWcm −2 along with the experimental values from Table 1.Note that (3)   gr χ and ( ) n I  can in general have the same or opposite signs, depending on the frequency, since (3)   gr χ is complex, the relationship between them is not just a simple proportionality factor.Despite the fact that the 2PA is neglected and the CW conductivity of Eq. ( 4) is used, it is remarkable to observe that the values extracted from the plot of Fig. 4(b) are in excellent qualitative agreement with Table 1.Most importantly, it can be seen that the nonlinear refraction coefficient is negative for a broad region of frequencies, as also pointed out by Ooi [28], albeit in the case of doped graphene and thus for non-vanishing Fermi energy.The imaginary part of ( ) n I  is also negative, which is natural since nonlinearity reduces the linear losses due to the saturation process.

Conclusions
In this paper we studied the nonlinear optical properties of multilayer graphene by Z-scan.The saturable absorption, two-photon absorption, and nonlinear refraction properties were studied in the femtosecond temporal regime for a broad range of wavelengths, namely 1150, 1550, 1900 and 2400 nm.The saturation irradiances, I sat , and the two-photon absorption coefficients, β, were measured in the operating wavelength range.Furthermore, a negative irradiance-dependent nonlinear refraction was observed.This irradiance-dependent nonlinear refraction was symbolized with ( ) n I  and discriminated from the conventional nonlinear refraction coefficient n 2, which is irradiance-independent.A theoretical analysis arising from the conductivity of the graphene monolayer along with a thermal analysis based on the laser parameters used, verify that the negative value of the nonlinear refraction arises from the graphene multilayer.This is the first time that the nonlinear refractive effect in graphene is studied over a broad wavelength range.The values obtained for ( ) n I  show that graphene presents giant nonlinear refraction in the order of ∼10 −9 cm 2 W −1 , approximately 8 orders of magnitude larger than bulk dielectrics.The work performed here provides a fast and accurate way for characterizing graphene-based materials for the development of graphene-based nonlinear photonic devices in the infrared up to 2.5 μm.
bulk dielectrics, further pointing out the giant nonlinear refraction in graphene.The value of(

Fig. 1 .
Fig. 1.Open aperture Z-scan traces at 5 consecutive pulse energies at 1550 nm showing the competitive saturable absorption and two-photon absorption processes.

Fig. 4 .
Fig. 4. (a) Total saturation irradiance Isat versus wavelength, in conditions of room temperature along with the experimental values from Table 1 (black dots); (b) Real (blue line) and imaginary (red line) parts of the nonlinear refractive index versus wavelength at 22 GWcm −2 along with the experimental values fromTable 1 (black dots).
function that gives the slope of the nonlinear refraction versus irradiance diagram, and is not connected to any perturbative expansion.It is ( )